Stark Structure of the Spectrum and Decay Kinetics of the Er3+ Photoluminescence in Pseudoamorphous A-Nc-Gan Films A

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SEMICONDUCTORS AND DIELECTRICS

Stark Structure of the Spectrum and Decay Kinetics of the Er3+ Photoluminescence in Pseudoamorphous a-nc-GaN Films A. A. Andreev Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia Received April 29, 2003

Abstract—Amorphous ﬁlms containing nanocrystalline inclusions, a-nc-GaN, were codoped with Er in the course of magnetron sputtering. Intense intracenter emission of Er3+ ions in the wavelength region λ = 1510Ð 1550 nm was obtained only after multistage annealing at temperatures of 650Ð770°C. The intracenter emission was excited indirectly through electronÐhole pair generation in the a-nc-GaN matrix by a pulsed nitrogen laser. Subsequent recombination via a number of localized states in the band gap transferred the excitation energy to the Er3+ ions. Measurements of photoluminescence (PL) spectra carried out with time resolution in various annealing stages at different temperatures in the range 77Ð500 K and in the course of decay permitted us to establish the Stark nature of the PL spectrum and to reveal the dynamic and nonequilibrium character of redistribution of the intracenter emission energy among the Stark modes in the course of the excitation pulse relaxation. The information thus obtained was used to interpret the dominant hot-mode contribution and the complex decay kinetics. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION energy of the electronÐhole pairs created in the matrix with Eg = 3.4 eV. In the course of interband recombina- The interest in studies of the luminescence of triva- tion involving a broad spectrum of localized states, the lent erbium ions in various semiconductor matrices energy of the pairs is transferred to the erbium ions non- stems primarily from the possibility of developing nar- resonantly via the GaN latticeÐEr3+ bonds; this row band, thermally stable emitters for the wavelength (1535 nm) corresponding to minimum losses in quartz accounts for the multitude of intracenter transitions, of fiber waveguides [1]. This subject is also of fundamen- which we are interested only in the transition connect- tal significance, because many attendant issues still ing the first excited state of the ion to the ground state, 4 4 λ ≅ remain unsolved. In particular, the specific features of i.e., the I13/2 I15/2 transition at the wavelength erbium photoluminescence (PL) at high concentrations 1535 nm. The nonresonant mechanism of energy trans- of optically active ions (≥1020 cmÐ3), the mechanism of fer to the ion is very poorly understood, but it undoubt- energy transfer from the pumped matrix to erbium ions, edly has specific features and should affect the Er3+ PL and the PL dynamics at high pulsed pump power levels pattern. Furthermore, the high excitation density 2 4 on the order of 100 kW/cm , i.e., in the conditions close reached in a pulse makes the population of the I13/2 to the superluminescence threshold [2], are very poorly Stark states a nonequilibrium process, a factor that sub- understood. stantially modifies the PL decay kinetics. While the present study deals to a certain extent with these issues, it does not claim to have solved them and should be considered rather as a step in this direction. 2. EXPERIMENTAL This study includes two new elements, namely, (1) the choice of an amorphous a-nc-GaN matrix for Er3+ ions, Films of amorphous a-GaN measuring 4Ð7 nm with which allows erbium doping to high concentrations (up a fairly uniform distribution of nc-GaN nanocrystallites to 5 × 1020 cmÐ3) [3], and (2) the use of a pulsed nitro- of hexagonal symmetry were prepared by magnetron gen laser with a pulse power of 1.6 kW to excite the sputtering of metallic Ga in a reactive gas mixture Ar + matrixÐerbium ions system. The laser radiation does N2. The critical parameter for the process is the sub- not coincide in energy (បν = 3.68 eV) with any of the strate temperature Ts, which was found to be optimal 4f 11 erbium ion shell levels and, therefore, cannot be around 300Ð350°C. It was established that, at higher ° transferred to the erbium ion through direct resonance values of Ts approaching 500 C, films tend toward absorption of a photon. Thus, the erbium ions can be uncontrollable crystallization under subsequent anneal- excited only indirectly, through dissipation of the ing. The erbium ion PL in such films is unstable.

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Codoping with erbium was achieved in the course of 3. RESULTS AND DISCUSSION film growth by placing small Er-metal platelets of cali- 3.1. Changes in the Er3+ Spectra in the Course brated area into the magnetron discharge. The concen- of Optical Activation tration thus introduced was varied in the range (1Ð10) × 1020 cmÐ3. The maximum intensity was achieved at an It is known [1, 3, 6] that the mechanism of optical Er concentration of 5 × 1020 cmÐ3. A further increase in activation consists in the formation of a local atomic the concentration resulted in a strong decrease in the PL environment around the erbium ion, generating a crys- lifetime. tal field of electrostatic nature without inversion symmetry. Such a crystal field results in Stark splitting of Structural characterization of the films was done by the erbium 4f 11 multiplets. In addition, because there is means of x-ray diffraction, Raman spectral measure- no inversion site symmetry, the allowed fÐd transitions ments, and direct HRTEM observation of the micro- become admixed to the fÐf transitions forbidden by structure. The HRTEM method permitted us to detect selection rules, thus making the fÐf intracenter transi- inclusions of nanocrystals and their belonging to the c- tions possible, although with a low probability, and this GaN hexagonal phase and determine their dimensions manifests itself, as will be seen from what follows, in (5Ð7 nm), the fractional volume of the phase (≈10%), the low PL decay rate. and its variation under annealing, namely, the increase The most efficient crystal field is generated by in the volume fraction of the nanocrystals without an atoms of highly electronegative elements, such as nitro- increase in their size [4, 5]. The HRTEM data were gen and oxygen. The absence of any experimentally kindly provided by Prof. H.P. Strunk (Erlangen, Ger- observable optical activity of the erbium ion in as-pre- many). While the vibrational mode structure in the pared films implies that nitrogen does not create the Raman spectra obtained by V.Yu. Davydov (Ioffe Insti- 3+ tute, Russian Academy of Sciences, St. Petersburg) is required configuration in the Er ion environment. fairly diffuse, the maxima of the A (TO) + E + E and Such a configuration does not form even though there 1 1 2 is an excess of chemically active nitrogen in the form of A1(LO) + E1(LO) modes nevertheless coincide in posi- plasma ions during the film growth. Our experiment tion with those for the c-GaN. The Raman spectrum of shows that optical activity appears only at temperatures the films sharpens with annealing, which is evidence favoring the generation of nitrogen vacancies in the there is a growing contribution from the nanocrystalline GaN structural network [7]. This takes place in the tem- fraction, and is in agreement with the HRTEM data. perature interval 650Ð770°C. This allows two alterna- Despite this trend, the a-nc-GaN films retain their two- ° tives, namely, either the released nitrogen becomes phase (a + nc) microstructure up to at least 775 C, dem- chemically bonded to Er3+ or the process of nitrogen onstrating also high adhesion to such substrates as c-Si generation is conducive to the diffusion of residual oxy- and fused quartz. gen (which is always present in the matrix in dissolved Directly after deposition, Er-doped films do not form) to the erbium ions. This residual oxygen is get- exhibit the characteristic erbium PL at λ ≅ 1535 nm. To tered efficiently by the erbium to form the ErOx com- obtain this emission, one has to optically activate the plex, whose erbium ion is acted upon by the field of erbium ions, a process consisting in a series of high- negative oxygen charges. Irrespective of the way in ° temperature anneals with Tann in the range 650Ð770 C. which this process evolves, one can be confident that it To achieve the maximum Er3+ PL intensity, several does not occur quickly and is extended both in time and 3+ cycles with Tann increased judiciously in each subse- temperature. By measuring the Er PL spectra in dif- quent cycle will suffice. Annealing at a temperature in ferent stages of optical activation, one can follow the ° mechanism of crystal field formation, as well as draw excess of the maximum Tann, 775 C, initiates a clearly pronounced macrocrystallization, which brings about some conclusions concerning the field symmetry and, film turbidity and a complete or partial disappearance hence, understand the nature of the Er3+ PL spectra. ° of the erbium PL. The first anneal (Tann = 650 C) gives rise to a very 3+ PL measurements were carried out with an optical weak Er PL signal with a fine structure consisting of resolution of 5 nm. The InGaAs-based photodiode with a set of narrow, well-resolved peaks. Curve 1 in Fig. 1 a sensitivity of 25 µV/mW provided by L.B. Karlina displays this spectrum, obtained by S.B. Aldabergen- (Ioffe Institute, RAS, St. Petersburg) had a low internal ova on a high-resolution double monochromator resistance, which favored reaching a low level of elec- (0.85-m Spex 1404). After the next anneal (Tann = tromagnetic noise induction from the pulsed-laser 700°C), the intensity increases by a factor of 2Ð3, the power supply and permitted us to reach high instrument spectral lines broaden, and the line structure still per- ° sensitivity over a broad enough dynamic range of vari- sists. Only in the last stage (Tann = 750Ð770 C) is there ation of the signal amplitude (up to two orders of mag- a significant growth in intensity of up to one-hundred- nitude). Time-resolved spectra were obtained with a fold the initial level and does the fine structure in the gate width of down to (3Ð5) × 10Ð2 of the PL decay spectrum disappear (curve 2, Fig. 1). time. Decay curves were recorded directly with an The PL displayed in Fig. 1 (curve 1) is obviously a oscillograph. Stark spectrum. It is specific in that the Stark manifold

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STARK STRUCTURE OF THE SPECTRUM AND DECAY KINETICS 1003 consists of “hot” transitions only, i.e., of transitions Er3+PL intensity, arb. units 4 4 from the Stark levels of the I13/2 manifold to the I15/2 ground state. Only the weak line at 1598 nm can be 120 assigned to “cold” transitions, i.e., transitions from the 4 4 lowest I13/2 state to the I15/2 Stark manifold. 2 The spectrum obtained in the final stage of optical activation (curve 2) is fairly diffuse and does not con- 80 ×4 tain distinct Stark lines. Nevertheless, taking into account the dynamics of the transformation of spec- 4 trum 1 to spectrum 2, as well as a certain similarity in the distribution of averaged intensity (which also man- 3 40 ifests itself in the dominant contribution of the high- ×8 energy wing), one can assume the final-state spectrum to derive from a sum of Stark transitions. The Stark 1 peaks should be broadened compared to the ones in curve 1, because they originate from the considerably 0 higher density of elementary emitters, which have an inherent configurational diffuseness [8]. To verify the 1450 1500 1550 1600 1650 possibility itself of assigning the spectrum to Stark Wavelength, nm transitions, we performed a numerical modeling by fitting the radiation originating from the Stark levels by Fig. 1. Photoluminescence spectra of Er3+ ions in the a-nc- Gaussians. In this procedure, the positions of the Stark GaN pseudoamorphous matrix: (1) initial stage of optical levels for the high-energy wing of the experimental activation following annealing at 650°C, (2) spectrum of the maximum intensity reached after multistage annealing with spectrum were prescribed in accordance with those in a final temperature of 770°C (solid line), (3) manifold of Fig. 1 (curve 1). As for the low-energy wing, it Stark emitters whose total spectrum is depicted by open cir- appeared reasonable to assume an averaged distribution cles on curve 2, and (4) spectrum obtained at the onset of of Stark levels spaced about 10 nm apart. This simplifi- macrocrystallization. cation does not introduce fundamental errors, because, first, the difference in the Stark level distribution between the major types of crystal field symmetries is Si : H amorphous silicon, which contains no nitrogen at of the order of or less than this quantity and, second, the all [10]. This suggests that it is the ErOx oxygen com- tendency of ions of rare-earth metals (REM) to form plex that produces the emission spectrum. The matrix multipole positions at high densities [9], i.e., positions plays a secondary role, which is easy to understand with symmetries alternating from one site to another, physically. Because it is inherently labile, it adjusts inevitably brings about an averaging of the spectrum. itself, as it were, to the rigid ErOx complex. Figure 1 presents a modeled Stark spectrum (curve 3) (2) The above observations and conclusions (see the and its envelope (open circles on the experimental previous comment) give rise to the legitimate question curve 2). We readily see that a Stark representation of of the structure of the ErOx complex, which was raised the diffuse spectrum is reasonable. Subsequently, we more than once before. It was found that the complex will consider additional arguments for the Stark nature contains six oxygen atoms making up a somewhat of the broad and diffuse experimental spectrum of the deformed octahedron, on whose C axis the erbium Er3+ ions. It would be appropriate to make a few com- atom is located in a slightly off-center position, and this ments here. is what accounts for there being no center of inversion (1) At first glance, the narrowness of the Stark lines [11, 12]. This complex has a C4v-type tetragonal sym- in the initial annealing stage might seem strange. This metry. Some authors, however, tend to believe that the observation can, however, be accounted for by the weak erbium site has a lower symmetry and that the ion fea- interaction of the erbium ion and its nearest environ- tures have, accordingly, a lower oxygen coordination ment (to be more precise, the oxygen ions) with the [13, 14]. We compared an experimental set of Stark amorphous network of the matrix. The strong chemical lines (for hot transitions) with the calculations available bonding of the ErOx complex governs its structure, and for two crystal field symmetry types, namely, the high- the weaker external interaction with the matrix does not est cubic symmetry Td and the lower rhombohedral affect the structure of this complex noticeably. This symmetry C3v. No one-to-one correspondence was results in the formation of a set of structurally identical found for either symmetry group. The only thing one emitting centers, which do not interact with one another can maintain is that the experimental spectrum contains because of their low concentration. This is what elements of both symmetries simultaneously. In princi- accounts for the fine structure of the spectrum. This ple, this is possible, because the C3v configuration, just idea is argued for by the fact that a nearly identical as C4v, is actually the result of a deformation of cubic Stark spectrum of Er3+ was obtained by us earlier on a- symmetry along the (111) and (001) directions. If we

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Er3+PL intensity, arb. units Er3+PL intensity, arb. units 100 120

80 2

60 80 3 40 1 ×4 20 123 40

1450 1500 1550 1600 1650 0 Wavelength, nm 1460 1500 1540 1580 1620 Wavelength, nm Fig. 2. Time-resolved spectra obtained at 300 K under pulsed excitation by a nitrogen laser with a pulse power of ≈100 kW/cm2 for time delay (1) ≈5, (2) ≈150, and Fig. 3. Time-resolved spectra obtained at 77 K. The notation (3) ≈450 µs. is the same as that in Fig. 2. allow now that the extent of this deformation varies are displayed in Fig. 2, and for 77 K, in Fig. 3. Measure- from one emitting center to another, we come to the ments were also performed at 473 K. The relations multipole conﬁguration of the erbium centers and, recorded did not reveal any qualitative differences; for accordingly, to a structure of the spectrum that does not this reason, the data for 473 K are omitted. Neverthe- lend itself to unambiguous assignment. less, to illustrate the pattern of the temperature behav- (3) Curve 4 in Fig. 1 presents the spectrum of a sam- ior, Fig. 4 presents integral spectra, i.e., spectra mea- ple that was not properly annealed to perform optical sured without gating, for the three temperatures. As for activation; more speciﬁcally, the sample was overan- nealed, which led to macrocrystallization. In addition the temperature dependence of the amplitude of the to the overall weakening of the PL intensity, one observes an additional spectral broadening and a noticeable contribution from the Stark line at 3+ ≈1695 nm. The weakening of the PL intensity is most Er PL intensity, arb. units probably due to a sizable fraction of the erbium ions being expelled to intergrain voids, followed by their 120 precipitation. The broadening of the spectrum should be assigned to the impact of the rigid crystalline matrix 2 environment, which is capable of straining the ErOx complexes that became incorporated in the GaN micro- 80 crystals. 1

3.2. Er3+ Photoluminescence in a-nc-GaN as Derived 3 from Its Time-Resolved Spectra and Decay Kinetics 40 The pulsed excitation technique offers the possibility of studying the Er3+ emission kinetics. We obtained both PL spectra at various gate delays ∆ after the termination of the pump pulse and PL decay curves at a number of emission wavelengths. This series of measure- 0 ments was carried out on samples with the highest PL 1440 1480 1520 1560 1600 1640 intensity. The PL decay in such samples lasted as long Wavelength, nm as 500Ð800 µs. The time-resolved spectra were taken by properly delaying the 20-µs-wide gate to ﬁxed times Fig. 4. Erbium photoluminescence spectra obtained at tem- ∆ = 5, 150, and 450 µs. The results obtained for 300 K peratures of (1) 77, (2) 300, and (3) 473 K.

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 STARK STRUCTURE OF THE SPECTRUM AND DECAY KINETICS 1005 intensity, it is seen to be practically absent over the 4fn 5d range 77Ð300 K. 100 The main and most essential ﬁnding that follows First 5 excited from the above data is that the peak of maximum emis- state sion intensity shifts toward higher energies with both a 80 decrease in the delay time and an increase in tempera- 3 4 ture. The magnitude of the shift amounts to 10 nm. Because the position of the emission lines of REM ions 60 2 3+ and, speciﬁcally, of Er does not depend on the method Stark manifolds 6 of measurement or temperature, one may conclude that 40 the observed shift is only apparent and is due to a redis- Ground 7 state tribution of intensity between the emission peaks in the PL intensity, arb. units

3+ λ → λ manifold of the Stark modes. This conclusion offers an 20 min max additional and compelling argument for the validity of Er 1 the earlier model, following which the broad PL spec- 8 trum is represented by a set of narrower band emitters 0 (the Stark states). 0 200 400 600 800 Figure 5 displays the PL decay kinetics measured Decay time, µs for different wavelengths at 300 K. The curves obtained at 77 and 473 K do not differ radically from the data Fig. 5. Erbium photoluminescence decay curves obtained at collected for 300 K and, therefore, are omitted in the room temperature and various values of wavelength λ: figure. Figure 5 reveals the presence of two competing (1) 1490, (2) 1500, (3) 1510, (4) 1520, (5) 1535, (6) 1550, (7) 1575, and (8) 1595 nm. Inset: diagram illustrating the processes, namely, a rise in intensity and a decay, which radiative and nonradiative transitions between the Stark occur with different characteristic times τ. The intensity manifold states. rise and decay curves can be approximated generally by τ a sum of three exponentials, ÐA1exp(Ðt/ 1) + − τ τ τ A2exp( t/ 2) + A3exp(Ðt/ 3), where 1 is the time of the τ τ rise in intensity and 2 and 3 are the decay times. Fig- Calculated solid line: ures 6a and 6b illustrate the fitting of experimental –130exp(–t/30)+30exp(–t/40)+100exp(–t/180) curves with three and two exponentials, respectively. 70 The first case applies always to decay curves obtained (a) at wavelengths of the high-energy wing of the spec- 50 trum, and the second case, to all other wavelengths. As follows from an analysis of the complete set of data obtained for the decay curves, the characteristic times 30 vary, depending on the actual wavelength and temperature, within the following limits: τ ≈ 10Ð30 µs, τ ≈ PL decay, arb. units

1 2 3+ µ τ ≈ µ λ ≈ 10 10Ð60 s, and 3 70Ð200 s. For 1535 nm (the Er wavelength corresponding to emission from the lowest 0 4 τ I13/2 state), the times of the intensity rise, 1, and decay, τ µ Calculated solid line: 3, reach maximum values of 60 and 200 s, respec- –250exp(–t/60)+250exp(–t/180) tively. We can add that the decay kinetics follows a 100 monomolecular law, as should be expected for intrac- (b) enter emission. 60 The totality of the experimental data available on the PL decay kinetics and on the emission peak shift as a function of delay time ∆ can be understood in terms of 4 4 PL decay, arb. units 20 the Stark splitting scheme of the I13/2 and I15/2 states 3+ proposed earlier. Consider the relaxation of the exter- Er 0 nally excited matrixÐoptically active erbium center sys- 0 200 400 600 800 tem. The electronÐhole pairs created by the laser Decay time, µs undergo thermalization to the conduction band edge, their energy being transferred subsequently to the localized-state distribution in the conduction band tail. The Fig. 6. Modeling of the intensity rise and decay curves by dominant role played by localized states in the energy (a) a sum of three exponentials for hot transitions at λ ≈ transfer to the erbium ions is supported by measure- 1500 nm and (b) a sum of two exponentials for transitions ments of the PL excitation (PLE) function in the at λ ≈ 1535 nm.

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4 4 I13/2 I15/2 transitions. These measurements per- ative until the population of the upper Stark levels formed on c-GaN [15] show that the PLE function con- decreases to the equilibrium value determined by the tains not only the resonance peaks originating from Boltzmann distribution. Now, considering emission direct resonance photon absorption by the 4f 11 multip- kinetics from lower energy Stark states and, in the limit, 4 let states but also an exponentially decaying tail, which from the lowest I13/2 state, we note that these states extends from the edge of the conduction band deep into receive excitation energy in the nonsteady-state and the band gap. This tail is due to nonresonant energy nonequilibrium process in the course of its nonradiative transfer to ions, apparently through dÐf hybridization relaxation from the upper Stark levels. Because this and excitation thermalization in the f-multiplet system. energy is transferred in a multistage pattern, the popu- τ Rather than trying to delve into the physics underlying lation of the lower levels slows down and 1 should this process, which remains largely unknown theoreti- increase, accordingly, by a factor of 2Ð4, exactly what cally, we stress only the point of most interest for the is observed experimentally. Furthermore, the lowest 4 present study, namely, that the energy received by the I13/2 state is no longer capable of transferring the 4 matrix reaches the first excited state I13/2 and dissipates energy in a nonradiative manner. This results in the loss subsequently in radiative transitions to the ground state. of the second term in the sum of the exponentials, so the In the crystal field, the first excited state 4I represents emission process is now described by one characteristic 13/2 τ τ a manifold of Stark levels; hence, the excitation energy constant only, 3. Note that the values of 3 are approx- 4 is absorbed by these levels. Moreover, if the energy dis- imately the same for all the I13/2 Stark levels. It should sipates through thermalization, the first to be reached be stressed that the peak of emission from the lowest by excitation are the upper Stark levels. If we accept state is located at ≈1535 nm and is seen as the narrowest this scenario, the appreciable contribution from the hot line with a half-width of ≈25 nm. Considered within Stark transitions to the PL spectra observed by us, as our model, this comes as no surprise, because emission 4 well as the dominant part played by them in time- from the lowest I13/2 state corresponds to the conditions resolved spectra for ∆ 0, become clear; namely, closest to equilibrium, in which the contribution of the this is the result of a nonsteady-state process in which hot transitions is minimal. We may also add that the the population of the upper Stark states immediately emission spectra measured with long delay times are after the termination of the pump pulse is nonequilib- very close to those observed under stationary and equi- rium and approaches equilibrium only as the excitation librium conditions realized under continuous low- undergoes thermalization over the Stark manifold of power excitation (≈20Ð50 mW). 4 the I13/2 state. Thus, immediately after the termination of the excitation pulse, two competing processes occur The temperature behavior of both the time-resolved simultaneously, namely, radiative transitions from the spectra and the decay kinetics also agrees with the 4 4 above scenario. As the temperature increases, the pro- I13/2 Stark states to the I15/2 ground state and nonradi- cess of nonequilibrium occupation of the upper Stark 4 ative transitions from the upper I13/2 Stark states to levels is paralleled by an increase in the population of 4 lower lying levels of the I13/2 Stark manifold. these levels through thermal excitation. Combined The inset to Fig. 5 shows a simplified scheme of action of these two mechanisms gives rise to an these transitions. This scheme offers the following increase in the intensity of the high-energy PL wing. explanation for the complex rise in PL intensity and decay kinetics observed at various wavelengths. For the shortest wavelengths in the high-energy wing of the 4. CONCLUSIONS spectrum, 1500Ð1520 nm, the intensity rise and decay (1) The main emission line of Er3+ (4I 4I ) process can be described by a sum of three exponen- 13/2 15/2 τ in the a-nc-GaN matrix has been shown to be of Stark tials. The first of them, with a characteristic time 1, nature. reflects the transfer of excitation energy to the erbium ion, including thermalization of the excitation over the (2) The relaxation of pulsed excitation energy for a 4 pulse power of 100 kW/cm2 was established to be a f shell to the I13/2 Stark states. The second exponential, τ nonequilibrium process for times of at least up to with characteristic time 2, describes thermalization of µ 4 100 s. the pump energy over the system of I13/2 Stark levels through successive nonradiative transitions from the (3) The PL intensity rise and decay process occur- uppermost Stark level of this manifold to a lower lying ring in nonequilibrium conditions is described, in a first 4 approximation, by three exponentials, the first of which one, down to the lowest I13/2 state. The third exponen- τ characterizes the mechanism of excitation transfer; the tial, with characteristic time 3, describes the radiative process itself. The second term in the sum is actually second, the mechanism of nonradiative energy relax- responsible for the apparent acceleration of decay in the ation from the upper to lower Stark levels; and the third, initial stage of the PL decay caused by the decrease in the radiative process itself. the population of the upper Stark levels resulting from The characteristic time of the radiative process for nonradiative transitions. This mechanism remains oper- optimized samples can be as long as ≈200 µs.

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REFERENCES 9. S. Kim, S. J. Rhee, D. A. Turnbull, E. E. Reuter, X. Li, J. J. Coleman, and S. G. Bishop, Appl. Phys. Lett. 71 (2), 1. A. Polman, J. Appl. Phys. 82 (1), 1 (1997). 231 (1997). 2. M. S. Bresler, O. B. Gusev, E. I. Terukov, I. N. Yass- 10. S. B. Aldabergenova, M. Albrecht, H. P. Strunk, J. Viner, ievich, B. P. Zakharchenya, V. I. Emel’yanov, P. C. Taylor, and A. A. Andreev, Mater. Sci. Eng. B 81 B. V. Kamenev, P. K. Kashkarov, E. A. Konstantinova, (1Ð3), 29 (2001). and V. Yu. Timoshenko, Mater. Sci. Eng. B 81, 52 (2001). 11. M. Ishii, Y. Tanaka, I. Ishikawa, S. Komuro, T. Mori- 3. A. A. Andreev, Fiz. Tverd. Tela (St. Petersburg) 44 (2), kawa, and Y. Aoyagi, Appl. Phys. Lett. 78 (2), 183 239 (2002) [Phys. Solid State 44, 248 (2002)]. (2001). 12. M. Ishio and Y. Komukai, Appl. Phys. Lett. 97 (7), 934 4. S. B. Aldabergenova, M. Albrecht, A. A. Andreev, (2001). C. Ingleﬁeld, J. Viner, V. Yu. Davydov, P. C. Taylor, and H. P. Strunk, J. Non-Cryst. Solids 283 (1Ð3), 173 (2001). 13. H. Przybylinska, W. Jantsch, Yu. Suprun-Belevitch, M. Stepikhova, L. Palmetshofer, G. Hendorfer, A. Koza- 5. A. A. Andreev, Fiz. Tverd. Tela (St. Petersburg) 45 (3), necki, R. J. Wilson, and B. J. Sealy, Phys. Rev. B 54 (4), 395 (2003) [Phys. Solid State 45, 419 (2003)]. 2532 (1996). 6. A. Terrasi, G. Franzo, S. Coffa, F. Priolo, F. D. Acapito, 14. D. E. Wortman, C. A. Morrison, and J. L. Bradshaw, J. and S. Mobilio, Appl. Phys. Lett. 70 (13), 1712 (1997). Appl. Phys. 82 (5), 2580 (1997). 7. S. Nakamura and G. Fasol, The Blue Laser Diode. GaN- 15. Myo Thaik, U. Hommerich, R. N. Schwartz, R. G. Wil- Based Light Emitters and Lasers (Springer, Berlin, son, and J. M. Zavada, Appl. Phys. Lett. 71 (18), 2641 1997), p. 129. (1997). 8. A. A. Kaminskiœ, Laser Crystals (Nauka, Moscow, 1975), pp. 25Ð27. Translated by G. Skrebtsov

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SEMICONDUCTORS AND DIELECTRICS

Frequency Dispersion of Dielectric Coefficients of Layered TlGaS2 Single Crystals S. N. Mustafaeva Institute of Physics, National Academy of Sciences of Azerbaijan, pr. Narimanova 33, Baku, 1143 Azerbaijan e-mail: [email protected] Received June 30, 2003

δ ε σ Abstract—Frequency dependence of the dissipation factor tan , the permittivity , and the ac conductivity ac × 4 × 7 across the layers in the frequency range f = 5 10 Ð3 10 Hz was studied in layered TlGaS2 single crystals. A signiﬁcant dispersion in tanδ was observed in the frequency range 106Ð3 × 107 Hz. In the range of frequencies × 4 6 studied, the permittivity of TlGaS2 samples varied from 26 to 30. In the frequency range 5 10 Ð10 Hz, the 0.8 6 σ 2 ac conductivity obeyed the f law, whereas for f > 10 Hz ac was proportional to f . It was established that × the mechanism of the ac charge transport across the layers in TlGaS2 single crystals in the frequency range 5 104Ð106 Hz is hopping over localized states near the Fermi level. Estimations yielded the following values of × 18 Ð1 Ð3 the parameters: the density of states at the Fermi level NF = 2.1 10 eV cm , the average time of charge carrier hopping between localized states τ = 2 µs, and the average hopping distance R = 103 Å. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION described. Quite recently, observations of the dispersion of the permittivity and conductivity of TlGaS2 TlGaS2 single crystals are typical representatives of crystals measured at frequencies of 102, 103, 104, and layered semiconductors and attract a lot of attention 6 10 Hz were reported [5]. Whereas in [2] the TlGaS2 ac due to their interesting physical properties. These prop- conductivity measured in the frequency range 105Ð erties include strong anisotropy of the electronic 7 0.8 σ parameters related to special features in the crystalline 10 Hz was described by a power law (f ), the ac(f) structure. Layered crystals usually contain structural dependence measured in [5] in the frequency range defects, such as vacancies and dislocations. The pres- 102Ð106 Hz was rather weak. In view of these inconsis- ence of these defects results in a high density of local- tencies, we experimentally studied the frequency- ized states near the Fermi level. The states localized in dependent dielectric parameters of layered TlGaS2 sin- the band gap are responsible for most electronic pro- gle crystals and the mechanisms of ac charge transport. cesses occurring in semiconductors. Both dc and ac The layered TlGaS2 single crystals studied had a charge transport proceeds via these localized states. In rather high electrical resistance ρ at room temperature. [1], it was experimentally established that, in layered The value of ρ was 2 × 107 Ω cm along the natural lay- ≤ TlGaS2 single crystals at temperatures T 200 K, the dc ers and ρ = 7 × 109 Ω cm across the layers. conductivity both along and across the natural layers have variable-range hopping over localized states near the Fermi level. Using the experimental results on the 2. EXPERIMENTAL TECHNIQUES dc conductivity of TlGaS2 single crystals along and across Measurements of the dielectric coefficients of the layers, the density of states at the Fermi level N was F TlGaS2 single crystals were performed at fixed frequen- × 18 × 18 Ð1 Ð3 estimated to be 2.5 10 and 2.0 10 eV cm , cies in the range 5 × 104Ð3 × 107 Hz by the resonant respectively; i.e., the density of states is rather high. method using a TESLA BM560 Q meter. TlGaS2 sam- From this point of view, TlGaS2 single crystals are ples formed flat capacitors whose plane was perpendic- interesting objects for study in ac electric fields. Exper- ular to the crystalline C axis. The thickness of a TlGaS2 imental data on the frequency-dependent conductivity single crystalline layer was 0.05Ð0.06 cm, and the of TlGaS2 single crystals have been reported in the lit- capacitor plate area was 0.5Ð0.7 cm2. For electrical × 18 Ð1 Ð3 erature [2, 3]. In [2], a value of 9.0 10 eV cm was measurements, the samples were placed in a specially obtained for NF, which exceeds the value of NF that we constructed screened cell. An ac electric field was σ found by measuring dc across the TlGaS2layers [1]. In applied across the natural layers of TlGaS2 single crys- [4], the results of permittivity measurements for a tals. The amplitude of the applied field corresponded to 5 TlGaS2 single crystal at a frequency of 10 Hz were the Ohmic region of the currentÐvoltage characteristics

FREQUENCY DISPERSION OF DIELECTRIC COEFFICIENTS 1009

δ × 4 ε σ Ω–1 –1 tan 10 ac, cm 200 10–5 30 10–6 σ 2 150 ac ~ f 2 10–7 100 25 σ 0.8 10–8 ac ~ f 50 1 20 10–9 0 105 106 107 108 5 × 104 105 106 107 108 f, Hz f, Hz

Fig. 1. Frequency dependences of (1) the dissipation factor and Fig. 2. Frequency dependence of the ac conductivity of a (2) the permittivity of a TlGaS2 single crystal at T = 300 K. TlGaS2 single crystal at T = 300 K.

4 of TlGaS2 samples. All measurements were performed 10 Hz the permittivity ε was practically frequency- at T = 300 K. The accuracy in determining the reso- independent. nance capacitance and the quality factor Q = 1/tanδ of the measuring circuit was limited by errors related to Figure 2 shows the experimentally measured frequency dependence of the ac conductivity of a TlGaS2 the resolution of the device readings. The accuracy of σ the capacitor graduation was ±0.1 pF. The reproducibil- single crystal at T = 300 K. The ac conductivity ac var- ity of the resonance position was ±0.2 pF in capacitance ies as f 0.8 in the frequency range 5 × 104Ð106 Hz and σ 2 6 × and ±(1.0Ð1.5) scale divisions in quality factor. follows a quadratic law ac ~ f in the range f = 10 Ð3 7 σ 0.8 10 Hz. The ac ~ f dependence indicates that the mechanism of charge transport is hopping over local- 3. EXPERIMENTAL RESULTS ized states near the Fermi level [6]. The magnitude of AND DISCUSSION this conductivity is much greater than that of the dc Figure 1 shows the experimental frequency depen- hopping conductivity of TlGaS2 [1]. This charge trans- δ dence of the dissipation factor tan for a TlGaS2 single port mechanism is characterized by the following crystal (curve 1). The tanδ(f) curve has two branches: a expression obtained in [7]: steadily descending one (in the frequency range 5 × 4 6 6 σ ()() π3 2 2 5 []()ν 4 10 Ð10 Hz) and a rising one at f > 10 Hz. A signiﬁcant ac f = /96 e kTNFa f ln ph/ f , (1) dispersion in tanδ is observed for f > 107 Hz. At room where e is the elementary charge, k is the Boltzmann temperature, where TlGaS2 single crystals exhibit appreciable ac conductivity, conductivity loss becomes constant, NF is the density of localized states at the the main dielectric loss mechanism. Fermi level, a = 1/α is the localization length, α is the decay parameter of the wave function of a localized We also measured the electric capacitance of Ψ Ðar ν × 4 × charge carrier, ~ e , and ph is the phonon fre- TlGaS2 samples in the frequency range 5 10 Ð3 7 quency. Using expression (1), we can calculate the den- 10 Hz; the capacitances were 21Ð24 pF. The maximum sity of states at the Fermi level from the measured val- capacitance of TlGaS2 samples corresponded to a fre- ues of the conductivity σ (f). For T = 300 K, ν = 7 ac ph quency of ~3 × 10 Hz. Using the measured capacities 12 6 ε 10 Hz, and f = 10 Hz, we obtain from Eq. (1) the fol- of TlGaS2 samples, we calculated the permittivity at lowing relation: different frequencies; the permittivity varied from 26 to 30 (curve 2 in Fig. 1); i.e., no appreciable dispersion of N2 = 2.1× 1050σ ()f aÐ5, (2) ε was observed in the whole frequency range investi- F ac gated. where the localization length is measured in angstroms, σ ΩÐ1 Ð1 Ð1 Ð3 It was shown in [2] that the permittivity of TlGaS2 ac in cm , and NF in eV cm . The quantity NF for 4 × 18 Ð1 Ð3 samples is frequency-independent in the interval 10 Ð TlGaS2 calculated using Eq. (2) is 2.1 10 eV cm 7 ε ± 10 Hz, = 22 2. In [4], the TlGaS2 permittivity at a with the localization radius chosen as 14 Å, in analogy frequency of 105 Hz was found to be ε = 25. In [5], the with the GaS single crystal [8], which is a double ana- ε ≈ TlGaS2 permittivity varied in the limits 20Ð38 in the log of TlGaS2. This value of NF agrees well with the 2 6 × 18 Ð1 Ð3 frequency range 10 Ð10 Hz; the dispersion of ε was value NF = 2 10 eV cm found in experiments on 2 4 observed in the range 10 Ð10 Hz, whereas for f > the dc conductivity across the TlGaS2 layers [1].

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1010 MUSTAFAEVA

× 18 Ð1 Ð3 The theory of ac hopping conductivity provides an (NF = 2.0 10 eV cm ). The average hopping dis- τ σ σ opportunity to determine the average time of charge tances are 108 and 103 Å for dc and ac, respectively. carrier hopping from one localized state to another The values of the TlGaS2 permittivity agree with the using the formula [6] results from [2, 4, 5], and the frequency dependences σ 0.8 σ 2 τÐ1 = ν exp()Ð2Rα , (3) ac ~ f and ac ~ f observed for TlGaS2 in the ranges ph f = 5 × 104Ð106 Hz and f > 106 Hz, respectively, are sim- where R is the average hopping distance. Using the ilar to those obtained for the isostructural TlInS2 com- σ τÐ1 × experimental ac(f) dependence, is found to be 5 pound in [9]. 105 Hz (τÐ1 is deﬁned as the average frequency for σ 0.8 which ac obeys the f law). This corresponds to the REFERENCES average hopping time τ = 2 µs. The average hopping distance calculated by the formula 1. S. N. Mustafaeva, V. A. Aliev, and M. M. Asadov, Fiz. Tverd. Tela (St. Petersburg) 40 (4), 612 (1998) [Phys. ()να () Solid State 40, 561 (1998)]. R = 1/2 ln ph/ f (4) 2. A. M. Darvish, A. É. Bakhyshov, and V. I. Tagirov, Fiz. is R = 103 Å, which is approximately seven times Tekh. Poluprovodn. (Leningrad) 11 (4), 780 (1977) [Sov. greater than the average distance between localization Phys. Semicond. 11, 458 (1977)]. centers. From the measurements of the TlGaS2 dc con- 3. A. É. Bakhyshov, R. S. Samedov, S. Bules, and ductivity, the average value of R was found to be 108 Å V. I. Tagirov, Fiz. Tekh. Poluprovodn. (Leningrad) 16 [1]. (1), 161 (1982) [Sov. Phys. Semicond. 16, 98 (1982)]. σ 4. R. A. Aliev, K. R. Allakhverdiev, A. I. Baranov, In contrast to the results from [2], where ac follows 0.8 5 7 N. R. Ivanov, and R. M. Sardarly, Fiz. Tverd. Tela (Len- the f law in the whole frequency range 10 Ð10 Hz ingrad) 26 (5), 1271 (1984) [Sov. Phys. Solid State 26, investigated, in our experiments, as indicated above, the 775 (1984)]. σ 2 quadratic dependence ac ~ f was observed for f > 5. A. U. Sheleg, K. V. Iodkovskaya, and N. F. Kurilovich, 106 Hz (Fig. 2). It was shown in [6] that the conductiv- Fiz. Tverd. Tela (St. Petersburg) 45 (1), 68 (2003) [Phys. ity proportional to f 2 is related to optical transitions in Solid State 45, 69 (2003)]. semiconductors and is dominant at high frequencies. 6. N. F. Mott and E. A. Davis, Electronic Processes in Non- σ 2 Previously, we also observed the quadratic law ac ~ f Crystalline Materials (Clarendon, Oxford, 1971; Mir, 6 Moscow, 1974). in TlInS2 single crystals for f > 10 Hz [9]. 7. M. Pollak, Philos. Mag. 23, 519 (1971). 8. V. Augelli, C. Manfredotti, R. Murri, R. Piccolo, and 4. CONCLUSIONS L. Vasanelli, Nuovo Cimento B 38 (2), 327 (1977). Thus, the results of the study of the dc [1] and ac 9. S. N. Mustafaeva, M. M. Asadov, and V. A. Ramazan- conductivities across the TlGaS layers are in good zade, Fiz. Tverd. Tela (St. Petersburg) 38 (1), 14 (1996) 2 [Phys. Solid State 38, 7 (1996)]. agreement. The dc conductivity at T ≤ 200 K and the ac conductivity for f ≤ 106 Hz are both due to charge carrier hopping over localized states near the Fermi level Translated by I. Zvyagin

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

SEMICONDUCTORS AND DIELECTRICS

New Data on RC Radiation Centers Obtained from a Photoluminescence Study of Cubic Boron Nitride Irradiated with Electrons of Near-Threshold Energies E. M. Shishonok* and J. W. Steeds** *Institute of Solid State and Semiconductor Physics, Belarussian Academy of Sciences, ul. Brovki 17, Minsk, 220072 Belarus e-mail: [email protected] **Bristol University, Bristol, Great Britain Received October 28, 2003

Abstract—New data on radiation centers of the RC series are obtained by studying the photoluminescence of boron nitride cubic crystals irradiated with electrons of energies of 150 to 300 eV and annealed in vacuum at temperatures of 400 to 800°C. Changes in the fine structure of no-phonon lines from these centers were observed as the electron energy reached the “threshold” value or after annealing. These changes are likely to be associated with the response of a vacancy (which is a structural element of RC defects) to differently directed uniaxial stresses. The fine structure of the no-phonon line from the RC3 center is less sensitive to threshold effects and relaxes more easily during annealing in comparison with those from the RC1 and RC2 centers. New data are obtained on the fine structure of phonon replicas of no-phonon lines from RC centers; this structure is due to interaction of the corresponding electron transitions with lattice phonons of energies of 0.1 to 0.125 eV. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION ted by three Gaussians. The reason for no-phonon line splitting was associated with the JahnÐTeller effect. Cubic boron nitride (cBN) is well known as a wide- Another luminescence center, RC4, was observed in band-gap semiconductor; this compound is similar to CL spectra of cBN only after irradiation of samples diamond but excels it in many characteristics, such as with ions and was not included in the RC series men- chemical resistance and ability to emit electrons and to tioned above. form materials exhibiting both types of electrical con- The RC2 and RC2ÐRC3 centers were detected in ductivity. Cubic boron nitride offers higher resistance CL spectra of polycrystals [3] and (111) faces of cBN to radiation than does diamond; therefore, it can be single crystals [4] after the samples were subjected to a used for devices operating under high-radiation condi- pressure of up to 3.5 GPa. tions. Unfortunately, the properties of this compound have not been adequately studied. The effect of irradiation with electrons of near- threshold energies (150Ð300 keV) on the photolumi- Studies into the effect of irradiation with 4.5-MeV nescence (PL) of single-crystal cubic boron nitride was electrons, γ rays, and ions on the defect structure of studied in [5], and a new radiation-induced (apparently cubic boron nitride were performed in [1, 2]. Cathod- interstitial) defect, BN1, was detected. This defect was oluminescence (CL) spectra of irradiated samples characterized by a no-phonon luminescence line at revealed the presence of a series of luminescence cen- 376.5 nm [5], and its total photoluminescence spectrum ters (irrespective of the quality of the samples) charac- was similar in structure to the CL band of the center terized by no-phonon lines at 545 (RC1), 577 (RC2), observed at 3.188 eV in diamond [6]. The formation of and 624 nm (RC3). These centers were found to be BN1 defects in cBN was observed to occur after the Frenkel pairs differing in the mutual orientation of samples were irradiated with electrons with an energy nitrogen vacancies and nitrogen interstitials. The tem- of 230 keV, which is equal to the threshold energy for peratures of annealing after which the RC lines in CL displacements of intrinsic atoms. spectra of cBN samples irradiated with ions and elec- In this work, we studied the photoluminescence of trons disappeared completely were different. For sam- cubic boron nitride irradiated with electrons of near- ples irradiated with electrons, the annealing tempera- threshold energies. New data are obtained on the structure was approximately 1000 K. It was also shown in ture of the luminescence spectra of RC centers in cBN, [1, 2] that all no-phonon lines of RC centers in samples which are of both fundamental and experimental inter- irradiated with ions can be only doublets. The experi- est. Particular attention is given to the differences in the mental proﬁles of these doublets were most closely ﬁt- defect structure between cBN samples irradiated with

1012 SHISHONOK, STEEDS

IPL, arb. units the samples were annealed in vacuum at temperatures of 400 to 800°C. RC2 (a) 20000 3. RESULTS AND DISCUSSION RC3 RC1 It was established that cubic boron nitride crystals BN1 irradiated with electrons of near-threshold and high RC4 energies differed not only in the luminescence centers formed but also in the inﬂuence of irradiation on the parameters of no-phonon lines of RC centers, including 0 the energy position of the maxima and the structure of the lines. 400 500 600 700

RC1 RC3 RC2 (b) 3.1. The Formation of RC Defects 1 It was found in [5] that irradiation of nitrogen- 20000 enriched cBN single crystals with low-energy electrons does not bring about the formation of RC defects. This fact can be explained by radiation-induced defects 2 3 4 RC4 being compensated for by the defects that were in cBN (due to departures from stoichiometry) before the irradiation. An analogous phenomenon was observed in 0 5 electron-irradiated diamond; more speciﬁcally, the 550 600 650 700 GR-1 luminescence center was observed in CL spectra Wavelength, nm of diamond [7] only after a critical radiation dose was reached or after the electron energy was increased in Fig. 1. (a) PL spectrum obtained from a region of a cubic order to increase the defect production rate. For exam- boron nitride single crystal irradiated with 230-keV elec- ple, RC defects were observed in all cBN samples irre- trons and (b) the lines of RC centers in PL spectra obtained spective of their quality when the electron energy was from different regions of a cBN crystal irradiated with elec- increased up to 4.5 MeV. trons with energies of (1) 300, (2) 230, (3) 200, (4) 170, and (5) 150 keV. All centers of the RC series (RC1, RC2, RC3, and RC4, corresponding to PL lines at 543.9, 576.8, 623.1, and 667.3 nm, respectively) were represented in PL electrons of low (150Ð300 keV) and high (4.5 MeV) spectra from ﬁve regions of a boron-enriched crystal energies. irradiated with electrons of various energies (150, 170, 200, 250, 300 keV; dose of 2 × 1020 cmÐ2); however, the intensity distributions of no-phonon lines of these cen- 2. EXPERIMENTAL ters and their energy positions were different in these PL spectra of samples of cubic boron nitride were regions (Fig. 1). recorded with a Renishaw-1000 spectrometer at tem- The association between the formation of RC perature T = 6 K using a helium cryostat and a laser defects and the stoichiometry of cBN substantiates the operating at a wavelength of 488 nm. The surface of a assumption (made in [1, 2, 5]) that these defects are crystal was scanned in the discrete mode to form a map intrinsic. and in the line-scan mode along directions crossing the Figure 1a shows a typical PL spectrum from a trans- irradiated and unirradiated areas of the surface, which parent boron-enriched cBN single crystal irradiated made it possible to study the irradiated and nonirradi- with 230-keV electrons at room temperature. No- ated materials simultaneously. phonon lines and their phonon replicas corresponding Optically transparent cBN single crystals 200 to to the BN1, RC1, RC2, RC3, and RC4 centers are 400 µm in size were studied. The crystals were synthe- observed in the spectrum. sized from hexagonal boron nitride (hBN) at high pres- In this study, the RC4 center was observed to appear sures and temperatures using catalysts that ensured an in the PL spectrum only after cubic boron nitride was excess of boron or nitrogen in the crystals (light smoky irradiated with electrons of energies of 150 to 230 keV gray or light yellow in color, respectively). Different at room temperature, which contradicts the CL data [2]. areas of the single-crystal surfaces were irradiated with This center was not observed in PL spectra of cBN sam- electrons (directly in a Philips EM430 TEM electron ples irradiated with low-energy electrons at liquid- microscope) of different energies in a single experi- helium temperatures and with 300-keV electrons at ment (300, 230, 200, 170, 150 keV) [5] at room and liq- room temperature, which can be due to differences in uid-helium temperatures (only at 230 keV), and then structure and stability between the RC1ÐRC3 and RC4

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

NEW DATA ON RC RADIATION CENTERS 1013 defects. We note that the latter center is represented by IPL, arb. units a weak no-phonon line. LO TO 3.2. The Structure of Phonon Replicas 20000 BN1 of No-Phonon Lines of the RC Centers 132 125 64 The Renishaw-1000 spectrometer enabled us to resolve and analyze the structure of the PL spectra of 48 75 107 RC centers in more detail than that of the CL spectra 0 117 obtained earlier. RC2 It was found [2] that electron transitions in the RC1Ð 0 0.05 0.10 0.15 , eV RC4 centers involve phonons with an energy of 64 meV E presumably localized on a nitrogen vacancy. In this Fig. 2. PL spectra of the BN1 (first phonon replica) and RC2 study, we established that phonon replicas of the no- centers in which the energy (in millielectronvolts) is reck- phonon lines of all RC centers are similar in structure; oned from the position of the maximum of the no-phonon however, their structure was found to be more complex line. than assumed earlier. Each of the four no-phonon lines is accompanied by a phonon wing consisting of two 3.3. The Structure of the No-Phonon Lines wide bands, which also have a structure (see, e.g., the of RC Centers PL spectrum of the RC2 center in Fig. 2). It was found that the structure of the first band of each of the four In this study, we established that the structure of the phonon wings is associated not only with a phonon no-phonon lines of RC centers in electron-irradiated cBN is sensitive to the electron energy and annealing with an energy of 64 meV but also with RC phonons temperature. with energies of 47Ð48 and 71Ð75 meV, whose interaction with electron transitions is less strong. Note that The no-phonon lines of RC centers are observed in electron transitions occurring in the GC-1 and GC-2 the form of doublets in PL spectra of cBN, in contrast to CL spectra, after cBN is irradiated with electrons of centers (nitrogen and boron vacancies bound to identi- even near-threshold energies, which is likely due to the cal defects of unknown nature [8]) interact not only differences in the conditions of excitation of PL and with phonons with energies of 64 and 56 meV, respec- CL, as was the case with the GR-1 center in diamond tively, but also with phonons with an energy of 48 meV. [9]. However, there is a relation between the splitting of The second band in the phonon wings of the no- a no-phonon line and the electron energy in the energy phonon lines of all RC centers has a fine structure, range under study (Fig. 3). As the electron energy increases from 150 to 200 keV (in this range, BN1 whose resolution depends on the sample. In addition to defects form in small proportion or do not form at all), the PL spectrum of the RC2 center, Fig. 2 shows the no- the no-phonon lines of the RC2, RC3, and RC4 centers phonon line and its phonon wing (the first phonon rep- (in the form of singlets) shift to higher energies by 2.3 lica) for the BN1 center [5], with the energy measured (RC2) and 2.2 meV (RC3). Irradiation with 230-keV from the maximum of the no-phonon line of the corre- electrons (producing BN1 defects in quantity) causes a sponding center. It can be seen that the fine structures higher energy (wider) component to split off from the of the phonon wings correlate well with each other in no-phonon lines of the RC2 and RC3 centers by the range 0.1Ð0.125 eV. 2.1 meV. The no-phonon line of the RC1 center does not shift noticeably, but a component splits off from it It is likely that the high-energy part of the phonon by ∆ = 1.9 meV at the same electron energy. As the wing of the BN1 no-phonon line reproduces the phonon electron energy is increased up to 300 keV (where BN1 density of states of cubic boron nitride, with the fea- defects no longer form), the no-phonon lines of all RC tures of the fine structure corresponding to singular centers tend to shift to lower energies. A 400°C anneal- points of the phonon spectrum, as was the case for an ing of a cBN single crystal irradiated with electrons of analog of the BN1 center in the luminescence spectrum threshold energy (230 keV) causes broadening of all of diamond (at 3.188 eV) [6]. Therefore, electron tran- no-phonon lines and smoothes their structure but does sitions in the RC centers strongly interact with phonons not change the energy position of the maxima of their with energies of 100 to 125 meV, which are very likely components. Subsequent heat treatment at 650°C lattice phonons. makes the no-phonon lines narrower than their initial profiles by decreasing the distance between the energy Annealing of the single crystals at temperatures of positions of their components. Annealing at 800°C 400 to 800°C did not change the structure of the phonon increases the distance between the components of the wings of the RC center no-phonon lines. no-phonon line of the RC1 center, and this line splits;

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

1014 SHISHONOK, STEEDS

IPL, arb. units IPL, arb. units 30000 1 20000 RC1 RC3 2

20000 3

2 1 10000 10000

4 4 0 540 542 544 546 548 622 623 624 625 Wavelength, nm RC2 1 Fig. 4. No-phonon line of the RC3 center in PL spectra 20000 obtained from a region of a cBN single crystal (1) irradiated with electrons with an energy of 230 keV and then annealed 2 at temperatures of (2) 400, (3) 650, and (4) 800°C. 10000 3 Since the no-phonon lines of all RC centers were 4 asymmetric, we not only considered their doublet struc- 5 0 ture but also performed a thorough analysis of their fine structures by decomposing them into components using 574 576 578 580 582 the ORIGIN-5 software package. The optimal decom- ∆ 30000 position corresponded to the minimal error I in fitting RC3 1 the experimental no-phonon line profile and to the minimum possible number of components used for the given line profile. 20000 The number of components used to fit the no- 4 2 phonon lines of RC centers (6, 4, 3, 2) depended on the 10000 RC center, electron energy, and annealing temperature 3 (Fig. 5). Six components were used to decompose the no-phonon line of the RC1 center observed in PL spec- 5 tra after irradiation with electrons with energies of 300 0 622 623 624 625 626 and 230 keV, and three components, for electron ener- Wavelength, nm gies of 170 and 150 keV or after annealing at 650Ð 800°C. After irradiation with electrons with energies of Fig. 3. No-phonon lines of the RC1ÐRC3 centers in PL 300, 230, and 170 keV, the no-phonon line of the RC2 spectra obtained from different regions of a cBN single center was decomposed into 4, 4, and 3 components, crystal irradiated with electrons with energies of (1) 300, (2) respectively, and after annealing at T = 400, 650, or 230, (3) 200, (4) 170, and (5) 150 keV. 800°C, into 3, 2, and 2 components, respectively. The number of components used to decompose the no- phonon line of the RC3 center was 3, 2, and 2 after irra- the low-energy components of the no-phonon lines of diation with electrons with energies of 230, 200, and the RC2 and RC3 centers increase in intensity, whereas 170 keV, respectively. For samples irradiated with elec- the high-energy components decrease in intensity (as trons with an energy of 230 keV and then annealed at an example, Fig. 4 shows the no-phonon line of the 400, 650, or 800°C, the fine structure of the no-phonon RC3 center). In contrast to an earlier report, annealing line of the RC3 center was decomposed into 3, 3, and 2 at 850°C does not cause a noticeable decrease in the components, respectively. intensity of the no-phonon lines of any RC centers. Note that the case where only low-energy compo- 4. DISCUSSION nents are observed in the structure of no-phonon lines The symmetry of defects can be lower than the sym- (RC2, RC3) is characteristic of the crystal regions irra- metry of their position in the perfect crystal, e.g., diated with electrons of energies below or above the because of a spontaneous distortion lowering the sym- threshold value (i.e., in the absence of BN1 defects) and metry and resulting in a more stable configuration of a of crystals annealed at 800°C. defect (the JahnÐTeller effect). The symmetry group of

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

NEW DATA ON RC RADIATION CENTERS 1015

10000 RC1-230 (a) RC1-170 (a') 2000 9000

8000 1500 7000

6000 1000

5000 500 541 542 543 544 545 541 542 543 544 545 546 20000 (b) RC1-230 4000 RC2-170 (b')

15000

3000

10000

PhL intensity, arb. units 2000

5000 574 575 576 577 578 574 575 576 577 578 579 16000 RC1-230 (c) 15 000 RC3-170 keV (c') 14000

12000 10 000 10000

8000 5000

6000 622.0 622.5 623.0 623.5 624.0 624.5 623.5 624.0 624.5 625.0 625.5 626.0

Fig. 5. Experimental and calculated shapes of the no-phonon lines of (a, a') the RC1, (b, b') RC2, and (c, c') RC3 centers in PL spectra obtained from regions of a cBN single crystal irradiated with electrons with energies of (a, c) 230 and (a', c') 170 keV. The Gaussians into which the spectral lines were decomposed are shown by thin lines.

a distorted conﬁguration is a subgroup of the original The splittings of energy levels of a defect of symme- group. For example, in the case of diamond, the Td try Td caused by tetragonal, trigonal, and orthorhombic group (vacancy GR1) can reduce to its (tetragonal) D2d, distortions can be obtained by applying uniaxial stresses to the defect along the 〈100〉, 〈111〉, and 〈110〉 (orthorhombic) C2v, and (trigonal) C3v subgroups. A distortion causes a splitting of degenerate electronic directions, respectively. states of the defect (depending on the character of the Each of the subgroups mentioned above is charac- distortion) or a shift of the corresponding energy levels. terized by electronic states that are different from the The magnitude of the shift is a measure of the amount ground and excited states corresponding to the Td of distortion [10]. group. For example, in the case of an orthorhombic dis-

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1016 SHISHONOK, STEEDS tortion of the GR1 vacancy, the ground state E is doubly split and that the defect centers are affected by stresses degenerate and the excited state T2 is triply degenerate, randomly distributed over cBN. The changes in the fine while in the case of tetragonal and trigonal distortions structure of no-phonon lines caused by “threshold” the excited state T2 is doubly degenerate [11]. In the effects characterize the end effect of directed stresses former case, the fine structure of the no-phonon line on the vacancies belonging to RC defects. Judging from (GC1) consists of a sextuplet of lines, while in the latter the number of components into which the no-phonon case it consists of a quadruplet, as observed in the lines were decomposed, the vacancy belonging to the absorption and cathodoluminescence spectra of dia- RC3 center is affected by threshold and subthreshold mond [9, 11]. stresses to a smaller extent than the vacancies in other In addition to the JahnÐTeller effect, the state of a RC centers. defect can be affected by other defects randomly dis- Annealing at temperatures of 650 to 800°C causes tributed over the lattice. In this case, the no-phonon line RC centers in the “threshold” state to relax to the “sub- associated with electron transitions between the degen- threshold” state, the RC3 center being more mobile in erate (unsplit) ground and excited states of a defect this respect (Fig. 4). belonging, say, to the Td group should be broadened. After irradiation with electrons with an energy of This conclusion is supported by the fact that the no- 4.5 MeV, stresses are randomly distributed over the phonon line of the RC3 center consists of two compo- material due to secondary displacements, with the con- nents centered at the same energy and having different sequence that all RC defects can be in the unperturbed half-widths (Fig. 5c') and that all no-phonon lines in the states for which the no-phonon lines are in the low- PL spectrum are highly broadened after annealing of energy position. These lines are either unsplit or their cBN single crystals at 400°C (curve 2 in Fig. 4). profiles can be closely fitted by three Gaussians [2]. Based on the results of the analysis of the structure of the no-phonon lines of vacancy-type RC centers, the known data on the vacancy defect V° (GR1) in irradi- 5. CONCLUSIONS ated diamond, and the similarity between crystal lat- (1) The RC1ÐRC4 centers can be produced in cBN tices of diamond and cubic boron nitride, the following by irradiating a sample with electrons of near-threshold conclusions can be drawn regarding radiation-induced energies (cathodoluminescence spectra reveal RC4 centers in cubic boron nitride. centers only after irradiating a sample with ions). The fine structure of the no-phonon lines of RC cen- (2) The no-phonon lines of RC centers in PL spectra ters changes after irradiation with electrons of the of cBN irradiated with electrons of near-threshold ener- threshold energy for atomic displacement from their gies can be doublets, in contrast to CL spectra, where lattice sites over a distance smaller than the lattice the no-phonon lines are not doublets. parameter of cBN. Vacancies that are located at sites of (3) The electron energy during irradiation affects the Td symmetry and belong to RC defects are affected differently (depending, in particular, on the charge state) position of the no-phonon lines of the RC1ÐRC4 cen- by directed and uniaxial stresses arising in the cBN lat- ters; lines with the longest wavelength are observed tice, which can be considered a manifestation of differ- after irradiation by electrons with energies above the ent types of distortion of the vacancies (or other defects threshold value. substituting for them) [10]. As in the case of diamond (4) Electron transitions in RC centers strongly inter- [11], the RC1 no-phonon line can split into six lines act with the 64-meV phonon and lattice phonons with when the vacancy is located along a 〈110〉 direction energies of 0.1 to 0.125 eV and weakly interact with the with respect to the source of its distortion and reduces 48- and 75-meV phonons. its symmetry to orthorhombic. The RC2 no-phonon (5) The no-phonon lines of the RC1ÐRC3 centers line can split into four components when the vacancy is have a fine structure, which changes as the electron located along the 〈100〉 or 〈111〉 directions relative to energy reaches a threshold value; it is likely that, at this the source of its distortion and reduces its Td symmetry threshold energy, the symmetry of the vacancies to tetragonal and trigonal, respectively. The source of belonging to RC defects is reduced to orthorhombic distortion that reduces the symmetry of a vacancy symmetry with the axis along 〈110〉 (for the RC1 cen- belonging to an RC defect can be another element of ter) and to tetragonal or trigonal symmetry with the axis the structure of the defect situated near the vacancy. along 〈100〉 and 〈111〉, respectively (for the RC2, RC3 The no-phonon lines corresponding to unperturbed centers); this change in the vacancy symmetry is due to vacancies of Td symmetry should be single lines. The directed stresses, which can be caused by an element of triplet structure of the no-phonon lines of the RC1 and the RC defect structure situated near the vacancy. These RC2 centers (observed after irradiation of cBN with conjectures can be useful in modeling the structure of electrons of energies below the threshold value) and of RC defects. the RC3 center (observed when the electron energy is (6) For samples of cubic boron nitride irradiated by equal to the threshold value) indicates that the doubly electrons with threshold energies, annealing at temper- degenerate ground states of defects with Td symmetry atures of 650 to 800°C causes relaxation of the defect

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 NEW DATA ON RC RADIATION CENTERS 1017 structure and the structure of the RC center no-phonon 4. V. B. Shipilo, E. M. Shishonok, A. M. Zaœtsev, N. G. Ani- lines to their “subthreshold” state. chenko, and L. S. Unyarkha, Neorg. Mater. 26 (8), 1651 (7) The ﬁne structure of the no-phonon line of the (1990). RC3 center is less sensitive to threshold and subthresh- 5. E. M. Shishonok and J. W. Steeds, Diamond Relat. old effects and relaxes more easily during annealing Mater. 11 (10), 1774 (2002). than do the structures of the RC1 and RC2 no-phonon 6. A. M. Zaitsev, Phys. Rev. B 61 (19), 12909 (2000). lines. 7. C. D. Clark and J. Walker, Proc. R. Soc. London, Ser. A 334, 241 (1973). ACKNOWLEDGMENTS 8. V. B. Shipilo, A. M. Zaœtsev, E. M. Shishonok, and This study was supported by the Royal Society Fel- A. A. Mel’nikov, Zh. Prikl. Spektrosk. 45 (4), 601 lowship (2001). (1986). 9. A. T. Collins, J. Phys. C: Solid State Phys. 20 (13), 2027 REFERENCES (1978). 1. A. M. Zaitsev, A. A. Melnikov, E. M. Shishonok, and 10. M. Lanno and J. Bourgoin, Point Defects in Semiconduc- V. B. Shipilo, Phys. Status Solidi A 94, 125 (1985). tors, Experimental Aspects (Springer, New York, 1981; Mir, Moscow, 1985). 2. A. M. Zaœtsev, Doctoral Dissertation (Belorus. Gos. Univ., Minsk, 1992). 11. G. Davies, Proc. R. Soc. London, Ser. A 36, 507 (1974). 3. V. B. Shipilo, E. M. Shishonok, A. M. Zaitsev, A. A. Mal- nikov, and A. I. Olekchnovich, Phys. Status Solidi A 108, 431 (1988). Translated by Yu. Epifanov

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Physics of the Solid State, Vol. 46, No. 6, 2004, pp. 1018Ð1023. Translated from Fizika Tverdogo Tela, Vol. 46, No. 6, 2004, pp. 989Ð994. Original Russian Text Copyright © 2004 by Bakaleœnikov, Zamoryanskaya, Kolesnikova, Sokolov, Flegontova. SEMICONDUCTORS AND DIELECTRICS

Silicon Dioxide Modification by an Electron Beam L. A. Bakaleœnikov, M. V. Zamoryanskaya, E. V. Kolesnikova, V. I. Sokolov, and E. Yu. Flegontova Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia e-mail: [email protected] Received October 13, 2003

Abstract—The overheating temperature of a microvolume of silicon dioxide produced by bombardment by a high specific-power electron beam has been estimated. Calculations showed that the maximum temperature to which a microvolume of silicon dioxide is overheated can be as high as 1200°C for an electron beam current of 100 nA. The variation in the cathodoluminescence characteristics of amorphous silica with different contents of hydroxyl groups was studied for various electron beam specific-power levels. The impact of a high specific- power electron beam was shown to create additional lattice defects up to the formation of silicon clusters. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION neighboring oxygen atoms, i.e., a twofold-coordinated silicon atom [4, 5]. These emission bands are excited in The SiO2/Si structure is a basic system currently a CL spectrum, because the electron beam energy (1Ð employed in the development of modern microelec- 30 keV) is many times larger than the band-gap width tronics, and it is unlikely to be replaced by a different of silicon dioxide. system in the near future. The CL properties of the SiO2/Si system vary To achieve a higher efficiency of SiO2/Si systems in strongly in the presence of silicon clusters. Silicon is an optoelectronics, various methods of their fabrication indirect-gap semiconductor with weak luminescence. It are currently being tried. The radiative properties of this emits in the IR region at 1.1 eV, but silicon microcrys- system depend on the structural features and geometric tals with linear dimensions on the order of a few parameters of its components. Such systems can be nanometers produce luminescence bands from 1.4 to produced by a variety of means, for instance, by depo- 1.8 eV [6]. The reason for this lies in the fact that the sition of off-stoichiometric silicon dioxide films, oxida- band gap width of silicon depends on the geometric size tion of porous silicon, etc. Among these techniques is of the silicon crystals [6]. The presence of silicon clus- modification of silicon dioxide by a high-energy elec- ters in silicon dioxide is always accompanied by a tron beam of a high specific power [1]. In this case, sil- strong variation in the luminescence properties and the icon clusters are produced from silicon dioxide irradi- appearance of an intense emission in the green region ated by electrons. Silicon clusters form as a result of of the spectrum with a maximum at 2.2Ð2.3 eV [7]. irradiation and local heating of silicon dioxide by the electron beam, which brings about a variation in the In CL studies, one can vary the electron beam diam- cathodoluminescence (CL) characteristics of the eter and electron current over a broad range. This microvolume. enables one to vary the specific power of the electron beam to within several orders of magnitude. If the spe- This communication reports on a study of the varia- cific power of the electron beam is high enough, the tion in the CL properties of amorphous silica bom- sample microvolume heats. Heating can produce addi- barded by a high-energy electron beam at different spe- tional structural defects in the irradiated microvolume. cific-power levels. Investigation of the temperature fields generated by CL spectra of impurity-free amorphous silicon an electron beam in various samples provides informa- dioxide contain bands related to structural defects, such tion of value for interpreting experimental data as the non-bridging oxygen atom and the twofold-coor- obtained in CL studies. Because the experiments were dinated silicon atom. The non-bridging oxygen atom performed at the maximum attainable focusing of the produces a luminescence band peaking at 1.9 eV, which electron beam (the beam diameter did not exceed corresponds to the absorption bands at 4.75 and 2.0 eV 0.1 µm), the region of heat release was small, which [2, 3]. The other intrinsic defect in silicon dioxide is makes experimental measurement of the overheating responsible for the 4.3- and 2.65-eV luminescence temperature a very difficult problem. This necessitated bands, which are excited at 5.0 eV. The most probable a theoretical estimation of the silicon dioxide overheat- model of this defect is a silicon atom with only two ing temperature under electron beam impact.

SILICON DIOXIDE MODIFICATION BY AN ELECTRON BEAM 1019

z z (a) (b)

m 2.4 2.4 ( z)

µ q p,

2.3444E16 Depth, 1.2 1.2 6.445E16 1.875E17 pp8.438E17 z z (c) (d) 1.5E18 2.4 1.275E19

m 2.4 µ 2.4E19

Depth, 1.2 1.2

Fig. 1. Monte Carlo calculation of the spatial stationary energy loss distribution in silica for various beam energies: (a) 20, (b) 15, (c) 10, and (d) 5 keV. The energy loss densities are in units of J/(m3 s K).

2. THEORETICAL ESTIMATION provides an estimate for the maximum overheating OF TEMPERATURE FIELDS IN SILICON temperature [9]. DIOXIDE We considered the problem of uniform heat genera- To estimate the effect of sample heating under elec- tion in the volume of a half-ellipsoid with semi-axes a tron beam irradiation, we calculated the temperature and b. The temperature ﬁeld is given by the equation distribution in silicon dioxide by the technique proposed in [8, 9]. The distribution of the energy loss den- ∆Tq= Ð /κ, (1) sity per electron was calculated by the Monte Carlo method in terms of the single-scattering model. For the where ∆ is the Laplace operator, κ is the thermal con- differential cross section of elastic scattering, the Mott ductivity coefﬁcient, and q is the energy loss density cross section was taken [10], while the inelastic interac- (Fig. 1), tion was computed based on optical data with the algo- 1 rithm proposed in [11].  ρ2 2  z ≤ The code followed the trajectories of only the pri- q0, -----2 + -----2 1 mary electrons, which resulted in a certain error in the  b a q =  (2) distribution of the energy loss but did not affect its total 2 2  ρ z value. The simulation took into account the carbon 0, ----- + ----- > 1. layer present on the sample surface.  b2 a2 We made use of the technique proposed in [9] to obtain the spatial distribution of the energy loss density The semi-axes a and b were twice the half-widths σ ρ ρ z q( , z) (here, z is the depth, and is the radius) for an and σρ, respectively, of the Gaussian distribution electron beam current of 100 nA and beam energies 2 2 2 2 Aexp(Ðz /σ Ð ρ /σρ ) approximating the energy loss E0 = 5, 10, 15, and 20 keV (Fig. 1). density distribution. By approximating the shape of the heat generation region with a half-ellipsoid, one can take into account We obtained estimates for the maximum overheat- the differences between the lateral and longitudinal ing temperature for the case of heat generated in a pro- source dimensions. While the real shape naturally does late ellipsoid of revolution (a > b), not coincide with this approximation, this approach 2 2 2 1 The data on the differential electron scattering cross sections q0 ab aa+ Ð b T max = ------ln------(3) computed by the present authors can be found at 4k 2 2 2 2 www.ioffe.ru/ES. a Ð b aaÐ Ð b

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

1020 BAKALEŒNIKOV et al.

100 100 nA. We also studied the dependence of the intensity of the main CL bands on exposure time to a 100-nA 30° electron beam. 75 70° 3. EXPERIMENTAL TECHNIQUE 260° The effect produced by a focused electron beam 50 1000° incident on silica was studied with a Camebax electron microprobe analyzer equipped with two optical spec-

Depth, nm 2100° trometers [14]. To broaden the range of probing the CL properties 25 of a sample, measurements were conducted in various operational regimes of the analyzer: (1) stationary regime, in which a beam bombards the 0 25 50 75 100 sample continuously to obtain an emission spectrum; Radius, nm (2) time-resolved mode, in which two spectra are measured at each point of the sample, with the first Fig. 2. Temperature field isotherms for sources of semiellip- spectrum being obtained under irradiation of the sam- tical shape. The figures at the isotherms are the overheating ple by an electron beam and the second, with a given temperatures relative to the sample temperature that were µ reached in interaction with an electron beam with an energy time delay (0.5 s to a few seconds) after the deflection of 5 eV and with a current of 100 nA. of the beam; (3) study of the emission band decay dynamics; this mode permits measurement of the emission band inten- and in an oblate ellipsoid of revolution (a < b), sity versus time following the termination of the excita- 2 tion; q0 ab a T max = ------arccot------. (4) (4) measuring the dependence of luminescence band 2k 2 2 2 2 b Ð a b Ð a intensity on the sample exposure time to the electron beam. Figure 2 shows the spatial distribution of the overheating temperature for a focused beam with an elec- The stationary and band decay dynamics spectra tron energy of 5 keV and a beam current of 100 nA. We were obtained at an electron energy of 15 keV and a readily see that, at the point of electron beam impact on beam current of 100 nA. The dependence of the lumi- the sample, the temperature rises 2100°C above the nescence band intensity on exposure time was studied sample temperature. We calculated the overheating at the same electron energy and beam current. The time temperature taking into account the finite beam diame- resolved spectra were taken at an electron energy of µ ° 15 keV and beam current of 15 nA at a gating delay of ter (0.1 m); it was found to be 1200 C for a region µ about 40 nm in diameter. Indeed, as the electron beam 200 s. current is increased to 100 nA, the optical microscope We studied CL spectra of two types of SiO2 glasses reveals irreversible surface changes in the silicon diox- containing different amounts of the hydroxyl group ide sample. A crater forms in the region of electron OH: KU with an OH content of 2400Ð3000 ppm and KI interaction with the sample. It is known that, for silicon with an OH content of 1Ð2 ppm. islands to form, it is sufficient to heat silicon oxide in The content of other impurities in the glasses stud- vacuum to 400Ð600°C [12]. It thus follows that, at an ied was below 3 ppm. Before the measurements, the electron current of 100 nA bombarding silicon oxide, glass samples were polished and a carbon layer about defects associated with oxygen deficiency should form 200 Å thick was applied to them to improve the sample in it up to the creation of silicon clusters. conductivity. Since the overheating temperature depends linearly, in a first approximation, on the probing electron current (the relation of Castaing [13]), it may be conjectured 4. MAIN RESULTS AND DISCUSSION that the structure of silicon oxide should not undergo Characteristics of the CL emission bands were pronounced modification at currents on the order of 10– obtained for the KU glass. Figure 3 presents typical 15 nA, because the overheating temperature would not time-resolved CL spectra of amorphous silica mea- be in excess of 200°C. sured at an electron beam current of 15 nA and energy In this connection, we studied primarily the CL of 15 keV with a time delay of 200 µs. spectra of silicon dioxide obtained at electron beam The CL spectra of the sample exhibit bands charac- currents of 15 nA, which contained emission lines of teristic of such defects as the non-bridging oxygen weakly modified silicon dioxide. The next in line were atom [the (1.90 ± 0.05) eV band] and the twofold-coor- the changes in the CL spectra observed with currents of dinated silicon atom [the (2.65 ± 0.05) eV band]. The

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

SILICON DIOXIDE MODIFICATION BY AN ELECTRON BEAM 1021

60 (‡) 100 (a) 40

20 0 0 0.2 0.4 0 (b) 20 50 (b) CL intensity, arb. units

10 0

CL intensity, arb. units 0 0.5 1.0

250 0 (c) 1.5 2.0 2.5 3.0 3.5 Energy, eV

Fig. 3. Time-resolved CL spectra of KU-type glass: (a) spectrum obtained under sample irradiation with an electron beam and (b) spectrum measured with a delay of 036 200 µs after termination of the sample excitation. t, ms spectrum also has a shoulder in the green region. To Fig. 4. Intensity decay dynamics obtained for some bands in the CL spectrum of KU-type glass: (a) band at 1.9 eV, derive the spectral characteristics of the bands, the CL (b) band at 2.3 eV, and (c) band at 2.65 eV. spectrum was unfolded into three constituent bands using the ORIGIN code. The observed bands were assumed to have Gaussian shape. Such a deconvolution and its half-width is (0.13 ± 0.04) eV. For the 2.2-eV allows us to interpret the green shoulder as a band peak- band, these parameters are (59 ± 3) µs and (0.40 ± ing at (2.20 ± 0.05) eV and having a half-width of 0.05) eV. The band at 2.65 eV is comparatively the µ 0.4 eV. The spectrum taken with a time delay of 200 s longest lived; its decay time was found to be (4.9 ± clearly reveals a band at 2.65 eV and a shoulder in the 0.2) ms and its half-width, (0.30 ± 0.05) eV. Because green region. Deconvolution of this spectrum showed that this shoulder can also be traced to a band with a the half-widths and decay times for different bands are maximum at 2.2 eV. The 1.9-eV band in this spectrum different, this can only mean that they are due to differ- is not observed. This implies that this band decays in a ent defects. time shorter than the gate delay time. The strongest Increasing the speciﬁc excitation power brings band, peaking at 2.65 eV, has the longest decay time; about a noticeable change in the CL spectra. We raised therefore, its intensity in the delayed spectrum is maximum. the excitation power in this study by increasing the electron beam current by an order of magnitude. Figure 5 dis- These conclusions were supported by measure- plays spectra of two types of glass obtained at an elec- ments of the decay time dynamics of the bands observed in the spectrum. As seen from Fig. 4, the band tron current of 100 nA and an energy of 15 keV. decay dynamics has an exponential character, I = A comparison of the spectra taken at 15 and 100 nA τ reveals that an increase in beam current results in a sub- I0exp(Ðt/ ), where I0 is the band intensity during the sample irradiation by the electron beam, t is the time stantial growth in the intensity of the 2.2-eV band rela- elapsed from the instant the beam deﬂects from the tive to the other bands. Comparing CL spectra of sample, and τ is the decay time of the luminescence glasses of different types shows that the KU glass (OH band. group content 2400Ð3000 ppm) produces a stronger Thus, we obtained the following spectral character- luminescence than does the KI glass, which indicates istics of the main CL bands for amorphous silicon diox- that the number of defects is larger in SiO2 with a ide. For the 1.9-eV band, the decay time is (28 ± 1) µs higher concentration of OH groups. This accounts for

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1022 BAKALEŒNIKOV et al.

150 (‡) 3 (a) 100 5 50

0 4 1.0 1.2 1.4 1.6 1.8 2.0 3 75 (b) 50 2 25 2 1 0 1.0 1.2 1.4 1.6 1.8 2.0

300 (c)

CL intensity, arb. units 1 (b) 150 CL intensity, arb. units 3

0 2.0 2.5 3.0 3.5 4.0 4.5 2 2 1000 (d) 3 500 5 1 0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 4 Energy, eV 0 200 400 600 t, s Fig. 5. CL spectra of glasses obtained in the IR and visible regions: (a) KI glass, IR spectrum; (b) KU glass, IR spec- Fig. 6. CL intensity plotted vs. exposure time to an electron trum; (c) KI glass, visible spectrum; and (d) KU glass, vis- beam for (a) KI glass and (b) KU glass: (1) band peaking at ible spectrum. 2.65 eV, (2) 1.9 eV, (3) 2.2 eV, (4) 1.4 eV, and (5) 1.1 eV. the lower intensity of the KI glass, because its structure bands at 1.35 and 2.3 eV (after 300 s). This can be contains less defects. attributed to the formation of crystalline or amorphous The spectral changes are the most pronounced in the silicon clusters of small size. The last band to reach IR region of glass emission. At low specific-excitation maximum intensity (in 350 s) is the one at 1.1 eV, powers, no emission bands appear in the IR range (1.0Ð which signals the formation of silicon clusters of a 1.8 eV). As the electron current is increased, weak fairly large size. emission bands peaking at 1.1 and 1.35 eV emerge in the spectra. The positions of the maxima were found by The pattern is different for the KU glass with a high unfolding the spectra into two constituent bands. The OH content. All bands grow in intensity with increasing shoulder with the maximum at 1.35 eV may be due electron irradiation time in the same way, except for the either to the emission of amorphous silicon or to the band at 1.9 eV. However, for all bands, it takes a longer formation of silicon clusters [15]. time to reach the maximum than in the case of the sili- Figure 6 presents the dependences of the CL band con dioxide with a low OH group content. The intensity intensity on the sample exposure time to the electron of the 1.9-eV band decreases and comes to a minimum beam. The 1.9- and 2.65-eV bands in the KI glass (with 200 s after the modification begins. This may mean that a low OH group content) reach maximum intensity 200 s first the defects associated with the high OH content are after the modification begins, thus indicating an healed and only after that do new defects form. The increase in the number of defects in the volume under band at 1.1 eV attains a maximum 400 s after the mod- study. The next to attain maximum intensity are the ification begins.

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 SILICON DIOXIDE MODIFICATION BY AN ELECTRON BEAM 1023

5. CONCLUSIONS 4. H.-G. Fitting, T. Barfles, A. N. Trukhin, B. Schmidt, A. Gulans, and A. von Czarnowsky, J. Non-Cryst. Solids We thus see that, when exposed to a high-power 303, 218 (2002). electron beam, glass with a low hydroxyl group content passes through three stages of modification [16, 17]: 5. L. N. Skuja, A. N. Streletsky, and A. B. Pakovich, Solid State Commun. 50 (12), 1069 (1984). (1) an increase in the fraction of intrinsic defects associated with either a deficiency or excess of oxygen, 6. M. S. Bresler and I. N. Yassievich, Fiz. Tekh. Polupro- vodn. (St. Petersburg) 27 (5), 873 (1993) [Semiconduc- which is caused by the rupture of the oxygenÐsilicon tors 27, 475 (1993)]. bonds in the glass; 7. M. Takeguchi, K. Furuya, and K. Yoshinara, Jpn. J. Appl. (2) a fast rise in the intensity of the band with a max- Phys., Part 1 38 (12B), 7140 (1999). imum at 2.2Ð2.3 eV, which originates from the formation of intrinsic oxygen-deficient defects, as well as the 8. S. K. Obyden, G. A. Perlovskiœ, G. V. Saparin, and S. I. Popov, Izv. Akad. Nauk SSSR, Ser. Fiz. 48 (12), appearance and growth in intensity of the 1.4-eV band 2374 (1984). associated with silicon nanoclusters or islands of amorphous silicon; 9. L. A. Bakaleœnikov, E. V. Galaktionov, V. V. Tret’yakov, and É. A. Tropp, Fiz. Tverd. Tela (St. Petersburg) 43 (5), (3) appearance and growth in intensity of the band 779 (2001) [Phys. Solid State 43, 811 (2001)]. peaking at 1.1 eV, which should be assigned to the for- 10. E. D. Palik, Handbook of Optical Constants of Solids II mation of relatively large islands of crystalline silicon. (Academic, New York, 1991). Interaction of an electron beam with KU-type glass 11. Z.-J. Ding and R. Shimizu, Surf. Sci. 222, 313 (1989). brings about, first of all, a decrease in the fraction of intrinsic defects associated with an oxygen excess. The 12. N. A. Kolobov and M. M. Samokhvalov, Diffusion and formation of islands or clusters of silicon takes more Oxidation in Semiconductors (Metallurgiya, Moscow, 1975). time. 13. R. Castaing, Thesis (Univ. de Paris, 1951). 14. M. V. Zamoryanskaya, A. N. Zamoryanskiœ, and ACKNOWLEDGMENTS I. A. Vaœnshenker, Prib. Tekh. Éksp., No. 4, 192 (1987). This study was supported by the Russian Founda- 15. M. V. Zamoryanskaya and V. I. Sokolov, Mater. Res. tion for Basic Research, project no. 03-02-16621. Soc. Symp. Proc. 737, F3.40 (2003). 16. M. V. Zamoryanskaya, V. I. Sokolov, and A. A. Sytnik- REFERENCES ova, Solid State Phenom. 78Ð79, 349 (2001). 17. M. V. Zamoryanskaya, V. N. Bogomolov, S. A. Gurevich, 1. G. Allan, C. Delerue, and M. Lannoo, Phys. Rev. Lett. 78 V. I. Sokolov, A. A. Sitnikova, and I. Smirnova, Fiz. (16), 3161 (1997). Tverd. Tela (St. Petersburg) 43 (2), 357 (2001) [Phys. 2. L. N. Skula and A. R. Silin, Phys. Status Solidi A 70, 43 Solid State 43, 373 (2001)]. (1982). 3. C. Delerue, G. Allan, and M. Lannoo, J. Lumin. 80, 65 (1999). Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

Physics of the Solid State, Vol. 46, No. 6, 2004, pp. 1024Ð1026. Translated from Fizika Tverdogo Tela, Vol. 46, No. 6, 2004, pp. 995Ð997. Original Russian Text Copyright © 2004 by Stefanovich, Suslikov, Gad’mashi, Peresh, Sideœ, Zubaka, Galagovets. SEMICONDUCTORS AND DIELECTRICS

Optical Phonons in Rb2TeBr6 and Cs2TeBr6 Crystals V. A. Stefanovich, L. M. Suslikov, Z. P. Gad’mashi, E. Yu. Peresh, V. I. Sideœ, O. V. Zubaka, and I. V. Galagovets Uzhgorod National University, Uzhgorod, 88000 Ukraine e-mail: [email protected] Received October 28, 2003

Abstract—Raman scattering in Rb2TeBr6 and Cs2TeBr6 crystals is studied. The phonon spectra of the crystals are calculated using the factor group method. The number of Raman-active modes, their symmetries, and selection rules are found. Observed Raman spectrum lines are identiﬁed with atomic vibration modes of the crystal. © 2004 MAIK “Nauka/Interperiodica”.

I VI VII Crystals of alkali metal chalcogenides A2 B C6 10.713 Å for Rb2TeBr6. The primitive unit cell contains (where AI is K, Rb, or Cs; BVI is S, Se, or Te; CVII is Cl, 5 nine atoms (fcc Bravais lattice). The space group Oh is Br, or I) represent a vast class of semiconductors whose characterized by the following types of atomic posi- physical properties are virtually unexplored [1Ð3]. X- tions: 2O (1), T (2), D (6), C (6) (6), C (8), ray structure analysis shows that alkali metal chalco- h d 2h 4v 3v genides crystallize into a cubic lattice with space group 3C2v(12), 2Cs(24), and C1(48). X-ray analysis reveals 5 that Rb and Cs atoms occupy sites with point symmetry Oh ÐFm3m (except K2TeBr6) [4, 5]. Therefore, these group Td, Te atoms occupy sites with symmetry Oh, and materials are promising for use in optical modulation Br atoms occupy sites with symmetry C4v. The results and scanning devices [6, 7] and comprehensive study of of factor group analysis of vibration spectra of their optical properties is of considerable interest. Rb2TeBr6 and Cs2TeBr6 crystals based on these data are In this paper, the phonon spectra of Rb2TeBr6 and presented in Table 1. Cs2TeBr6 crystals is studied by means of Raman spec- It follows from this table that lattice vibrations of the troscopy. In addition to independent determination of crystals under study are distributed over irreducible the crystal structure of the materials, this technique pro- representations of the factor group Oh of the crystal at vides valuable information about interactions of atoms the center of the Brillouin zone in the following way: in the crystal and about the nature of their chemical Γ bonding. Our experiments were performed at room = A1g +++++Eg F1g 2F2g 4F1u F2u. temperature. Γ Acoustic modes have symmetry ac = F1u; therefore, Rb2TeBr6 and Cs2TeBr6 single crystals were grown using the directional solidification method (Bridgman Γ ()k = 0 = A ()R ++E ()R 2F ()R +3F ()IR . technique). The solidification front velocity was 0.1Ð opt 1g g 2g 1u 0.3 mm/h, and the temperature gradient in the solidifi- Here, R stands for Raman scattering and IR, for infra- cation zone was 2Ð4 K/mm. Crystals were annealed at red reflection. a temperature of 473 K for three days and then cooled to room temperature at a rate of 30 K/h. The Rb2TeBr6 Because the crystals under study have a center of inversion, there are selection rules according to which and Cs2TeBr6 single crystals thus obtained were 25 to 55 mm long and 15 to 22 mm in diameter. the Raman spectra contain only four vibrational modes: one A mode, one E mode, and two F modes. The phonon spectra of the crystals under study were 1g g 2g calculated beforehand using the positional symmetry The experimental studies of the Raman spectra were method [8, 9]. As mentioned above, Rb2TeBr6 and performed on a Raman spectrometer DFS-24 with two Cs2TeBr6 compounds have cubic structure (space group identical diffraction gratings (replicas) serving as dis- 5 persive elements. The gratings had 1200 lines per mil- Oh ÐFm3m) and contain four formula units per unit limeter and worked in the first order. A photomultiplier cell. Atoms occupy sites as follows: 8 Rb (Cs) atoms FÉU-79 was used as a detector of light. The instanta- are in position d (0, 1/2, 1/4), 4 Te atoms are in position neous light intensity (averaged over the specified char- a (0, 0, 0), 8 Br atoms are in position e (0, 0, 0.2509), acteristic time) was measured using electronic circuit and 16 Br atoms are in position h (0.2341, 0.2756, 0). and registered with a PDA-1 chart recorder. The spectra The lattice constant a is 10.918 Å for Cs2TeBr6 and obtained were processed on computer.

OPTICAL PHONONS IN Rb2TeBr6 AND Cs2TeBr6 CRYSTALS 1025

Table 1. Factor group analysis of lattice vibrations of Rb2TeBr6 and Cs2TeBr6 crystals Correlation Site symmetry with Number and Number of inclusion of translation Factor group O Symmetry Activity type of atoms h modes (groups C4v, Td, Oh) α α α A1g 1 xx + yy + zz R A1 6Br E 1 α + α Ð 2α , R g αxx αyy zz xx Ð yy

E F1g 1 α α α F2g 2 xy, yz, zx R 2Cs F2 µ µ µ F1u 4 x, y, z IR Te F1u F2u 1

An LGN-215 HeÐNe laser (λ = 0.6328 µm, W = The Raman spectrum of this molecule contains two ν ν 50 mW) working in a single-mode regime was used to stretching vibrations 1 and 2 with A1g and Eg symme- excite Raman spectra. We utilized the 90° scattering ν tries and one bending vibration 5 with F2g symmetry. geometry, which provides for maximal collections of 2Ð scattered light. This geometry offers the best reduction Frequencies of these modes were measured for TeBr6 of parasitic Tyndall and Rayleigh scattering. A polariz- ions in a water solution: ν = 166, ν = 151, and ν = ing prism was used to check the polarization of the scat- 1 2 5 tered light. The spectral resolution was 1 cmÐ1. The rel- 174 ative error in measuring the intensity for the maxima of (a) spectral lines and of their half-width distortion in comparison to the true half-width of the spectral lines did not exceed 1%. We used single crystals of good optical 91 quality oriented along the {100} planes. Because of the cubic symmetry of the crystals and the small number of lines in the Raman spectra, it is possible to identify observed vibrations correctly by means of unpolarized 150 , arb. units light measurements. I

The experimental Raman spectra of Rb2TeBr6 and 45 Cs2TeBr6 crystals are presented in Fig. 1, and the symmetries and frequencies of all observed modes are listed in Table 2. The experimental results agree with theoretical calculations; the Raman spectra do have 90 four vibrational modes, in contrast to those reported in (b) [10], where only three modes were observed. Vibra- tions with the F1u symmetry could be observed in the IR-transmission (reﬂection) spectra only. Hence, Raman spectroscopy data conﬁrm the structure of 180 Rb2TeBr6 and Cs2TeBr6 crystals established previously. Our interpretation of the experimental results is 155 , arb. units

based on the features of the crystal structure. The lat- I tices of these compounds consist of closely packed 54 {Rb(Cs)Br3} layers, with half of the [Br6] octahedral voids ﬁlled with Te atoms. Hence, it is possible to single out an isolated group of atoms TeBr6 with an octahedral symmetry Oh. The vibrational spectrum of the 0 50 100 150 200 250 crystal can be considered to consist of internal vibra- ν, cm–1 tions of these atomic complexes and their external oscillations relative to Rb(Cs) atoms. Normal modes of Fig. 1. Raman spectra of (a) Cs2TeBr6 and (b) Rb2TeBr6 the free octahedral molecule are shown in Fig. 2 [11]. crystals.

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1026 STEFANOVICH et al.

Table 2. Raman-active modes of Rb2TeBr6 and Cs2TeBr6 mode to external oscillations of the TeBr6 complex rel- crystals ative to the Rb and Cs atoms situated in the octahedral Ð1 Frequency, cmÐ1 voids. The frequency of these vibrations is 45 cm for Ð1 Symmetry the Cs2TeBr6 crystal and 54 cm for the Rb2TeBr6 crys- Rb2TeBr6 Cs2TeBr6 tal (Fig. 1, Table 2). The minor difference in electrone- A 180 174 gativity between the Te and Br atoms indicates the 1g covalent nature of the TeÐBr bond. This feature is the Eg 155 150 cause of noticeable changes in polarizability during rel- F2g 90 91 ative oscillations of these atoms and of the considerable F2g 54 45 intensity of the Raman scattering lines corresponding to these oscillations (Fig. 1). At the same time, the RbÐ (TeBr6) bond has a pronounced ionic nature and the Ð1 2Ð Raman scattering line corresponding to this stretching 74 cm [11, 12]. Because the TeBr6 ion and crystals under study are similar in structure, we can assume that vibration has a low intensity. Note that the replacement the crystal field does not cause significant distortion of of Rb atoms in these crystals by the heavier Cs atom has only a slight effect both on the frequencies of the 2Ð the TeBr6 ion and does not break the selection rules phonon modes at the center of the Brillouin zone for its vibrational spectrum. The crystal field can give (Table 2) and on the intensities of the corresponding rise to an insignificant frequency shift of the modes, Raman spectral lines. and this is actually observed experimentally. Hence, for the Rb2TeBr6 and Cs2TeBr6 crystals, the vibrational REFERENCES modes at 180 and 174 cmÐ1, respectively, could be 1. V. A. Grin’ko, V. V. Safonov, V. I. Ksenzenko, and attributed to the fully symmetric A1g stretching vibra- B. G. Korshunov, Zh. Neorg. Khim. 17 (6), 1755 (1972). Ð1 tion of bromine atoms; the modes at 155 and 150 cm , 2. G. M. Serebryannikova, V. V. Safonov, G. R. Allakhver- respectively, to the antisymmetric Eg stretching vibra- dov, V. A. Grin’ko, B. D. Stepin, and V. I. Ksenzenko, tion; and the modes at 90 and 91 cmÐ1, respectively, to Zh. Neorg. Khim. 18 (1), 155 (1973). the triply degenerate F2g bending vibration of the TeBr6 3. V. V. Safonov, T. V. Kuzina, L. G. Titov, and V. G. Kor- shunov, Zh. Neorg. Khim. 21 (4), 585 (1976). group. The Raman spectrum of the free TeBr6 molecule has only one F mode. In Rb TeBr and Cs TeBr com- 4. A. K. Das and I. D. Brown, Can. J. Chem. 44 (7), 939 2g 2 6 2 6 (1966). pounds, another Raman active F2g mode arises because of the presence of the alkali atoms and the appearance 5. W. Abriel and J. Ihringer, J. Solid State Chem. 52 (2), of a crystal field (Table 1). It is natural to attribute this 274 (1984). 6. Yu. K. Rebrin, Controlling of an Optical Beam in Space (Sovetskoe Radio, Moscow, 1977). ω ω ω 2 3 7. E. R. Mustel’ and V. N. Parygin, Methods of Light Mod- 1 ulation and Scanning (Nauka, Moscow, 1970). 8. A. I. Grigor’ev, Introduction to Oscillation Spectroscopy of Inorganic Compounds (Mosk. Gos. Univ., Moscow, 1977). 9. G. N. Zhizhin, B. N. Mavrin, and V. F. Shabanov, Optical Oscillation Spectra of Crystals (Nauka, Moscow, 1984). 10. K. I. Petrov, A. V. Konov, Yu. M. Golovin, V. A. Grin’ko, ω ω ω V. V. Sazonov, and V. I. Ksenzenko, Zh. Neorg. Khim. 19 4 5 6 (1), 82 (1974). 11. K. W. F. Kohlrausch, in Hand- und Jahrbuch der Che- mischen Physik (Edwards, Ann Arbor, 1944), Vol. 9, Part 6 [Raman Scattering Spectra (Inostrannaya Liter- atura, Moscow, 1952)]. 12. The Raman Effect, Vol. 2: Application, Ed. by A. Ander- sen (Marcel Dekker, New York, 1973; Mir, Moscow, 1977).

Fig. 2. Normal modes of the TeBr6 octahedral molecule. Translated by G. Tsydynzhapov

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

Physics of the Solid State, Vol. 46, No. 6, 2004, pp. 1027Ð1029. Translated from Fizika Tverdogo Tela, Vol. 46, No. 6, 2004, pp. 998Ð1000. Original Russian Text Copyright © 2004 by Parfen’eva, Shelykh, Smirnov, Prokof’ev, Assmus.

SEMICONDUCTORS AND DIELECTRICS

Electrical Conductivity and Permittivity of the One-Dimensional Superionic Conductor LiCuVO4 L. S. Parfen’eva*, A. I. Shelykh*, I. A. Smirnov*, A. V. Prokof’ev**, and W. Assmus** *Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia e-mail: [email protected] **Institute of Physics, Goethe University, Frankfurt am Main, 60054 Germany Received October 24, 2003

σa εa εb εc Abstract—The electrical conductivity and permittivities , , and of a LiCuVO4 single crystal have been measured along the a, b, and c crystallographic axes, respectively, in the temperature range 300Ð390 K at a frequency of 103 Hz. The temperature dependences σ(T) and ε(T) were found to be typical for superionics. © 2004 MAIK “Nauka/Interperiodica”.

Measurements of the electrical conductivity σ of A chemical analysis of a large set of such high-tem- LiCuVO4 performed in the temperature interval 300Ð perature LiCuVO4 single crystals revealed, despite a 500 K and the anomalous behavior of thermal conduc- certain scatter of data, that their average composition is tivity in the range 100Ð300 K led to the conclusion [1] Li Cu VO , i.e., that they are off-stoichiometric that this compound is a quasi-one-dimensional superi- 0.92 1.03 4 onic. ε When the temperature is increased, superionic conductors exhibit, in addition to an exponential growth in 15000 A electrical conductivity, an exponential increase in the permittivity ε, which can reach fairly large values at V high enough temperatures [2, 3]. For instance, ε of the 13000 Cu Li2B4O7 quasi-one-dimensional lithium superionic [2] (which belongs to materials with the 4mm tetragonal Li symmetry) measured along the polar c axis of the crys- 11000 2 tal at ~10 Hz increases from 20 at 300 K to 4200 at c B 500 K. a 9000 The present study was aimed primarily at measuring b ε (T) for LiCuVO4 in order to verify the conclusion from Li Cu [1] that this material is a fairly good superionic. 7000

LiCuVO4 crystallizes in an orthorhombic distorted 5+ inverted spinel structure, in which the nonmagnetic V 5000 ions occupy tetrahedral voids and the nonmagnetic Li+ a and magnetic Cu2+ (S = 1/2) ions are arranged in an ordered manner in octahedral voids of the anion sublat- 3000 tice [4, 5]. The CuO6- and LiO6-octahedra make up “magnetic” and “nonmagnetic” chains, respectively, b aligned in LiCuVO4 with the b and a crystallographic 1000 directions (see inset to Fig. 1). c 0 The LiCuVO4 single crystals intended for permittiv- 300 325 350 375 400 ity and electrical conductivity measurements were T, ä grown using the technique described in [6]. They were grown from solution in a LiVO3 melt under slow cool- Fig. 1. Temperature behavior of the permittivity of a ing in the range 650Ð580°C. Such crystals are conven- LiCuVO4 single crystal in the a, b, and c crystallographic directions. Inset shows (A) the crystal structure [4] and (B) tionally called [1] “high-temperature” LiCuVO4 single the rod model [5] proposed for LiCuVO4. The mutually per- crystals. pendicular Li and Cu rod layers alternate.

1028 PARFEN’EVA et al.

dimensional superionic [2], LiCuVO4 exhibits exponential growth in ε with increasing temperature. The 4.5 increase in the permittivity is the strongest for εa(T) (Fig. 2); it scales with temperature as εa(T) ~ –1.5 exp(−∆Ea/kT) with ∆Ea ~ 0.46 eV, a ﬁgure close in 4.0 magnitude to the value of ∆Ha obtained for σa(T) (Fig. 2). K]

–2.0 –1 εb(T) and εc(T) also grow with temperature but not a 3.5 cm ε ε as strongly as (T) does (Fig. 1). This behavior of –1 εa, b, c Ω log (T) can apparently be explained by the observation

–2.5 ) [ that ionic conduction is realized most easily over the T

3.0 σ defected lithium sublattice in LiCuVO4, i.e., along the a axis. This conclusion is corroborated by our measure- log( ments of σ(T) in LiCuVO performed with an ac current –3.0 4 2.5 at ~103 Hz at 390 K in the a, b, and c directions, for which we obtained the following values: 1.5 × 10Ð4, 7 × 10Ð6, and 4.5 × 10Ð6 ΩÐ1 cmÐ1, respectively.

2.0 The permittivity of LiCuVO4 in the region 2Ð300 K was extracted earlier [9] for the a(εa) and b(εb) crystal- 2.0 2.4 2.8 3.2 3.6 lographic directions from optical high-frequency mea- 1000/T, K–1 surements in the IR region; εa varied from 10.9 at 2 K to 11.4 at 300 K, while εb remained practically constant Fig. 2. Temperature dependences of the electrical conduc- in the same temperature region (10.4 at 2 K and 10.6 at tivity and permittivity of a LiCuVO4 single crystal measured along the a crystallographic direction. 300 K). Analysis of experimental data on σ(T), ε(T), and other kinetic coefﬁcients in materials with ionic con- both in lithium and in copper [1]. The main type of duction is complicated by the need to take into account defects in these crystals is, however, vacancies on the and evaluate parameters that are determined not only by lithium sublattice. the volume of the material but also by its near-electrode The electrical conductivity σa(T) and the permittiv- regions [7, 8]. Applying an external voltage may bring εa εb εc about the formation of a space charge region to a depth ity (T), (T), and (T) were measured in the range Ð4 300Ð390 K at a frequency ω ~ 103 Hz. The experimen- d0 ~ 10 cm near one of the electrodes in this material tal data obtained on σa(T) and εa, b, c(T) for LiCuVO are [7], and it is this region which will govern the magni- 4 tude of the capacitance C and electrical conductivity presented graphically in Figs. 1 and 2. being measured. In a uniform dielectric (with no near- Consider the results obtained on σa(T). As is evident contact charged layer present), ε is determined from Fig. 2, the experimental points fall on a straight uniquely by the measured capacitance of the material line, which is described, in the case of ionic conduction, ﬁlling the space between the electrodes (a distance d by the Arrhenius relation [7, 8] apart). Therefore, the error in determining ε, with a Ð4 a near-contact space-charge layer d ~ 10 cm thick and σT = σ exp()Ð∆H /kT , (1) 0 0 for sample thicknesses d ~ 0.2 cm, would constitute a 3 where ∆H is the activation enthalpy for electrical con- d/d0 ~ 10 , which may result in an overestimated mag- duction. nitude of ε calculated under the assumption of the ∆ a absence of a near-contact space-charge layer in the H for LiCuVO4 was found to be ~0.40 eV, which is in good agreement with the values of ∆H quoted for sample under study. lithium-based superionics [8]. We attempted several experimental approaches to answering the question of whether such a near-contact Because the LiCuVO4 single crystals at our disposal σa charged layer does indeed form in our case in the mea- were fairly small, the values of may have been deter- surement of ε(T) and σ(T) in LiCuVO . mined with some error, but the slope of the log(σa) = 4 3 (1) A near-contact space-charge layer should have f(10 /T) straight line was found with good accuracy. made the dependence of the voltage drop along the The increase in σa(T) with temperature should be layer thickness d nonlinear. We observed no such effect assigned primarily to the diffusion of Li ions over on the samples studied. vacancies on the lithium sublattice in LiCuVO4. (2) The voltage on the samples on which the capac- Consider the data obtained on εa(T), εb(T), and εc(T) itance and electrical conductivity were measured with 3 (Figs. 1, 2). As in the case of the Li2B4O7 quasi-one- an ac current (ω ~ 10 Hz) was 4.3 V. We also per-

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 ELECTRICAL CONDUCTIVITY AND PERMITTIVITY 1029 formed measurements in which an additional dc volt- ACKNOWLEDGMENTS age of 20 V was applied simultaneously across the sam- This study was supported by the Russian Founda- ple, which could have produced a noticeable impact on tion for Basic Research (project no. 02-02-17657) both the magnitude and distribution of the near-elec- within the framework of a bilateral agreement between trode potential and affected the capacitance and electri- the Russian Academy of Sciences and Deutsche Fors- cal conductivity measurements. We did not observe any chungsgemeinschaft. differences between the measurements of C(T) and σ(T) with and without the dc bias applied.

(3) The concentration of the electric field (E0) in a REFERENCES thin near-contact layer of the material should have gen- 1. L. S. Parfen’eva, A. I. Shelykh, I. A. Smirnov, A. V. Pro- 4 5 erated a field E0 ~ 10 Ð10 V/cm capable of causing kof’ev, W. Assmus, H. Misiorek, J. Mucha, A. Jezowski, electric breakdown in the near-contact region or current and I. G. Vasil’eva, Fiz. Tverd. Tela (St. Petersburg) 45 instability. No such effect was observed by us experi- (11), 1991 (2003) [Phys. Solid State 45, 2093 (2003)]. mentally. 2. A. É. Aliev, Ya. V. Burak, and I. T. Lyseœko, Neorg. Thus, the above experiments permit the conclusion Mater. 26, 1991 (1990). that, most likely, no thin near-contact layer was present 3. A. A. Abramovich, A. Sh. Akramov, A. É. Aliev, and ε σ in LiCuVO4 in the (T) and (T) measurements. L. N. Fershtat, Fiz. Tverd. Tela (Leningrad) 29 (8), 2479 (1987) [Sov. Phys. Solid State 29, 1426 (1987)]. Not all factors capable of affecting the behavior of ε(T) in the LiCuVO superionic were, however, taken 4. M. A. Lafontaine, M. Leblanc, and G. Ferey, Acta Crys- 4 tallogr. C 45, 1205 (1989). into account by us (we disregarded the effect of the contact material, the existence of electronic conduction, 5. M. O. Keefe and S. Andersson, Acta Crystallogr. A 33, etc.). It would therefore be more prudent to call the ε(T) 914 (1977). obtained by us experimentally the “effective permittiv- 6. A. V. Prokofiev, D. Wichert, and W. Assmus, J. Cryst. Growth 220, 345 (2000). ity” of LiCuVO4. We emphasize in conclusion that in carrying out this 7. A. B. Lidiard, in Handbuch der Physik, Ed. by S. Flugge (Springer, Berlin, 1957; Inostrannaya Literatura, Mos- study we were not interested in the question of why cow, 1962), p. 246. εa, b, c(T) in LiCuVO have such high values or of 4 8. A. K. Ivanov-Shits and I. V. Murin, Ionics of Solids whether these values could be assigned to the existence (S.-Peterb. Gos. Univ., St. Petersburg, 2000), Vol. 1. of a near-contact space-charge region, thus invalidating our calculation of ε(T). The only thing we can be confi- 9. B. Gorshunov, P. Haas, M. Dressel, V. I. Torgashev, V. B. Shirokov, A. V. Prokofiev, and W. Assmus, Eur. dent of is that LiCuVO4 features ionic conduction, Phys. J. B 23, 427 (2001). because ε grows markedly with temperature. Furnish- ing supportive evidence for this conclusion was the main purpose of this experiment. Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

Physics of the Solid State, Vol. 46, No. 6, 2004, pp. 1030Ð1036. Translated from Fizika Tverdogo Tela, Vol. 46, No. 6, 2004, pp. 1001Ð1007. Original Russian Text Copyright © 2004 by Randoshkin, Vasil’eva, Plotnichenko, Pyrkov, Lavrishchev, Ivanov, Kiryukhin, Saletskiœ, Sysoev. SEMICONDUCTORS AND DIELECTRICS

Optical Absorption by Nd3+ and Gd3+ Ions in Epitaxial Films Grown on Gd3Ga5O12 Substrates from a Lead-Containing Solution Melt V. V. Randoshkin*, N. V. Vasil’eva*, V. G. Plotnichenko**, Yu. N. Pyrkov**, S. V. Lavrishchev**, M. A. Ivanov*, A. A. Kiryukhin*†, A. M. Saletskiœ***, and N. N. Sysoev*** * Institute of General Physics, Russian Academy of Sciences, ul. Vavilova 38, Moscow, 119991 Russia ** Research Center of Fiber Optics, Institute of General Physics, Russian Academy of Sciences, ul. Vavilova 38, Moscow, 119991 Russia *** Moscow State University, Vorob’evy gory, Moscow, 119992 Russia e-mail: [email protected] Received July 15, 2003; in ﬁnal form, November 4, 2003

Abstract—Epitaxial films of composition (Gd,Nd)3Ga5O12 or (Gd,Y,Nd)3Ga5O12 with a neodymium content varying from 0.3 to 15 at. % are grown by liquid-phase epitaxy from a supercooled PbOÐB2O3-based solution melt on Gd3Ga5O12(111) substrates. The optical absorption spectra of the epitaxial films grown are measured in the wavelength range 0.2Ð1.0 µm. The results of interpreting the absorption bands observed in the spectra are used to construct the energy level diagrams of Nd3+ and Gd3+ ions in the matrices of the epitaxial films. © 2004 MAIK “Nauka/Interperiodica”.

† 1. INTRODUCTION by van der Ziel et al. [3]. In 1973, Bonner [4] succeeded in generating stimulated emission at a wavelength of Neodymium-doped lasers with various solid matri- µ ces, especially with those made up of single-crystal gar- 1.06 m at 300 K in Y3Al5O12 epitaxial films containing nets (for example, gadolinium gallium garnet 2 at. % Nd3+. Gd3Ga5O12), have been extensively used in quantum In this work, we investigated the optical absorption electronics [1]. The basic problem of increasing the in the wavelength range 0.2Ð1.0 µm in neodymium- efficiency of these lasers is associated with the impos- containing single-crystal garnet films grown on sibility of growing optically homogeneous crystals Gd3Ga5O12(111) substrates through liquid-phase epit- with a high concentration of doping rare-earth ions, in axy from a supercooled PbOÐB O -based solution 3+ 2 3 particular, Nd ions, whose distribution coefficient is melt. considerably less than that for the ions replacing them in the matrix [2]. As a result, the concentration of doping rare-earth elements changes significantly along the 2. THE GROWTH OF FILMS axis of the growing single crystal. The above disadvantage can often be eliminated Epitaxial single-crystal gallium garnet films with a upon changing over to single-crystal films grown by neodymium content C(Nd) ranging from 0.3 to 15 at. % liquid-phase epitaxy from a supercooled solution melt. were grown through liquid-phase epitaxy from a super- In this case, the difference between the distribution cooled PbOÐB2O3-based solution melt [5]. For a coefficients of the rare-earth elements introduced into neodymium content C(Nd) ≥ 1.7 at. %, yttrium was the epitaxial film can be decreased through appropriate introduced into the epitaxial single-crystal films with choice of the ratio between garnet-forming components the aim of matching the lattice parameters of the film in the solution melt. Since only an insignificant amount and the substrate. The composition of the batch was of these components passes from the solution melt into characterized by the following molar ratios (R1, R2, R3): the film, the concentration of rare-earth elements in the []()Σ[]≈ bulk of the film remains almost unchanged over the film R1 = Ga2O3 / Ln2O3 14.4, thickness. [][]≈ The first mention of a thin-film single-crystal laser R2 = PbO /B2O3 16.0, based on an Y1.25Ho0.1Er0.55Tm0.5Yb0.6Al5O12 garnet ()Σ[][](Σ[][] film grown on an Y3Al5O12 substrate was made in 1972 R3 = Ln2O3 + Ga2O3 / Ln2O3 + Ga2O3 † [][]) ≈ Deceased. + PbO + B2O3 0.08,

OPTICAL ABSORPTION 1031

0.074 0.926 1 000 2+ 1 2 Pb λ = 280 nm 3 PbO–B O 0.075 0.925 3+ 2 4 Nd 0 5 –Gd 100 3 2 O O λ = 276 nm 2 0.076 0.924 3 3

Ga –1 10 0.077 0.923 , cm α

3+ 0.078 0.922 Gd 0 0.001 0.002 0.003 0.004 1

Nd2O3–Y2O3 0.1 200 400 600 Fig. 1. Fragment of the concentration triangle of the λ, nm (Nd O ÐY O )Ð(Gd O ÐGa O )Ð(PbOÐB O ) pseudoter- 2 3 2 3 2 3 2 3 2 3 Fig. 2. Optical absorption spectra α(λ) for (0) the nary system. Mole fractions of oxides contained in the batch Gd Ga O substrate; (1) the neodymium-free Gd Ga O are plotted along the sides of the triangle. Closed circles 3 5 12 3 5 12 indicate the compositions of the solution melts used to grow film; (2Ð4) Gd3Ga5O12 epitaxial film samples (2) 5, (3) 7, epitaxial single-crystal films of gallium garnets with a and (4) 10 with different neodymium contents; and (5) the neodymium content ranging from 0.3 to 15 at. % (parame- Nd3Ga5O12 single crystal. The numbering of the films is the ters of the films are given in Table 1). same as in Table 1.

Σ where [Ln2O3] = [Gd2O3] + [Nd2O3] + [Y2O3] and orless and optically homogeneous, except the films brackets designate the oxide content in the batch in with a neodymium content C(Nd) ≈ 15 at. %, which mole percent. were yellowish brown and had microcracks. The growth experiments are illustrated in Fig. 1, The solution melt was homogenized in a platinum which shows a fragment of the concentration triangle of crucible at a temperature T = 1100°C for 4 h. Then, the the ((Nd2O3ÐY2O3)Ð(Cd2O3ÐGa2O3)Ð(PbOÐB2O3) sys- temperature was reduced step-by-step to the crystalli- tem. In order to prepare an epitaxial single-crystal film zation temperature of the solution melt. At each tem- of the specified composition, the ratio between rare- perature step (i.e., at the growth temperature Tg = earth oxides in the batch was determined with due const), we carried out the growth of the films. As the regard for the known distribution coefficients of the neodymium content increased, the temperature of satu- rare-earth elements [2]. Epitaxial films were grown ration Ts of the solution melt decreased from 1030 to under supercooling conditions ensuring the absence of 920°C. The maximum thickness of the epitaxial films additional absorption associated with the intervalence grown amounted to 50 µm. Thick films with a thickness pair transitions of Pb2+ and Pb4+ impurity ions [6, 7]. h ≥ 25 µm were grown after an additional homogeniza- The epitaxial single-crystal films thus grown were col- tion of the solution melt.

Table 1. Characteristics of (Gd,Y,Nd)3Ga5O12 epitaxial single-crystal ﬁlms ° µ µ α Ð1 ∆λ Sample no. C(Nd), at. % Tg, C2h, m fg, m/min 806.9, cm , nm 1 0.3 999 105.0 0.44 3.8 0.84 2 0.9 979 76.8 1.28 13.7 0.86 3 0.9 976 33.4 0.56 16.7 0.87 4 1.7 950 95.8 0.40 Ð Ð 5 2.3 964 73.8 0.37 30.6 0.92 6 3.3 950 53.8 0.90 47.8 0.92 7 5.0 930 67.9 0.28 76.2 0.92 8 10.0 944 12.0 0.20 125.0 0.96 9 10.0 935 68.7 0.19 123.0 0.96 10 15.0 887 99.8 0.16 176.3 1.01 α Note: Tg is the growth temperature, 806.9 is the absorption coefﬁcient without regard for scattering at the maximum of the spectral line, and ∆λ is the half-width of the spectral line at a wavelength of 806.9 nm.

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1032 RANDOSHKIN et al. 5/2 3/2 7/2 1/2 9/2 15/2 13/2 11/2 17/2 9/2 7/2 3/2 5/2 7/2 { { { { { { { { { { 7/2 D I P S 6 6 6 8 7 6 5 4 3 2 1 0 41 40 39 38 37 36 33 32 28 27 23 22 21 20 14 13 11 10 ~ ~ 1 0 40800 40400 40000 39600 39200 36600 36400 36200 36000 35800 35600 33000 32800 32600 32400 32200 32000 31800 E, cm–1 E, cm–1 E, cm–1

3+ Fig. 3. Diagram of Stark levels of the Gd ion in the Gd3Ga5O12 single crystal at room temperature.

For the above ratio of components in the batch, the 3. EXPERIMENTAL TECHNIQUE growth of films on the surface of the solution melt and The total thickness 2h of the films grown on both platinum auxiliaries was accompanied by spontaneous sides of the substrate and the growth rate f of these crystallization of garnet single crystals in the form of g tetragonal trioctahedra with {211} facets. films (Table 1) were determined by weighing the substrate prior to the epitaxial growth and the film–substrate–film structure after the growth. In this case, we ignored the difference in the quantitative compositions 30 of the film and the substrate. 12 (a) The transmission spectra of the films were measured on a Lambda 900 (Perkin-Elmer) spectrophotometer in the wavelength range λ = 0.2Ð1.0 µm in steps of 2 nm 20 at room temperature.

–1 The absorption spectra of the ﬁlms were calculated

, cm from the transmission spectra as follows. Initially, the α 10 transmission spectrum of the substrate, which was measured prior to the growth, was divided by the transmission spectrum of the substrate with the grown films. Then, the natural logarithm of this ratio was divided by 0 the total thickness of the films grown on both sides of the substrate. 4 4 4 4 D + I + D + D 3/2 11/2 5/2 1/2 (b) 3 4. RESULTS AND DISCUSSION 2 P1/2 Figure 2 shows the optical absorption spectra α(λ) 200 2 D5/2 2 for the Gd Ga O substrate (curve 0), the neodymium- 2 K15/2 3 5 12 free Gd Ga O epitaxial film (curve 1), Gd Ga O –1 3 5 12 3 5 12 2G + 2K 9/2 13/2 4F + 4S 4 epitaxial films with different neodymium contents 2 4 7/2 3/2 F5/2

, cm P G 3/2 7/2 4G 2 (curves 2Ð4), and the Nd Ga O single crystal (curve 5) α 2 5/2 H9/2 3 5 12 2 G7/2 100 D3/2 grown by the Czochralski technique. As can be seen 4 G 2 4 11/2 H F3/2 from Fig. 2, the optical absorption spectra of neody- 1 4 11/2 4 G9/2 F9/2 mium-containing films exhibit absorption bands associated with Nd3+ ions that are characteristic of Nd3Ga5O12 single crystals (curve 5). Moreover, these 0 400 600 800 spectra contain an absorption band with a maximum at λ, nm a wavelength of 280 nm, which, as in the case of Gd3Ga5O12 epitaxial films (Fig. 2, curve 1) [6], is attrib- 2 1 3 2+ α λ uted to the (6s ) S P transition of the Pb ions Fig. 4. Optical absorption spectra ( ) for Gd3Ga5O12 epi- 0 1 taxial films with different neodymium contents C(Nd): [8]. It should be noted that all absorption bands of (a) (1) 0.3 and (2) 0.9; and (b) (1) 5, (2) 10, and (3) 15 at. %. neodymium ions that are characteristic of the

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 OPTICAL ABSORPTION 1033

3+ Table 2. Stark levels of the Gd ion in the Gd3Ga5O12 crystal at 300 K Number of Stark components Nos. of Stark levels Positions of Stark levels ∆E, Term in the term (Fig. 4) in the term, cmÐ1 experi- [10] [11] cmÐ1 theory 3+ ment YAG : Gd GdCl3 á 6H2O 8 S7/2 00 4ÐÐÐÐ 6 P7/2 1Ð3 31942, 31972, 32020 4 3 4 4 78 6 P5/2 4, 5 32545, 32575 3 2 3 3 30 6 P3/2 6 33150 2 1 2 2 6 I7/2 7Ð10 35694, 35731, 35748, 35834 4 4 4 4 140 6 I9/2 11Ð13 36044, 36061, 36071 5 3 5 5 27 6 I17/2 14Ð20 36125, 36140, 36150, 36166, 36177, 77 3 7 69 36184, 36194 6 I11/2 21, 22 36320, 36335 6 2 3 5 15 6 I13/2 23Ð27 36413, 36438, 36448, 36456, 36482 7 5 Ð 7 69 6 I15/2 28Ð32 36515, 36534, 36549, 36569, 36594 8 5 Ð 8 79 6 D9/2 33Ð36 39320, 39425, 39464, 39505 5 4 Ð 5 185 6 D1/2 37 40465 1 1 Ð 1 Ð 6 D7/2 38, 39 40602, 40645 4 2 Ð 2 43 6 D3/2 40 40748 2 1 Ð 2 Ð 6 D5/2 41 40860 3 1 Ð 3 Ð

3+ Nd3Ga5O12 single crystal are observed in the absorption complete splittings ∆E of the electron terms of the Gd spectra of neodymium-containing films. ion in the Gd3Ga5O12 single crystal at room temperature are listed in Table 2. Since the ground state of the Gd3+ The optical absorption spectrum of the substrate ion at room temperature corresponds to a single level (Fig. 2, curve 0) exhibits narrow bands in the wave- 3+ (level 0 in Fig. 3), the absorption spectrum of the length range 244Ð314 nm due to the presence of Gd Gd Ga O single crystal contains bands associated ions in the substrate. In this range, the transmission 3 5 12 with the transitions from the ground state to excited spectra were additionally measured in steps of 0.1 nm 3+ with a spectral resolution of 0.1 nm. As a result, we states of the Gd ion. revealed 41 maxima in the absorption bands of the Moreover, we revealed that the short-wavelength spectra. Figure 3 shows the energy level diagram of the edge of the absorption band of Gd3+ ions in the Gd3+ ion in the Gd Ga O single crystal, which was 3 5 12 Gd3Ga5O12 epitaxial films with a neodymium content constructed taking into account the data available in the of 15 at. % is shifted by 0.014 nm toward the longer literature [9Ð11]. Although Hellwege et al. [11] studied wavelength range with respect to the absorption band of GdCl3 á 6H2O and Y(Gd)Cl3 á 6H2O matrices with a the Gd3+ ion in the substrate. In order to determine this nongarnet structure, we used their data for the interpre- shift, we measured the absorption bands associated tation of our experimental results. This is explained by with the substrate and films in the wavelength range the fact that those authors obtained more complete from 270 to 285 nm with a step of 0.2 nm and a spectral information as compared to Azamatov et al. [10], who resolution of 0.05 nm. This shift can be attributed to the 3+ investigated the Y2.95Gd0.05Al5O12 matrix (YAG : Gd ). distortion of the crystal lattice at a high neodymium However, our results of the measurements of the wave- content, which is confirmed by the formation of micro- lengths at the maxima of the absorption bands proved cracks in the epitaxial films at C(Nd) = 15 at. %. There- to be in closer agreement with the data obtained in [10]. fore, we can state that the energy level diagram of the 6 6 3+ 3+ Note also that the diagram of the I17/2 and I11/2 electron Gd ion in the grown films differs from that of the Gd terms obtained in [11] (and, consequently, in our work) ion in the Gd3Ga5O12 single crystal (Fig. 3) in terms of differs from that proposed in [10]. The Stark levels the magnitude of the aforementioned shift. For epitax- shown in Fig. 3 are numerated sequentially starting ial films with a lower neodymium content, these energy 8 from the fundamental term S7/2. The level positions and levels coincide with each other.

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1034 RANDOSHKIN et al. + 1/2 13/2 3/2 11/2 I D K S 2 4 2 4 + + + + 3/2 5/2 3/2 11/2 9/2 9/2 7/2 7/2 5/2 11/2 9/2 15/2 1/2 9/2 7/2 5/2 3/2 9/2 D D P D K G G G G G G H F F H F F I 2 4 4 4 4 2 4 2 4 4 2 4 4 2 2 4 4 2 { { { { { { { { { { { { { { 8 7 6 5 1 61 60 59 58 57 52 46 45 37 34 33 29 28 23 22 19 18 15 14 10 ~ ~ 0 500 28000 26000 24000 22000 20000 18000 16000 15000 14000 13000 12000 E, cm–1 E, cm–1

3+ Fig. 5. Diagram of Stark levels of the Nd ion in the Gd3Ga5O12 : Nd epitaxial ﬁlm at room temperature.

Figure 4 shows the absorption bands associated with The energy level diagram of the Nd3+ ion in the the presence of neodymium in the epitaxial films and Gd3Ga5O12 : Nd film is depicted in Fig. 5. The experi- the conventional notation for electron terms of the Nd3+ mental data on the position of the Stark levels and the ion (Fig. 4b) [12, 13]. It turned out that the positions of complete splitting ∆E of the electron terms of the Nd3+ these bands are independent of the neodymium content ion in the Gd3Ga5O12 : Nd film at a temperature of and completely coincide with those in the optical 300 K are presented in Table 3. In this case, the trans- absorption spectrum of the Nd Ga O single crystal mission spectra of the epitaxial films were additionally 3 5 12 measured with a step of 0.5 nm and a spectral resolution (Fig. 2). In the general case, both the intensity of each of 0.5 nm in the wavelength range from 200 to 950 nm. absorption peak and the intensity of absorption at any As a result, we revealed 61 maxima of the absorption wavelength in the range under investigation increase in bands. For the purpose of determining the positions of proportion to the neodymium content C(Nd). the Stark levels more exactly, the spectral lines were

(a) (b) –1 (a) (b) 1' cm –1 4 1' 3' cm F3/2 11484 12519 249 cm–1 11438 2' 4 F5/2 12423 178 761 12383 95 4' 4 249 2' I9/2 4' 3' 178 178 95 4I 95 5' 0 9/2 0 6' 1'2'3'4'5'6'7'8'9'10' 1' 2'354' ' 6' 7' 7' 8' 761 1' 6' 9' 10' 5' 7' 11600 11400 11200 11000 10800 10600 ν, cm–1 12500 12400 12300 12200 12100 ν, cm–1

4 4 Fig. 6. (a) Optical absorption spectrum of the I9/2 Fig. 7. (a) Optical absorption spectrum of the I9/2 4 4 F3/2 transition and the curves of decomposition into com- F5/2 transition and the curves of decomposition into components and (b) diagram of the crystalline splitting of the ponents and (b) diagram of the crystalline splitting of 4 4 3+ 4 4 3+ I9/2 and F3/2 terms of the Nd ion in the Gd3Ga5O12 : Nd the I9/2 and F5/2 terms of the Nd ion in the Gd3Ga5O12 : Nd ﬁlm at room temperature. ﬁlm at room temperature.

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 OPTICAL ABSORPTION 1035

3+ Table 3. Stark levels of the Nd ion in the Gd3Ga5O12 : Nd ﬁlm at 300 K

Nos. of Stark Number of Stark components Positions of Stark levels ∆E, Term levels in the in the term, cmÐ1 experi- [12] [13] cmÐ1 term (Fig. 5) theory ment Nd : YGaG Nd : Gd3Ga5O2 4 I9/2 1Ð5 0, 95, 178, 249, 761 5 5 4 5 761 4 F3/2 6, 7 11438, 11484 2 2 2 2 46 4 F5/2 8Ð10 12383, 12423, 12519 3 3 3 3 136 2 H9/2 11Ð14 12565, 12596, 12598, 12810, … 5 4 5 5 Ð 4 4 F7/2 + S3/2 15Ð18 13375, 13415, …, 13560, 13597, … 6 4 5 6 Ð 4 F9/2 19Ð22 14638, 14661, 14785, 14884, … 5 4 4 4 Ð

2H11/2 23Ð28 15779, 15840, 15883, 15993, 16088, 6 6 6 6 316 16095 4 G5/2 29Ð31 16888, 16991, 17053 3 3 3 3 165 2 G7/2 32, 33 …, 17266, …, 17551 4 2 4 4 Ð 4 G7/2 34Ð37 18771, 18828, 18872, 18986 4 4 4 4 215 2 2 G9/2 + K13/2 38Ð45 19248, 19324, 19383, 19494, 12 8 10 10 743 19570, 19692, 19821, 19991 4 G9/2 46, 47 …, …, 20813, 20825 5 2 3 4 Ð 4 G11/2 48Ð52 21012, …, 21089, 21116, 21171, 21190 6 5 5 6 178 2 K15/2 53Ð56 21602, 21679, …, …, 21768, 21867 8 4 6 6 265 2 D3/2 57 21978 2 1 Ð 1 Ð 2 P1/2 58 23172 1 1 1 1 Ð 2 D5/2 ÐÐ 3Ð 3 3 Ð 2 P3/2 ÐÐ 2Ð 2 Ð Ð 4 2 D3/2 + I11/2 59Ð61 27718*, 28270*, 28854* 12 3 Ð Ð Ð 4 4 + D5/2 + D1/2 Note: The ellipsis … indicates the position of the Stark levels, which we failed to determine in the grown films. The level positions marked by an asterisk require further refinement. decomposed into components with the use of a spe- (CNGG) and calcium lithium niobium gallium garnet cially developed program. The results of this decompo- (CLNGG) crystals doped with neodymium ions at sition with numbering of the maxima of the absorption C(Nd) = 1 at. %, the full width at half-maximum is bands are presented in Figs. 6aÐ8a. The diagrams of the equal to 4 nm. For yttrium aluminum garnet (YAG) crystalline splitting of the electron terms of the Nd3+ ion crystals with the same neodymium content, the full in the Gd3Ga5O12 : Nd film are depicted in Figs. 6b–8b. width at half-maximum is approximately equal to 1 nm. The absorption coefficient at a wavelength of 807.5 nm In order to determine the absorption coefficient at is equal to 4 cmÐ1 for CNGG and CLNGG and 12 cmÐ1 the maximum of the absorption band at a wavelength of for YAG [14]. 806.9 nm (Table 1), we additionally measured the transmission spectra for the films and substrates in the It should be noted that, during double passage of a wavelength range from 770 to 860 nm with a step of semiconductor diode laser at pumping wavelengths of 0.1 nm and a spectral resolution of 0.1 nm. As the 805 and 808 nm, epitaxial films with a neodymium con- neodymium content increases, the absorption coeffi- tent C(Nd) = 1.7Ð3.3 at. % absorb approximately 1/3 of cient in the absence of scattering increases from the power. Therefore, these films can be used in cross- 3.8 cmÐ1 at C(Nd) = 0.3 at. % to 176.3 cmÐ1 at C(Nd) = beam lasers. Moreover, epitaxial films with a neody- 15 at. %. The half-width (the full width at half-maxi- mium content up to 3.3 at. %, for which the lumines- mum) of this band also increases from 0.84 to 1.01 nm, cence lifetime can be as long as 60 µs [5], hold promise respectively. For calcium niobium gallium garnet for lasing.

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1036 RANDOSHKIN et al.

(a) (b) ACKNOWLEDGMENTS cm–1 1' We would like to thank M.I. Belovolov, 2 23172 2' P1/2 A.V. Vasil’ev, V.F. Lebedev, and S.N. Ushakov for their 249 assistance in performing this work and participation in 4I 178 helpful discussions. 9/2 95 0 1' 2' 3' 4' REFERENCES 3' 4' 1. A Handbook on Lasers, Ed. by A. M. Prokhorov (Sovetskoe Radio, Moscow, 1978), Vol. 1, p. 237. 2. V. V. Randoshkin, V. I. Chani, and A. A. Tsvetkova, Pis’ma Zh. Tekh. Fiz. 13 (4), 839 (1987) [Sov. Tech. Phys. Lett. 13, 348 (1987)]. 3. J. P. van der Ziel, W. A. Bonner, L. Kopf, and L. G. van Uitert, Phys. Lett. A 42A (1), 105 (1972). 23300 23200 23100 23000 22900 22800 ν, cm–1 4. W. A. Bonner, J. Electron. Mater. 3 (1), 193 (1974). 5. V. V. Randoshkin, M. I. Belovolov, N. V. Vasil’eva, 4 K. A. Zykov-Myzin, A. M. Saletskiœ, N. N. Sysoev, and Fig. 8. (a) Optical absorption spectrum of the I9/2 4 A. N. Churkin, Kvantovaya Élektron. (Moscow) 31 (9), P1/2 transition and curves of the decomposition into com- 799 (2001). ponents and (b) diagram of the crystalline splitting of the 4 4 3+ 6. V. V. Randoshkin, N. V. Vasil’eva, A. V. Vasil’ev, I9/2 and P1/2 terms of the Nd ion in the Gd3Ga5O12 : Nd V. G. Plotnichenko, S. V. Lavrishchev, A. M. Saletskiœ, film at room temperature. K. V. Stashun, N. N. Sysoev, and A. N. Churkin, Fiz. Tverd. Tela (St. Petersburg) 43 (9), 1594 (2001) [Phys. Solid State 43, 1659 (2001)]. 5. CONCLUSIONS 7. V. V. Randoshkin, N. V. Vasil’eva, A. M. Saletskiœ, In this work, we obtained the following results. K. V. Stashun, N. N. Sysoev, and A. N. Churkin, Fiz. Mysl’ Ross., No. 2, 27 (2000). (i) Neodymium-containing single-crystal gallium 8. G. B. Scott and J. L. Page, J. Appl. Phys. 48 (3), 1342 garnet films with a neodymium content varying from (1977). 0.3 to 15 at. % were grown by liquid-phase epitaxy 9. A. A. Kaminskiœ, Laser Crystals (Nauka, Moscow, 1975). from a supercooled PbOÐB2O3-based solution melt on Gd Ga O (111) substrates. 10. Z. T. Azamatov, P. A. Arsen’ev, and M. V. Chukichev, 3 5 12 Opt. Spektrosk. 28 (2), 289 (1970). (ii) The energy diagram of Stark levels was con- 11. K. H. Hellwege, S. Hufner, and H. Schmidt, Z. Phys. 172 structed for different electron terms of the Gd3+ ion in (4), 460 (1963). 12. J. A. Koningstein, J. Chem. Phys. 44 (10), 3957 (1966). Gd3Ga5O12 at room temperature. 13. Kh. S. Bagdasarov, G. A. Bogomolova, M. M. Gritsenko, (iii) It was found that the short-wavelength edge of A. A. Kaminskiœ, A. M. Kevorkov, A. M. Prokhorov, and the absorption band of Gd3+ ions in epitaxial films at a S. É. Sarkisov, Dokl. Akad. Nauk SSSR 216 (5), 1018 neodymium content of 15 at. % is shifted by 0.014 nm (1974) [Sov. Phys. Dokl. 19, 353 (1974)]. with respect to that in the substrate. 14. Yu. K. Voron’ko, N. A. Es’kov, A. S. Podstavkin, P. A. Ryabochkina, A. A. Sobol’, and S. N. Ushakov, (iv) The positions of the Stark levels of the Nd3+ ion Kvantovaya Élektron. (Moscow) 31 (6), 531 (2001). were determined for (Gd,Nd)3Ga5O12 and (Gd,Y,Nd)3Ga5O12 epitaxial films at room temperature. Translated by N. Korovin

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

Physics of the Solid State, Vol. 46, No. 6, 2004, pp. 1037Ð1042. Translated from Fizika Tverdogo Tela, Vol. 46, No. 6, 2004, pp. 1008Ð1012. Original Russian Text Copyright © 2004 by Glinchuk, Prokhorovich.

DEFECTS, DISLOCATIONS, AND PHYSICS OF STRENGTH

Effect of Isolated Defects and Defect Pairs on the Spectrum of the Edge Luminescence of Solids K. D. Glinchuk and A. V. Prokhorovich Institute of Semiconductor Physics, National Academy of Sciences of Ukraine, pr. Nauki 45, Kiev, 03028 Ukraine e-mail: [email protected] Received May 5, 2003; in ﬁnal form, August 26, 2003

Abstract—Expressions for the intensities of bands in the edge luminescence spectrum of a solid containing both isolated defects (shallow acceptors and donors) and defect pairs are given. Conditions under which the contribution of defects to the edge luminescence bands is negligible or dominant are found. Analysis of the edge luminescence spectrum of semi-insulating GaAs shows that situations in which the intensities of the edge luminescence bands are determined by the states of isolated shallow acceptors and donors and donorÐacceptor pairs are quite probable. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION acceptors and donors be NA1 and ND1 and the concentrations of acceptors and donors in the pairs be N and It is known that, in the luminescence spectra of sol- A2 ids, intense edge (impurity and excitonÐimpurity) ND2, respectively (the total acceptor concentration is bands are observed; their appearance is due to recombi- NA = NA1 + NA2, and the total donor concentration is nation of electrons and holes at defects (shallow accep- 0 + ND = ND1 + ND2). We denote by cnA and cnD the capture tors and donors) and annihilation of excitons bound to coefficients of free electrons by acceptors and donors, defects. When calculating the intensities of edge lumi- Ð 0 nescence bands, it is usually assumed that a solid con- respectively, and by c pA and c pD the capture coeffi- tains isolated defects (shallow acceptors and donors; cients of free holes by acceptors and donors, respec- we call them acceptors and donors in what follows) [1Ð tively. Let the average coefficient of interimpurity 5] or acceptors and donors forming donorÐacceptor * pairs [1, 6, 7]. However, it is obvious that there are both recombination be cn . The probabilities of occupation isolated acceptors and donors and donorÐacceptor of the acceptors in the pairs (at low interimpurity transition rates) and isolated acceptors by holes are ϕ 0 = pairs. In this study, we calculate the intensities of near- A edge luminescence bands in solids containing both iso- Ð δ Ð δ 0 δ lated defects (shallow acceptors and donors) and defect c pA p/(c pA pc + nA n) , the probability of donors pairs. We show that, in this case (in contrast to existing + + 0 filling by electrons is ϕ 0 = c δn/(c δnc + δp) , concepts), the contribution of isolated impurities and D nD nD pD pairs to the edge luminescence spectra is not always and the probability of donors filling by holes is 1 – ϕ 0 = D determined only by the relationship between their con- 0 δ + δ 0 δ centrations. In particular, situations are quite probable c pD p/(cnD nc + pD p) [5, 7]. Free X excitons (with in which isolated impurities and impurity pairs give 0 concentration nX) can be bound to neutral acceptors A substantially different contributions to the intensities of and ionized D+ and neutral D0 donors forming excitonÐ edge luminescence bands even if their concentrations 0 + 0 are of the same order of magnitude. The relationships impurity complexes A X, D X, and D X (the binding coefficients are b 0 , b + , b 0 , and the probabilities obtained are used to clarify the role of isolated impuri- A X D X D X ties and impurity pairs in forming the edge lumines- are b 0 N 0 , b + N + , b 0 N 0 , respectively). Below, cence spectrum of semi-insulating gallium arsenide A X A D X D D X D crystals. we write expressions for intensities of edge luminescence bands corresponding to free-electron (e) transi-

tions to neutral acceptors (I 0 ), free-hole (h) transi- 2. MODEL AND BASIC ASSUMPTIONS eA tions to neutral donors (I 0 ), donorÐacceptor transi- We consider a solid at low temperatures, where there hD 0 tions (I 0 0 ), and annihilation of bound excitons A X, are no thermally stimulated processes and the conduc- D A tivity is determined by uniformly distributed photoelec- + 0 D X, and D X (I 0 , I + , and I 0 , respectively) (see trons (with concentration δn) and photoholes (with A X D X D X concentration δp). Let the concentrations of isolated Fig. 1) under the assumption that recombination transi-

1038 GLINCHUK, PROKHOROVICH

E 0 c I 0 = c ()δN 0 + N 0 p, (2) hD pD D 1 D 2

IhD0 IeA0 IA0X ID+X ID0X I 0 0 = c*N 0 N 0 , (3) D A n D 2 A 2 ID0A0

I 0 = b 0 ()N 0 + N 0 n , (4) A X A X A 1 A 2 X E v DDAAAD0X +XD0X I + = b + ()N + + N + n , (5) D X D X D 1 D 2 X Fig. 1. Radiative (wavy lines) and nonradiative (straight I 0 = b 0 ()N 0 + N 0 n . (6) lines) transitions to isolated donors D and acceptors A, tran- D X D X D 1 D 2 X sitions in donorÐacceptor pairs DA, and bound exciton transitions A0X, D+X, and D0X. It follows from Eqs. (1)Ð(6) that the relative effects of isolated acceptors and donors and of donorÐacceptor pairs on the intensities of edge luminescence bands tions are mainly radiative. Moreover, the intensities of depend on the relationship between the values of N 0 the edge spectrum bands are usually determined using A 1 and N 0 , N + and N + , and N 0 and N 0 (the the most probable relationships between the quantities A 2 D 1 D 2 D 1 D 2 + δ Ð δ ӷ ӷ cnD ND2 nc = b and pA NA2 p = d (b d or d b) and intensities are determined by isolated acceptors and + donors if N 0 ӷ N 0 , N + ӷ N + , and N 0 ӷ ӷ δ A 1 A 2 D 1 D 2 D 1 for the case of low [cn* (b + d) (cnD n + N 0 and by donorÐacceptor pairs if N 0 ӷ N 0 , 0 δ Ð δ 0 δ ӷ D 2 A 2 A 1 c pD p )(c pA p + cnA nc ) = a] and high [a n* (b + N + ӷ N + , and N 0 ӷ N 0 ). It also follows from d)] values of δp and δn. The reason for this is that one D 2 D 1 D 2 D 1 can obtain convenient analytical expressions for the Eqs. (1)Ð(6) that the intensities I 0 and I 0 are concentrations of acceptors and donors in pairs in dif- eA A X ferent charge states only in these cases [5, 7]. related to the same state of acceptors (isolated if N 0 ӷ N 0 and belonging to pairs if N 0 ӷ N 0 ) A 1 A 2 A 2 A 1

and that the intensities I 0 and I 0 are determined by 3. THEORETICAL EXPRESSIONS hD D X

FOR THE INTENSITIES OF EDGE the same state of donors (isolated if N 0 ӷ N 0 and LUMINESCENCE BANDS IN D 1 D 2 belonging to pairs if N 0 ӷ N 0 ). SEMICONDUCTORS CONTAINING BOTH D 2 D 1 ISOLATED IMPURITIES AND DONORÐACCEPTOR PAIRS In what follows, we use expressions (1)Ð(6) to deﬁne the intensities of edge luminescence bands in 3.1. General Relationships semiconductors containing both isolated acceptors and donors and donorÐacceptor pairs at low and high con- Obviously, in the case under study, the spectrum of centrations of photoholes and photoelectrons (the quan- edge luminescence is formed by electronÐhole transi- tities N 0 , N + , and N 0 in these expressions tions both to isolated acceptors and donors and to A 12, D 12, D 12, are given in [5, 7] for different relationships between b donorÐacceptor pairs and also by annihilation of bound δ δ excitons (Fig. 1). The band intensities are determined and d and for low and high values of p and n). by the concentrations of neutral acceptors and ionized We represent each of the above expressions for I 0 , eA and neutral donors both that are isolated (N 0 , N + , A 1 D 1 I 0 , I 0 , I + , and I 0 as the sum of two terms, one hD A X D X D X and N 0 , respectively) and form pairs (N 0 , N + , D 1 A 2 D 2 of which describes the contribution of isolated accep- and N 0 , respectively; we have N + + N 0 = ND1 tors and donors to the intensities of edge luminescence D 2 D 1 D 1 bands and the second represents the contribution from and N + + N 0 = N ), as well as by the concentra- D 2 D 2 D2 donorÐacceptor pairs. tions of photoelectrons, photoholes, and free excitons (δn, δp, and n , respectively, which are possibly depen- X 3.2. Low Values of δp and δn dent on NA and ND). The intensities are given by (see Fig. 1)1 For simplicity, we restrict ourselves to the typical + δ ӷ Ð δ Ð δ ӷ 0 cases cnD ND2 n c pA NA2 pc and pA NA2 p I 0 = c ()δN 0 + N 0 n, (1) eA nA A 1 A 2 + δ cnD ND2 n . 1 When writing expressions (4)Ð(6), we assumed that only a small + ӷ Ð number of neutral acceptors and ionized and neutral donors are 3.2.1. cnD ND2dn c pA NA2dp. If this condition on occupied by bound excitons. the recombination parameters of shallow defects is sat-

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 EFFECT OF ISOLATED DEFECTS AND DEFECT PAIRS ON THE SPECTRUM 1039 ϕ δ ϕ isﬁed, then the intensities of the impurity, interimpurity, (3) I 0 ~ 0 n and I 0 ~ 0nX if the radiation and exciton luminescence are given by eA A A X A is due to isolated acceptors (then I 0 ~ δp and I 0 ~ eA A X Ð 0 c δp δ δ ϕ Ӷ δ δ ϕ pA ( p/ n)nX if 0 1), and I 0 ~ p n/0 and I 0 ~ I 0 = c ϕ N + ------N δn, (7) A eA D A X eA nA A0 A1 ϕ A2 cn* 0 ND2 D n δp/ϕ 0 if the radiation is due to acceptors of the pairs X D 0 δ 2 δ 2 δ ϕ Ӷ (then I 0 ~ p and I 0 ~ ( p / n)nX if 0 1); I 0 = c ϕ 0()δN + N p, (8) eA A X D hD pD D D1 D2 ϕ δ ϕ (4) I 0 ~ 0 p, I + ~ (1 Ð 0 )nX, and I 0 ~ I 0 0 = c N δp, (9) hD D D X D D X D A pA A2 ϕ 0 n if the radiation is due to isolated donors or to D X Ð δ c pA p acceptors and donors of the pairs [then I + ~ I 0 = b 0 ϕ 0 N + ------N n , (10) D X A X A X A A1 ϕ A2 X cn* 0 ND2 D (δp/δn)n if ϕ 0 Ӎ 1 and I 0 ~ δn and I 0 ~ X D hD D X

(δn/δp)n if ϕ 0 Ӷ 1].  X D ()ϕ I + = b +  1 Ð 0 ND1 --- D X D X D Ð ӷ + 3.2.2. c pA NA2dp cnD ND2dn. If this inequality is (11) satisﬁed, then the intensities of edge luminescence Ð δ  c pA NA2 p bands are determined by +1()Ð ϕ 0 N + ------n , D D2 + δ 0 δ X cnD nc+ pD p  0 I 0 = c ϕ 0()δN + N n, (13) eA nA A A1 A2

I 0 = b 0 ϕ 0()N + N n (12) D X D X D D1 D2 X + δ 0 cnD n Ð ϕ δ I 0 = c pD 0 ND1 + ------ND2 p, (14) (here, obviously, c δpc /*ϕ 0 N Ӷ 1, hD D pA n D2 c*ϕ 0 N D n A A2 Ð δ + δ 0 δ Ӷ c pA NA2 pc/(nD nc + pD p ) ND2, and ND1 + ND2 = + N ). I 0 0 = c N δn, (15) D D A nD D2 In this case, we have the following:2 I 0 = b 0 ϕ 0()N + N n , (16) 0 Ð A X A X A A1 A2 X (1) I 0/I 0 0 = c ϕ 0 N δn/c N δpI if 0 is eA D A nA A A1 pA A2 eA []()ϕ determined by isolated acceptors [therefore, I 0 Ӷ I + = b + 1 Ð 0 ND1 + ND2 nX, (17) eA D X D X D

I 0 0 if N ≥ N and ϕ 0 Ӎ 1; I 0 /I 0 0 = N /N D A A2 A1 A eA D A A1 A2 + δ cnD n (I 0 > I 0 0 for N > N , and I 0 < I 0 0 for N < I 0 = b 0 ϕ 0 N + ------N n (18) eA D A A1 A2 eA D A A1 D X D X D D1 ϕ D2 X cn* 0 NA2 0 A N ) if ϕ 0 Ӷ 1], and I 0 /I 0 0 = c δn/c*ϕ 0 N Ӷ A2 A eA D A nA n D D2 + 1 if I 0 is determined by acceptors of the pairs; (here, obviously, c δn/c*ϕ 0 N Ӷ 1 and N + eA nD n A A2 A1 0 Ð NA2 = NA). In this case, we have the following: (2) I 0 /I 0 0 = c ϕ 0 N /c N if I 0 is deter- hD D A pD D D1 pA A2 hD 0 + (1) I 0 /I 0 0 = c ϕ 0 N /c N if I 0 is determined by isolated donors, and I 0 /I 0 0 = nA A1 nD D2 hD D A eA D A A eA 0 Ð mined by isolated acceptors, and I 0 /I 0 0 = c ϕ 0 N /c N if I 0 is determined by donors of eA D A pD D D2 pA A2 hD 0 + c ϕ 0 N /c N if I 0 is determined by acceptors the pairs (typically, I 0 Ӷ I 0 0 , since one might nA A2 nD D2 hD D A A eA 0 Ð of the pairs (typically, I 0 Ӷ I 0 0 , since one might Ӷ eA D A expect that c pD c pA ); 0 Ӷ + expect that cnA cnD ); 2 Ð 0 Obviously, ϕ 0 Ӎ 1 if c δpc ӷ δn ; ϕ 0 = A pA nA A 0 ϕ δ + δ Ð 0 Ð 0 + (2) I 0 /I 0 0 = c pD 0 ND1 p/cnD NA2 n if I 0 is c δpc/δnc Ӷ 1 if δpc Ӷ δn ; ϕ 0 Ӎ 1 if c δn ӷ hD D A D hD pA nA pA nA D nD 0 0 + determined by isolated donors [then I 0 Ӷ I 0 0 if c δp (then 1 Ð ϕ 0 Ӎ c δpc/δn Ӷ 1); and ϕ 0 = hD D A pD D pD nD D ≥ ϕ Ӎ + 0 + 0 ND2 ND1 and 0 1 and I 0 /I 0 0 = ND1/ND2 (i.e., c δnc/δpc Ӷ 1 if δnc Ӷ δp . Of course, ϕ 0 , 1 Ð D hD D A nD pD nD pD A ϕ 0 , ϕ 0 ≠ (δn, δp) if δp ~ δn. I 0 > I 0 0 for N > N and I 0 < I 0 0 for N < D D hD D A D1 D2 hD D A D1

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1040 GLINCHUK, PROKHOROVICH

0 3.4. Discussion N ) if ϕ 0 Ӷ 1], and I 0 /I 0 0 = c δp/c*ϕ 0 N Ӷ D2 D hD D A pD n A A2

1 if I 0 is determined by donors of the pairs; 3.4.1. Low values of dp and dn. When analyzing hD the above expressions for the intensities of edge lumi-

(3) I 0 ~ ϕ 0δnI and 0 ~ ϕ 0n if the radiation nescence bands at low photohole and photoelectron eA A A X A X concentrations, we can make the following comments. is due to isolated acceptors or acceptors of the pairs Only in certain cases do the contributions from isolated [then I 0 ~ δp and I 0 ~ (δp/δn)n if ϕ 0 Ӷ 1]; eA A X X A acceptors and donors and from donorÐacceptor pairs to the intensities of edge luminescence bands of a semi- (4) I 0 ~ ϕ 0δp , I + ~ (1 Ð ϕ 0 )n , and I 0 ~ hD D D X D X D X conductor depend only on the relative concentrations ϕ (they are proportional to the concentrations if δn, δp, 0 nX if the radiation is due to isolated donors [then D and nX are independent of NA and ND). These cases cor- δ δ ϕ Ӎ δ I + ~ ( p/ n)nX if 0 1 and I 0 ~ n and I 0 ~ respond to the situation in which the quantities N 0 , D X D hD D X A 2 δ δ ϕ Ӷ δ δ ϕ ( n/ p)nX if 0 1], and I 0 ~ p n/,0 I + ~ nX, N + , and N 0 are only slightly affected by interimpu- D hD A D X D 2 D 2 and I 0 ~ n δn/ϕ 0 if the radiation is due to donors of X rity transitions, i.e., N 0 , N + , and N 0 ≠ ϕ(c* ). In D X A A 2 D 2 D 2 n δ 2 δ 2 δ the pairs [then I 0 ~ n and I 0 ~ ( n / p)nX if these cases, the edge luminescence spectrum is deter- hD D X ӷ mined by isolated acceptors and donors if NA1 NA2 ϕ 0 Ӷ 1]. A ӷ and ND1 ND2, and, on the contrary, by acceptors and ӷ ӷ donors of the pairs if NA2 NA1 and ND2 ND1. For

b ӷ d, this statement refers to the intensities I 0 and 3.3. High Values of δn and δp hD

I 0 , and for d ӷ b, to the intensities I 0 , I 0 , and Obviously, at high δn and δp, the intensities of edge D X eA A X luminescence bands are I + [see Eqs. (8), (12), (13), (16), (17)]. However, in D X 0 most cases (where interimpurity transitions essentially I 0 = c ϕ 0()δN + N n, (19) eA nA A A1 A2 affect the quantities N 0 , N + , and N 0 ), the contri- A 1 D 2 D 2 0 butions from isolated acceptors and donors and from I 0 = c ϕ 0()δN + N p, (20) hD pD D D1 D2 donorÐacceptor pairs to the intensities of the edge luminescence bands of the semiconductor depend not only I 0 0 = c*N N , (21) D A n A2 D2 on their relative concentrations but also on the recom-

bination characteristics. Thus, for ϕ 0 N ӷ A A1 I 0 = b 0 ϕ 0()N + N n , (22) A X A X A A1 A2 X Ð (c δp/)c*ϕ 0 N N and ϕ 0 N ӷ pA n D D2 A2 D D1 I + = b + ()1 Ð ϕ 0 ()N + N n , (23) + D X D X D D1 D2 X (/c δn c*ϕ 0 N )N , the luminescence intensities nD n A A2 D2

I 0 = b 0 ϕ 0()N + N n , (24) I 0 and I 0 (for b ӷ d) and I 0 and I 0 (for d ӷ b) D X D X D D1 D2 X eA A X hD D X can be determined by isolated acceptors and donors where, as noted above, NA1 + NA2 = NA and ND1 + ND2 = 3 even if their concentrations NA1 and ND1 are much lower ND. than the concentrations of acceptors NA2 and donors

In this case, (1) I 0 , I 0 ӷ I 0 0 and (2) the form ND2 in the pairs [see Eqs. (7), (10), (14), (18)]. At the eA hD D A same time, the intensity of exciton luminescence I + of the dependences I 0 , I 0 , I 0 , I + , and I 0 on D X eA hD A X D X D X ӷ δ δ can be determined by acceptors of the pairs (for b d) p and n is the same for the cases where the radiation even if their concentration N is lower than the donor is related to isolated acceptors and donors and to A2 Ð ӷ 0 donorÐacceptor pairs [I + ~ (δp/δn) for ϕ 0 Ӎ 1; concentration ND [for cpANA2 c pD ND , see Eq. (11)] D X D ӷ δ δ δ ϕ Ӷ and by donors of the pairs (for d b) even if their con- I 0 ~ p and I 0 ~ ( p/ n)nX for 0 1; and I 0 ~ eA A X A hD centration ND2 is much lower than the concentration of ӷ δn and I 0 ~ (δn/δp)n for ϕ 0 Ӷ 1]. isolated donors ND1 [this is the case for ND2 (1 Ð D X X D ϕ 0 )N ; see Eq. (17)]. Obviously, the intensity of D D1 3 In Eq. (23) for I + , we took into account that N + ≈ (1 Ð D X D 2 interimpurity luminescence is determined only by ϕ 0 )N at high δp and δn [7]. As follows from the general acceptors and donors of the pairs [see Eqs. (9), (15)]. D D2 expression for dN + /dt = Ð dN 0 /dt = 0 (see [7, Eq. (2)]), this D 2 D 2 Typically, under the conditions considered, we have 0 Ӷ is justiﬁed for c δpc ӷ *N 0 ≈ c*ϕ 0 N , i.e., for high δp. I 0 , I 0 I 0 0 (see Subsection 3.2). Only in certain pD n A 2 n A A2 eA hD D A

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 EFFECT OF ISOLATED DEFECTS AND DEFECT PAIRS ON THE SPECTRUM 1041

I, arb. units cases (if the intensities I 0 and I 0 are determined by eA hD 4 1 2 0 ϕ Ӷ 10 D X isolated acceptors and donors, NA1 > NA2, 0 1, and A 3 + 4 D X N > N , ϕ 0 Ӷ 1) can the inequalities I 0 , I 0 > D1 D2 D eA hD 2 10 5 I 0 0 be satisﬁed. D A Of course, at low photohole and photoelectron con- 0 eA0 1 A X centrations, the illuminationÐbrightness characteristics 1017 1018 1019 D0A0 2 for the intensities I 0 and I 0 and for I 0 , I + , and L, quanta/cm s eA hD A X D X ×20 I 0 coincide (I 0 , I 0 ~ δn or δp, and I 0 , I + , D X eA hD A X D X

I 0 ~ n ) if they correspond to isolated acceptors and D X X ~ ~ donors and δp ~ δn [i.e., ϕ 0 , ϕ 0 ≠ ϕ(δp, δn) (see Sub- A D 1.48 1.50 1.510 1.515 section 3.2). At the same time, the illuminationÐbright- Photon energy, eV ness characteristics for the intensities I 0 and I 0 or eA hD Fig. 2. The spectrum of edge luminescence of semi-insulat- for I 0 and I 0 are substantially different if they are ing gallium arsenide (different dashed lines correspond to A X D X the decomposition of the spectrum into elementary compo- determined by acceptors and donors of the pairs. nents). The inset shows the dependence of the intensities of

edge luminescence bands (1) I 0 0 , (2) I 0 , (3) I 0 , 3.4.2. High dp and dn. Clearly, at high photohole D A eA A X and photoelectron concentrations, the contributions of (4) I + , and (5) I 0 on the excitation intensity (the isolated acceptors and donors and of donorÐacceptor D X D X pairs to the intensities of edge bands of impurity and ratios of the band intensities are arbitrary; the actual rela- exciton luminescence of a semiconductor depend only tions between the band intensities can be seen from the spectrum). Measurements were performed at T = 4.2 K for on their relative concentrations (the intensities are pro- δ δ δ δ low values of p and n; the spectrum was recorded at L = portional to the concentrations if n, p, and nX do not 1018 quanta/cm2 s. depend on NA and ND) [see Eqs. (19), (20), (22)Ð(24)]. Obviously, the intensity of interimpurity luminescence in this case, as well for low values of δp and δn, is deter- isolated acceptors (band I 0 ), and (c) radiative annihi- mined only by donorÐacceptor pairs [see Eq. (21)]. In A X the case considered, the illuminationÐbrightness char- lation of excitons bound to isolated donors (bands I + D X acteristics for the intensities I 0 and I 0 or for I 0 , eA hD A X and I 0 ). The intensities I 0 , I + , and I 0 depend I + , and I 0 coincide (I 0 , I 0 ~ δn or δp, and D X A X D X D X D X D X eA hD on the excitation intensity L in a similar way; namely, I 0 , I + , I 0 ~ n ) irrespective of their being deter- A X D X D X X 2 I 0 , I + , and I 0 ~ L (see Fig. 2 and Subsection 3.4); mined by isolated acceptors and donors or by donorÐ A X D X D X this is possible if δp ~ δn. The similar excitation inten- acceptor pairs and δp ~ δn [i.e., ϕ 0 , ϕ 0 ≠ ϕ(δp, δn)] A D sity dependence indicates the dominant role of isolated (see Subsection 3.3). acceptors and donors in forming the corresponding luminescence bands. The dominant contribution of iso- 4. ANALYSIS OF THE SPECTRUM OF EDGE lated acceptors to the band intensity I 0 is conﬁrmed LUMINESCENCE OF SEMI-INSULATING GaAs eA by the fact that the defects responsible for the band Figure 2 shows the spectrum of edge luminescence intensity I 0 were established to be isolated acceptors and the dependence of the spectral-band intensities on A X the excitation intensity L measured for undoped semi- (as noted in Subsection 3.1, the bands with intensities insulating gallium arsenide at 4.2 K at low δp and δn I 0 and I 0 are related to the same acceptor state). (the luminescence was excited by HeÐNe laser radia- eA A X tion, λ = 632.8 nm, hν = 1.96 eV; the conductivity was This conclusion is also conﬁrmed by the close values of controlled by excess holes and electrons, δp, δn ~ L). the intensities I 0 and I 0 0 at any value of L (,I 0 The shape of the edge luminescence spectrum and eA D A eA intensities of spectral bands are determined by (a) radi- I 0 0 ~ L; i.e., the ratio I 0 /I 0 0 does not depend on ative transitions in the donorÐacceptor pairs (lumines- D A eA D A L) (see Fig. 2 and Subsection 3.4). Therefore, both iso- cence band I 0 0 ), (b) radiative recombination due to D A lated acceptors and donors and donorÐacceptor pairs transitions of free electrons to isolated acceptors (band contribute to the spectrum of edge luminescence in I 0 ) and radiative annihilation of excitons bound to eA semi-insulating gallium arsenide.

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1042 GLINCHUK, PROKHOROVICH

5. CONCLUSIONS REFERENCES 1. A. A. Bergh and P. J. Dean, Light-Emitting Diodes (Clar- The contribution from shallow isolated defects and endon, Oxford, 1976, Mir, Moscow, 1979). defect pairs to the intensities of low-temperature edge 2. O. Brandt, J. Ringling, and K. H. Ploog, Phys. Rev. B 58 luminescence bands can be both additive (depending (24), R15977 (1998). only on the relative defect concentrations) and nonad- 3. S. Seto, K. Suzuki, M. Adachi, and K. Inabe, Physica B ditive (depending both on the relative concentrations (Amsterdam) 302Ð303, 307 (2000). and on the recombination characteristics of the defects, 4. I. Brousell, J. A. H. Stotz, and M. L. W. Thewalt, J. Appl. as well as on the photoelectron and photohole concen- Phys. 92 (10), 5913 (2002). trations in a solid). This conclusion is based on the fact 5. K. D. Glinchuk and A. V. Prokhorovich, Fiz. Tekh. Polu- that the ﬁlling of shallow defects by electrons and provodn. (St. Petersburg) 36 (5), 519 (2002) [Semicon- holes, i.e., the values N 0 , N + , and N 0 , is affected ductors 36, 487 (2002)]. A 2 D 2 D 2 differently by interimpurity transitions. The above rela- 6. T. Schmidt, K. Lischka, and W. Zulehner, Phys. Rev. B tions between the intensities of impurity, interimpurity, 45 (16), 8989 (1992). and exciton radiation bands, as well as the dependence 7. K. D. Glinchuk and A. V. Prokhorovich, Fiz. Tekh. Polu- of the intensities of edge luminescence bands on the provodn. (St. Petersburg) 37 (2), 159 (2003) [Semicon- ductors 37, 148 (2003)]. excitation level, enable one to experimentally establish the contributions of shallow isolated defects and defect pairs to the spectrum of edge luminescence in solids. Translated by I. Zvyagin

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

DEFECTS, DISLOCATIONS, AND PHYSICS OF STRENGTH

Study of the IR Photoionization Spectrum of Electronic States Generated under Plastic Deformation of Colored NaCl Crystals E. V. Korovkin Institute of Solid-State Physics, Russian Academy of Sciences, Chernogolovka, Moscow oblast, 142432 Russia e-mail: [email protected] Received June 6, 2003; in ﬁnal form, October 13, 2003

Abstract—This paper reports on the results of an experimental study of IR photoconductivity induced in NaCl single crystals colored under gamma irradiation and subjected to plastic deformation after preliminary exposure to F-center exciting light. The IR photoconductivity spectrum is decomposed into main components, among which the broad band observed in the range 0.64Ð2.20 eV upon plastic deformation is of special interest. The assumption is made that this band can be caused by photoionization of electrons in the dislocation band and deeper localized electronic states associated with the dislocation. © 2004 MAIK “Nauka/Interperiodica”.

1. The electronic properties of dislocations in semi- trons have a long lifetime at room temperature and conductors have been intensively studied in recent remain stable at low temperatures. Therefore, the expo- years (see, for example, [1, 2]). Similar investigations sure of a sample to light at a specific wavelength in dielectrics, in particular, alkali halide crystals, are (according to [14], IR light) should lead to photoioniza- also of great interest. These investigations go back to tion of dislocation electrons and, consequently, the pos- the discovery of the phenomenon of luminescence in sibility of observing IR photoconductivity. Our first colored alkali halide crystals under deformation [3Ð5]. attempts to carry out such experiments enabled us to More recently, it was found that luminescence is also detect an intense signal due to polaron-type electronic observed under plastic deformation of alkali halide states at liquid-helium temperatures (see [17] and refer- crystal samples without fracture of the material. These ences therein). These experimental investigations findings provided a basis for the development of model revealed that polaron states in NaCl disappear at tem- concepts regarding the existence of a dislocation elec- peratures T > 90Ð100 K. On the other hand, according tronic level (band) [6Ð13] (see also references in [13]). to our data, the deformation of NaCl crystals brings A considerable contribution to the elaboration of the about the formation of a large number of defects that model concepts was made by the discovery of IR have a dipole moment and are responsible for the strong quenching of the photoplastic effect in alkali halide signal detected at T > 180 K. In our more recent work crystals [14]. Ermakov and Nadgornyœ [14] interpreted [18], these findings were taken into account when the measured IR quenching spectrum in terms of the choosing the temperature of the sample and the experi- photoionization spectrum of electrons in a dislocation ments were performed in the temperature range from level (band). Subsequent investigations of the photo- 110 to 120 K. It was found that, upon plastic deforma- plastic effect in alkali halide crystals have actually tion of NaCl crystals colored under gamma irradiation, proved not only the participation of “dislocation” elec- the spectrum of IR photoconductivity induced by F trons in the photoplastic effect itself and in its manifes- light contains a broad band with two sharp maxima at tations (such as IR quenching, aftereffect, photoplastic photon energies of 0.74 and 0.88 eV. It is known that memory) but also the motion of electrons along a dislo- plastic deformation leads to the formation of a large cation (i.e., the existence of dislocation-induced elec- number of point defects and their complexes in the tron bands in alkali halide crystals) and with a disloca- sample; part of these defects can capture electrons from tion [15, 16] (see also references in [16]). the conduction band and thus initiate a signal of IR con- Infrared photoconductivity induced after exposure ductivity induced by F light. Therefore, it can be of a sample to F-center exciting light (F-induced IR assumed that the observed band consists of individual photoconductivity) is an efficient tool for studying elec- peaks (forming the aforementioned two sharp maxima) tronic levels in colored alkali halide crystals. Electrons corresponding to electron traps of different types. Since released from F centers under exposure to F light the position of the maximum at 0.74 eV is close to the occupy electron traps, including the dislocation level. position of the maximum at 0.7 eV in the IR quenching The observed IR quenching of the photoplastic effect at spectrum [14], we can also assume that the former max- room temperature [16] indicates that dislocation elec- imum is attributed to the photoionization of dislocation

1044 KOROVKIN

I, pA from the same spectrometer. This cycle was repeated two times at different polarities of the voltage applied to the sample for the purpose of preventing electrical 1.2 1 2 polarization. Then, the measurements were carried out 3 at the next point of the IR photoconductivity spectrum. 4 0.8 Figure 1 shows the spectra measured in one of these experiments (test experiment; strain of the sample, 12.65%; T = 113 K). Initially, the IR photoconductivity 0.4 spectrum was measured for the sample without additional exposure to monochromatic light (Fig. 1, curve 4). Then, the time of additional monochromatic treatment E E E was chosen so that the intensity of the signal at a wave- 0 2 1 3 0.6 0.8 1.0 1.2 length corresponding to the monochromatic light used E, eV in this treatment was approximately halved (the time required to accomplish the additional monochromatic Fig. 1. IR photoconductivity spectra induced by F light for treatment turned out to be 10 s). The same procedure plastically strained NaCl crystals (1Ð3) with and (4) without was performed when measuring the other spectra with additional exposure to monochromatic light at photon ener- additional exposure of the sample to the monochro- gies (1) E1 = 0.91 eV, (2) E2 = 0.74 eV, and (3) E3 = 1.07 eV matic light. Thereafter, we measured first the IR photo- (indicated by arrows E1ÐE3). conductivity spectrum for the sample additionally exposed to monochromatic light with photon energy E = 0.91 eV (Fig. 1, curve 1) and then the spectrum for electrons. In the present work, an attempt was made to 1 decompose the F-induced photoconductivity spectra of the sample subjected to additional exposure to mono- plastically strained NaCl samples into its main compo- chromatic light with photon energy E2 = 0.74 eV nents with the aim of separating and analyzing the con- (Fig. 1, curve 2). As can be seen, spectrum 2 coincides, tribution from a signal caused by the photoionization of to within a constant factor, with spectrum 1 over the dislocation electrons. For this purpose, prior to each entire range of measurements, including the range measurement, the sample was additionally exposed to between the photon energies E1 and E2. In the range E2Ð monochromatic light at a specific wavelength. After E1, the difference between these spectra, if it exists, subtracting the spectrum of the sample subjected to should be the most pronounced. Therefore, spectrum 2 additional exposure to monochromatic light from the was not measured at E > E1 (in this range, the spectra spectrum of this sample measured prior to additional should be all the more identical to each other). More- exposure, we obtained the spectrum of electronic states over, it should be noted that the decrease in the number sensitive only to the additional monochromatic treat- of measurements was dictated by the necessity of ment (which effectively depletes these states). By vary- reducing the bleaching of the crystal as much as possi- ing the wavelength of monochromatic light used in ble (for the recording of each point of the spectrum, the additional treatment, it is possible to reveal the main sample was twice exposed to F light for 30 s). For the components of the spectrum under consideration. same reason, all the spectra were measured with a rela- 2. The experiments were performed with NaCl crys- tively large step. Then, we measured the IR photocon- tal samples 2 × 2 × 12 mm in size. The NaCl crystal ductivity spectrum for the sample additionally exposed samples were irradiated with gamma rays at a dose of to monochromatic light with photon energy E3 = 107 rad and were then subjected to plastic deformation 1.07 eV (Fig. 1, curve 3). at a rate of 1.8 × 10Ð5 sÐ1 to a strain of 12Ð14% at room 3. The processing of the measured spectra was per- temperature. The experimental technique and condi- formed as follows. The spectra recorded for different tions for measuring the IR photoconductivity induced additional monochromatic treatments were sequentially by F-center exciting light were described in our previ- subtracted from the total spectrum (Fig. 1, curve 4). As a ous work [18]. Initially, the sample was exposed to light result, we obtained, to within a constant factor, the at wavelengths in the range 450Ð800 nm (SI-8 lamp, spectra of electronic states destroyed upon these addi- S3S25 optical filters) for 30 s in order to excite centers tional treatments and then compared them with each (F, M, etc.) donating electrons to the conduction band other. The calculated spectra are shown in Fig. 2. The and to deplete “undesirable” traps (see [18]) in the difference between spectra 4 and 1 in Fig. 1 is repre- range 600Ð800 nm. Then, these traps were further sented by spectrum 1 in Fig. 2. The difference between depleted for 20 s with the use of an OS14 optical filter spectra 4 and 2 in Fig. 1 was multiplied by 0.9 and also (without excitation of the F and M centers) added to the depicted in Fig. 2 (spectrum 2). As can be seen from S3S25 filters. Thereafter, if required, the sample was Fig. 2, spectra 1 and 2 virtually coincide with each additionally exposed to monochromatic light from an other. The difference between spectra 4 and 3 in Fig. 1 IKS-21 spectrometer. Finally, the IR photoconductivity (multiplied by 1.23) is represented by spectrum 3 in was measured under exposure to monochromatic light Fig. 2. It is seen from Fig. 2 that spectrum 3 almost

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

STUDY OF THE IR PHOTOIONIZATION SPECTRUM 1045 coincides with spectra 1 and 2 in the range between E1 I, pA and E2 but deviates from them at E > E1. The difference 1 between spectra 3 and 1 (or 2) in Fig. 2 is depicted by 0.4 2 curve 4. From analyzing the results obtained, we can 3 make the inference that, in the plastically strained NaCl 4 crystal, there exist single-type electron traps, for which the photoionization spectrum is represented by curve 1 in Fig. 2. This spectrum remains unchanged with varia- 0.2 tions in the photon energy in the range 0.74Ð1.07 eV; i.e., it cannot be further decomposed into components by the method employed. Moreover, the plastically E strained NaCl crystal contains electron traps of another 3 E2 E1 type, for which the photoionization spectrum is repre- 0 0.6 1.0 1.2 sented by spectrum 4 in Fig. 2. This spectrum most 0.8 , eV likely corresponds either to the low-energy tail of the E peak located at E > 1.2 eV or to the sum of similar Fig. 2. Photoionization spectra of electronic states peaks. Furthermore, this spectrum can be associated destroyed under additional exposure of the samples to with the electronic states that remained destroyed under monochromatic light at photon energies (1) E1 = 0.91 eV, additional exposure of the sample to nonmonochro- (2) E2 = 0.74 eV, and (3) E3 = 1.07 eV. (4) Photoionization matic light (orange light; see [18]). A comparison of spectrum of electronic states destroyed under additional spectrum 1 in Fig. 2 with the spectra obtained in [18] exposure to monochromatic light at photon energy E3 and allowed us to draw the conclusion that the electronic electronic states not destroyed at photon energies E1 and E2. states responsible for spectrum 1 are absent in the unstrained samples and appear in the strained samples. The spectrum of these states extends to photon energies I, pA E > 1.2 eV, which cannot be measured with our experimental setup. 1.2 1 Spectra 1 and 4 in Fig. 2 are the spectra of electronic 2 states that are effectively destroyed under additional 3 exposure of the sample to the monochromatic light used. If the spectrum of these states is subtracted from 0.8 the total spectrum (Fig. 1, curve 4), we obtain the spectrum of electronic states that are not destroyed upon the monochromatic treatments with a pulse duration of 0.4 10 s. The latter states are responsible for the IR photoconductivity signal and, hence, are destroyed by IR light; however, they remain undestroyed in our experi- 0 ment. This situation occurs only with F and M centers. 0.6 0.8 1.0 1.2 Their concentration is too high to be changed signiﬁ- E, eV cantly by the action of a short IR light pulse (under exposure to F light from the same source for 1 h, the Fig. 3. (1) Total IR photoconductivity spectrum of the intensity of the F band decreases by only 10Ð20%). In strained NaCl crystal and photoionization spectra of electronic states (2) destroyed and (3) undestroyed under addi- this case, only the long-wavelength tails of the spectra tional exposure to monochromatic light. of the electronic states under consideration lie in the IR range. It should be noted that the photoconductivity induced by a light pulse with an intensity identical to the intensity of IR light used in the additional mono- ments (curve 2 is the sum of spectra 1 and 4 in Fig. 2), chromatic treatment, but with a wavelength 475 nm and their difference (curve 3). As expected, the spec- (F light), is at least four orders of magnitude higher trum of undestroyed states consists predominantly of than the photoconductivity observed in the above long-wavelength tails whose maxima are located in the experiments. Thus, the high concentration of these cen- visible range. Moreover, there is one more signal at E = ters and the high sensitivity of the equipment make it 0.67 eV. Since this signal is absent at E > E , it cannot possible to observe their photoionization under expo- 2 be destroyed under exposure to monochromatic light at sure to IR light without a noticeable decrease in the concentration. any energy. It is not surprising, then, that the electronic states responsible for this signal appear to be unde- Figure 3 depicts the total spectrum (curve 1 is simi- stroyed in our experiment. The above signal can be lar to curve 4 in Fig. 1), the spectrum of electronic associated with any traps (not necessarily generated states destroyed upon additional monochromatic treat- under plastic deformation).

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1046 KOROVKIN

I, pA through different mechanisms depending on whether or 10 not they capture an electron) located at immobile dislocations under exposure to light. This dependence can be explained by a complex mechanism of the action of the 8 1 2 light used. On the one hand, the exposure to the light corresponding to the photosensitivity spectrum of traps 6 results in their depletion. On the other hand, the light excites F centers (possibly, also M centers) donating 4 electrons for the filling of traps, expels electrons from them, and thus provides repeated filling of the traps under consideration. Consequently, there is a limit 2 below which traps cannot be destroyed. Since the light wavelength of 562 nm is closer to the light wavelength 0 of the F band (475 nm) as compared to the light wave- 0 102030 length of 1000 nm, the regenerating effect of the light t, s at a wavelength of 562 nm is more pronounced (see Fig. 4), which provides a higher equilibrium concentra- Fig. 4. Dependences of the intensity of the IR photoconduc- tion of filled traps. tivity signal induced by IR light at a photon energy of 0.74 eV on the time of additional exposure to monochro- Thus, the above experiment showed that the spec- matic light at wavelengths of (1) 1000 and (2) 562 nm. The trum of electron traps created under plastic deformation fitted curves represent exponential functions plus constants. (Fig. 2, curve 1) extends from 0.64 eV to photon energies of no less than 2 eV. It is difficult to perceive the As can be seen from Fig. 3, the IR photoconductiv- origin of such a broad spectrum that cannot be decom- ity signal at E = 0.74 eV is almost completely provided posed into individual components. The photoionization by the electronic states generated under deformation of spectra of electron traps produced by point defects and the sample and characterized by a spectrum in the form their simple complexes, as a rule, have a smaller width. of a broad band (spectrum 1 in Fig. 2 will be referred to Such a broad photoionization spectrum can be hereafter as the broad spectrum). Therefore, the signal attributed to electrons in a sufficiently narrow disloca- at E = 0.74 eV can serve as a measure of concentration tion band. In this case, the mass md of a dislocation of these states. electron should be considerably greater than the mass In the next (second) experiment, this signal was mc of an electron in the conduction band. As a result, the used to determine the high-energy edge of the broad substantial difference between the slopes of their para- spectrum. The wavelength of the monochromatic light bolic dispersion curves leads to a large width H of the ∆ ∆ ≈ used in the additional treatment varied in the range photoionization spectrum: H = E(md/mc), where E 1000Ð562 nm (1.2Ð2.2 eV, SI-8 lamp, interference fil- (1Ð2)kT is the energy width of the band of filled states ters). The rate of depletion of the electronic states in the dislocation band. responsible for the broad spectrum upon exposure to Moreover, the results of the second experiment can monochromatic light can be estimated from the inten- be explained in another way that does not require sity of the IR photoconductivity signal induced by IR extending the spectrum to an energy of 2 eV or higher. light at a photon energy of 0.74 eV, which, in turn, Actually, localized electronic levels of defects located makes it possible to determine the high-energy edge of in the vicinity of dislocations, for example, photostop- the broad spectrum. pers (see [15]) with a maximum in the photoionization It was assumed that the broad spectrum does not spectrum at approximately 620 nm, can be associated extend to wavelengths longer than 700 to 800 nm. Oth- with the dislocations. The transfer of electrons from erwise, the states under consideration would also be these levels to the conduction band under exposure to destroyed upon depletion of undesirable states with light will initially lead not to a decrease in the number orange light. However, it turned out that even the light of electrons in these levels but to a decrease in the num- at a wavelength of 562 nm (2.2 eV) effectively depletes ber of electrons in the dislocation band. Indeed, the the former states. The dependences of the intensity of electrons lost by defects will be immediately compen- the IR photoconductivity signal on the time of addi- sated for by dislocation electrons. Only after depleting tional exposure to monochromatic light at wavelengths the dislocation band will the levels of these defects of 1000 and 562 nm are plotted in Fig. 4. It can be seen begin to lose their electrons. that, in both cases, the electronic states are depleted Consequently, the width of the photoionization according to an exponential law, however not to zero spectrum can be little more than 1 eV. In our case, we but to a particular level (differing for different filters). have ∆E ≈ 0.01 eV and the above width of the spectrum A similar dependence was observed in our earlier will be observed for dislocation electrons with a mass ≈ work [15] concerned with the destruction of photostop- md (100Ð150)mc. The following factors can be pers (electron traps that can interact with a dislocation responsible for this large mass of dislocation electrons.

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 STUDY OF THE IR PHOTOIONIZATION SPECTRUM 1047

(i) The distance between equivalent sites in an edge 3. J. Trinks, Sitzungsber. Akad. Wiss. Wien, Math.-Natur- dislocation is larger than the analogous distance in an wiss. Kl., Abt. 2A 147, 217 (1938). ideal NaCl lattice by a factor of 2 . This leads to a 4. F. J. Metz, R. N. Schweiner, H. R. Leider, and G. A. Gir- ifalko, J. Phys. Chem. 61, 86 (1957). threefold or fourfold decrease in the overlap integral. 5. C. T. Butler, Phys. Rev. 141 (2), 750 (1966). (ii) The structure factor, i.e., the number of equiva- 6. V. V. Korshunov, F. D. Senchukov, and S. Z. Shmurak, lent sites surrounding a given site in the NaCl lattice, is Pis’ma Zh. Éksp. Teor. Fiz. 13, 289 (1971). six times larger than that in the dislocation (twelve and two, respectively). As a result, the probability of the 7. M. I. Molotskiœ, Fiz. Tverd. Tela (Leningrad) 26 (4), 1204 (1984) [Sov. Phys. Solid State 26, 731 (1984)]. dislocation electron leaving the occupied site is six times less than the same probability for the electron in 8. V. A. Zakrevskiœ and A. V. Shul’diner, Fiz. Tverd. Tela (Leningrad) 27 (10), 3042 (1985) [Sov. Phys. Solid State the conduction band, all other factors being equal. 27, 1826 (1985)]. (iii) In an edge dislocation, a negative chlorine ion is 9. A. A. Kusov, M. I. Klinger, and V. A. Zakrevskiœ, Fiz. located between two equivalent sites (sodium ions). Tverd. Tela (Leningrad) 31 (7), 67 (1989) [Sov. Phys. This affects the electron wave function and drastically Solid State 31, 1136 (1989)]. decreases the overlap integral. 10. M. V. Gol’dfarb, M. I. Molotskiœ, and S. Z. Shmurak, Fiz. If the above hypothesis explaining the large width of Tverd. Tela (Leningrad) 32 (8), 2398 (1990) [Sov. Phys. the photoionization spectrum holds true, the width of Solid State 32, 1392 (1990)]. the spectrum should substantially depend on the tem- 11. V. A. Zakrevskiœ, T. S. Orlova, and A. V. Shul’diner, Fiz. perature. This offers possibilities for experimentally Tverd. Tela (St. Petersburg) 37 (3), 675 (1995) [Phys. verifying the hypothesis, even though low-temperature Solid State 37, 367 (1995)]. investigations involve problems due to the presence of 12. V. A. Zakrevskiœ and A. V. Shul’diner, Fiz. Tverd. Tela a strong IR signal caused by the bound polaron states (St. Petersburg) 42 (2), 263 (2000) [Phys. Solid State 42, (studied in our earlier work [17]) at temperatures T < 270 (2000)]. 78 K. It should also be noted that the real situation is 13. M. I. Molotskii and S. Z. Shmurak, Phys. Lett. A 166 (3Ð much more complex, because the electron captured by 4), 286 (1992). the dislocation is most likely to be a dislocation polaron 14. G. A. Ermakov and É. M. Nadgornyœ, Pis’ma Zh. Éksp. than a dislocation electron. Teor. Fiz. 14, 45 (1971) [JETP Lett. 14, 29 (1971)]. 15. G. A. Ermakov, E. V. Korovkin, and Ya. M. Soœfer, Fiz. Tverd. Tela (Leningrad) 16 (3), 697 (1974) [Sov. Phys. ACKNOWLEDGMENTS Solid State 16, 457 (1974)]. I would like to thank V.Ya. Kravchenko for his par- 16. E. V. Korovkin, Fiz. Tverd. Tela (Leningrad) 24 (2), 524 ticipation in discussions of the results and for helpful (1982) [Sov. Phys. Solid State 24, 294 (1982)]. remarks. 17. E. V. Korovkin and T. A. Lebedkina, Fiz. Tverd. Tela (St. Petersburg) 42 (8), 1412 (2000) [Phys. Solid State REFERENCES 42, 1451 (2000)]. 18. E. V. Korovkin and T. A. Lebedkina, Fiz. Tverd. Tela 1. H. F. Matare, Defect Electronics in Semiconductors (St. Petersburg) 44 (12), 2155 (2002) [Phys. Solid State (Wiley, New York, 1971; Mir, Moscow, 1974). 44, 2257 (2002)]. 2. Electronic Properties of Dislocations in Semiconduc- tors, Ed. by Yu. A. Osip’yan (Editorial URSS, Moscow, 2000). Translated by O. Borovik-Romanova

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Physics of the Solid State, Vol. 46, No. 6, 2004, pp. 1048Ð1050. Translated from Fizika Tverdogo Tela, Vol. 46, No. 6, 2004, pp. 1018Ð1020. Original Russian Text Copyright © 2004 by Pronina, Aristova, Mazilkin.

DEFECTS, DISLOCATIONS, AND PHYSICS OF STRENGTH

Polygonization in High-Purity Rolled (001)[110] Tungsten Single Crystals L. N. Pronina, I. M. Aristova, and A. A. Mazilkin Institute of Solid-State Physics, Russian Academy of Sciences, Chernogolovka, Moscow oblast, 142432 Russia e-mail: [email protected], [email protected], [email protected] Received October 14, 2003

Abstract—Electron microscopy is used to study changes in the dislocation structure of high-purity rolled (001)[110] tungsten single crystals during short-term high-temperature annealings. The effects of the annealing temperature and time on the formation of low-angle boundaries are investigated. Local defects, which are similar to those detected earlier upon annealing in the structure of molybdenum single-crystal ribbons, are found to form and dissociate upon annealing. These defects are concluded to have a dislocation nature. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION stages, which includes the formation and dissociation of the so-called “local” defects [3] located in disloca- It was shown in [1, 2] that, under certain conditions, tion lines and at subgrain-boundary dislocation nodes. oriented rolling of refractory bcc single crystals can produce single-crystal ribbons of metals, such as Mo, It is of interest to compare the data on the formation W, and Nb. Upon long-term high-temperature anneal- and dissociation of local defects in molybdenum sin- ing (at temperatures above 0.8T , where T is the melt- gle-crystal ribbons with analogous data for other m m refractory metals, in particular, tungsten. ing temperature), the single-crystal ribbons produced do not recrystallize and retain the orientations of the initial single crystals. 2. EXPERIMENTAL The evolution of rolled and annealed single crystals As the initial materials for our investigations, we has a number of peculiar features. For example, a used cylindrical [110] tungsten single crystals grown detailed electron-microscopic study of polygonization by electron-beam zone melting. The purity of the mate- in (001)[110] single-crystal ribbons of molybdenum rial was characterized by the residual resistivity ratio having different degrees of purity (with the resistivity RRR = ρ /ρ , which was 50000 and 200000 for ρ ρ 273 K 4.2 K ratio RRR = 273 K/ 4.2 K = 1000 and 140000) showed the single crystals under study. Single-crystal ribbons that a stable dislocation structure forms in several were fabricated by rolling along the (001) plane in the

µ µm (‡) 2 m (b) 0.5

Fig. 1. Local defects in subgrain boundaries in tungsten single crystals with RRR = 50000 after annealing at 2200°C for 10 s: (a) hexagonal network consisting of two families of (a/2)〈111〉 dislocations (dislocations with Burgers vector a[001] form at the points of intersection of these families) and (b) vertical wall of a[001] edge dislocations.

POLYGONIZATION IN HIGH-PURITY ROLLED (001)[110] TUNGSTEN SINGLE CRYSTALS 1049

[110] direction. High-temperature annealings of various duration were performed by passing an electric current through the ribbons in an USU-4 vacuum device in an oil-free vacuum of 10Ð6 Torr at temperatures varied from 2000 to 2300°C. Electron-microscopic studies were carried out on JEM-100CX, JEM-1000, and JEM- 2000FX electron microscopes.

3. RESULTS Local defects in the structure of tungsten single crystals with RRR = 50000 are detected after annealing 1 µm at 2200°C for 10 s. This annealing results in the formation of subgrain boundaries; some of the boundaries are dislocation networks. Figure 1a shows a hexagonal network consisting of two families of (a/2)〈111〉 disloca- Fig. 2. Structure of the tungsten with RRR = 50000 after tions; at the points of intersection of these families, dis- annealing at 2200°C for 60 s. locations with the Burgers vector a[001] are formed. Figure 1b shows a subgrain boundary, which is a vertical wall of a[001] edge dislocations. As seen from Fig. 1, local defects form in dislocations with b = a[001] and in the nodes of dislocation networks. These defects are also visible after annealing at the same temperature for 20 s and disappear when the annealing time increases to 40 s. After 60 s of annealing, the structure is made up predominantly of walls of edge dislocations with b = a[001] almost without dislocation networks (Fig. 2). This structure is stable; it does not change as the holding time increases at this annealing temperature. It should be noted that the diffraction contrast of local defects is extremely complex and cannot be interpreted unambiguously; the presence of these defects does not cause any additional reflections in the electron diffraction pattern. The tungsten single crystals with RRR = 200000 2 µm have no local defects upon annealing at 2200°C for even the shortest time (3 s). However, by decreasing the annealing temperature, we could find that local defects Fig. 3. Local defects in subgrain boundaries in the tungsten single crystals with RRR = 200000 after annealing at appear in these crystals in subgrain boundaries upon ° annealing at 2000°C for 10 s (Fig. 3) and disappear 2000 C for 10 s. when the annealing time at this temperature exceeds 30 s. The stacking fault energy for refractory bcc metals is very high, 0.3 J/m2 in molybdenum [5] and 0.5 J/m2 4. DISCUSSION OF THE RESULTS in tungsten [6]. However, it should be taken into It was shown in [3] that the local defects forming in account that a large number of excess vacancies (as molybdenum have a dislocation nature. It was assumed compared to the equilibrium concentration) are present that they can form by the mechanism proposed by Cot- in the material under study due to the high degrees of trell and Beilby for bcc crystals [4]. According to this deformation. If the velocity of vacancies moving to dis- mechanism, part of a perfect lattice dislocation can split locations is higher than the rate of their disappearance into two partial dislocations, one of which is a sessile at dislocation jogs, Cottrell vacancy clouds can form dislocation and the other is a glide dislocation. The around a dislocation [7]. The high concentration of region between the two partials is a twin stacking fault. point defects can result in a significant decrease in the The cross slip of a partial dislocation leads to the for- stacking fault energy and make the CottrellÐBeilby mation of new stacking faults lying in the plane of mechanism operative. motion. One partial dislocation can turn around a pole dislocation to form a spiral. The result of the disloca- When the vacancy concentration decreases with tion restructuring is likely to be visible as local defects. increasing annealing time, the reverse process occurs:

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

1050 PRONINA et al. partials transfer into perfect dislocations, which are REFERENCES more stable for bcc crystals. This process is accompa- 1. L. N. Pronina, M. V. Bayazitova, and A. A. Mazilkin, Fiz. nied by the disappearance of stacking faults. Tverd. Tela (St. Petersburg) 38 (3), 441 (1996) [Phys. This explanation implies that the rate of the disloca- Solid State 38, 437 (1996)]. tion restructuring should noticeably depend on the 2. L. N. Pronina, I. M. Aristova, and A. A. Mazilkin, Mate- stacking fault energy. Pronina et al. [8] showed that rialovedenie, No. 4, 7 (1999). annealing of tungsten single crystals at 2300°C did not cause the formation of local defects in subgrain bound- 3. L. N. Pronina and I. M. Aristova, Fiz. Tverd. Tela aries. However, annealing of molybdenum at the same (St. Petersburg) 35 (10), 2701 (1993) [Phys. Solid State homologous temperature brought about the formation 35, 1336 (1993)]. of local defects about 0.1 µm in size, which existed for 4. J. P. Hirth and J. Lothe, Theory of Dislocations a time that was sufﬁcient for their detection and obser- (McGraw-Hill, New York, 1967; Atomizdat, Moscow, vation. 1972). These parameters should also strongly depend on 5. Ya. D. Vishnyakov, Stacking Faults in Crystalline Struc- the presence of impurities, which can also decrease the ture (Metallurgiya, Moscow, 1970). stacking fault energy; the rate of the process in a pure 6. C. S. Hartley, Acta Met. 14 (9), 1193 (1966). material might be expected to be higher. 7. P. Coulomb and J. Friedel, in Dislocation and Mechani- A signiﬁcant increase in the purity of tungsten (to cal Properties of Crystals (Wiley, New York, 1957; Inos- RRR = 200 000) leads to an increase in the rate of the trannaya Literatura, Moscow, 1960). dislocation restructuring such that we can detect local defects only at low annealing temperatures (as low as 8. L. N. Pronina, A. A. Mazilkin, and I. M. Aristova, Fiz. ° Tverd. Tela (St. Petersburg) 40 (3), 498 (1998) [Phys. 2000 C). Solid State 40, 458 (1998)]. Thus, our experiments support the assumption [3] that the aggregates termed local defects have a dislocation nature. Translated by K. Shakhlevich

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

Physics of the Solid State, Vol. 46, No. 6, 2004, pp. 1051Ð1054. Translated from Fizika Tverdogo Tela, Vol. 46, No. 6, 2004, pp. 1021Ð1024. Original Russian Text Copyright © 2004 by Glebovskiœ, Petrov. DEFECTS, DISLOCATIONS, AND PHYSICS OF STRENGTH

Kinetic Interpretation of the Structural-Time Criterion for Fracture P. A. Glebovskiœ and Yu. V. Petrov St. Petersburg State University, Petrodvorets, 198504 Russia e-mail: [email protected], [email protected] Received September 19, 2003

Abstract—The incubation-period-based criterion for fracture is considered in terms of the Zhurkov kinetic model of fracture. Within the kinetic model, fracture is treated as a continuously developing process, which starts immediately after the application of a tensile load to a sample and consists in breaking of the interatomic bonds and gradual accumulation of broken bonds in the material in the course of a fracture test. For certain materials, the inclusion of the thermal-fluctuation mechanism for fracture in the incubation-period-based criterion significantly affects the position of the static branch of the time dependence of strength. Time dependences of strength are calculated for a number of materials. The experimental data are analyzed using the structural- time criterion for fracture, which allows one to obtain a unified time dependence of strength for quasi-static and high-rate short-term loadings. The temperature dependence of the incubation period (latent time) is calculated analytically, and a relation is found between the latent fracture time and the thermal vibration frequency of atoms. © 2004 MAIK “Nauka/Interperiodica”.

In classical criteria, it is assumed that the energy and the assumption that τ = d/c, where c is the maximum momentum that go into forming new surfaces and frac- 2 2 K ture regions are expended continuously. Allowance for velocity of elastic waves and d = ------lc is a length char- π σ2 the discreteness of the process of dynamic fracture c leads to a dynamic generalization of the linear mechan- acterizing an elementary fracture cell on the given scale ics of fracture. (Klc is the toughness of the material). At the present In particular, in [1Ð4], a structural-time approach is time, the physical meaning of the length d is not clearly developed in which the pulse characteristics of the understood. The problem of interpreting d is discussed stress field and the structural features of materials are in [6, 7]. taken into account. The structure is considered on both It is shown in [4] that τ can be interpreted as the min- space and time scales [1, 2]. The corresponding time imum time introduced in [8]. Within this approach, the scale τ is referred to as the latent (structural) time of σ τ parameters c and describe the strength of the defect- fracture. free material. The structural fracture time τ character- It is assumed that fracture occurs when the impulse izes the dynamic features of brittle fracture and should of a force acting over time τ reaches a critical value, be determined experimentally for each material. As for ≤ σ J(t) Jc. In terms of the continuum mechanics, we can c, the physical meaning of this parameter needs clari- write fication and can be determined using the approach t developed by Zhurkov and coworkers. The present paper is devoted to this problem. σ()t' dt' ≤ σ τ, (1) ∫ c Studies of the physical basis of strength of solids t Ð τ have shown that there are universal features character- σ izing the temperature and time dependences of where c is the ultimate stress for the defect-free material, i.e., a material that does not contain intentionally strength. It was established in [9] that the time t∗ elaps- produced defects and concentrators, such as cracks and ing between the application of a tensile breaking stress sharp notches. σ and fracture of the test sample is described well by σ Applying a stress c is not enough to destroy an ele- the formula ment; the critical stress must act over a certain period of t = t exp[]()U Ð γσ /kT , (2) time. The time τ is defined as the minimum time neces- * 0 0 γ sary for fracture of a material subjected to a critical where T is the absolute temperature and U0, t0, and are σ stress c. It is shown in [3Ð5] that, in the case of defect- constants characterizing the strength properties of the free materials, the experimental data on high-rate frac- material: U0 is closely related to the energy for inter- ture (chipping-off) can be adequately explained under atomic bond breaking and characterizes the activation

1052 GLEBOVSKIŒ, PETROV The tensile-stress maximum ﬁrst arises at the point x = cti/2. The minimum amplitude A∗ at which fracture

occurs for a given pulse duration ti can be found from 1 t A the condition max I = σ , where I = --- σ (t')dt'. t c τ∫()t Ð τ As a result, we obtain the following expression

Amplitude describing the time dependence of strength for any loading duration with allowance for the latent fracture time: σ  c ------, t ≥ τ Pulse duration ti  τ i 1 Ð ------ 2ti Fig. 1. Tensile stress pulse. A* = (3)  σ τ 2 c ≤ τ ------, ti . Ð13 ti energy for fracture, t0 = 10 s is a fundamental characteristic of the material coinciding with the period of The time t∗ elapsing between the application of a stress γ thermal oscillations of atoms, and is a coefficient and fracture is given by that is proportional to the excess stress applied to interatomic bonds (relative to the average stress in the ti t = τ + ---, sample). * 2 According to the kinetic model developed in [9Ð11], τ σ where is the structural fracture time, c is the static fracture is a process that starts immediately after the strength, t is the stress pulse duration, and A∗ is the application of a load to the sample and continuously i develops in time. This process consists in the breaking threshold amplitude. σ of interatomic bonds and the gradual accumulation of Since the static tensile strength c is defined ambig- broken bonds in the material subjected to a fracture uously and has no clear physical interpretation [9], we load. It was shown in [9] that the time factor is a funda- transform Eq. (3) using the Zhurkov formula (2) and mental characteristic of strength. One cannot gain a obtain the time dependence of strength (Fig. 2). The deep insight into the phenomenon of strength without fracture of a material subjected to external forces is not considering the time taken by fracture. Equation (2) a purely mechanical phenomenon; thermal fluctuations shows that tensile breaking is an activation process are of fundamental importance here, and tensile whose rate is determined by the mean thermal fluctua- stresses only favor the fluctuation process by lowering tion frequency (or waiting time) and that there is an the energy barrier and making the fracture more proba- energy barrier U0 that must be overcome in order to ble. Let us represent Eq. (2) in the form break the bonds responsible for the strength of the σ = ()U Ð kTln() t /t /γ . (4) material. This barrier depends on the nature of these 0 * 0 bonds, and, according to Eq. (2), a certain mechanism Substituting Eq. (4) into Eq. (3) gives operates that decreases the initial barrier U by γσ when 0 U Ð kTln() t /t tensile stresses are applied. ------0 * 0-, t ≥ τ  τ i Let us analyze the fracture of a defect-free material.  1 Ð ------/γ The one-dimensional problem of fracture of an elastic 2t A* =  i (5) rod subjected to a tensile stress pulse triangular in shape  ()τ() (Fig. 1) was solved in [12] using the incubation-period- 2 U0 Ð kTln t*/t0 ≤ τ ------γ -, ti . based criterion (1). ti Let us consider the reflection of a compressive stress γ The parameters t0, U0, and can be determined pulse that is triangular in shape from the free end of a experimentally and are listed in the table. All calcula- semi-infinite rod aligned with the x axis and located in σ tions are performed for the same temperature, equal to the region x > 0. The incident pulse is written as Ð = 300 K. For comparison, Fig. 2 also shows the results − A(1 Ð t/ti Ð x/cti)[H(ct + x) Ð H(ct + x Ð cti)], where A obtained without regard for the Zhurkov correction is the amplitude of the pulse, ti is its duration, H(t) is the (dashed lines). The experimental data are taken from Heaviside step function, and c is the maximum velocity [15], where brittle fracture of solids subjected to tensile of acoustic waves. The rise time of the pulse is zero. stress waves was studied using transverse impacts of a σ The pulse reflected from the free end of the rod is + = plate-shaped target and a parallel striking plate. A(1 Ð t/ti + x/cti)[H(ct Ð x) Ð H(ct Ð x Ð cti)]. The total It should be noted that earlier attempts have also σ σ σ stress is = Ð + +. been made to analyze the kinetic nature of strength. For

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 KINETIC INTERPRETATION OF THE STRUCTURAL-TIME CRITERION FOR FRACTURE 1053 instance, the kinetics of fracture was discussed in [16] logt* [s] in terms of the Bailey criterion. However, this approach 6 cannot be extended to the case of high-rate pulse tests. (a) Using the incubation-period-based criterion with cor- 4 rection (4) makes it possible to describe the entire range of loading durations (covering both quasi-static and dynamic tests). It can also be seen from Fig. 2 that the σ 0 substitution of Eq. (4) for c has a signiﬁcant effect on the position of the static branch of the strength time dependence. The kinetic model of fracture describes the experimental data more adequately. However, using –4 Eq. (2) has no effect on the dynamic branch, whose position is mainly determined by the incubation period of fracture. –8 Let us show that, in the limiting case, the latent time 0 100 200 300 can be expressed in terms of t0. This condition must be Threshold amplitude, MPa satisﬁed, because the parameters of the structural–time logt* [s] criterion should be chosen so that the results obtained 6 within the kinetic model of fracture are reproduced in (b) the limiting case. Therefore, we will analyze the 4 strength in structural-time terms within the kinetic model in the transient region of the time dependence of ≥ τ strength, i.e., for loading pulse durations ti and ti ~ 0 τ, where the effect of the incubation period is significant. For this purpose, we use Eq. (3), which corresponds to the period of time before fracture characterized by the time t∗. Using Eq. (2), the latent (structural) –4 time is found to be t τ = t exp[]()U Ð γσ /kT Ð ---i, –8 0 0 2 0 200 600 1000 Threshold amplitude, MPa where t0 is associated with the period of thermal oscil- σ lations of atoms and is determined by Eq. (5) or is Fig. 2. Relation between the fracture time and strength for chosen as the stress corresponding to the given pulse (a) PMMA and (b) aluminum calculated with (solid lines) duration in the strengthÐtime curve. By introducing the and without (dashed lines) regard for the Zhurkov formula. τ Points are experimental data [16]. notation ti = n , the latent time can be written as τ []()γσ () = t0 exp U0 Ð /kT /1+ n/2 . (6) The dependence of the latent time on the stress pulse duration is shown in Fig. 3. The time τ is seen to be con- –6 1 –4 stant over almost the entire range of pulse durations. [s] [s]

Therefore, the latent time is independent of the loading τ τ parameters and can be considered a parameter of the 2 material. –6 According to Eq. (6), τ can be expressed in terms of τ ≅ λ λ the parameter t0 as t0, where = exp[(U0 Ð –9 γσ)/kT]/(1 + n/2). This equation can be used to analyze –8 the temperature dependence of the latent time. Incubation time log Thus, the following conclusions can be made: Incubation time log (1) In the case of slow loadings, the structural-time –10 criterion given by Eq. (1) in which the static strength is 20 60 100 calculated from the Zhurkov formula (2) allows one to Pulse duration obtain a more correct uniﬁed time dependence of strength [in comparison with that calculated without regard for Eq. (2)]. Fig. 3. Dependence of the latent fracture time at a constant temperature on the relative duration of a threshold pulse cal- (2) Taking into account the thermal-ﬂuctuation culated with allowance for the Zhurkov formula for (1) alu- mechanism for fracture and using the incubation period minum and (2) PMMA.

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1054 GLEBOVSKIŒ, PETROV

Parameters for calculating the time dependence of strength 4. N. F. Morozov and Yu. V. Petrov, Dokl. Akad. Nauk 324 (5), 964 (1992) [Sov. Phys. Dokl. 37, 285 (1992)]. γ Ð28 3 τ µ Material U0, kJ/mol , 10 m t0, s , s [6] 5. Y. V. Petrov and N. F. Morozov, ASME Trans. J. Appl. Al [13] 195 14.7 10Ð13 0.75 Mech. 61, 710 (1994). PMMA [14] 150 13.2 10Ð13 0.65 6. N. V. Morozov and Yu. V. Petrov, Dynamics of Fracture (Springer, Berlin, 2000). 7. R. V. Gol’dshteœn and N. M. Osipenko, Dokl. Akad. model makes it possible to more correctly determine Nauk SSSR 240 (4), 829 (1978) [Sov. Phys. Dokl. 23, the breaking load in dynamic fracture tests. 421 (1978)]. (3) A relation is found between the latent time and 8. J. F. Kalthoff and D. A. Shockey, J. Appl. Phys. 48 (3), the thermal oscillation period. 986 (1977). 9. S. N. Zhurkov and S. A. Abbasov, Vysokomol. Soedin. 3 (4) Allowance for the kinetic nature of fracture (3), 450 (1961). makes it possible to ﬁnd the temperature dependence of the latent fracture time, which can be used in further 10. S. N. Zhurkov, V. I. Betekhtin, and A. N. Bakhitbaev, Fiz. Tverd. Tela (Leningrad) 11 (3), 690 (1969) [Sov. Phys. analysis of the fracture of materials. Solid State 11, 553 (1969)]. 11. S. N. Zhurkov, Izv. Akad. Nauk SSSR, Neorg. Mater. 3 ACKNOWLEDGMENTS (10), 1767 (1967). This work was supported by the Russian Foundation 12. N. F. Morozov, Yu. V. Petrov, and A. A. Utkin, Dokl. for Basic Research (project nos. 03-01-39010, 02-01- Akad. Nauk SSSR 40 (4), 1233 (1991). 01035) and the Ministry of Education (project no. A03- 13. A. M. Molodets and A. N. Dremin, Fiz. Goreniya 2.10-270). Vzryva, No. 5, 154 (1983). 14. B. Tsoœ, É. M. Kartashov, and V. V. Shevelev, Strength and Fracture of Polymer Films and Fibers (Khimiya, REFERENCES Moscow, 1999). 1. Yu. V. Petrov, Preprint No. 139 (Inst. of Problems in 15. É. N. Bellendir, Candidate’s Dissertation (Physicotech- Machine Science, Russian Academy of Sciences, nical Inst., USSR Academy of Sciences, Leningrad, St. Petersburg, 1996). 1990). 2. N. F. Morozov and Yu. V. Petrov, Problems of the Frac- 16. V. R. Regel’, A. I. Slutsker, and É. E. Tomashevskiœ, ture Dynamics of Solids (S.-Peterb. Gos. Univ., Kinetic Nature of the Strength of Solids (Nauka, Mos- St. Petersburg, 1997). cow, 1974). 3. N. F. Morozov, Yu. V. Petrov, and A. A. Utkin, Dokl. Akad. Nauk SSSR 313 (2), 276 (1990) [Sov. Phys. Dokl. 35, 646 (1990)]. Translated by Yu. Epifanov

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

DEFECTS, DISLOCATIONS, AND PHYSICS OF STRENGTH

Molecular Dynamics Simulation of Plastic Effects upon Spalling A. M. Krivtsov Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, Vasil’evskiœ Ostrov, Bol’shoœ pr. 61, St. Petersburg, 199178 Russia e-mail: [email protected] Received June 2, 2003; in ﬁnal form, September 26, 2003

Abstract—The molecular dynamics method is applied to simulate spalling during the plane shock interaction between plates. The effect of lattice defects in a material on the propagation of a shock wave and the process of spalling is studied. The plastic effects are described using a model of imperfect particle packing with defects (vacancies). The model proposed can describe the separation of the shock-wave front into an elastic precursor and a plastic front and give velocity proﬁles for the free target surface close to the experimental proﬁles. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION particles interacting through a pair potential Π(r). The equations of particle motion have the form In the last decades, the molecular dynamics (MD) method has been widely used to simulate the deforma- N () f rk Ð rn () tion and fracture of solids [1Ð5]; the representation of a mrúúk = ∑ ------rk Ð rn , (1) rk Ð rn material as a set of interacting particles allows one to n = 1 describe its mechanical properties at both the micro- and macrolevel [6, 7]. To construct computer models where rk is the radius vector of the kth particle, m is the for high-strain-rate fracture of materials, it is conve- particle mass, N is the total number of particles, and nient to use the results of full-scale spalling tests, in f(r) = ÐΠ'(r) is the interparticle interaction force. Con- which extremely high loads are created in a material trary to [13], dissipative forces do not act in this system. under the conditions of uniaxial deformation [8Ð10]. In We use the following notation: a is the equilibrium dis- most works dealing with the MD simulation of spalling, tance between two particles, f(a) ≡ 0, C is the stiffness ideal single-crystal particle packing was studied [11Ð of the interatomic bond in equilibrium, and T0 is the 13]. However, the mechanical properties of materials period of vibrations of the mass m under the action of a are rather sensitive to structural defects [14, 15]. More- linear force with stiffness C: over, in many processes where real materials exhibit Π ()≅ () π plastic properties, single crystals behave elastically C =='' a Ð f ' a , T 0 2 m/C. (2) until fracture. Although plastic behavior manifests itself upon MD simulation of the propagation of shock We will use the quantities a and T0 as microscopic dis- waves in single crystals [16], it differs substantially tance and time scales. from the plastic deformation of real materials. In partic- For a particle of mass m that is in equilibrium in the ular, in full-scale tests, a shock wave is clearly divided potential field Π(r) to be at infinity, its minimum veloc- into an elastic precursor and a plastic front, whereas ity (dissociation velocity) must be equal to this division is weak or even absent in ideal single crystals. To solve this problem and describe the phenomena v mentioned above, we propose a model of a crystalline d = 2D/m, (3) material with artificially introduced structural defects Π (vacancies). where D = (a) is the binding energy. The velocity of propagation of long-wavelength waves in an infinite chain is determined by the formula

2. SIMULATION PROCEDURE 2 v 0 = a C/m. (4) The simulation procedure applied in this work is identical to that used in [13, 17] and is described in In this work, we analyze a two-dimensional material. In detail in [18]. The material is represented by a set of the case where particles are packed to form an ideal tri-

1056 KRIVTSOV

a b z x

Fig. 1. Setup of the computer experiment: (a) impactor and (b) target. Fig. 2. Formation of a spalling crack.

angular (close-packed) crystal lattice, the velocity vl of particles moving away from each other, and the absence propagation of long longitudinal waves is of interaction at large distances between them. In this work, we study the principal possibility of describing 9 the plastic effects upon spalling rather than simulate the v = ---v ≈ 1.06v . (5) l 8 0 0 behavior of a certain material; therefore, we will use only the LennardÐJones potential. Our results can eas- We will illustrate these formulas for the case of the ily be extended to more complex potentials describing classical LennardÐJones potential: the properties of materials more exactly. a 12 a 6 Π()r = D --- Ð,2 --- (6) r r 3. SIMULATION OF SPALLING where D and a are the binding energy and equilibrium Figure 1 shows the setup of the computer experi- interatomic distances introduced earlier, respectively. ment. Particles form two rectangles lying in the xz The corresponding interaction force f(r) = ÐΠ'(r) has plane; they simulate the sections of an impactor (a) and the form a target (b). The impactor and target consist of the same particles, whose interaction is described by Eq. (6). The 12D a 13 a 7 particles are ordered to form a triangular lattice, which fr()= ------Ð.--- (7) a r r is identical for the impactor and target; the lattice is oriented so that one of its basis vectors is directed along In the case of the LennardÐJones potential, the stiffness the x axis. Free boundary conditions are used at all 2 C and binding energy D obey the relation C = 72D/a ; outer boundaries. The lattice parameter ae in the initial therefore, the velocity of propagation of long waves and conﬁguration is chosen so as to ensure the absence of the dissociation velocity are connected by the simple internal stresses at a cutoff radius of 2.1a [18].1 relation v0 = 6vd. The LennardÐJones potential is the Initially, the target has a zero velocity and the simplest potential that allows one to take into account impactor velocity is directed along the z axis toward the the general properties of interatomic interaction: repul- target. Moreover, at the initial instant of time, each par- sion of particles that approach each other, attraction of ticle in the impactor and target is given an additional random velocity that is chosen from a two-dimensional uniform random distribution with a given variance σ, Calculation parameters deﬁned as Experiment n n 1 2 1 Parameter σ ==--- ∑ ()V Ð V , V --- ∑ V . (8) ABC n k n k k = 1 k = 1 Approximate number of 100000 500000 1000000 particles N Here, Vk is the projection of the velocity of the kth par- Impactor velocity v 1.05v v 1.1v ticle along the shock direction and k runs over the val- imp d d d ues specifying a certain set of particles that enters the Particle-velocity 0.001v 00 ∆ d total system of particles to be analyzed in the experi- deviation V0 ment. We also use the standard deviation of the particle Vacancy concentration p, %061 velocities ∆V = σ . Cutoff radius a 2.1a 2.1a 2.1a cut The states of the impactor and target after simulation are shown in Fig. 2. A spalling crack formed in the Impactor (target) width w 708a 1584a 2238a target is clearly visible. However, the interface between Impactor thickness h1 35a 78a 111a the impactor and target is invisible because of the ideal Target thickness h2 88a 196a 277a coincidence of the contacting surfaces (matching of the crystal lattices in the impactor and target). The param- Number of particle layers Nz 142 319 450 ∆ eter values for this case are given in the table (experi- Integration step t 0.03T0 0.03T0 0.03T0 1 Calculation time tmax 3ts 3ts 3ts ae = 0.9917496a.

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 MOLECULAR DYNAMICS SIMULATION 1057

v, m/s v, m/s to use the time ts as a time scale, since it has a clear physical meaning and can easily be determined. The 500 (a) 500 (b) quantity Nz in the table is equal to the total number of particle layers (in the impactor + target system) along 300 300 the shock direction. Figure 3a shows the simulated time dependence of 100 100 the velocity of the free target surface. To measure the velocity of the free surface, we used two lower particle 0 0 –0.5 0 0.5 1.5 2.5 –0.5 0 0.5 1.5 2.5 layers, more speciﬁcally, only the central part of these layers of width w = w Ð 2(h + h ), to eliminate edge t, µs t, µs 0 1 2 effects. The dependence obtained was averaged over time to exclude high-frequency vibrations. For compar- Fig. 3. Time dependence of the velocity of the free target surface for (a) single crystal (calculation) and (b) titanium ison, Fig. 3b shows an analogous experimental depen- alloy (experiment). dence obtained during a full-scale spalling test of a titanium alloy at an impactor velocity of 602 m/s [9]. The time and velocity scales in Fig. 3a are chosen according ment A). Computer experiments show that the mini- to the full-scale experiment. The shapes of the curves mum impactor velocity at which spalling begins is are seen to be similar: the main maximum correspond- approximately equal to the dissociation velocity vd; ing to the instant at which the shock wave reaches the therefore, it is worthwhile to compare the velocities free surface and vibrations in the target are clearly used in the calculation with this value (see table). The deﬁned in both dependences. However, there are sub- dimensions of the impactor and target in this computer stantial differences. First, the shapes of the shock-wave ≈ ≈ experiment are such that h1/w 1/20 and h1/h2 2/5. fronts are different. In the full-scale experiment, the The spalling time ts can be estimated from the formula shock front is clearly divided into an elastic precursor ts = (h1 + h2)/vl, where h1 and h2 are the thicknesses of and the subsequent plastic front, whereas this division the impactor and target, respectively, and v is deter- is absent in the simulation. In the latter case, only the l elastic component of the front exists and the plastic mined from Eq. (5). In this case, ts = 18.4T0. Of course, this estimation is very rough, since v is the velocity of front is virtually absent; this behavior is caused by the l ideal (defect-free) structure of the model single crystal. longitudinal waves according to linear theory; i.e., this velocity corresponds to small-amplitude waves. To simulate plastic deformation at the shock front, According to nonlinear theory, the larger the amplitude we consider a crystalline material with lattice defects. of waves in a single crystal, the higher their velocity. As the initial material, we choose the single crystal However, in a crystalline material with defects, the mentioned above. To produce an imperfect material, we wave velocity can be lower than vl. Moreover, a wave randomly remove atoms from the initial material to has a certain extent in space, which can also introduce achieve the required defect (vacancy) concentration, error in the spalling time. Nevertheless, it is convenient which is calculated as the percentage ratio of the num-

v, m/s v, m/s v, m/s 400 400 400 (a) (b) (c)

300 300 300

200 200 200

100 100 100

0 0 0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 t, µs t, µs t, µs

Fig. 4. Time dependence of the velocity of the free target surface for (a) a perfect single crystal (calculation), (b) a single crystal with defects (calculation), and (c) VT-20 titanium alloy (experiment).

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1058 KRIVTSOV

1.2 (a) (b) 1

2 8 0.8 3 4 7 d

v / 6 v 5 5 0.4 4

3 6 2 1 0 0.4 0.6 0.8 1.0 1.2 0.6 0.8 1.0 1.2 t/ts t/ts

Fig. 5. Velocity profiles for the free target surface (simulation): (a) at vimp = vd and a vacancy fraction of (1) 0, (2) 1, (3) 2, (4) 4, (5) 6, and (6) 9%; (b) at a vacancy fraction of 6% and vimp/vd equal to (1) 0.1, (2) 0.2, (3) 0.4, (4) 0.6, (5) 0.8, (6) 1.0, (7) 1.2, and (8) 1.4. ber of removed atoms to the initial number of atoms. the plastic-front amplitude, which leads to their distinct When a shock wave moves in such a material, vacan- separation. Figure 5b shows the results of analogous cies initiate irreversible changes in its structure, which simulations at the same defect concentration (6%) and lead to plastic deformation of the medium. various values of the impactor velocity vimp. At low vimp v ≤ v The initial values of the parameters are given in the values ( imp 0.2 d), the shock-wave front contains table (experiment B). The model contains ~5 × 105 par- only an elastic component. However, as the impactor ticles, with the ratios of the impactor and target dimen- velocity increases, a plastic component appears, which sions remaining as before. Figures 4a and 4b show the increases much faster than the elastic component with time dependences of the velocity of the free surface a further increase in vimp. when the shock wave reaches the surfaces of the perfect single crystal and the imperfect material (vacancy fraction 6%), respectively. For comparison, Fig. 4c shows the experimental curve recorded during a full-scale spalling test of a VT-20 titanium alloy at an impactor velocity of 365 m/s [9]. The time and velocity scales in Figs. 4a and 4b are chosen according to the experimen- d tal scales. In Fig. 4a, the shock-wave front is virtually vertical and carries only elastic deformation. In con- c trast, Figs. 4b and 4c demonstrate a pronounced elastic a precursor followed by a plastic front. The relation between the amplitudes of the elastic and plastic fronts depends on the defect concentration and the impactor velocity, which is illustrated in b Figs. 5a and 5b. Figure 5a shows the velocity profiles of the target free surface simulated at the same parameters as those in Fig. 3b but at different defect concentrations. The time and velocity are measured in units of the calculated spalling time t and the dissociation velocity Fig. 6. Structure of the model material before passing a s shock wave: (a) microvoid and (bÐd) pairs of vacancies. vd. As the defect concentration increases, the elastic- The particles that have less than six neighbors at a distance precursor amplitude decreases significantly faster than of 1.1a are dark.

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 MOLECULAR DYNAMICS SIMULATION 1059

1.4

1.0 d d

v /

a v c 0.6

0.2 b 0 0.5 1.0 1.5 2.0 2.5 3.0 t/ts

Fig. 7. Changes in the material structure upon passing a shock wave: (a) microvoid, (b) dislocation nucleation, and Fig. 8. Velocity proﬁles of the free target surface for the (c, d) decomposition of pairs of vacancies into dislocation vacancy concentrations of 0 (thin line) and 0.1% (heavy pairs. line).

4. DISCUSSION OF THE RESULTS causes shear strains at the shock front upon defect To elucidate the behavior of defects at the shock- motion. wave front, we consider an element of the model mate- Note that even an insignificant number of defects rial before and after passing a shock wave (Figs. 6, 7). results in a significant change in the velocity profile of These pictures are obtained as a result of simulating the free surface. Figure 8 illustrates the results of simu- spalling at the parameters given in the table (experi- lating a perfect single crystal and a similar crystal ment C). The material element shown in Figs. 6 and 7 where one atom per one thousand atoms is removed contains about 4800 particles, which is less than 0.5% of the total volume to be simulated. To study the behav- (the defect concentration is 0.1%). The impactor veloc- v v ior of defects, we analyzed the total volume; this ele- ity is imp = 1.1 d, and the values of the other parame- ment is shown only for illustration. In Figs. 6 and 7, the ters correspond to experiment B (see table). For illus- particles located near lattice defects are dark.2 In the tration, the curves were not averaged over time. There- undeformed material (Fig. 6), the defects are vacancies. fore, the profile of the perfect single crystal However, due to their random distribution, some of demonstrates clearly visible high-frequency vibrations them are in immediate vicinity to each other and some (induced by the discreteness of the MD representation) vacancies combine to form microvoids (Fig. 6a). Let us at the shock front. These vibrations virtually disappear choose several closely spaced vacancies (Figs. 6, in the material with defects. Moreover, the vibrations in vacancies aÐd) and follow their evolution upon passing the target strongly change in character and the rate of a shock wave (Fig. 7, vacancies aÐd). Microvoid a in their damping increases significantly. At the low defect Fig. 7 is seen to change its orientation under the effect concentration used in the simulation, the elastic and of the shock wave. As for the other chosen vacancy plastic fronts cannot be separated. However, these groups, they decompose into pairs of moving disloca- fronts will likely be separated in this material as the tions (Fig. 7, groups c, d; the arrows show the directions crystal thickness along the shock direction increases. of dislocation motion). The shock front in Fig. 7 moves from the top down; therefore, different stages of the decomposition of the vacancy groups are visible in 5. CONCLUSIONS Fig. 7 (groups bÐd): nucleation of dislocations (Fig. 7, group b) and their recession (Fig. 7, groups c, d). A Thus, when loaded with a shock wave, a single crys- comparison of single vacancies in Figs. 6 and 7 shows tal consisting of particles that interact through the Len- that a shock wave of the given intensity cannot cause nardÐJones potential and containing a sufficient con- their migration or transformation into dislocations. centration of lattice defects demonstrates pronounced Thus, plasticity in the imperfect material under study plastic effects, such as separation of the shock-wave occurs mainly via the decomposition of closely spaced front into elastic and plastic components. As a result, vacancies to form dislocation-like defects, which we could obtain time dependences of the velocity of the 2 Specifically, the particles that have less than six neighbors at a free target surface very close to the experimental distance of 1.1a are dark (in the ideal triangular lattice, the num- curves, in particular, to the curves recorded upon spal- ber of neighbors for each atom is equal to six). ling of titanium alloys.

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1060 KRIVTSOV

ACKNOWLEDGMENTS 8. A. M. Rajendran and D. J. Grove, Int. J. Impact Eng. 18 The author is grateful to Yu.I. Meshcheryakov for (6), 611 (1996). experimental data and useful discussions. 9. Yu. I. Mescheryakov, A. K. Divakov, and N. I. Zhi- gacheva, Shock Waves 10, 43 (2000). This work was supported by the Ministry of Educa- 10. G. I. Kanel’, S. V. Razorenov, and V. E. Fortov, in Prob- tion of the Russian Federation, project no. E02-4.0-33. lems in Mechanics of Deformed Solids (S.-Peterb. Gos. Univ., St. Petersburg, 2002), pp. 159Ð165. REFERENCES 11. N. J. Wagner, B. L. Holian, and A. F. Voter, Phys. Rev. A 45 (12), 8457 (1992). 1. B. L. Holian, Shock Waves 5 (3), 149 (1995). 12. W. C. Morrey and L. T. Wille, Comput. Mater. Sci. 10 2. A. I. Lobastov, V. E. Shudegov, and V. G. Chudinov, Zh. (1Ð4), 432 (1998). Tekh. Fiz. 70 (4), 123 (2000) [Tech. Phys. 45, 501 (2000)]. 13. A. M. Krivtsov, Int. J. Impact Eng. 23 (1), 466 (1999). 3. V. A. Lagunov and A. B. Sinani, Fiz. Tverd. Tela 14. S. P. Nikanorov and B. K. Kardashev, Elasticity and Dis- (St. Petersburg) 43 (4), 644 (2001) [Phys. Solid State 43, location Inelasticity of Crystals (Nauka, Moscow, 1985). 670 (2001)]. 15. V. A. Lagunov and A. B. Sinani, Fiz. Tverd. Tela 4. F. F. Abraham, R. Walkup, H. Gao, M. Duchaineau, (St. Petersburg) 45 (3), 542 (2003) [Phys. Solid State 45, T. D. De La Rubia, and M. Seager, Proc. Natl. Acad. Sci. 573 (2003)]. USA 99 (9), 5783 (2002). 16. B. L. Holian and P. S. Lomdahl, Science 280 (5372), 5. A. M. Krivtsov, in Problems in Mechanics of Deformed 2085 (1998). Solids (S.-Peterb. Gos. Univ., St. Petersburg, 2002), 17. A. M. Krivtsov and Y. I. Mescheryakov, Proc. SPIE pp. 173Ð178. 3687, 205 (1999). 6. A. M. Krivtsov and N. F. Morozov, Dokl. Akad. Nauk 18. A. M. Krivtsov and N. V. Krivtsova, Dal’nevost. Mat. 381 (3), 825 (2001) [Dokl. Phys. 46, 825 (2001)]. Zh. 3 (2), 254 (2002). 7. A. M. Krivtsov and N. F. Morozov, Fiz. Tverd. Tela (St. Petersburg) 44 (12), 2158 (2002) [Phys. Solid State 44, 2260 (2002)]. Translated by K. Shakhlevich

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

MAGNETISM AND FERROELECTRICITY

Refraction of Surface Spin Waves in Spatially Inhomogeneous Ferrodielectrics with Biaxial Magnetic Anisotropy S. A. Reshetnyak Institute of Magnetism, National Academy of Sciences of Ukraine, Kiev, 03142 Ukraine e-mail: [email protected] Received August 13, 2003

Abstract—The refractive index of surface spin waves propagating in a ferromagnetic medium with a nonuniform distribution of the parameters of uniaxial and orthorhombic magnetic anisotropies and exchange coupling is determined within the spin-density formalism. The coefficients of reflection and transmission of spin waves at the interface between two homogeneous magnets with different constants of uniaxial and orthorhombic magnetic anisotropies, exchange coupling, and saturation magnetization are calculated. The dependences of the intensity of a reflected wave and the refractive index on the wave frequency and the strength of an external dc homogeneous magnetic field are determined. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION spin wave from the interface between two homoge- Considerable progress achieved in the field of nano- neous ferromagnets with different constants of uniaxial technologies and nanoelectronics over the last decade and orthorhombic magnetic anisotropies, exchange has brought to the fore the necessity of designing new coupling, and saturation magnetization. materials and devices based on the use of unique properties of high-frequency waves. In this respect, it is of 2. EQUATIONS OF MAGNETIZATION interest to investigate the specific features of surface DYNAMICS spin waves from the practical standpoint. A theoretical description of the features of propaga- Let us consider an infinite ferromagnet consisting of tion of spin waves, as a rule, has been performed within two semi-infinite parts that are in contact along the yOz the wave approach, which, for example, has been suc- plane. In the corresponding half-spaces, these parts of cessfully used to determine the spectral characteristics the ferromagnet are characterized by the saturation of magnetic materials [1Ð5]. magnetizations M01 and M02, as well as by continuously (or piecewise continuously) and slowly varying param- In this work, the behavior of surface spin waves α α propagating in a ferromagnetic medium with a nonuni- eters of exchange coupling ( 1, 2), uniaxial magnetic β β form distribution of magnetic parameters was anisotropy ( 1, 2), and orthorhombic magnetic anisot- ρ ρ described using mathematical tools of geometrical ropy ( 1, 2). The easy magnetization axis of the ferro- optics. Within this approach, the required change in the magnet and the external dc magnetic field are directed direction of propagation of surface spin waves (in par- parallel to the Oz axis. ticular, focusing) can be achieved both through the cre- For a magnet of the above configuration, the energy ation of artificial inhomogeneities of the magnetic density in the exchange approximation has the form parameters of a medium of the specified configuration and by varying the strength of an external magnetic 2 field. Θ[]()j δ() w = ∑ Ð1 x w j + A x M1M2. (1) Earlier [6, 7], we calculated the refractive indices of j = 1 bulk and surface spin-wave rays and investigated their behavior at the interface between two homogeneous Here, magnets with different parameters of exchange cou- α ∂ 2 β pling and uniaxial magnetic anisotropy. The purpose of m j ()ρ2 2 2 w j = --- ------∂ - ++--- m jx + m jy m jx Ð H0M jz, (2) the present work was to analyze (i) the case of a contin- 2 xk 2 uous distribution of the aforementioned parameters of exchange coupling and uniaxial magnetic anisotropy Θ(x) is the Heaviside step function, A is the parameter and (ii) the case of a continuous distribution of the characterizing the exchange coupling between the half- parameter of orthorhombic anisotropy in a magneti- spaces at x = 0, and Mj = M0jmj (where mj are the unit cally biaxial medium. Moreover, I calculated the refrac- vectors aligned parallel to the magnetization vector and tive index and the intensity of reflection of a surface j = 1, 2).

1062 RESHETNYAK

In the framework of the spin-density formalism [8], In this case, the boundary condition satisfied at the sur- the magnetization can be represented in the form face z = 0 can be written in the form [9] (), Ψ +(), Ψ (), , ∂χ M j r t ==M0 j j r t s j r t , j 12, (3) j(),,, χ (),,, ------xy0 t Ð0,L j j xy0 t = (8) Ψ ∂z where j are the quasi-classical wave functions, which play the role of an order parameter of the spin density; where Lj is a parameter characterizing the spin pinning r is the radius vector in the Cartesian coordinate sys- on the surface of the magnet. After the Fourier transfor- tem; t is the time; and s is the Pauli matrix. mation, we obtain the dispersion relation for the expo- The Lagrange equations for the quasi-classical wave nential damping of a surface spin wave along the Oz Ψ functions j take the form axis of a homogeneous uniaxial magnetic material [7]: ∂Ψ (), 2 2 2 j r t Ω []α β ˜ α iប------= Ðµ H ()r, t sΨ ()r, t , (4) j = ()r⊥ k⊥()r⊥ ++()r⊥ H0 j Ð ()r⊥ L j ∂t 0 ej j (9) ×α[]2 ρ β ˜ α 2 ()r⊥ k⊥()r⊥ +++()r⊥ ()r⊥ H0 j Ð ()r⊥ L j , ∂w where µ is the Bohr magneton and H = Ð------j + 0 ej ∂ where Ω = ωប/2µ M , ω is the frequency, k = (k⊥, k ) M j j 0 0j z ∂ is the wave vector, and r⊥ = (x, y). ∂ w j ------∂ ------∂∂()∂ - . xk M j/ xk Making allowance for the fact that, in the ground 3. GEOMETRICAL-OPTICS APPROXIMATION state, the material is magnetized parallel to the vector ez Now, we will attempt to simplify the equation of and assuming that M 2 (r, t) = const in each half-space, magnetization dynamics (7). For this purpose, we will j use the WentzelÐKramersÐBrillouin method described we will seek the solution to the Lagrange equations (4) in [10, 11]. in the following form: In the equation of magnetization dynamics (7), we χ Ψ (), ()µ ប 1 introduce the function j(r⊥, t) = Cexp[i(k0sj(r⊥) Ð j r t = exp i 0 H0t/ χ (), , (5) ω j r t t)], where k0 is the magnitude of the wave vector of the surface wave, for example, at an infinitely large dis- χ where j(r, t) is a small addition accounting for the tance from the boundary x = 0 as viewed from the inci- deviation of the magnetization from the ground state. dent wave [the precise definition of this quantity, as will After linearization of the Lagrange equations (4) with be shown below, is necessary only for the relative mea- due regard for expression (5), we obtain surement of the wave vector k(r⊥) when determining ∂χ (), the refractive index] and C is the slowly varying ampli- iប j r t Ð------µ ------∂ tude. From the dispersion relation (9), we obtain 2 0M0 j t α()2 () α()2 ρ() β() ρ ρ r⊥ k⊥ r⊥ = r⊥ L j Ð r⊥ /2 Ð r⊥ α ∆β ()r ˜ χ , ()r χ* , = ()r Ð ()r Ð ------Ð H0 j j()r t Ð ------j (),r t 2 2 2 2 Ð H˜ ± Ω +.ρ ()r⊥ /4 (6) 0 j j ∂χ (), iប *j r t Next, we assume that the length of a spin wave λ sat------µ ------∂ isfies the geometrical-optics condition: 2 0M0 j t ρ ρ λ Ӷ a, (10) α ∆β ()r ˜ χ* , ()r χ , = ()r Ð ()r Ð ------Ð H0 j j ()r t Ð ------j(),r t 2 2 where a is the characteristic size of inhomogeneities ˜ involved in the medium. Then, from the equation of where H0 j = H0/M0j and j = 1, 2. magnetization dynamics (7), we deduced an equation By separating out the quantity χ*(r, t) from one of analogous to the classical HamiltonÐJacobi equation, the equations in system (6) and substituting it into the that is, other equation of this system, we derive the equation of ()()2 2() magnetization dynamics: —⊥s j r⊥ = n j r⊥ , (11) ប2 ∂2χ (), ρ where j r t 2 2 ˜ Ð------= α ∆ Ð 2αβ++--- H0 j ∆ ()µ 2 ∂ 2 2 2 0M0 j t ∂ ∂ —⊥ = ex∂------+ ey∂----- , (7) x y (12) --- ()βρβ ˜ ()χ˜ (), + + H0 j ++H0 j j r t . 2() 2()2 n j r⊥ = k j r⊥ /k0 .

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As has been done in optics [12], we assume that the We will consider the situation where a spin wave is right-hand side of expression (11) is the square of the incident on the interface between two homogeneous refractive index, magnets that are in contact along the yz plane and are α β ρ characterized by the parameters 1, 1, 1, M01, and L1 ±() 1 [(α()2 ρ() β() ˜ α β ρ n j r⊥ = ---- r⊥ L j Ð r⊥ /2 Ð r⊥ Ð H0 j and 2, 2, 2, M02, and L2, respectively. Note that the k0 (13) spin wave is incident on the interface as viewed from ±Ω[]2 ρ2() 1/2 ) α()]1/2 the first magnet. A spin-wave ray propagates from the j + r⊥ /4 / r⊥ , point (x1, y1, z1) located in medium 1 with a refractive because the ratio between the magnitudes of the wave index n0 = 1 to the point (x2, y2, z2) in medium 2 with a vectors at two different points of the space character- refractive index n±, which, according to expression (13), izes the change in the direction of propagation of the can be written in the form spin wave. Therefore, the pinning of spins on the surface of a α α 2 β ρ ˜ ± Ω 2 ρ 2 ± 1 2 L2 Ð 2 Ð 2/2 Ð H02 2 + 2 /4 magnetic material makes it possible to observe the n = α------. 2 α 2 β ρ ˜ ± Ω 2 ρ 2 effect of birefringence of a surface spin wave. It should 1 L1 Ð 1 Ð 1/2 Ð H01 1 + 1 /4 be noted that, for bulk spin waves propagating in a magnetic material, there exists only one branch that corre- The spin-wave ray passes through the interface at the sponds to the positive sign in expression (13), as is the point (0, y, z). In this case, from the extremum condi- case in the absence of spin pinning on the surface. If the tions of the function s, we have 2 value of α(r⊥)L is relatively large, there appears ± ± j sinθ k another branch that corresponds to the negative sign in ------1 = ----2- θ ± ± expression (13). sin 2 k0 Equation (11) can be represented in the form (15) α α 2 β ρ ˜ ± Ω 2 ρ 2 1 2 L2 Ð 2 Ð 2/2 Ð H02 2 + 2 /4 ± 1 2 2 = ------= n , H ==---[]p Ð n ()r⊥ 0, α 2 α 2 β ρ ˜ ± Ω 2 ρ 2 2 1 L1 Ð 1 Ð 1/2 Ð H01 1 + 1 /4 where p = —⊥s. From this relation, we derive the equa- θ ± θ ± tions describing the ray path in the Hamiltonian form: where 1 is the angle of incidence and 2 is the angle of refraction. rú = p, The same result can also be obtained from other 1 2 considerations. In the equation of magnetization pú = ---—⊥n ()r⊥ . 2 dynamics (7), we introduce the following designations: | | χ ()()ω Hence, it follows from expression (11) that —⊥s⊥(r⊥) = I = exp i k0r Ð t (16) 2 2 ds(r⊥)/dτ = n(r⊥), where dτ = dx Ð dy is the ele- for an incident wave, ment of the ray path. Therefore, the function s can be χ ()()ω determined as the curvilinear integral taken along the R = Riexp k1r Ð t (17) ray path; that is, for a reflected wave, and B χ = Diexp()()k r Ð ωt (18) sn= ∫ dτ. (14) D 2 A for a transmitted wave. Minimizing the curvilinear integral (14) with the In expressions (16)Ð(18), R is the complex ampli- use of Fermat’s principle [13], we deduce the following tude of reflection of the spin wave from the interface; equation of the ray path [10]: D is the transmission amplitude; and k0, k1, and k2 are the wave vectors of the incident, reflected, and trans- d dr⊥ mitted waves, respectively. -----τn------τ = —⊥n. d d Taking into account the boundary condition (8), we can write the following relations: 4. REFRACTION OF A SPIN-WAVE RAY ()± 2 ()± 2 AT THE INTERFACE BETWEEN TWO k0 ⊥ = k1 ⊥ HOMOGENEOUS MAGNETS 2 ˜ 2 2 = ()α L Ð β Ð ρ /2 Ð H01 ± Ω + ρ /4 /α , Formula (14) can be applied to the case of a semi- 1 1 1 1 1 1 1 infinite magnet composed of two contacting homoge- ()± 2 ()α 2 β ρ ˜ ± Ω 2 ρ 2 α neous parts. k2 ⊥ = 2 L2 Ð 2 Ð 2/2 Ð H02 2 + 2 /4 / 2.

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1064 RESHETNYAK

Here, The critical angle of total reflection of the spin wave is determined from the expression k = Ð ()k 2 Ð,k 2 1x 1 ⊥ 1y α α 2 β ρ ˜ ± Ω 2 ρ 2 1/2 θ ± 1 2 L2 Ð 2 Ð 2/2 Ð H02 2 + 2 /4 sin 0 = ------. 2 2 α 2 2 2 k = ()k ⊥ Ð k , 2 α β ρ ˜ ± Ω ρ 2x 2 2y 2 L2 Ð 1 Ð 1/2 Ð H01 1 + 1 /4 where the sign “–” in the expression for k1x corresponds to a spin wave propagating from the interface and k0y = 5. REFLECTION OF SPIN WAVES k1y = k2y are the magnitudes of the wave vectors at the FROM THE INTERFACE BETWEEN TWO interface. It follows from these equations that the inci- HOMOGENEOUS MEDIA dent, reflected, and transmitted waves, as well as the In all cases where waves of arbitrary nature undergo normal to the surface at the point of incidence, lie in the reflection and refraction, it is necessary to estimate the same plane. In this case, the angle of incidence is equal intensity ratio of the transmitted and reflected waves. If to the angle of reflection (by analogy with the law of the intensity of the reflected wave is substantially reflection of light waves in optics [12]). higher than the intensity of the transmitted wave, the For real values of k2x, i.e., under the condition structure under investigation can be used for designing mirrors of different types (for example, plane, convex, ()2 > 2 k2 ⊥ k2y , or concave mirrors of spherical or cylindrical type). Otherwise, when the intensity of the reflected wave is which is equivalent to the condition considerably lower than the intensity of the transmitted wave, the studied structure can be used for fabricating α α 2 β ρ ˜ ± Ω 2 ρ 2 1 2 L2 Ð 2 Ð 2/2 Ð H02 2 + 2 /4 > 2 θ + lenses with specified parameters. α------sin 1 , 2 α L 2 Ð β Ð ρ /2 Ð H˜ ± Ω 2 + ρ 2/4 Let us derive appropriate expressions for the ampli- 2 2 1 1 01 1 1 tudes of reflection and transmission of a spin wave. For we obtain formula (15). this purpose, we will use the boundary conditions for χ(r, t) at the interface, which follow from formulas (1) ()2 2 2 2 If the condition k2 ⊥ < k2y (k1 > 0, k2 < 0) is sat- and (2): isfied, we have []γχ()αχ χ A 2 Ð 1 + 1 1' x = 0 = 0, (19) 2 ()2 k2x ==Ðik1y Ð k2 ⊥ Ði/2h, []()χ χ γα χ A 1 Ð 2 Ð 2 2' x = 0 = 0, χ (), ()()()ω γ D r⊥ t = Dxexp Ð /2h exp ik2yt Ð t . where = M02/M01. χ This means that the quantity h is the depth of penetra- Upon substituting the functions (r, t) in the expo- tion of the spin wave into the second material: nential form [see formulas (16)Ð(18)] into the system of equations (19), we obtain the following expressions for ± 1 the amplitudes of reflection and transmission of a spin ------h = . wave (for simplicity, the signs “±” are omitted from the ()± 2 θ ± ()± 2 k0 ⊥ sin 1 Ð n formulas):

k α α γθcos n2 Ð sin2 θ Ð iA()α cosθ Ð α γ 2 n2 Ð sin2 θ R = ------0 1 2 1 1 1 1 2 ------1 , k α α γθcos n2 Ð sin2 θ Ð iA()α cosθ + α γ 2 n2 Ð sin2 θ 0 1 2 1 1 1 1 2 1 (20) Ð2iAα cosθ D = ------1 1 ------. α α γθ2 2 θ ()α θ α γ 2 2 2 θ k0 1 2 cos 1 n Ð sin 1 Ð iA 1 cos 1 + 2 n Ð sin 1

6. ESTIMATION OF THE PARAMETERS the lens. Recall that the intensity of a reﬂected wave is OF SPIN LENSES AND MIRRORS determined by the square of the reﬂection amplitude | |2 ≈ α Now, we estimate the parameters of a material of a and, as follows from expression (20), R [( 1 Ð α γ2 α α γ2 2 thin lens in the case when the angles of incidence of 2 n)/( 1 + 2 n)] (for small angles of incidence and spin-wave rays with respect to the optical axis of the A ∞). On this basis, under the condition |R|2 < η lens are small and ensure the required transparency of (where η is a parameter specifying the required degree

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 REFRACTION OF SURFACE SPIN WAVES 1065

|R+|2 n+ 0.6 1.0

0.4 0.6

0.2

0 0 56 8 10 46810 ω, 1011 s–1 ω, 1011 s–1

Fig. 1. Dependence of the reflection coefficient |R+|2 on the Fig. 2. Dependence of the refractive index n+ on the fre- frequency ω of a surface spin wave for the following param- quency ω of a surface spin wave for the following parame- α Ð7 2 α × Ð7 2 β β α Ð7 2 α × Ð7 2 β β eters: 1 = 10 cm , 2 = 2 10 cm , 1 = 50, 2 = 100, ters: 1 = 10 cm , 2 = 2 10 cm , 1 = 50, 2 = 100, ρ ρ 4 Ð1 ρ ρ 4 Ð1 1 = 10, 2 = 20, L1 = L2 = 10 cm , M01 = 100 G, M02 = 1 = 10, 2 = 20, L1 = L2 = 10 cm , M01 = 100 G, M02 = θ π 125 G, A = 100, 1 = /80, and H0 = 1870 Oe. 125 G, and H0 = 1870 Oe. of smallness of the reflection coefficient), we obtain the Figures 1 and 2 show the dependences of the + 2 + following restrictions on refractive index n and, conse- reflected intensity I + = |R | and the refractive index n quently, on the parameters α, β, ρ, ω, L, M , and H : R 0 0 on the frequency of a surface spin wave for characteris- η α η tic parameters of the material [14]. It is clearly seen 1 Ð <<2 1 + ------α----- n ------. from these figures that, at a chosen frequency, the 1 + η 1 1 Ð η α α In particular, at 1 = 2, M01 = M02, and L1 = L2, the | +|2 reflection coefficient does not exceed 10%, provided R the refractive index falls in the range 0.52 < n < 1.92. 1.0 For a mirror, the restrictions are as follows: α η α η 2 < 1 Ð 2 > 1 + α----- n ------or α----- n ------. 1 1 + η 1 1 Ð η 0.6 For example, the reflection amplitude squared |R|2 > α α 0.9 can be reached in the case when 1 = 2, M01 = M02, and L1 = L2 at n < 0.03 or n > 37.97. The geometrical-optics condition (10) is satisfied when the thickness of the lens (or mirror) is limited by 0.2 the inequality

2 ˜ 2 2 a ӷ 2πα/()αL Ð β Ð ρ/2 Ð H0 ± Ω + ρ /4 . (21) 0 0.51.0 1.5 2.0 Η0, kOe As can be seen from expressions (15), (20), and (21), the choice of the parameters for a lens or a mirror | +|2 does not present any problems, even with a large variety Fig. 3. Dependence of the reflection coefficient R on the external dc magnetic field H0 for the following parameters: of magnetic materials [14]. Specifically, in the case of α Ð7 2 α × Ð7 2 β β ρ garnet-type ferrites, from condition (21) for a thin lens, 1 = 10 cm , 2 = 2 10 cm , 1 = 50, 2 = 100, 1 = ρ 4 Ð1 it follows that the characteristic size of inhomogeneities 10, 2 = 20, L1 = L2 = 10 cm , M01 = 100 G, M02 = 125 G, Ð4 Ð6 θ π ω × 11 Ð1 should fall in the range a > 10 Ð10 cm. A = 100, 1 = /80, and = 5.55 10 s .

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1066 RESHETNYAK

n+ |R–|2 0.6 1.0

0.8 0.4 0.6 246810 ω, 1011 s–1 0.2 Fig. 5. Dependence of the reflection coefficient |RÐ|2 on the frequency ω of a surface spin wave for the following param- α Ð7 2 α × Ð7 2 β β eters: 1 = 10 cm , 2 = 2 10 cm , 1 = 50, 2 = 100, ρ ρ × 4 Ð1 1 = 10, 2 = 20, L1 = L2 = 8 10 cm , M01 = 100 G, M = 125 G, A = 100, θ = π/80, and H = 750 Oe. 0 0.51.0 1.5 2.0 02 1 0 Η0, kOe tional lines of the total reflection, even though, in this Fig. 4. Dependence of the refractive index n+ on the external case, the incident wave is actually absent because of the homogeneous magnetic field H0 for the following parame- fast damping in the first material. α Ð7 2 α × Ð7 2 β β ters: 1 = 10 cm , 2 = 2 10 cm , 1 = 50, 2 = 100, Furthermore, the material parameters, if required, ρ ρ 4 Ð1 1 = 10, 2 = 20, L1 = L2 = 10 cm , M01 = 100 G, M02 = can be chosen so that the spin waves corresponding to 125 G, and ω = 5.55 × 1011 sÐ1. only one of the two branches will be transmitted through the interface between the two materials, whereas the waves of the second branch will be com- required intensity ratio of the reflected and transmitted pletely filtered out. Obviously, the material parameters waves can be achieved by appropriately choosing the can also be chosen so that a situation will occur in material parameters. Moreover, as can be seen from which the “negative” branch of spin waves will corre- Fig. 3, the reflected intensity substantially depends on spond to the forbidden band of the second material with the strength of the external homogeneous magnetic an increase in the frequency (or in the external mag- field. Therefore, the intensity of a reflected wave can be controlled over a wide range by varying only the strength of the external magnetic field for constant n– parameters of the material. The corresponding depen- 20 dence of the refractive index n+ on the external homogeneous magnetic field is depicted in Fig. 4. Thus, the coefficient of reflection of spin waves from an inhomogeneous inclusion that plays the role of 15 a lens or a mirror can be controlled by varying the strength of the external magnetic field. It should be noted that, in this case, the reflection coefficient can 10 change significantly, even though the parameters of the medium remain constant. Consequently, the same inhomogeneities can be used both as lenses and as mirrors for the same parameters of the structure. 5 Ð 2 The dependences of the reflected intensity I Ð = |R | R and the refractive index nÐ on the frequency of a surface 0 spin wave and the strength of the external magnetic 246810 field for parameters of the material that allow for the ω 11 –1 existence of this branch of spin waves are shown in , 10 s Figs. 5Ð8. It is worth noting that the refractive index becomes infinite in the ranges of spin-wave frequencies Fig. 6. Dependence of the refractive index nÐ on the fre- and external magnetic fields that correspond to the edge quency ω of a surface spin wave for the following parame- α Ð7 2 α × Ð7 2 β β of the band gap in the first material (Figs. 6, 8). In ters: 1 = 10 cm , 2 = 2 10 cm , 1 = 50, 2 = 100, ρ ρ × 4 Ð1 Figs. 5 and 7, these ranges of spin-wave frequencies 1 = 10, 2 = 20, L1 = L2 = 8 10 cm , M01 = 100 G, and external magnetic fields are indicated by conven- M02 = 125 G, and H0 = 750 Oe.

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 REFRACTION OF SURFACE SPIN WAVES 1067

|R–|2 neous media is accompanied by birefringence due to the pinning of spins on the surface of the material. In 1.0 this situation, each of the two branches is characterized by a specific dependence of the coefficient of reflection of a spin wave from the interface between these two 0.8 media. Consequently, it becomes possible to control the ratio between the intensities of the waves transmitted through the interface between two materials and corre- 0.6 0123sponding to different branches. This ratio can be H , kOe changed to the extent that one of the branches will be 0 eliminated completely. The regularities revealed in this | Ð|2 work can be used in designing devices for spin-wave Fig. 7. Dependence of the reflection coefficient R on the microelectronics, including filters and spin-wave ana- external dc magnetic field H0 for the following parameters: α Ð7 2 α × Ð7 2 β β ρ logs of optical devices. 1 = 10 cm , 2 = 2 10 cm , 1 = 50, 2 = 100, 1 = ρ × 4 Ð1 10, 2 = 20, L1 = L2 = 8 10 cm , M01 = 100 G, M02 = θ π ω × 11 Ð1 ACKNOWLEDGMENTS 125 G, A = 100, 1 = /80, and = 5.55 10 s . The author would like to thank Yu.I. Gorobets for

Ð his participation in discussions of the results and help- n ful remarks. 20 This work was supported by the National Academy of Sciences of Ukraine and the Ministry of Education and Science of Ukraine. 15 REFERENCES 1. A. L. Sukstanskii, E. P. Stefanovskii, S. A. Reshetnyak, 10 and V. N. Varyukhin, Phys. Rev. B 61 (13), 8843 (2000). 2. J. Y. Gan, F. C. Zhang, and Z. B. Su, Phys. Rev. B 67 (14), 4427 (2003). 3. M. Buchmeier, B. K. Kuanr, R. R. Gareev, D. E. Bürgler, 5 and P. Grünberg, Phys. Rev. B 67 (18), 4404 (2003). 4. J. Fransson, E. Holmström, O. Eriksson, and I. Sandalov, Phys. Rev. B 67 (20), 5210 (2003). 5. V. P. Antropov, J. Magn. Magn. Mater. 262 (2), L192 0 123 (2003). H0, kOe 6. Yu. I. Gorobets and S. A. Reshetnyak, Zh. Tekh. Fiz. 68 (2), 60 (1998) [Tech. Phys. 43, 188 (1998)]. Fig. 8. Dependence of the refractive index nÐ on the external 7. Yu. I. Gorobets and S. A. Reshetnyak, in Proceedings of homogeneous magnetic field H0 for the following parame- International SchoolÐSeminar on New Magnetic Materi- α Ð7 2 α × Ð7 2 β β ters: 1 = 10 cm , 2 = 2 10 cm , 1 = 50, 2 = 100, als in Microelectronics (Moscow, 2002), p. 175. ρ = 10, ρ = 20, L = L = 8 × 104 cmÐ1, M = 100 G, 8. V. G. Bar’yakhtar and Yu. I. Gorobets, Bubble Domains 1 2 1 2 01 and Their Lattices (Naukova Dumka, Kiev, 1988). M = 125 G, and ω = 5.55 × 1011 sÐ1. 02 9. A. I. Akhiezer, V. G. Bar’yakhtar, and S. V. Peletminskiœ, Spin Waves (Nauka, Moscow, 1967; North-Holland, netic field), whereas the relevant band of the first mate- Amsterdam, 1968). rial will remain allowed. It is this situation that actually 10. Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of arises with the total reflection of spin waves from the Inhomogeneous Media (Nauka, Moscow, 1980). interface. 11. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953; Inostrannaya Literatura, Moscow, 1960), Vol. 2. 7. CONCLUSIONS 12. M. Born and E. Wolf, Principles of Optics, 4th ed. (Per- gamon, Oxford, 1969; Nauka, Moscow, 1973). Thus, the geometrical-optics approximation, as 13. R. Courant, Partielle Differentialgleichungen (Göttin- applied to the case of propagation of surface spin waves gen, 1932; Mir, Moscow, 1964). in biaxial ferromagnets, made it possible to determine 14. A. H. Eschenferder, Magnetic Bubble Technology, 2nd the restrictions imposed on the material parameters at ed. (Springer, Berlin, 1980; Mir, Moscow, 1983). which the spin waves possess properties specific to rays. It was demonstrated that the refraction of a surface spin wave at the interface between two homoge- Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

Physics of the Solid State, Vol. 46, No. 6, 2004, pp. 1068Ð1072. Translated from Fizika Tverdogo Tela, Vol. 46, No. 6, 2004, pp. 1038Ð1042. Original Russian Text Copyright © 2004 by Ryabinkina, Abramova, Romanova, Kiselev.

MAGNETISM AND FERROELECTRICITY

Hall Effect in the FexMn1 Ð xS Magnetic Semiconductors L. I. Ryabinkina, G. M. Abramova, O. B. Romanova, and N. I. Kiselev Kirensky Institute of Physics, Siberian Division, Russian Academy of Sciences, Akademgorodok, Krasnoyarsk, 660036 Russia e-mail: [email protected] Received July 16, 2003

≤ Abstract—The electrical properties and the Hall effect in the FexMn1 Ð xS magnetic semiconductors (0 < x 0.5) have been experimentally studied in the range 77–300 K in magnetic fields of up to 15 kOe. The cation- substituted sulfides with 0.25 ≤ x ≤ 0.3 possessing colossal magnetoresistance (CMR) were established to be narrow-gap semiconductors with carrier concentrations n ~ 1011Ð1015 cmÐ3 and high carrier mobilities µ ~ 102Ð 104 cm2 VÐ1 sÐ1. It is believed that the CMR effect in these sulfides can be explained in terms of the model of magnetic and electron phase separation, which is analogous to the percolation theory in the case of heavily doped semiconductors. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION 3. EXPERIMENTAL RESULTS Despite an impressive number of experimental and At room temperature, manganese monosulfide is a theoretical studies into the colossal magnetoresistance p-type semiconductor with carrier concentration n = 18 Ð3 µ 2 Ð1 Ð1 effect (CMR) [1Ð5], clear understanding of its mecha- 10 cm and carrier mobility = 0.065 cm V s nism is still lacking and the criteria governing the real- [10]. When in the paramagnetic phase, the conduction ization of this effect in a substance are unknown. An activation energy of manganese monosulfide is 0.3 eV efficient method to gain insight into the mechanisms of in the range 150Ð300 K. Below the Néel temperature conduction is based on investigating the Hall effect; (TN ~ 150 K), conduction has a practically nonactivated this approach makes it possible to determine the type, character (similar to LaMnO3 [1]). concentration, and mobility of charge carriers. Com- Increasing the cation substitution x in the FexMn1 Ð xS bined study of the Hall effect and the magnetoresis- sulfide compounds prepared from α-MnS changes the tance may help us to understand the mechanism of type of magnetic order from antiferromagnetic to ferro- CMR in magnetic semiconductors [6Ð8]. magnetic [2], as occurs in manganese oxide compounds synthesized from LaMnO3 and possessing CMR [1]. Similar to the LaMnO3-based cation-substituted sys- 2. SAMPLES AND EXPERIMENTAL tems, the FexMn1 Ð xS sulfides exhibit CMR for some TECHNIQUES concentrations x [2]. The present communication reports on the results of Figure 1 plots the temperature dependences of the electrical resistivity ρ of the cation-substituted a study on the electrical and galvanomagnetic proper- ≤ FexMn1 Ð xS sulfides (0 < x 0.5) measured with no ties of solid solutions in the FexMn1 Ð xS sulfide system (0 < x ≤ 0.5). The technology of synthesized polycrys- magnetic field applied. As the cation substitution x increases, the electrical resistivity of the FexMn1 Ð xS talline FexMn1 Ð xS samples and the results of their x-ray characterization are described elsewhere [2, 3]. X-ray sulfides changes behavior with temperature in the range 77Ð300 K. One observes a decrease in the electrical diffraction analysis showed the FexMn1 Ð xS samples α resistivity and activation energy for conduction (the prepared from -MnS by cation substitution to have a ρ NaCl-type fcc lattice deformed slightly in the region slope of the log vs. 1/T curves), a feature characteristic ~150 K. Experimental studies on the electrical and gal- of the Anderson-type concentration transitions [11]. vanomagnetic properties of the samples were carried The critical concentration at which the FexMn1 Ð xS sul- out using the standard four-probe dc method at temper- fides switch from the semiconductor to semimetal state atures of 77 to 300 K in magnetic fields H = 5, 10, and is xc ~ 0.4 [12]. 15 kOe oriented normal to the sample plane. To avoid Figure 2 displays the temperature dependences of possible side effects, the Hall voltages were measured the electrical resistivity measured for the compositions for two field and current directions in a sample [9]. The with x ~ 0.29 and ~0.3 in magnetic fields of up to samples prepared for measurements were 3 × 5 × 30 kOe. The variation in the electrical resistivity in the 10 mm parallelepipeds pelletized from the starting temperature interval from 4.2 to 300 K with magnetic powder material and annealed at 1000°C for 2 h. field (for a given composition x) is similar in character

HALL EFFECT IN THE FexMn1 Ð xS MAGNETIC SEMICONDUCTORS 1069

4 6 1 (a) 1 2 3 4 2

cm] 2

Ω 3 [

ρ 2

log 5 0 4 0 cm] Ω

–2 [ 4 0 4 8 12 ρ 3 –1 (b)

10 /T, K log 2 1 Fig. 1. Temperature dependences of the electrical resistivity 2 3 of samples of the FexMn1 Ð xS sulfide system obtained for compositions with (1) x = 0.3, (2) 0.33, (3) 0.36, (4) 0.4, and (5) 0.5 in zero magnetic field. 4 0 ρ to the variation in (T) observed in FexMn1 Ð xS with increasing cation substitution x without a magnetic field; in other words, application of a magnetic field –2 brings about a simultaneous decrease in the electrical 4 12 20 resistivity and conduction activation energy, as is seen 103/T, K–1 as the substitution concentration x increases. Negative Fig. 2. Temperature dependences of the electrical resistivity δ ≠ ≠ × magnetoresistance H = (R(H 0) Ð R(H = 0))/R(H 0) of FexMn1 Ð xS samples obtained in different magnetic 100% is observed in the FexMn1 Ð xS sulfide system in fields: (a) x ~ 0.29 and (1) H = 0, (2) 5, and (3) 30 kOe; ferromagnetic compositions in the range 0.25 ≤ x ≤ 0.4. (b) x ~ 0.3 and (1) H = 3, (2) 7, (3) 8, and (4) 14 kOe. The value of the magnetoresistance depends on the concentration x and reaches a maximum for the compo- 40 δ sition x ~ 0.29 ( H = –450% in a magnetic field of (a) 30 kOe at a temperature of 50 K) [2, 3]). Figure 3 shows temperature dependences of the Hall

constant, carrier concentration, and mobility obtained cm)/Oe 20 for x ~ 0.25 in a magnetic field of 10 kOe. The Hall con- Ω , ( stant RH grows with decreasing temperature and is pos- H itive. When measured at temperature T = 110 K in a R field of 10 kOe, the carrier concentration and mobility for x ~ 0.25 are n = 0.32 × 1011 cmÐ3 and µ = 1.2 × 0 104 cm2 VÐ1 sÐ1. Figure 4 presents graphs of the temperature depen- (b) dences of the Hall constant, carrier mobility, and con- 12 centration measured in different magnetic fields on the –1

FexMn1 Ð xS sulfide with x ~ 0.29, the composition with s maximum magnetoresistance. In magnetic fields H = 5 –1 V and 10 kOe at temperatures of 77–300 K, our sulfide 2 4

samples with x ~ 0.29 are p-type semiconductors, cm which is illustrated by the positive sign of the Hall con- 3 stant R (Fig. 4a). The Hall constant changed in sign to , 10 0

H µ negative in a magnetic field of 15 kOe for T < 180 K, with electrons becoming the majority carriers in this –4 temperature region. For a given field H = 5 kOe, the 100 125 150 175 mobility of the p-type carriers in this sample increases T, K and their concentration decreases with decreasing tem- Fig. 3. Temperature dependences of (a) the Hall constant perature, similar to the way this occurs in the x ~ 0.25 and (b) carrier concentration and mobility for FexMn1 Ð xS composition in a field of 10 kOe. An increase in mag- (x ~ 0.25) obtained in a field H = 10 kOe.

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1070 RYABINKINA et al.

0.2 1 (a) 0 (a)

0.1 –5 1 cm)/Oe cm)/Oe 2 Ω Ω –10 2 , ( , ( H H 0 R R 3 –15 3

9 ] (b) –3 4 –1 (b) s cm –1

6 11 V 2 1 3 [10 n cm

3 3

, 10 2 µ 0 2 4 (c) ] 2 (c) 5 ] log –3

3 –3

cm 1 11 cm 4

2 11 [10 n

[10 3 n

log 1 log 2 50 150 250 300 50 100 200 300 , K T, K T

Fig. 4. Temperature dependences of (a) the Hall constant and Fig. 5. Temperature dependences (a) of the Hall constant in (b) carrier mobility and (c) concentration for FexMn1 Ð xS magnetic fields H (1) 5, (2) 10, and (3) 15 kOe and (b, c) of (x ~ 0.29) obtained in different magnetic fields H: (1) 5, (2) the carrier concentration in magnetic fields H (b) 5 and (c) 10, and (3) 15 kOe. 15 kOe for FexMn1 Ð xS (x ~ 0.3). netic field (up to 10 kOe) brings about a decrease in the Figure 6 shows field dependences of the electrical mobility of p carriers and lead to a weak temperature resistivity, carrier concentration, and magnetization for dependence in the sample with x ~ 0.29 (Fig. 4b). The the x ~ 0.29 composition measured at 160 K. We see p-carrier concentration increases (Fig. 4c). For T = that the decrease in the electrical resistivity and the 110 K and H = 10 kOe, the carrier concentrations and growth in the magnetization are related to the increas- mobilities in the sample with x ~ 0.29 are n = 8.6 × ing carrier concentration. 1012 cmÐ3 and µ = 1.3 × 103 cm2 VÐ1 sÐ1. Figure 5 illustrates the temperature behavior of the 4. DISCUSSION OF THE RESULTS Hall constant and carrier concentration in different The CMR effect was first discovered in NaCl-type magnetic fields studied for the composition with x ~ fcc magnetic semiconductors (such as europium chal- 0.3. The majority carriers in this sulfide are negatively cogenides EuSe [13]), which are isostructural to man- charged particles, electrons. In a magnetic field of ganese monosulfide. Just as in the europium chalco- 5 kOe, the temperature dependence of n(T) is similar to genides, the magnetic transition from the paramagnetic that for the x ~ 0.25 composition (Fig. 3b), with the to antiferromagnetic state in manganese monosulfide is temperature at which n(T) undergoes a sharp change accompanied by a rhombohedral-type lattice deforma- shifting toward high temperatures with increasing field. tion caused by the exchangeÐstriction mechanisms [14]. The antiferromagnetic transition in the cation- At T = 110 K and H = 10 kOe, the carrier concentration × substituted sulfides FexMn1 Ð xS is apparently of the and mobility for the sample with x ~ 0.3 are n = 6.94 same nature. As the cation substitution in Fe Mn S 13 Ð3 µ × 2 2 Ð1 Ð1 x 1 Ð x 10 cm and = 1.92 10 cm V s . As the mag- increases from x = 0 to ~0.3, the Néel temperature netic field is increased to 15 kOe, the carrier concentra- increases from 150 K (x = 0) to 230 K (x ~ 0.3) [2, 3], tion increases to n ~ 1015 cmÐ3. The carrier mobility for which indicates expansion of the temperature region in this composition depends only weakly on temperature which the antiferromagnetic state exists. As follows and was measured to be ~102 cm2 VÐ1 sÐ1. from Hall effect data, the concentration of the p carriers

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 HALL EFFECT IN THE FexMn1 Ð xS MAGNETIC SEMICONDUCTORS 1071 decreases and their mobility increases. This suggests 250 that the nature of the antiferromagnetic transition in (a) the FexMn1 Ð xS solid solutions is related to a decrease in the p-carrier concentration and an increase in their 200

mobility. cm Ω ,

Many researchers relate the origin of the CMR in ρ 150 magnetic semiconductors (for instance, in EuSe, CdCr2Se4, HgCr2Se4) to a red shift of the conduction band bottom and carrier localization in ferron-type 100 12 impurity states [6]. The dependence of the carrier con- (b) centration and mobility on magnetic field observed in these compounds is assigned to the delocalization of –3 8 electrons in the ferron states, which gives rise to a cm reversal of the carrier sign and an increase in the elec- 13 4 tron concentration and mobility. , 10 n The above results obtained in an experimental study 0 of the galvanomagnetic properties of the Fe Mn S 8 x 1 Ð x (c) sulfides permit us to conclude that, in magnetic fields H ≤ 10 kOe at temperatures from 77 to 300 K, the ≤ FexMn1 Ð xS sulfides (x 0.29) are p-type semiconductors, similar to manganese monosulfide. As x is 4 , emu/g increased from 0 to ~0.29, the p-carrier concentration σ decreases by five orders of magnitude and their mobility rises by the same factor (relative to α-MnS, x = 0). In the composition region of x ~ 0.3, the carriers reverse 0 sign, with n-type carriers becoming the majority spe- 0 5 10 15 Ð1 H, kOe cies. The conduction activation energy Ea ~ 10 eV, the 11 15 Ð3 carrier concentration n ~ 10 Ð10 cm , and the mobil- Fig. 6. Magnetic field dependences of (a) the electrical µ 3 2 Ð1 Ð1 ity ~ 10 cm V s found for the FexMn1 Ð xS sulfides resistivity, (b) carrier concentration, and (c) magnetization in the range 0.25 ≤ x ≤ 0.3 are typical of semiconductors for the x ~ 0.29 composition measured at T = 160 K. with narrow-band gaps and small (m*/m ~ 10Ð2) effective carrier masses (for instance, PbS, Ge, Si) [11]. duction character, where both holes and electrons act as The Hall resistivity in magnetic semiconductors is carriers. The ratio of the electron to hole concentrations ρ defined as H = R0B + RsM, where R0 and Rs are the nor- determines the type of the carrier, which varies depend- mal and anomalous (spontaneous) Hall coefficients, ing on the temperature and composition. A magnetic B is the induction in the sample, and M is the magneti- field apparently exerts a similar impact on the x ~ 0.29 zation. For a sample geometry with a demagnetizing composition in the range 77Ð200 K, where CMR and factor of unity, we have B = H. Studies of the Hall effect the coexistence of the antiferromagnetic and ferromag- in manganese monosulfide [10] showed that the Hall netic phases are observed [15]. The sharp rise in the p- constant R0 in the range 77Ð300 K is so small that it lies type carrier mobility as the degree of cation substitution within the experimental error. Our studies demon- x in the FexMn1 Ð xS sulfides is increased compared to strated a similar situation to exist in a FexMn1 Ð xS sam- the starting manganese monosulfide indicates either the ple with x ~ 0.25 at temperatures of ~200Ð300 K. formation of a light-carrier impurity band or a weaken- Below 200 K, the Hall constant in the antiferromag- ing of the polaron or ferron effects, as is the case with netic phase grows sharply (Fig. 3a). In ferromagnetic the magnetic semiconductors La1 Ð xSrxMnO3 and samples with x ~ 0.29 and ~0.3, the Hall effect is HgCr2Se4 [6Ð8]. observed throughout the temperature range 77Ð300 K. It may be conjectured that the Hall constant observed in To explain the concentration-driven transition from the interval from 200 to 300 K is actually the anoma- an antiferromagnetic semiconductor to a ferromagnetic metal occurring in FexMn1 Ð xS for zero magnetic field, lous component Rs. It has positive sign for the composition x ~ 0.29 and negative sign for x ~ 0.3. The rever- an electronic level diagram was proposed, which sal of the carrier sign from the hole to the electronic includes a pÐd hybridized valence band E1 and a narrow sign in manganese monosulfide (x = 0) is observed to impurity band E2 [16]. It was shown that as x is occur above ~480 K [10]. In view of this observation increased, an Anderson transition takes place in and the results of studies into the Hall effect in cation- FexMn1 Ð xS through a shift of the mobility edge Ec and substituted sulfides, the compounds investigated by us crossing of the Fermi level EF. The similarity observed could be assigned to semiconductors with mixed con- in the behavior of the electrical resistivity as a function

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1072 RYABINKINA et al. of concentration x (Fig. 1) and of magnetic field H 6. N. I. Solin and N. M. Chebotaev, Fiz. Tverd. Tela (St. (Fig. 2) suggests that the mechanism of the concentra- Petersburg) 39 (5), 848 (1997) [Phys. Solid State 39, 754 tion transition and the decrease in the electrical resistiv- (1997)]. ity in a magnetic field (the negative CMR effect) have 7. R. I. Zaœnullina, N. G. Bebenin, V. V. Mashkautsan, the same (percolation) character. This conjecture fits V. V. Ustinov, V. G. Vasil’ev, and B. V. Slobodin, Fiz. the model of electronic and magnetic phase separation Tverd. Tela (St. Petersburg) 40 (11), 2085 (1998) [Phys. [1], which is analogous to the theory of percolation in Solid State 40, 1889 (1998)]. heavily doped semiconductors [17]. 8. V. V. Mashkautsan, R. I. Zaœnullina, N. G. Bebenin, V. V. Ustinov, and Ya. M. Mukovskiœ, Fiz. Tverd. Tela (St. Petersburg) 45 (3), 468 (2003) [Phys. Solid State 45, ACKNOWLEDGMENTS 494 (2003)]. 9. V. V. Chechernikov, Magnetic Measurements (Mosk. This study was supported, in part, jointly by the Gos. Univ., Moscow, 1969). Russian Foundation for Basic Research and Krasno- 10. H. Heikens, C. F. van Bruggen, and C. Haas, J. Phys. yarsk Kraœ Science Foundation, project no. 02-02- Chem. Solids 39 (8), 833 (1978). 97702 “r 2002 Eniseœ a.” 11. N. F. Mott and E. A. Davis, Electron Processes in Non- Crystalline Materials, 2nd ed. (Clarendon, Oxford, 1979; Mir, Moscow, 1982), Vol. 2. REFERENCES 12. L. I. Ryabinkina, Candidate’s Dissertation (Inst. of Phys- 1. É. L. Nagaev, Usp. Fiz. Nauk 166 (8), 833 (1996) [Phys. ics, Siberian Division, Russian Academy of Sciences, Usp. 39, 781 (1996)]. Krasnoyarsk, 1993). 13. Y. Shapira, S. Foner, and T. B. Reed, Phys. Rev. B 8 (5), 2. G. A. Petrakovskiœ, L. I. Ryabinkina, G. M. Abramova, N. I. Kiselev, D. A. Velikanov, and A. F. Bovina, Pis’ma 2299 (1973). Zh. Éksp. Teor. Fiz. 69 (12), 895 (1999) [JETP Lett. 69, 14. B. Morosin, Phys. Rev. B 1 (1), 236 (1970). 949 (1999)]. 15. G. Petrakovskii, B. Roessli, L. Ryabinkina, G. Abram- ova, D. Balaev, and O. Romanova, in Book of Abstracts: 3. G. A. Petrakovskiœ, L. I. Ryabinkina, G. M. Abramova, A. D. Balaev, D. A. Balaev, and A. F. Bovina, Pis’ma Zh. Moscow International Symposium on Magnetism (Mos- Éksp. Teor. Fiz. 72 (2), 99 (2000) [JETP Lett. 72, 70 cow, 2002), p. 167. (2000)]. 16. G. V. Loseva, L. I. Ryabinkina, and S. G. Ovchinnikov, Fiz. Tverd. Tela (Leningrad) 33 (11), 3420 (1991) [Sov. 4. P. Chen and Y. W. Du, J. Phys. Soc. Jpn. 71 (1), 209 Phys. Solid State 33, 1929 (1991)]. (2001). 17. B. I. Shklovskiœ and A. L. Éfros, Electronic Properties of 5. G. A. Petrakovskii, L. I. Ryabinkina, G. M. Abramova, Doped Semiconductors (Nauka, Moscow, 1979; Springer, N. I. Kiselev, D. A. Balaev, O. B. Romanova, G. I. Mak- New York, 1984). ovetskii, K. I. Yanushkevich, A. I. Galyas, and O. F. Demidenko, Phys. Met. Metallogr. 93 (1), 82 (2002). Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

Physics of the Solid State, Vol. 46, No. 6, 2004, pp. 1073Ð1080. Translated from Fizika Tverdogo Tela, Vol. 46, No. 6, 2004, pp. 1043Ð1050. Original Russian Text Copyright © 2004 by Shkar’, Nikolaev, Sayapin, Poœmanov. MAGNETISM AND FERROELECTRICITY

High-Frequency Susceptibility and Ferromagnetic Resonance in Multidomain Thin Films of the YttriumÐIron Garnet Type V. F. Shkar’*, E. I. Nikolaev*, V. N. Sayapin*, and V. D. Poœmanov** * Donetsk Physicotechnical Institute, National Academy of Sciences of Ukraine, Donetsk, 83114 Ukraine ** Donetsk National University, Universitetskaya ul. 24, Donetsk, 83114 Ukraine e-mail: [email protected], [email protected] Received May 26, 2003; in ﬁnal form, October 13, 2003

Abstract—The field dependence of the high-frequency susceptibility and the ferromagnetic resonance were experimentally studied in a thin (d ≈ 0.1 µm) (111)-oriented single-crystal film of substituted yttrium–iron garnet with the factor q Ӷ 1. It was shown that the anomaly in the high-frequency susceptibility observed in a mag- ≈ netic field H parallel to the normal to the film surface in the magnetization saturation region (H Hs) has a dual nature; more specifically, this anomaly is associated with an abrupt collapse of the stripe domain structure and a ferromagnetic resonance in the experimental configuration H || [111] and h ⊥ H. In this case, the film transi- ≈ tion from the inhomogeneous multidomain state to the homogeneous (single-domain) state at the point H Hs has no indications of a second-order phase transition. The experimental frequency–field dependence of ferro- ω magnetic resonance (FMR) in the sample under study, having a characteristic minimum at the point 0 = 5 MHz and HFMR = Hs, agrees qualitatively and quantitatively with calculations. The influence of the cubic magnetic anisotropy and the film thickness on the FMR spectrum and the orientation of the spontaneous magnetization in domains with respect to the film plane in the zero field H was theoretically studied. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION second-order orientational phase transitions (OPT-2) Thin films of yttrium–iron garnet (YIG) with partial were observed in a similar film at room temperature. rare-earth substitution and diamagnetic dilution and an This result seems to be unique, since well-established in-plane magnetic moment were previously used as thin OPTs were observed exclusively in the low-tempera- underlayers controlling the domain-wall structure in ture range (mostly at T = 4Ð150 K) in long-term studies the above-lying magnetically uniaxial garnet ferrite of both spontaneous and field-induced OPTs in crystals layer of a two-layer garnet structure [1, 2]. Recently, and films of rare-earth–iron garnets [8Ð10]. The exper- significant attention has been given to the domain struc- iments in [7] cast doubt on the interpretation of the ture (DS) of the thinnest YIG films due to new possibil- results obtained and the validity of the corresponding ities for their application. In particular, they can be a conclusions. With thin films similar to those studied in component of the layered periodic structure of photonic [7], we carried out experimental and theoretical studies; crystals containing a stripe DS with a period of the the results are considered below. order of the light wavelength [3, 4]. Since the Faraday rotation is small in such a film, it is not feasible to study 2. EXPERIMENTAL DSs using conventional magnetooptical methods. Previous studies [5, 6] of the spin-wave interactions 2.1. Sample under Study and Experimental Technique in the above-mentioned two-layer garnet ferrite struc- Experimental studies were performed using a sin- tures suggest that the ferromagnetic resonance (FMR) gle-crystal garnet film of composition ≈ µ method used in combination with high-frequency mag- (YLaGd)3(FeGa)5O12 with a thickness d 0.1 m netic susceptibility (HFMS) makes it possible not only grown on a single-crystal Gd3Ga5O12 substrate oriented to detect specific dynamic effects in ultrathin YIG films in the (111) plane such that the [111] crystallographic but also to obtain certain information on the “hidden” axis of the film was parallel to the normal n to the sur- DS. In this study, the HFMS and FMR spectra are mea- face. To grow such a thin film, we used modified liquid- sured in an ultrathin (d ≈ 0.1 µm thick) single-crystal phase epitaxy from a wetting melt-solution layer [11]. YIG-type film with a DS and theoretical estimates are According to the original destination of such films, made. their composition was selected with the intent to mini- Interest in this study was stimulated in some mize the growth-induced magnetic anisotropy along respects by paper [7], where it was reported that two the normal to the surface. In the demagnetized state,

1074 SHKAR’ et al. χ field H could be varied within 0Ð2000 Oe. The two rota- tors made it possible to rotate the sample about the normal n to its surface and to rotate the field H in the vertical plane with respect to the sample from the position H ⊥ n to H || n. The drive circuit voltage varied with the HF power absorbed by the sample. A change in this F = 5 MHz voltage was amplified and applied to the Y input termi- nals of an XÐY recorder. A voltage proportional to the external magnetic field was applied to the X input ter- minals. The FMR parameters were determined using a F = 20 MHz reflection-mode setup allowing study in the frequency range 0.005–10 GHz with magnetic field modulation. –500 0 500 1000 1500 As a measuring cell (sensitive element), a copper H, Oe microstrip 0.1 mm wide and 6 mm long was used at the first stage (operation at frequencies above 1 GHz); at Fig. 1. High-frequency susceptibility of a (111)-oriented the second stage, a copper meander on a polycore sub- garnet ferrite film in a field H || [111] at frequencies of 5 and 20 MHz. strate was used (operation in the range 0.005Ð1 GHz). A sample 6 mm in diameter was glued to the end of a brass axis of the rotator and was densely tightened to such a film should contain a stripe DS. The orientation the meander by a spring. The rotation axes of the sam- of the spontaneous magnetic moment vector M in ple and magnet were mutually perpendicular, which either of the two types of oppositely polarized magnetic allowed us to apply the external field along a required domains in such a film is mainly controlled by the com- direction. petition between the demagnetization field, which tends to align the vectors M in the film plane, and the cubic magnetic anisotropy field, which tends to deflect 2.2. High-Frequency Susceptibility the vectors M from the film plane toward one of the The studies presented in this paper were initiated by easy magnetization axes of the [111] type, three of ° ′ the observation of a significant change in the HFMS of which form an angle of 19 30 with the film surface. an YIG-based thin film in a quasi-static magnetic field The magnetic characteristics of the film were deter- || H n in the vicinity of the point H = Hs (Hs is the mag- mined by measuring the FMR parameters. The effec- netization saturation field of the film, generally defined π tive uniaxial anisotropy, which is equal to 4 Ms in this as a characteristic point (region) in the magnetization case (since there is almost no growth- and stress- curve). Figure 1 shows the dependence of the sample induced uniaxial magnetic anisotropy in the sample), HFMS on the field H || n at frequencies of 5 and was determined according to [12]; more specifically, 20 MHz. We can see that the first and second suscepti- the experimental and theoretical dependences of reso- bility jumps are observed near the points H = 0 and nant fields were constructed for the (110) plane perpen- ≈ H Hs, respectively. The second jump is reliably dicular to the film plane and, by setting various values observed only in the case of exact orientation of the π of 4 Ms, these dependences were brought into coinci- external field H along the [111] axis of the film and cor- dence. The gyromagnetic ratio γ was determined from responds to the film transition from the inhomogeneous the slope of the frequency–field dependence (straight multidomain to a homogeneous single-domain state. line) for the field direction perpendicular to the sample Based on published experimental data, we analyze plane [13]. The cubic anisotropy field HA was deter- whether or not a causal relation exists between these mined from the angular (azimuthal) dependence of the phenomena. θ ≈ ° resonant field for a fixed polar angle 52 , which is Studies of the static susceptibility of garnet ferrite the most accurate method for determining HA in a (111) films [14] have shown that the DS plays a part in the 7 Ð1 Ð1 film. The values determined are γ = 1.76 × 10 Oe s , formation of the sample response to an external field. It π HA = 45 Oe, and 4 Ms = 1350 Oe. is clear that the HFMS of YIG-type multidomain films The HFMS was studied using the autodyne tech- should also differ from the HF susceptibility of single- nique. The sample under consideration was placed on a domain samples [15]. However, the question arises of rotator in a high-frequency (HF) magnetic field h how sharply the DS rearrangement or disappearance induced by high-frequency Helmholtz coils incorpo- occurs. In this regard, study [16] is worth noting as it rated in a drive circuit of an autodyne oscillator, whose dealt with the field dependence of the YIG film HFMS frequency was set between 5 and 30 MHz. The external in an external magnetic field oriented in the film (111) dc magnetic field H was induced by an electromagnet plane (i.e., H ⊥ [111]). It was shown that an HFMS placed on an independent rotator. The electromagnet jump occurs only in the case of sharp DS rearrangement was power-supplied from a stabilized dc source. The in the narrow range of values of the field strength H

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 HIGH-FREQUENCY SUSCEPTIBILITY AND FERROMAGNETIC RESONANCE 1075 where distinctive features in the film magnetization 1000 curve are observed. No susceptibility anomalies at the 1 2 3 point H ≈ H were detected during magnetization rever- s 800 1a sal of the film along either the easy (Hs = 30 Oe) or hard (H = 50 Oe) axes. s 600 1c A similar feature should presumably also be 4 observed in the experimental configuration H || [111] if , MHz the magnetization reversal in a sample proceeds as was F 400 assumed in [7], i.e., via gradual rotation of the spontaneous magnetization vectors M toward the field H in 200 oppositely polarized stripe domains separated by 180° walls. However, near the field H = Hs, these vectors are almost aligned along the field H and domain walls are 0 500 1000 1500 2000 spread; therefore, such a film state (in terms of energy) H, Oe differs only very slightly from the single-domain state. This means that no sharp rearrangement of the film DS Fig. 2. Frequency–field dependences of FMR in a (111)-ori- occurs in the vicinity of H = H ; hence, there is no cause ented garnet ferrite film for field H || [111]: (1) curve calcu- s lated for transverse (h ⊥ H) FMR in a multidomain film, for the HFMS anomaly. (1a, 1c) FMR observed (1a) before and (1c) after modern- In this case, a question arises concerning the cause ization of the setup, (2) uniform FMR, and (3, 4) presum- ≈ ably nonuniform FMR in the saturated state of the sample. of the HFMS jump near H Hs in Fig. 1. According to [7], as mentioned above, an OPT-2 takes place at || H [111] at the point H = Hs in the sample under study. ≈ below. Figure 2 shows the experimental frequencyÐ However, in this case, for H Hs, there is a critical point field dependence of the FMR (curve 1a) in the sample passage through which should be accompanied by a set under study in a field H || [111] at h ⊥ H. Solid lines of characteristic features. Such features, observed in represent the theoretical dependence of the FMR corre- spin-orientational transitions in rare-earth magnets, are sponding to the type of oscillations excited in the sam- anomalies in the static magnetic susceptibility, specific ple. The calculation of these dependences is described heat, Young’s modulus, thermal expansion, magneto- in Section 4. As follows from Fig. 2, there is good striction, and others [8, 9]. However, in the case of a agreement between the experimental (curve 1a) and very thin film, measurement of many static characteris- calculated (curve 1c) data, except in the region of H ≈ tics is complicated in practice. It is more convenient to Hs. In this region, the resonance frequency does not observe critical dynamic phenomena, which should vanish, in contrast to the prediction from the theory, take place during an OPT-2 in cubic ferrimagnets, for which results in a “gap” ~200 MHz wide in the experi- example, the critical spin dynamics that is observed in mental curve; this gap is similar to that described in [7]. a YIG single crystal at the Curie temperature T (where C The fact that the FMR frequency ω = γH in the sam- a typical second-order phase transition occurs) and 0 eff ple must vanish in the field H ≈ H follows from general manifests itself in a characteristic change in the relax- s ation rate of uniform magnetization precession (spin considerations. With no external field H, the magnetic diffusion) [17]. This relaxation rate can be determined moment in an anisotropic ferromagnet is subject to the from the measured linewidth ∆H . resulting anisotropy field, which, in general, is due to FMR the uniaxial, cubic, orthorhombic, and other types of ∆ || The field dependence of HFMR for H [111] was crystallographic anisotropy, as well as the shape anisot- experimentally studied in the vicinity of the point H ≈ ropy of the sample and domains. This field is the effec- H . The quantity ∆H has been repeatedly measured ω s FMR tive field Heff that controls the frequency 0 at the point in the resonance frequency range 5Ð180 MHz. The H = 0 in the FMR spectrum. As an external field H is ∆ HFMR measurement accuracy was 0.1 Oe. The average applied along a hard magnetization axis of the sample, value of this quantity was 15 Oe, with the spread being the magnetic moment begins to rotate from the easy ± ∆ 1.5 Oe, whereas the change in H near TC in the YIG magnetization direction to the hard magnetization axis. crystal was tens of oersteds [17]. As follows from these This rotation lasts until the external magnetic field H data, the critical spin dynamics was not observed exper- become equal to the anisotropy field. At this point, the imentally. effective magnetic field Heff (acting on the magnetic ω The situation that arose in the above-described study moment) becomes zero and the FMR frequency 0 van- of HFMS called for additional analysis (see below). ishes. In the sample at hand, any direction in the (111) plane is easy, while the [111] axis, parallel to the normal n to the film surface, is the hard direction along 2.3. Ferromagnetic Resonance which the external field H is applied. When the mag- The first stage of this study was carried out using an netic moments cease to rotate in domains and become FMR setup before it was modernized as described directed along n || [111], the field H is the saturation

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1076 SHKAR’ et al. modernization significantly improved the sensitivity of M the measuring cell. Z 1 ϕ 10 The frequency–field dependence of FMR in the [111] ϕ sample under study, measured using the modernized H 20 – M [211] radio spectrometer, is shown in Fig. 2 (curve 1c). We θ M 2 X 0 can see that the FMR frequency almost vanishes in the saturation field, in full agreement with theory. Y – – [011] [211] 3. Theory and Calculations ϕ X Let us study the factors determining the shape of the experimental frequency–field dependence of FMR Y – shown in Fig. 2. To this end, we calculate the theoretical [011] curve with allowance for the magnetic properties, domain structure, and shape of the sample under study. Fig. 3. Coordinate system and (inset) the domain structure (i) We now consider an infinite plate of the cubic (schematic) accepted in the theoretical consideration of the problem. YIG single crystal with a negative anisotropy constant K. The [111] easy magnetization axis coincides with the normal to the plate. The external magnetic field is

field Hs exactly equal to the resulting anisotropy field also perpendicular to the film plane. Under these condi- H , which (in the absence of uniaxial anisotropy) is tions, saturation in such a sample takes place in the field k π | | π π Hs = 4 M Ð (4/3) HA , where HA = K/M. At H > Hs, the given by Hk = 4 M Ð (4/3)K/M, where 4 M is the demagnetizing field (shape anisotropy) for an sample is saturated and the FMR frequency can be unbounded film and K is the first cubic anisotropy con- determined immediately from the SmithÐSuhl equa- stant. Taking into account that the first term is an order tions of magnitude greater than the second for the sample at γ ∂U γ ∂U ≈ π θú ==Ð------, ϕú ------, (1) hand, we can write Hs 4 Ms. Hence, when the field M sinθ ∂ϕ M sinθ ∂θ magnetizing the sample along the hard magnetization axis varies from zero to a value above the magnetiza- where θ and ϕ are the polar and azimuthal angles of the tion saturation field, the FMR frequency must pass magnetization vector, respectively. The energy density through zero as the external field becomes equal to this for such a structure is given by anisotropy field. If the magnetic field H is not directed θ π 2 2 θ strictly along the hard magnetization axis, only the pro- UU==H ++UM U A Ð2MHcos + M cos jection of the anisotropy field onto the external field 4 4 (2) sin θ cos θ 2 3 direction (rather that the total anisotropy field) will be Ð K ------++------sin θθcos cos3ϕ , compensated. In this case, the magnetic moment con- 4 3 3 tinues to be in a certain nonzero field Heff and the FMR frequency will not reach zero. Another cause for the where UH, UM, and UA are the energy densities of the FMR frequency not reaching zero can be the difference magnetization M in the external field and of the demag- in the symmetry of the components of the effective netizing and anisotropy fields of the plate, respectively [13]. Substituting expression (2) into the second equa- anisotropy or insufficient sensitivity of the microstrip θ sensor at frequencies close to zero. tion of set (1) and setting = 0, we immediately derive the FMR frequency Taking into account the above factors, we modernized the FMR setup as follows. 4 ωϕ==ú γ H Ð 4πM + --- H . (3) (i) The electromagnet rotator was equipped with an 3 A optical laser device for orienting the external magnetic θ field with an accuracy of several arc minutes with The fact that is positive means that the absence of respect to the normal n to the film surface. domains is possible if H > Hs. (ii) The 6-mm-long microstrip used as a measuring (ii) Let us consider the case where the external field cell was replaced with a copper meander grown by pho- H is directed along the normal to the film plane and is tolithography on a polycore substrate (for operation in weaker than the saturation field, H < Hs. We assume that the range 0.005Ð1 GHz). The meander dimensions the sample contains two domain groups forming a peri- were as follows: the width and thickness of a strip were odic stripe DS oriented along the Y axis, as shown in 0.1 and 0.05 mm, respectively, and the distance Fig. 3 (the dashed lines indicate the projections of the between the strips was 0.1 mm. The total strip length in three other triad axes onto the film plane). the meander was 180 mm. For the range 1Ð10 GHz, the Following [12, 13], we write the Lagrangian func- distance between strips was chosen to be 2 mm. This tion for such a structure. With allowance for the energy

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 HIGH-FREQUENCY SUSCEPTIBILITY AND FERROMAGNETIC RESONANCE 1077 of domain wall oscillations, the Lagrangian takes on the 1.60 form 1.6 1.58 LTU= Ð Ð δU, (4) 1.4 1.56 1.2 1.54 where –10 0 10

, rad 1.0

θ 0.16 βν2 0.8 0.12 π 2 ú 1 ()νθϕ ()θν ϕ T = 4 M ------Ð ------cos1 ú 1 + 1 Ð cos 2 ú 2 (5) 2 ω Angle 0.6 0.08 0 0.04 0.4 is the kinetic energy density and 0 0.2 12701272 1274 π 2[]()νθ()θν ˜ 0 U = 4 M Ð cos1 + 1 Ð cos 2 H 0 200 400 600 800 1000 1200 1400 H, Oe 1()νθ()θν 2 + --- cos1 + 1 Ð cos 2 2 Fig. 4. Dependence of the polar angle θ on the field H. ν()ν There are positive-slope portions in the curves near zero 1 Ð ()θ ϕ θ ϕ 2 + q------sin 1 cos 1 Ð sin 2 cos 2 field and the magnetization saturation field, which cause the 2 (6) features in the magnetization curve and peaks in the mag- 4 4 netic susceptibility curve. sin θ cos θ 2 3 Ð d ν ------1 ++------1 ------sin θ cosθ cos3ϕ 4 3 3 1 1 1 from which, by differentiating Eq. (6), we immediately 4 θ 4 θ obtain ()ν sin 2 cos 2 2 3 θ θ ϕ +1Ð ------++------sin 2 cos 2 cos3 2 4 3 3 π ν 1 ϕ ϕ 2 θ θ θ 0 =====---, 10 Ð 20 ------, 10 20 0 (9) is the potential energy density, consisting of the energy 2 3 densities of the magnetization M in the external field (see Fig. 3 for the angle ϕ in domains). The equilibrium and of the demagnetizing and anisotropy fields of the angle θ is related to the external field as plate, as well as the energy density of domain walls (the 0 third term). Subscripts 1 and 2 indicate two domain θ ϕ ˜ θ θ 7 2 θ groups, i and i are the polar and azimuthal angles of H = cos 0 + d cos 01 Ð --- cos 0 ν 3 the magnetization vector in each domain, is the rela- (10) ω π γ β π 2 tive domain size, 0 = 4 M , and = mγb/4 M (mγ is 4 2 +2sinθ 1 Ð --- sin θ . the domain wall mass and b is the average thickness of 03 0 a domain). Here, we introduced the dimensionless | | π anisotropy constant d = HA /4 M and the dimension- The corresponding dependence is shown in Fig. 4. less external field H˜ = H/4πM. The parameter q We can see that the equilibrium angle of the deviation (depending on the film thickness) defines the contribu- of the vectors M1 and M2 from the film plane is greater tion from the domain demagnetizing field to the total than one degree in a zero field. We one can also see that the sample transfers to the saturation state in a jump, energy and should be determined experimentally (for θ an indefinitely thin domain, this parameter is equal to since the derivative of 0(H) changes sign in the region ≈ ° zero, which means that the stripe structure half-period of H Hs. This jump is approximately 3 . These fea- θ is much larger than the film thickness). The energy den- tures in the 0(H) dependence arise exclusively as a sity of the ac field is given by result of cubic magnetic anisotropy in the sample and are absent in the case HA = 0. δ π 2 ˜ [νθ( θ ()ϕ ϕ U = 4 M h sin h sin 1 cos 1 Ð h Writing the Lagrange equations θ θ )] ()θν ( θ ()ϕ ϕ (7) + cos h cos 1 + 1 Ð sin h sin 2 cos 2 Ð h d ∂L ∂L d ∂L ∂L ------θ θ )) ==∂θ(), ∂ϕ ∂ϕ , + cos h cos 2 , dt ∂θ()cos ú i cos i dt ú i i (11) d ∂L ∂L where h˜ = h/4πM. With no perturbation δU, the equi------= ------dt∂νú ∂ν librium orientation of the magnetization vectors can be determined from the equilibrium conditions for this system and expanding the derivatives near the equilibrium values (assuming the domain wall mass to Uθ =====Uθ Uϕ Uϕ Uν 0, (8) 1 2 1 2 be infinite, since the magnetization is changed due to

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1078 SHKAR’ et al. rotation processes), we solve Eqs. (11) and find the fre- tional to the amplitude or the frequency shift is mea- quency–field dependences of FMR: sured. It is of no importance what processes cause a change in the magnetic state of the sample; this change 1()± 2 y12, = --- c1 c1 Ð 4c2 , (12) will be measured only as a result of an increase in the 2 HF power absorption (magnetic loss in the sample). Hence, identification of the processes initiated in the where y1 and y2 are expressed in terms of the FMR frequency and the equilibrium angle as sample by the field H calls for not only HFMS measurements but also additional studies (microscopic θ ω 2 sin 0 12, observation of the DS, construction of the magnetiza- y , = ------. (13) 12 2 ω tion curve, FMR parameter measurements, etc.). 0 Ω ω γ If the condition = 0 = Heff is satisfied in the The coefficients c1 and c2 are given by study of a YIG-type film using the scheme described ()2 above, these devices will measure a jump in the HF sus- c1 ==uθ+uϕ+ + uθÐuϕÐ, c2 uθ+uϕÐ uθÐuϕ+ Ð 4uθϕ , ceptibility; however, the jump will be caused in this where case by resonant absorption of the HF power in the sample as a result of FMR, because FMR is the selec- uϕ± ==uϕϕ ± uϕ ϕ , uθ± uθθ ± uθ θ , 1 2 1 2 tive absorption (by the YIG film in this case) of the electromagnetic field energy at the frequency equal to the 1 3 2 1 2 ω θ ˜ θ θ frequency 0 of precession of magnetic moments of the uθθ = ---cos 0 H ++--- sin 0 Ð 1 ---qsin 0 2 2 8 electronic system in the internal effective magnetic field Heff [13].  2 θ 2 θ 4 θ 4 4 θ Ð d7sin 0 cos 0 Ð sin 0 Ð --- cos 0 In [16], the HFMS was studied in fields H < 100 Oe 3 in the frequency range ω = 5Ð25 MHz, which is much lower than the precession frequency ω for a thick (5Ð 8 2   (14) 0 +2 2sinθ cosθ 1 Ð --- sin θ , µ 0 03 0   7 m) YIG film in such a field at any relative orientation of the vectors H and h and the [111] axis. For this rea- 3 q son, that experiment was free of side FMR effects (in uϕϕ = --- sinθ ---2+ d sin2θ , 4 04 0 terms of the objectives pursued in [16]). This circumstance allowed the authors of [16] to observe the anomalous HFMS caused only by the critical DS rearrange- 12 θ 1 2 θ 3 2 θ uθ θ ==--- sin 0 Ð ---qcos 0 , uϕ ϕ ------qsin 0. ment in the YIG film under study. It was shown that the 1 2 44 1 2 16 transition of the film from the multidomain to the sin- ≈ By substituting the previously determined constants gle-domain (saturated) state at the point H Hs is not in γ × 7 Ð1 Ð1 π itself a critical process in the sense that it is not accom- ( = 1.76 10 Oe s , HA = 45 Oe, 4 M = 1350 Oe) and setting q = 0.01, we can derive the frequency–field panied by an HFMS anomaly. dependences shown for transverse oscillations in Fig. 2 The experimentally observed resonant absorption of (solid lines). the HF power (5Ð20 MHz) at the point H ≈ 1300 Oe (Fig. 1) corresponds to the frequency range of uniform magnetization precession for the sample under study, as 4. DISCUSSION follows from Fig. 2; therefore, this absorption must be Let us compare these results and the data obtained in considered in connection with FMR. other studies of YIG [16] and substituted YIG [7] films As already noted, the above-mentioned HFMS where the autodyne technique was used to study the anomaly was interpreted in [7] as a magnetic suscepti- HFMS and FMR, as in this work. bility jump due to the OPT-2. In this case, an important We recall that this technique makes it possible to role in the arguments in favor of this version was observe variations in the HF energy absorption by a assigned in [7] to the shape of the frequency–field sample in the drive oscillatory circuit of an HF-power dependence of the FMR, which corresponds to experi- signal generator as the sample is exposed to a mutually mental curve 1a in Fig. 2 in this paper. Special attention perpendicular or parallel external dc magnetic field H was paid to distinct minima in weak fields and in a field and HF field h. The frequency Ω of the field h was set of the order of 1300 Oe, i.e., at the starting (the first between 5 and 25 MHz [16] or was 5 MHz [7]. In both OPT) and ending point (the second OPT) of reorienta- cases, the sensitive element of the measuring circuit, tion, and it was postulated that the observed spectrum whether it be a Helmholtz coil or a planar multiturn corresponds to soft quasi-ferromagnetic modes, since coil, is the inductance in the oscillatory circuit, whose they always have a minimum frequency at OPT-2 ≈ ν ≈ Q factor varies under the influence of the sample. These points. The energy gap in the region H 1300 Oe ( 0 variations give rise to a change in the amplitude and to 200 MHz) is also interpreted in [7] in favor of the OPT a frequency shift of HF oscillations of the signal gener- version by analogy with rare-earth orthoferrites, in ator. It makes no difference whether the signal propor- which such a gap is observed at the phase transition

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 HIGH-FREQUENCY SUSCEPTIBILITY AND FERROMAGNETIC RESONANCE 1079 point as “a result of the dynamic interaction of various axis of the sample and to measure the actual FMR spec- oscillatory systems of a magnet.” Moreover, the results trum (containing no gap) at frequencies 5Ð200 MHz ν ≈ calculated in [7] agree with the experimental value 0 (Fig. 2, curve 1c). According to [7], the second OPT-2 200 MHz. occurs when the angle θ between the vector H and the θ ≠ Our counterarguments are based on the above [111] axis is zero: “at 0, the transition disappears.” As follows from Fig. 2, the gap in the FMR spectrum results of the experimental and theoretical studies per- θ ≠ formed for an identical garnet ferrite film and are as fol- simply indicates the misorientation ( 0) at which (as lows. argued in [7]) the OPT-2 does not take place. In this case, there is no reason to interpret the minimum at H ≈ (i) A number of factors were determined that possi- 1300 Oe in the FMR spectrum as “softening … of the bly control the HFMS jumps observed in the garnet fer- soft mode at the end point of reorientation.” rite film under study regardless of the phase transition. (iv) It was shown that the existence of a minimum at One of these factors is irreversible rotation [18] of H ≈1300 Oe in the FMR spectrum follows from the the spontaneous magnetization vector M in stripe FMR theory as a result of the vanishing of the effective domains in the region H ≈ 1300 Oe. As follows from the θ || magnetic field, around whose direction the vector M dependence of the polar angle on the field H [111] precesses (Section 4, Fig. 2). The existence of the min- (Fig. 4), the gradual rotation of the vector M toward the || imum at H = 0 is also substantiated by the FMR theory field H [111] stops at the point H = Hs. As the field is for a cubic ferromagnet with a nonzero cubic magnetic further strengthened, the vector M abruptly transfers to || anisotropy energy (Fig. 2). The sample under study is the position M [111]. In [16], it was shown that such similar to such a ferromagnet under the conditions of an abrupt DS rearrangement is always accompanied by this experiment. The calculation carried out in [7] dis- an HFMS anomaly. regards the cubic magnetic anisotropy and the actual Another factor is FMR, which occurs, in particular, contribution from the demagnetizing fields of domains in an external field H || [111] for h ⊥ H at the point ω = in such a thin film and disagrees with the experiment, 5 MHz and H ≈ 1300 Oe corresponding to the coordi- which was incorrectly interpreted in [7] in favor of the nates of the second HFMS jump for the sample under occurrence of the OPT (namely, as the result of “soften- study (Figs. 1, 2). In this context, it should be noted that ing … of the soft mode at the start and end points of the the autodyne technique dealing with electromagnetic reorientation” [7]). oscillations with a frequency of 5 MHz is incorrectly (v) The first OPT-2 observed in the sample under interpreted in [7] to be a method of static magnetic sus- study is interpreted in [7] as the transition from the state ceptibility, whereas it is conventionally considered a where the magnetization in domains lies strictly in the method of HF susceptibility [16, 19]. Since such a film plane (i.e., M ⊥ [111]) at H = 0 to the state where purely dynamic effect as FMR is observed at this fre- the magnetization makes an angle with the film plane at quency, this cannot be statics. However, even an anom- H ≠ 0. In such a case, the initial phase M ⊥ [111] has aly in the static magnetic susceptibility is not an unam- no domain of existence in the magnetic phase diagram biguous attribute of the OPT, which is indicated, e.g., for the field H || [111], which should induce the OPT. by the shape of the magnetization curve of magneti- This is impossible from the viewpoint of the Landau cally uniaxial garnet ferrite films in the vicinity of the theory of phase transitions. Furthermore, the calcula- bubble domain collapse field, where the OPT is absent tion we carried out with allowance for the experimen- a priori [19]. tally measured cubic anisotropy constant shows that, The HFMS anomaly at the point H || [111] = 0 also even in the field H = 0, the magnetization M in domains results from irreversible processes of magnetization does not lie in the film plane and forms an angle α ≈ 1° rotation [18], which, in turn, is caused by cubic anisot- with it even in such a thin film (see the calculation per- ropy. formed in Section 4). In other words, the OPT-2 is (ii) It was shown that the critical spin dynamics, i.e., absent at the point H = 0 by definition. a characteristic change in the spin relaxation rate at the (vi) The second OPT-2 is defined in [7] as the end of boundary of the second-order phase transition, does not the transition from the state where the vector M makes manifest itself near the point H ≈ H (see Subsection 2.2). s an angle with the film plane (at H < Hs) to the state || || ≈ (iii) It was shown that the gap in the FMR spectrum where M H [111] at H Hs. Thus, it is assumed that of the sample is caused by incorrect measurement. In the sample contains two magnetic phases, with M ⊥ [7], measurements were carried out without precision [111] and M || [111], between which an induced orien- orientation of the external field with respect to the [111] tational transition via a third phase takes place. We axis; a microstrip sensor was used whose sensitivity is recall that, according to the phenomenological Landau insufficient for this film at frequencies below 200 MHz. theory, the phase transition is a transition between the A more sensitive sensor and an optical device for pre- magnetic phases possible in this crystal that correspond cise orientation of an electromagnet by a laser beam to the minimum of the thermodynamic potential [8, 9]. allowed us (for the same setup and an identical sample) The question concerning the magnetic phases that can to apply the external field accurately along the [111] exist in cubic ferromagnets (among which is included

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1080 SHKAR’ et al. garnet ferrite with a paramagnetic rare-earth sublattice) 5. A. M. Grishin, V. S. Dellalov, V. F. Shkar, E. I. Nikolaev, was studied theoretically and experimentally in [20]. and A. I. Linnik, Phys. Lett. A 140 (3), 133 (1989). The results are as follows. 6. A. M. Grishin, V. S. Dellalov, E. I. Nikolaev, and || V. F. Shkar’, Zh. Éksp. Teor. Fiz. 104 (4), 3450 (1993) (1) Only three high-symmetry magnetic phases M [JETP 77, 636 (1993)]. [100], M || [110], and M || [111] can exist in this sys- 7. V. D. Buchel’nikov, N. K. Dan’shin, A. I. Linnik, tem; there are no equilibrium canted phases, i.e., states L. T. Tsymbal, and V. G. Shavrov, Zh. Éksp. Teor. Fiz. in which the magnetization direction does not coincide 122 (1), 122 (2002) [JETP 95, 106 (2002)]. with these axes [8, 9]. 8. K. P. Belov, A. K. Zvezdin, A. M. Kadomtseva, and (2) In an external magnetic field H || [111], the tran- R. Z. Levitin, Orientational Transitions in Rare-Earth sition to the phase M || [111] is possible only from the Magnets (Nauka, Moscow, 1979). phase M || [100] or M || [110]; both transitions are first- 9. A. K. Zvezdin, V. M. Matveev, F. F. Mukhin, and order OPTs. These conclusions were experimentally F. I. Popov, Rare-Earth Ions in Magnetically Ordered verified for rare-earth–iron garnets in a magnetic field Crystals (Nauka, Moscow, 1985). by measuring the magnetization, susceptibility, and 10. T. N. Tarasenko, Candidate’s Dissertation (Physicotech- nuclear magnetic resonance [20]. nical Inst., Donetsk, 1986). Thus, the statement made by the authors of [7] about 11. E. I. Nikolaev and I. A. Krasin, Kristallografiya 33 (2), the occurrence of an OPT-2 in a thin garnet ferrite film 478 (1988) [Sov. Phys. Crystallogr. 33, 282 (1988)]. seems to be theoretically and experimentally 12. K. B. Vlasov and L. G. Onoprienko, Fiz. Met. Metall- unfounded. oved. 15 (1), 45 (1963). 13. A. G. Gurevich and G. A. Melkov, Magnetic Oscillations Some interesting results obtained in this study were and Waves (Nauka, Moscow, 1994). not properly analyzed, because they are beyond the 14. I. E. Dikshteœn, F. V. Lisovskiœ, E. G. Mansvetova, and scope of this work. In particular, the experimental E. S. Chizhik, Zh. Éksp. Teor. Fiz. 90 (2), 614 (1986) observation of magnetic-resonance modes correspond- [Sov. Phys. JETP 63, 357 (1986)]. ing to straight lines 3 and 4 in Fig. 2 should be the sub- 15. King-Ye Tang, J. Appl. Phys. 61 (8), 4259 (1987). ject of a separate study. We can only conjecture that 16. B. A. Belyaev, S. N. Kulinich, and V. V. Tyurnev, Preprint these modes are spin-wave resonances excited along No. 556F, IF SO AN SSSR (Inst. of Physics, Siberian the film surface by the inhomogeneous HF field of the Division, USSR Academy of Sciences, Krasnoyarsk, microstrip meander. 1989). 17. V. N. Berzhanskiœ and V. I. Ivanov, Preprint No. 405F, IF SO AN SSSR (Inst. of Physics, Siberian Division, USSR REFERENCES Academy of Sciences, Krasnoyarsk, 1986). 1. E. I. Nikolaev, A. I. Linnik, and V. N. Sayapin, Pis’ma 18. G. S. Krinchik, Physics of Magnetic Phenomena (Mosk. Zh. Tekh. Fiz. 17 (17), 85 (1991) [Sov. Tech. Phys. Lett. Gos. Univ., Moscow, 1985). 17, 639 (1991)]. 19. Bubble DomainsÐBased Elements and Devices: Refer- 2. E. I. Nikolaev, A. I. Linnik, and V. N. Sayapin, Mikroéle- ence Book, Ed. by N. N. Evtikhiev and B. N. Naumov ktronika 21 (6), 86 (1992). (Radio i Svyaz’, Moscow, 1987). 20. V. G. Bar’yakhtar, V. A. Borodin, and V. D. Doroshev, 3. M. Inoue, K. Arai, T. Fujii, and M. Abe, J. Appl. Phys. 83 Zh. Éksp. Teor. Fiz. 74, 600 (1978) [Sov. Phys. JETP 47, (11), 6768 (1998). 315 (1978)]. 4. I. L. Lyubchanskii, N. N. Dodoenkova, M. I. Lyubchan- skii, E. A. Shapovalov, Th. Rasing, and A. Lakhtakia, Proc. SPIE 4806, 302 (2002). Translated by A. Kazantsev

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

MAGNETISM AND FERROELECTRICITY

Influence of a Magnetic Two-Phase State on the Magnetocaloric Effect in the La1 Ð xSrxMnO3 Manganites R. V. Demin and L. I. Koroleva Lomonosov Moscow State University, Vorob’evy gory, Moscow, 119899 Russia e-mail: [email protected] Received June 3, 2003; in ﬁnal form, October 31, 2003

∆ Abstract—The magnetocaloric effect Tex and the magnetization in La1 Ð xSrxMnO3 single crystals (x = 0.1, ∆ 0.125, 0.175, 0.3) have been experimentally studied. The magnetic entropy and the magnetocaloric effect Tth ∆ were computed from magnetization curves. All the samples exhibited a maximum in the Tth(T) curve at T = ∆ ∆ Tmax' . A step was observed on the Tex(T) curve in the region of Tmax' , with the value of Tex on this step being ∆ ∆ substantially smaller than Tth. The step on the Tex(T) curve was followed by a maximum, which appeared at ∆ ∆ a temperature 20Ð40 K above Tmax' . This anomalous behavior of Tex and Tth is assigned to the coexistence ∆ of two magnetic (ferro- and antiferromagnetic) phases in the crystal. The calculated value of Tth is determined primarily by the ferromagnetic part of the crystal and disregards the negative contribution from the antiferro- ∆ magnetic part of the crystal to Tex. © 2004 MAIK “Nauka/Interperiodica”.

Studies of perovskite-type manganese compounds the conductive MTPS takes place in the latter. The R1 Ð xAxMnO3 (here, R stands for rare-earth elements MTPS should become manifest in specific features of and A = Ca, Sr, Ba) have been recently attracting con- the magnetocaloric effect, because this effect is nega- siderable attention. The interest in these materials tive for antiferromagnets and positive for ferromagnets. stems primarily from their colossal magnetoresistance We report here on a study of the temperature depen- (CMR), which is exhibited by some of them at room dence of the magnetocaloric effect and of the magneti- temperature. The giant volume magnetostriction found zation in La1 Ð xSrxMnO3 single crystals (x = 0.1, 0.125, in [1] in the vicinity of the Curie point in some manga- 0.175, 0.3). The single crystals were grown using the nite compositions makes them promising materials for crucibleless zone melting technique. The magnetiza- use in various magneto-mechanical devices. The above tion was measured with a vibrating magnetometer. The compounds also revealed a large change in magnetic magnetocaloric effect was studied with the use of a entropy near the Curie point [2, 3]. For some composi- copperÐconstantan thermocouple, whose one junction tions, this change is even larger than for Gd, for which was fitted tightly in the sample. The thermocouple EMF the magnetocaloric effect reaches 5 K in a field of 2 T; was measured with a microvoltmeter, thus providing therefore, they could be expected to feature a large sensitivity as high as 0.01 K. magnetocaloric effect. However, there are almost no According to classical thermodynamics, the change publications on direct measurement of the magnetoca- in the magnetic part of the entropy of a ferromagnet loric effect in manganites. induced by the adiabatic application of a magnetic field An ever increasing number of researchers have been can be written as recently tending to believe that the CMR in manganites H originates from the coexistence in them of two mag- ∆ dM SM = ∫------dH, (1) netic phases, ferro- and antiferromagnetic, caused by dT H the strong sÐd exchange interaction [4, 5]. The mag- 0 netic two-phase state (MTPS) can be insulating at low where M is the magnetization. The magnetocaloric doping levels, where ferromagnetic (FM) regions are effect is calculated from the relation embedded in an insulating antiferromagnetic (AFM) C ()T matrix, and conducting at higher doping levels, where ∆T = ------H ∆S ()T , (2) insulating AFM regions are incorporated in a conduct- th T M ing ferromagnetic matrix. In the La Sr MnO system, 1 Ð x x 3 where C is the specific heat at constant field. the compounds with x = 0.1 and 0.125 are semiconduc- H ∆ ∆ tors and the compounds with x = 0.175 and 0.3 are met- We calculated Tth from Eq. (2), with SM deter- als [6]. Therefore, it may be conjectured that the insu- mined with the use of Eq. (1) from magnetization mea- lating MTPS takes place in the former compounds and surements and with CH taken from [6]. Numerical inte-

1082 DEMIN, KOROLEVA

100 lie 20Ð40 K higher than those obtained from Eq. (2). x = 0.1 ∆ 0.175 Interestingly, the Tex(T) curves exhibit a step in the 0.125 ∆ 80 0.3 temperature region where the Tth(T) curves have a maximum. This observation, as well as the difference ∆ /g between the temperatures of the maxima on the Tex(T) 3 60 ∆ and Tth(T) curves, can be explained as being due to the cm

G coexistence of the FM and AFM phases in the samples. , 40 Invoking the entropy-based treatment [7], we write the M condition of adiabaticity of the process assuming the 20 magnet to consist of two subsystems, namely, a subsystem of magnetic moments with entropy SM and a 0 subsystem of lattice atoms with entropy SC. The condi- 100 150 200 250 300 350 tion of adiabaticity of the process can be cast as T, K ∆ ∆ SC +0.SM = (3) Fig. 1. Magnetization of La1 Ð xSrxMnO3 single crystals in In the case where the sample is in the MTPS and the a ﬁeld of 8.2 kOe plotted vs. temperature. FM and AFM parts can be considered to be decoupled, one can roughly assume SM to be the sum of two parts, the entropy of regions of ferromagnetically ordered 1.2 ∆ ∆ magnetic atoms, SFM, and the entropy of regions of anti- Tth Tex x = 0.3 ferromagnetically ordered atoms SAFM. In this case, the

, K 0.8

T condition of adiabaticity of the process for the MTPS ∆ 0.4 can be written in the form 0 ∆ ∆ ∆ 300 320 340 360 SC ++SAFM SFM = 0. (4) ∆ 0.6 Tex According to Eq. (4), the magnetocaloric effect can be

, K 0.4 ∆ x = 0.175 considered approximately to be the sum of two parts, T Tth ∆ 0.2 ∆T = ∆T + ∆T , (5) 0 ex AFM FM 180 200 220 240 260 280 300 where ∆T and ∆T are the changes in the sample 0.5 AFM FM ∆ temperature induced by a change in the magnetic order ∆ Tex Tth x = 0.125 in the FM and AFM regions of the crystal, respectively. , K 0.3

T Obviously enough, the value of the magnetocaloric ∆ ∆ 0.1 effect Tth calculated from Eq. (2) is determined prima- 0 100 120 140 160 180 200 220 240 rily by the change in the temperature of the sample caused by the change in the magnetic order in the FM 0.3 part of the sample, because the existence of the AFM ∆ ∆ x = 0.1 part should only provide an insigniﬁcant contribution to , K 0.2 Tth Tex

T ∆ SM near the Curie point TC. Therefore, if we subtract ∆ 0.1 the magnetocaloric effect calculated from Eq. (2) from 0 the experimental value, we obtain the contribution to 100 120 140 160 180 200 220 240 ∆ T, K Tex(T) due to the AFM part of the crystal. The graph of the temperature dependence of this AFM contribution Fig. 2. Theoretical and experimental magnetocaloric effects plotted in Fig. 3 has a shape characteristic of antiferro- obtained for La1 Ð xSrxMnO3 single crystals in a ﬁeld of magnets [8, 9]. 8.2 kOe. At the Néel point, the magnetocaloric effect vanishes, and above this point a positive magnetocaloric gration in Eq. (1) was performed following Simpson’s effect typical of paramagnets is observed. The above reasoning applies to compositions with x < 0.175, method. The dM/dT derivative was found by interpola- where the MTPS is either insulating (x = 0.1, 0.125) or tion through three points following numerical smooth- conducting (x = 0.175) but the FM and AFM phases are ing of the M(T) curves. Figure 1 plots the temperature comparable in volume. In the x = 0.3 composition, how- dependences of the magnetization M, and Fig. 2, the ever, the FM phase occupies nearly all of the sample ∆ ∆ Tex(T) and Tth(T) curves obtained in a field H = volume at low temperatures when there is no magnetic 8.2 kOe for all samples. As seen from Fig. 2, all the ﬁeld, although the contribution from the AFM part to ∆ curves have maxima but differ in terms of the tempera- Tex is fairly large, as is evident from Fig. 3. This fact ture positions: on the experimental curves, the maxima should apparently be assigned to the strong increase in

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 INFLUENCE OF A MAGNETIC TWO-PHASE STATE 1083

0.8 and the displaced hysteresis loops of samples cooled in a magnetic ﬁeld. Furthermore, the giant red shift = 0.3

, K x

T 0 ~0.4 eV observed by us in this system and associated ∆ with the FM order implies that the energy of mobile –0.8 holes decreases with increasing FM order, with the 290 300 310 320 330 340 350 360 370 result that localization in the FM part of the crystal is 0.6 preferable for the holes [14]. 0.4 x = 0.175 , K

T ACKNOWLEDGMENTS ∆ 0 The authors express their sincere gratitude to –0.4 180 200 220 240 260 280 300 A.M. Balbashov for preparation of the samples and their characterization. 0.2 x = 0.125 This study was supported by the Russian Founda- , K

T 0 tion for Basic Research, project no. 03-02-16100. ∆ –0.2 100 120 140 160 180 200 220 240 REFERENCES 0.2 1. R. V. Demin, L. I. Koroleva, R. Szymczak, and H. Szym- x = 0.1 czak, Pis’ma Zh. Éksp. Teor. Fiz. 75, 402 (2002) [JETP 0

, K Lett. 75, 331 (2002)]. T

∆ –0.2 2. Z. B. Guo, Y. W. Du, J. S. Zhu, H. Huang, W. P. Ding, and D. Feng, Phys. Rev. Lett. 78, 1142 (1997). –0.4 100 120 140 160 180 200 220 3. X. Bohigwas, J. Tejada, E. del Bbarco, X. X. Zhang, and T, K M. Sales, Appl. Phys. Lett. 73, 390 (1998). 4. É. L. Nagaev, Physics of Magnetic Semiconductors Fig. 3. Magnetocaloric effect induced by a change in mag- (Nauka, Moscow, 1979). netic order in antiferromagnetic regions of La1 Ð xSrxMnO3 5. É. L. Nagaev, Usp. Fiz. Nauk 166, 833 (1996) [Phys. single crystals. Usp. 39, 781 (1996)]. 6. G. L. Liu, J. S. Zhou, and J. B. Goodenough, Phys. Rev. B 64, 144414 (2001). the magnetic nonuniformity in this composition near 7. K. P. Belov, Thermomagnetic Phenomena in Rare-Earth TC. Indeed, data obtained from neutron diffraction [10], Magnets (Nauka, Moscow, 1990). EPR [11], absorption spectra [12], and other studies 8. B. V. Znamenskiœ and I. G. Fakidov, Fiz. Met. Metall- quoted in review [13] suggest the existence of a mixed oved. 13, 312 (1962). FMÐAFM state in compositions with 0.17 < x < 0.3 9. R. Rawat and I. Das, Phys. Rev. B 64, 52407 (2001). near T . Note that Eq. (4) does not take into account the C 10. M. Viret, H. Glattli, C. Fermon, A. M. Loen-Guevara, entropy of the interface separating the FM and AFM and A. M. Revcolevschi, Europhys. Lett. 42, 301 (1998). regions. However, it was shown in [1] that the exchange 11. V. A. Ivashin, J. Deisenhofer, H.-A. Krug von Nidda, interaction between the FM and AFM regions is sub- A. Loidl, A. A. Mukhin, A. M. Balbashov, and M. V. Ere- stantially weaker than that within these regions; there- min, Phys. Rev. B 61, 6213 (2000). fore, there should be practically no transition layer 12. A. Machida, Y. Moritomo, and A. Nakamura, Phys. Rev. between them. B 58, 12540 (1998). Thus, the existence of the MTPS can account for the 13. E. Dagotto, T. Hotta, and A. Moreo, Phys. Rep. 344, 1 anomalous behavior of the experimentally observed (2001). magnetocaloric effect. This is one more argument for 14. R. V. Demin, L. I. Koroleva, and A. M. Balbashov, the existence of the MTPS in the system under study. Pis’ma Zh. Éksp. Teor. Fiz. 70, 303 (1999) [JETP Lett. Earlier [1], we invoked the mixed FMÐAFM phase state 70, 314 (1999)]. in La1 Ð xSrxMnO3 to explain the analogous behavior of the volume magnetostriction and magnetoresistance Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

Physics of the Solid State, Vol. 46, No. 6, 2004, pp. 1084Ð1087. Translated from Fizika Tverdogo Tela, Vol. 46, No. 6, 2004, pp. 1054Ð1057. Original Russian Text Copyright © 2004 by Ognev, Samardak, Vorob’ev, Chebotkevich.

MAGNETISM AND FERROELECTRICITY

Magnetic Anisotropy of Co/Cu/Co Films with Indirect Exchange Coupling A. V. Ognev, A. S. Samardak, Yu. D. Vorob’ev, and L. A. Chebotkevich Institute of Physics and Information Technology, Far-East State University, ul. Sukhanova 8, Vladivostok, 690600 Russia e-mail: [email protected] Received November 4, 2003

Abstract—The effect of isothermal annealing on the magnetic anisotropy, bilinear and biquadratic exchange coupling energies, and domain structure of Co/Cu/Co trilayer films with dCo = 6 nm and dCu = 1.0 and 2.1 nm prepared by magnetron sputtering has been studied. It is shown that, under isothermal annealing, the biquadratic coupling energy decreases by more than an order of magnitude in films with dCu = 1.0 nm and increases in films with dCu = 2.1 nm. The fourth-order magnetic anisotropy is shown to be related to the existence of biquadratic exchange energy. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION in relation to the deposition time. The rate of Co and Cu Interlayer exchange coupling between ferromag- deposition was 0.1 and 0.08 nm/s, respectively. We netic (FM) layers separated by nonmagnetic spacers studied Co/Cu/Co films with a Co layer thickness dCo = plays a key role in many properties observed in multi- 6 nm and a copper spacer thickness dCu = 1.0 and layer films. For instance, the variation in the pattern of 2.1 nm. The film structure was probed by electron the magnetoresistance field dependence (or the magne- microscopy and electron diffraction techniques. The tization curves) under rotation of the film about the film magnetoresistance was measured by the four-probe plane normal, considered together with the energy of compensation method in magnetic fields varied from 0 indirect exchange coupling between the ferromagnetic to 10 kOe, and the magnetic hysteresis loops were layers, provides information about the order of the recorded with a computerized vibrating-sample magne- magnetic anisotropy of multilayer films [1]. Any multi- tometer. The isothermal annealing of the films was per- ° Ð5 layer structure in which the easy magnetization axes of formed at Tann = 250 C in a vacuum of 10 Torr. layers are noncollinear and the magnetization direction changes from one layer to another has a high order of anisotropy. Two exchange-coupled films with identical 3. RESULTS AND DISCUSSION thickness, magnetization, and anisotropy, whose easy magnetization axes are mutually perpendicular, reveal Electron microscope images showed all the films biaxial anisotropy of Ku/2Eb in weak magnetic fields, studied to be polycrystalline with a grain size of ~5 nm. where Eb is the coupling energy [2]. In strong magnetic The magnetic anisotropy of the trilayers was extracted fields, the anisotropy decreases in proportion to 1/H, from magnetization curves obtained in one cycle of with the system becoming gradually isotropic. Fields film rotation about the surface normal in a step of 10°. ϕ H 0 give rise to biaxial and four-axial anisotropy, Figure 1 shows polar diagrams M/Ms = f(H, ) of a which is small for high coupling energies between fer- trilayer film with dCu = 1.0 nm (the first antiferromag- romagnetic layers and decreases as 1/H3 in strong fields netic (AFM) maximum) obtained for several fixed val- [3, 4]. ues of the magnetic field. We readily see that, in mag- This communication reports on a study of the netic fields H = 100 and 250 Oe, the magnetization in behavior of the biquadratic exchange coupling, mag- the as-deposited film reaches the maximum value in all netic anisotropy, and domain structure under isothermal directions. In fields H = 20 and 60 Oe, the M(H, ϕ) annealing and the effect of biquadratic coupling energy curves exhibit four maxima, which indicates the pres- on the magnetic anisotropy. ence of biaxial anisotropy in the film. The pattern of the polar diagrams obtained in fields H = 100 and 250 Oe does not change under isothermal annealing (T = 2. EXPERIMENTAL TECHNIQUES ann 250°C). By contrast, the diagrams measured in fields Co/Cu/Co samples were prepared by dc magnetron H = 20 and 60 Oe vary substantially with annealing. A × Ð3 sputtering in an Ar atmosphere (PAr = 5 10 Torr). 30-min anneal reduces the two maxima in the diagram, The films were applied to (111)Si single crystals at with the polar diagram acquiring the shape of an elon- room temperature. The layer thickness was monitored gated lobe. Increasing the annealing time to 230 min

MAGNETIC ANISOTROPY OF Co/Cu/Co FILMS 1085

0 (a) 0 (b) 0 (c) 330 30 330 30 330 30 4 4 4 3 300 60 300 2 3 60 300 2 60 2 1 1 3 1 270 90 270 90 270 90 0 0.8 0.4 0.8 0.4 0.8

240 120 240 120 240 120

210 150 210 150 210 150 180 180 180

Fig. 1. Polar diagrams of the relative magnetization M/Ms of the Co/Cu(1 nm)/Co ﬁlm for (1) H = 20, (2) 60, (3) 100, and (4) 250 Oe and various values of tann: (a) 0, (b) 30, and (c) 230 min.

0 (a) 0 (b) 0 (c) 330 30 330 30 330 30

300 3 60 300 60 300 3 60 3 2 2 2 1 1 1 270 90 270 90 270 90 0 0.5 1 0 0.2 0.4 0 0.1 0.2

240 120 240 120 240 120

210 150 210 150 210 150 180 180 180

Fig. 2. Polar diagrams of the relative magnetization M/Ms of the Co/Cu(2.1 nm)/Co film for (1) H = 50, (2) 100, and (3) 150 Oe and various values of tann: (a) 0, (b) 30, and (c) 230 min. makes the lobe shape more distinct, which indicates the pling energy decreases gradually with increasing time 2 onset of uniaxial in-plane magnetic anisotropy. of annealing, to reach J2 = Ð0.003 erg/cm after 230 min The polar diagrams of relative magnetization of the of annealing. Co/(2.1 nm)Cu/Co film (the second AFM maximum), both as deposited and after annealing, are displayed in The bilinear and biquadratic exchange coupling Fig. 2. In a field of 150 Oe, the polar diagram of the as- energies in the film with dCu = 2.1 nm increased after deposited film is isotropic. The diagram obtained at isothermal annealing by a factor of 2.1 and 2.2, respec- 100 Oe (curve 2), however, reveals a weak high-order tively (see table). The increase in the biquadratic cou- ϕ pling energy brought about a clearly pronounced anisotropy. The M/Ms = f(H, ) curves measured in fields H = 100 and 150 Oe after 30 min of isothermal fourth-order anisotropy. annealing exhibit four clearly pronounced maxima, Thus, films with biquadratically coupled Co layers thus indicating the formation of in-plane fourth-order are biaxial and those in which J 0 are uniaxial. magnetic anisotropy, which becomes enhanced after an 2 additional t = 80 min of annealing. Indirect exchange coupling appears to induce biaxial ann anisotropy in the upper Co layer during condensation The parameters of the bilinear, J1, and biquadratic, (or under subsequent isothermal annealing), because J2, exchange coupling were found by fitting the theoret- the growth conditions of the bottom and top layers are ical magnetization curves to experimental data. It was different. The uniaxial anisotropy of the bottom Co revealed that the Co layers in the as-deposited films are layer is induced by the magnetron field. The anisotropy coupled both bilinearly and biquadratically (see table). in the top Co layer is induced by the magnetostatic field The bilinear coupling energy of the film with dCu = of the bottom layer and the field of indirect exchange 1.0 nm increases with increasing time of annealing, to coupling of the bottom layer with the growing top Co reach saturation for tann = 80 min. The biquadratic cou- layer through the Cu spacer.

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

1086 OGNEV et al.

Magnetoresistance ratio and the bilinear and biquadratic coupling energies in isothermally annealed Co/Cu/Co ﬁlms

2 2 ()2 3 3 ()1 3 3 ∆ρ ρ dCu, nm tann, min ÐJ1, erg/cm ÐJ2, erg/cm Hs, Oe Ku , 10 erg/cm Ku , 10 erg/cm / , % 1.0 0 0.05 0.017 550 60 10 1.4 30 0.14 0.005 750 30 60 3.3 80 0.14 0.04 790 25 110 3.4 230 0.135 0.03 760 20 105 3.3 2.1 0 0.04 0.018 450 60 30 1.2 30 0.075 0.043 570 70 25 2.2 80 0.082 0.042 620 80 24 2.4 230 0.085 0.04 630 85 22 2.4

()1 ()2 Note:Ku and Ku are magnetic anisotropy constants of second and fourth order, respectively, derived by ﬁtting theoretical relative ∆ρ ρ magnetization M/Ms = f(H) and the magnetoresistance ratio / = f(H) curves to experimental data.

Annealing initiates an increase in effective indirect spacer or of ferromagnetic atoms at the interface [6, 7], exchange coupling between the ferromagnetic layers and the magnetic dipole mechanism related to interface | | Hs = 2 Jeff /MsdCo, where Jeff = J1 + 2J2 [4] and Ms is the roughness [7]. In fine-grained polycrystalline trilayers, saturation magnetization (see table), and of the magne- it is the latter two mechanisms that are apparently real- toresistance in both films, but the actual relation ized. Biquadratic indirect exchange coupling between between the bilinear and biquadratic coupling compo- ferromagnetic layers in a film can result in noncollinear | | nents after annealing depends on the thickness of the magnetic order only under the condition J2 < Ð J1 /2 [8]. Cu spacer. As seen from Figs. 1 and 2, isothermal For our films with dCu = 1 and 2.1 nm annealed at Tann = ° annealing of the Co/(2.1 nm)Cu/Co film results in an 250 C, the conditions J1 Ð 2J2 < 0 and J1 < 0 are satis- increase in the biquadratic exchange coupling energy fied. This shows that the equilibrium state for H = 0 can and a more pronounced fourth-order anisotropy, be identified with antiferromagnetic ordering of mag- whereas in the Co/(1.0 nm)Cu/Co film the biquadratic netic moments in adjacent Co layers. exchange coupling energy decreases by an order of The difference in the behavior of biquadratic cou- magnitude (J2 0) and makes the film uniaxial. pling observed to occur as a result of isothermal anneal- Several mechanisms of biquadratic coupling ing between films with dCu = 1 and 2.1 nm should pos- between ferromagnetic layers in multilayer films have sibly be assigned to the impurity mechanism of biqua- ° been proposed to date. These are the model of bilinear dratic coupling. Annealing of films at Tann = 250 C is coupling fluctuations [5], the free-spin model based on accompanied by a growth in the size of grains (up to the presence of impurity atoms in the nonmagnetic 10 nm) and impurity diffusion toward sinks, which are grain boundaries in polycrystalline films. It may be conjectured that, in films with a thinner spacer, impu- ρ ρ ln( / 0) rity atoms escape faster from the spacer at the given 0 temperature of isothermal annealing, which results in a faster decrease in the J2/J1 ratio in the film with dCu = 1 nm. This conjecture is corroborated by the calculated diffusion coefficients D. If we accept the approximation –0.05 proposed for multilayer structures [9, 10], we can 1 derive the diffusion coefficient from the relation D = − Λ2 π2 ρ ρ ρ ( /8 )dln( t/ 0)/dt, where 0 is the initial electrical ρ resistivity, t is the electrical resistivity after isothermal 2 Λ –0.10 annealing for a time t, and = dCo + dCu. For this purpose, we derived a relation connecting the logarithm of the relative electrical resistivity with the time of anneal- ρ ρ ing (Fig. 3). The behavior of the ln( t/ 0) = f(t) relation –0.15 suggests that diffusion in multilayer Co/Cu films can 0 100 200 300 occur at two rates, depending on the isothermal anneal- t, min ing time. Within region 1 in Fig. 3, the diffusion rate is × Ð22 2 × D1 = 7 10 m /s, and within region 2, D2 = 0.2 Fig. 3. Logarithm of the relative electrical resistivity vs. − annealing time for a multilayer Co/Cu film. 10 22 m2/s. The relaxation processes are characterized

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 MAGNETIC ANISOTROPY OF Co/Cu/Co FILMS 1087

ciﬁcally, the domain walls are no longer preferentially oriented and become more twisted and the average domain size decreases to 1.8 µm (Fig. 4d).

(‡) (c) 4. CONCLUSIONS Studies of fine-grained, polycrystalline Co/Cu/Co trilayer films have shown the fourth-order magnetic anisotropy to be due to the presence of biquadratic exchange coupling between the ferromagnetic Co layers; the effect of isothermal low-temperature annealing 1.5 µm 3 µm (b) (d) on the energy of the bilinear and biquadratic exchange coupling between the Co layers depends on the thick- Fig. 4. Domain structure of Co/Cu/Co films with ness of the nonmagnetic spacer. Films with fourth- (a, b) dCu = 1 nm and (c, d) dCu = 2.1 nm; (a, c) as-depos- order magnetic anisotropy feature a complex domain ited, (b, d) after annealing. Arrows identify regions in which structure with no geometric regularity in the domain the Co layers are antiferromagnetically coupled. wall arrangement and whose domain walls are closed and domains small. by an activation energy Q = 0.15 eV, a figure typical of point defect annealing and internal stress relaxation. ACKNOWLEDGMENTS To be able to observe the domains, domain walls, This study was supported by the federal research and magnetization ripple simultaneously, the domain program “Studies in Topical Areas of Science and structure of films with dCu = 1 and 2.1 nm was studied Technology for Civilian Use” of the Ministry of Indus- by Lorentz microscopy. As seen from images of the try, Science, and Technology of the Russian Federation domain structure, the films have partially antiferromag- (project no. 3-02/DVGU under state contract netic and partially ferromagnetic coupling (Fig. 4). Fig- no. 40.012.1.1.1151) and by the Ministry of Education ure 4 reveals the presence of coupled Néel walls in the of the Russian Federation (project no. RD 02-1.2-50). films. The Néel walls in the top and bottom Co layers are displaced with respect to one another. In the photo- graphs, the AFM regions are identified by arrows. We REFERENCES readily see that the annealing-induced change in mag- 1. V. V. Ustinov, M. A. Milyaev, L. N. Romashev, netic anisotropy is accompanied by rearrangement of T. P. Krinitsina, and E. A. Kravtsov, J. Magn. Magn. Mater. 226Ð230, 1811 (2001). the domain structure. The as-deposited film with dCu = 1 nm (Fig. 4a), whose polar diagram exhibits biaxial 2. A. Yelon, in Physics of Thin Films: Advances in anisotropy, has a complex domain structure; namely, Research and Development, Ed. by M. H. Francombe the domain wall arrangement lacks geometric regular- and R. W. Hoffman (Academic, New York, 1971; Mir, ity, there are regions with closed domain walls, and the Moscow, 1973), p. 392. domains themselves are small (~0.43 µm). Such a 3. A. Yelon, J. Appl. Phys. 35, 770 (1964). domain structure is characteristic of films with strong 4. C. H. Marrows, B. J. Hickey, M. Herrmann, S. McVitie, indirect exchange coupling between the ferromagnetic J. N. Chapman, M. Ormston, A. K. Petford-Long, layers. The complex domain structure persists in the N. P. A. Hase, and B. K. Tanner, Phys. Rev. B 61, 4131 annealed film, as seen from the image shown in Fig. 4b. (2000). However, the domain size increased nearly twofold 5. J. C. Slonczewski, Phys. Rev. Lett. 67, 3172 (1991). (average domain dimension ~0.83 µm) and the domain 6. J. C. Slonczewski, J. Magn. Magn. Mater. 150, 13 walls became more extended. (1995). 7. S. O. Demokritov, E. Tsymbal, P. Grunberg, W. Zinn, In the film with dCu = 2.1 nm (where the polar dia- and I. K. Schuller, Phys. Rev. B 49, 720 (1994). gram reveals high-order anisotropy), the domain struc- µ 8. V. V. Ustinov, N. G. Bebenin, L. N. Romashev, ture is larger (average domain size ~3 m) and the V. I. Minin, M. A. Milyaev, A. R. Del, and A. V. Semer- domain walls are more extended (Fig. 4c). The overall ikov, Phys. Rev. B 54 (22), 15958 (1996). pattern of the domain structure suggests the presence of 9. P. M. Levy, J. Appl. Phys. 67 (9), 5914 (1990). a certain preferential magnetization orientation in the 10. A. V. Boltushkin, V. M. Fedosyuk, and O. I. Kasyutich, film. The film features regions with AFM and FM cou- Fiz. Met. Metalloved. 75 (6), 58 (1993). pling between Co layers. After the annealing, the film exhibits a clearly pronounced biaxial anisotropy (Fig. 2b) and the domain structure changes; more spe- Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

Physics of the Solid State, Vol. 46, No. 6, 2004, pp. 1088Ð1094. Translated from Fizika Tverdogo Tela, Vol. 46, No. 6, 2004, pp. 1058Ð1064. Original Russian Text Copyright © 2004 by Bayukov, Abd-Elmeguid, Ivanova, Kazak, Ovchinnikov, Rudenko.

MAGNETISM AND FERROELECTRICITY

Mössbauer Effect Study in Fe1 Ð xVxBO3 Solid Solutions O. A. Bayukov*, M. M. Abd-Elmeguid**, N. B. Ivanova***, N. V. Kazak*, S. G. Ovchinnikov*, and V. V. Rudenko* *Kirensky Institute of Physics, Siberian Division, Russian Academy of Sciences, Akademgorodok, Krasnoyarsk, 660036 Russia **Physikalisches Institut, Universität zu Köln, Köln, 50937 Germany ***Krasnoyarsk State Technical University, Krasnoyarsk, 660074 Russia Received September 16, 2003; in ﬁnal form, November 13, 2003

Abstract—The Mössbauer effect in the Fe1 Ð xVxBO3 solid solutions has been measured at 130 and 300 K. The Fe0.05V0.95BO3 composition was studied in the interval 4.2Ð300 K. The experimental data obtained are described in terms of the model of a dilute magnetic insulator in which atoms of the first coordination sphere provide a major contribution to the hyperfine field at iron nucleus sites. It was found that, at low temperatures, the field Hhf is generated primarily by the iron ion itself and depends only weakly on substitution. The hyperfine interaction parameters in the discrete configuration series 6Fe, 5Fe1V, 4Fe2V, 3Fe3V, and 2Fe4V were determined. The magnitude of the isomer shift suggests that iron in the crystal resides in the trivalent state. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION 2. SAMPLES AND EXPERIMENTAL TECHNIQUES Transition metal borates with the chemical formula The Fe V BO single crystals under study were MBO3 crystallize in the calcite structure [space group 1 Ð x x 3 grown by spontaneous crystallization from a melt solu- ()6 R3cD3d ]. Strong electronic correlations offer the tion [22]. The crystals were hexagonal platelets up to possibility of observing the rich variety of magnetic 4 × 4 mm in size and about 0.1 mm thick. The concen- structures and electronic properties in this series of tration x quoted in [22] was derived from the content of compounds governed by the presence of one or another components in the charge (xch). The exact amounts of 3+ transition metal M = Fe, V, Cr, or Ti. FeBO3 has been the elements were determined by EDAX ZAF quantifi- a subject of intense interest since the time it was first cation. The values of x thus obtained are listed in Table 1 prepared [1]; studies were made on its optical and mag- together with the earlier xch figures. netooptical properties [2Ð6], crystal structure [7, 8], Mössbauer measurements were performed on pow- magnetic properties [9Ð11], and NMR [12, 13] and dered samples of the Fe1 Ð xVxBO3 single crystals. The Mössbauer [14, 15] spectra. The FeBO3 iron borate is isomer shift δ was determined relative to metallic (α- transparent in the visible region, and its magnetic order- Fe) iron. ing temperature is above room temperature, TN = 348 K. Unfortunately, experimental information on The experiments on the Fe1 Ð xVxBO3 samples were other representatives of this class of borates is scarce. carried out at 130 and 300 K with a Co57(Cr) source. The isostructural compounds VBO3, CrBO3, and TiBO3 The optimum sample thickness was calculated with due account of the iron content and the absorption factor were first synthesized in 1964 [16]. VBO3 is known to and was found to be 5Ð10 mg Fe/cm2. The model spec- be a ferromagnetic semiconductor with TC = 32 K, and CrBO is a low-temperature antiferromagnet with T = tra were least squares fitted to experimental data under 3 N the assumption that the spectral lines have Lorentzian 15 K and an insulator [17]. TiBO3 was reported to be a shape. We took as an additional fitting parameter the weak ferromagnet (TN = 25 K) [18]. There are publications on studies of hyperfine interaction in iron borate– based solid solutions Fe1 Ð xMxBO3, where the ions M = Table 1. Vanadium concentration in Fe1 Ðx VxBO3 solid Al, Ga, and Cr act as substituting ions [19Ð21]. solutions

A coordinated investigation of the magnetic, electri- xch 0.25 0.5 0.6 0.75 0.95 cal, and optical properties of Fe1 Ð xVxBO3 solid solu- x 0.02 0.13 0.18 0.3 0.95 tions was performed earlier in [22]. This communica- Note: xch was derived from the content of components in the tion reports on a measurement of the Mössbauer effect charge, and x, from energy-dispersive analysis of x-rays in iron-containing samples of Fe1 Ð xVxBO3. (EDAX ZAF quantiﬁcation).

MÖSSBAUER EFFECT STUDY 1089 broadening of the outer sextet lines with respect to the a Γ Γ inner lines ( 16/ 34), which is proportional to the hyper- ﬁne ﬁeld at the nucleus site. Processing of the experimental spectra revealed the presence of an impurity phase, Fe3BO6, in the solid solutions, which should be assigned to the crystallization temperatures of these compounds being very similar and was described earlier in [21]. b

The spectra of the Fe0.05V0.95BO3 crystal in the range of 4.2 to 300 K were measured with a Co57(Rh) source of an initial activity of 25 mCi (925 MBq). The measurements were performed in the transmission geometry. The source was placed inside a 4He cryostat close to the sample. The maximum source velocity was c V max = CUeff, where Ueff is the effective amplitude of the sine signal and C is the calibration factor, equal to 0.041326 mm/s mV. The measurements were performed on polycrystalline samples containing 1 mg Fe/cm2 that were mixed thor- oughly with Al2O3. d 3. EXPERIMENTAL RESULTS

Figure 1 shows Mössbauer spectra of Fe1 Ð xVxBO3 solid solutions obtained at room temperature. The FeBO3 spectrum is a well-resolved sextet (Fig. 1, a). The hyperfine interaction parameters are in agreement with the data from [14]. The spectral line intensities are in the ratio 3 : 2 : 1 : 1 : 2 : 3. e Because of the compositional disorder characteristic of solid solutions, impurity atoms can be considered to be distributed randomly in the matrix. Thus, all the cation states are equivalent and can be occupied by iron and vanadium atoms in random fashion. Because the exchange interactions have short-range character, we analyze experimental spectra in the nearest neighbor –10–8 –6 –4 –2 0 2 4 6 8 10 approximation. The central iron atom has n atoms of V, mm s–1 vanadium and (6 Ð n) atoms of iron in its nearest environment. The probability of finding such atomic config- Fig. 1. Mössbauer spectra of (a) FeBO3, (b) Fe0.98V0.02BO3, urations in a crystal is described by the well-known (c) Fe0.87V0.13BO3, (d) Fe0.82V0.18BO3, and (e) Fe0.7V0.3BO3 binomial distribution obtained at 300 K.

6! n 6 Ð n P ()n = ------()1 Ð x x , 6 n!6()Ð n ! parameters in the solid solutions are displayed in Fig. 2 where x is the vanadium concentration. The local and listed in Table 2. It is assumed that the hyperfine molecular field theory was applied to interpret complex field in insulators is well described in the nearest neigh- Mössbauer spectra of mixed oxide systems in [23]. bor approximation; therefore, the fields of the 6Fe, Following this assumption, the spectrum of the 5Fe1V, 4Fe2V, etc., configurations are discrete and well Fe0.98V0.02BO3 crystal (Fig. 1, b) can be unfolded into resolved and the 6Fe configuration in a substituted two sextets with occupations of 76 and 24%. The crystal has the same hyperfine field as in an unsubsti- former sextet relates to the nearest iron environment tuted one. In our case, while the 5Fe1V configuration consisting of six iron atoms (6Fe), and the latter sextet, (Hhf = 323 kOe) differs noticeably from the 6Fe config- to the configuration in which there is one vanadium uration (H = 335 kOe), the field of 335 kOe is substan- atom among the nearest neighbors (5Fe1V). The prob- hf ability distributions of hyperfine fields at the iron tially smaller than Hhf = 345 kOe in the unsubstituted nucleus sites and the calculated hyperfine interaction borate FeBO3. Quite probably, the hyperfine field at

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1090 BAYUKOV et al.

P (a) a 0.20 0.15 0.10 0.05 0 b P (b) 0.20 0.15 0.10 0.05 0 c P (c) 0.20 0.15 0.10 0.05 0 50 100 200 300 400 500 600 d H, kOe

Fig. 2. Hyperﬁne ﬁeld probability distribution for iron nuclei at 300 K obtained for (a) FeBO3, (b) Fe0.98V0.02BO3, and (c) Fe0.87V0.13BO3. e iron nucleus sites in FeBO3 is contributed to markedly by next-to-nearest neighbors.

The spectrum of the Fe0.87V0.13BO3 has a complex shape (Fig. 1, c). The distribution of the hyperfine field probabilities allows us to isolate three regions (Fig. 2c). –10–8 –6 –4 –2 0 2 4 6 8 10 The region of 350–400 kOe corresponds to fields V, mm s–1 higher than the field in FeBO3 but lower than that in Fe3BO6, Hhf = 521 kOe [24]. This region can be Fig. 3. Mössbauer spectra of (a) FeBO3, (b) Fe0.98V0.02BO3, assigned to the vanadium-substituted Fe3BO6 phase. (c) Fe0.87V0.13BO3, (d) Fe0.82V0.18BO3, and (e) Fe0.7V0.3BO3 The broad region of 150–300 kOe can be identified obtained at T = 130 K. with the Fe1 Ð xVxBO3 crystal. The width of the hyperfine field distribution for the Fe1 Ð xVxBO3 crystal is to superparamagnetic regions in Fe1 Ð xVxBO3 or to the larger than that for the substituted Fe3BO6 phase. This substituted Fe3BO6. may be explained by the fact that the iron in Fe3BO6 has a larger number of magnetic bonds (6 for the 8d and 8 The room-temperature spectra of the Fe1 Ð xVxBO3 for 4c sites, symmetry group Pnma) [25] compared to solid solutions with vanadium concentrations x = 0.18, FeBO3 (6 bonds) and that the number of magnetic 0.3, and 0.95 represent quadrupole doublets with asym- bonds is a factor that stabilizes the hyperfine field. metry in the intensities of the 3/2 1/2 and 1/2 Moreover, the Néel temperature of FeBO3 (348 K) is 1/2 nuclear transitions. This could be assigned to the substantially lower than that of Fe BO (508 K) [26]. crystallite c axes being oriented preferentially parallel 3 6 γ Note also that the degree of vanadium substitution for to the -ray direction; however, we study here powder samples and the observed asymmetry is actually a man- the FeBO3 crystal is larger than that for Fe3BO6. The hyperfine field probability distribution in the 150- to ifestation of the Gol’danskiiÐKaryagin effect [27]. This 300-K region is almost symmetrical, with a slight devi- effect is small for monoclinic lattices and, hence, the ation toward higher fields. The experimental spectrum asymmetry is due to the rhombohedral lattice of the is fitted well by a set of sextets, whose parameters are Fe1 Ð xVxBO3 crystal. We may thus conclude that each listed in Table 2. In accordance with the binomial dis- doublet can be unfolded into two doublets, symmetric tribution, their assignment is specified by the number of and asymmetric. Fe and V neighbors. The weak peak in the hyperfine Figures 3 and 4 display the Mössbauer spectra and field probability distribution in the low field region is the distribution functions of the hyperfine field proba- due to a paramagnetic doublet, which probably belongs bility at iron nucleus sites, respectively, obtained for the

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 MÖSSBAUER EFFECT STUDY 1091

Table 2. Hyperﬁne interaction parameters in Fe1 Ð xVxBO3 solid solutions obtained at 300 K δ ∆ Γ Γ Γ Compound Hhf EQ 34 16/ 34 S Assignment

FeBO3 0.40 345 0.38 0.29 1 1 “FeBO3” Fe0.98V0.02BO3 0.40 335 0.39 0.30 1.04 0.76 6Fe 0.39 323 0.36 0.30 1.58 0.24 5Fe1V

Fe0.87V0.13BO3 0.39 278 0.41 0.43 1.72 0.20 6Fe 0.40 251 0.41 0.43 1 0.11 5Fe1V 0.39 231 0.38 0.41 1 0.14 4Fe2V 0.39 212 0.36 0.38 1 0.12 3Fe3V 0.39 189 0.41 0.30 1.91 0.13 2Fe4V 0.38 159 0.30 0.53 1.58 0.13 1Fe5V 0.44 1.31 0.06 Ð

0.36 375 0.54 0.20 5.26 0.11 “Fe3BO6” Fe0.82V0.18BO3 0.42 0.40 0.28 0.70 “FeBO3” (GKE = 1.1) 0.38 0.47 0.32 0.30 “Fe3BO6” Fe0.7V0.3BO3 0.41 0.42 0.32 0.71 “FeBO3” (GKE = 1.09) 0.40 0.45 0.30 0.29 “Fe3BO6” Fe0.05V0.95BO3 0.42 0.43 0.17 0.43 “FeBO3” (GKE = 1.67) 0.39 0.48 0.39 0.57 Note: δ is the isomer chemical shift relative to metallic iron (αFe), ±0.02 mm/s; H is the hyperfine field at an iron nucleus site, ±5 kOe; ∆ ± Γ hf ± Γ Γ EQ is the quadrupole splitting, 0.04 mm/s; 34 is the FWHM of inner sextet lines, 0.02 mm/s; 16/ 34 is the ratio of the outer to inner sextet line widths, ±0.04 mm/s; S is the fractional site occupation, ±0.05; and GKE stands for the Gol’danskiiÐKaryagin effect.

Fe1 Ð xVxBO3 solid solutions at 130 K. The temperature distribution passes through a maximum at the 5Fe1V dependences of the hyperfine field for FeBO3 and configuration (17%) for the x = 0.18 composition and at Fe3BO6 are known to cross in the temperature region the 6Fe configuration (15%) for the composition with close to 240 K, and the resolution of the sextets x = 0.3. The isomer shift is the same for all positions to becomes poorer as the temperature is lowered [21]. The within experimental error. The spectra obtained suggest hyperfine field probability distribution function does the existence of a vanadium-diluted superparamagnetic not permit identification of the individual iron positions phase of Fe3BO6. 6Fe and 5Fe1V (Fig. 4b) for the Fe0.98V0.02BO3 crystal. Deconvolution into two sextets makes it possible to The Fe0.05V0.95BO3 solid solution remains paramag- evaluate the probable occupations of these configura- netic down to 130 K. To gain deeper insight into the tions. For the same reason, one cannot reliably deter- hyperfine interactions in this crystal, we performed mine the occupations of the inequivalent positions in Mössbauer studies at temperatures spanning the range the Fe0.87V0.13BO3 composition (Fig. 4c). from 4.2 to 300 K. The results of calculations and measurements are listed in Table 4 and presented in graph- The Mössbauer spectra of Fe0.82V0.18BO3 and ical form in Fig. 5. The spectrum of the Fe0.05V0.95BO3 Fe0.7V0.3BO3 samples measured at 130 K are superposi- composition measured at 4.2 K is a sextet with an tions of several sextets and of a paramagnetic doublet admixture of the paramagnetic phase, which adds up to (Figs. 3d, 3e). This complexity of the hyperfine interac- not over 10% of the magnetically ordered phase tion pattern can originate from the nonuniform mag- (Fig. 5a). The relative sextet line intensities are close to netic state occurring in these crystals. Local deviations 3 : 2 : 1 : 1 : 2 : 3. The hyperfine field H at iron nucleus from stoichiometry are capable of affecting magnetic hf order in the sample. The temperature dependence of the sites in this sample is 507 kOe, which is 8.6% lower magnetization of these solid solutions follows a non- than that in the unsubstituted FeBO3 crystal (555 kOe) trivial pattern and exhibits two magnetic transitions in measured at the same temperature [14]. The nearest the temperature range from 30 to 200 K [22]. The broad neighbor contribution to the effective hyperfine field at hyperfine field distribution observed in the iron nucleus sites in FeBO3 is approximately 10% of Fe0.82V0.18BO3 and Fe0.7V0.3BO3 crystals (Figs. 4d, 4e) the ion-core spin polarization. The lower effective field allows isolation of a discrete set of the 6Fe, 5Fe1V, Hhf in Fe0.05V0.95BO3 compared to that in FeBO3 can be 4Fe2V, 3Fe3V, and 2Fe4V configurations, whose assigned to the perturbation introduced by vanadium hyperfine parameters are listed in Table 3. The binomial impurity atoms.

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1092 BAYUKOV et al.

P (a) 0.15 0.10 0.05 (a) P (b) –10–8–6–4–20246810 0.15 0.10 0.05

P (c) (b) 0.12 0.08 0.04 –4 –3 –2 –1 0 1 2 3 4

P 0.12 (d) 0.08 0.04 (c)

P –2 –1 0 1 2 0.08 (e) 0.06 0.04 0.02 0 50 100 200 300 400 500 600 H, kOe (d)

Fig. 4. Hyperﬁne ﬁeld probability distribution for iron nuclei obtained at T = 130 K for (a) FeBO3, (b) Fe0.98V0.02BO3, –4 –3 –2 –1 0 1 2 3 4 –1 (c) Fe0.87V0.13BO3, (d) Fe0.82V0.18BO3, and (e) Fe0.7V0.3BO3. V, mm s

Fig. 5. Mössbauer spectra of a Fe0.05V0.95BO3 crystal obtained at temperatures of (a) 4.2, (b) 70, (c) 130, and The Mössbauer spectra of the sample obtained at 70 (d) 300 K. and 130 K are asymmetric doublets characteristic of the paramagnetic state (Figs. 5b, 5c). Each spectrum can be unfolded into doublets relating to different atomic con- the inequivalent iron positions at room temperature figurations. According to the binomial distribution, for (Fig. 5d). a vanadium concentration x = 0.95, the probable occupations of the 6V and 1Fe5V configurations acquire maximum values of 76 and 25%, respectively. The two 4. DISCUSSION OF THE RESULTS doublets characterizing these configurations are identi- Thus, deconvolution of solid-solution spectra into fied in the figures by solid lines. The isomer shifts for constituent spectra of the components of these solu- these configurations are independent of temperature tions, which was made under the assumption of a ran- and are δ = 0.23 mm/s for iron in the 6V position and dom impurity distribution and of additivity of the con- δ δ = 0.29 mm/s for the 1Fe5V environment. These config- tributions due to impurity atoms to Hhf and and with urations have a constant quadrupole splitting (Table 4). due account of the influence of atoms in the first coor- Unfortunately, the poor resolution of the Fe0.05V0.95BO3 dination sphere, has permitted us to evaluate the hyper- spectrum does not allow isolation and identification of fine parameters for the 6Fe, 5Fe1V, 4Fe2V, 3Fe3V, and

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 MÖSSBAUER EFFECT STUDY 1093

Table 3. Hyperﬁne interaction parameters for Fe1 Ð xVxBO3 solid solutions obtained at 130 K δ ∆ Γ Γ Γ Compound Hhf EQ 34 16/ 34 S Assignment

FeBO3 0.50 540 0.46 0.47 1 1 “FeBO3” Fe0.98V0.02BO3 0.50 539 0.37 0.44 1 0.87 6Fe 0.49 528 0.33 0.32 1 0.13 5Fe1V

Fe0.87V0.13BO3 0.50 524 0.39 0.46 1.05 0.76 “FeBO3” 0.50 491 0.44 0.51 1.19 0.19 “FeBO3” 0.50 425 0.07 0.87 1.01 0.05 “Fe3BO6” Fe0.82V0.18BO3 0.51 514 0.41 0.28 2.54 0.10 6Fe 0.51 484 0.44 0.50 1.15 0.17 5Fe1V 0.51 462 0.45 0.44 1.13 0.14 4Fe2V 0.51 436 0.43 0.48 1 0.11 3Fe3V 0.52 403 0.54 0.53 1.11 0.08 2Fe1V

0.52 351 0.41 0.36 12.3 0.30 “Fe3BO6” 0.46 1.31 0.1 “Fe3BO6” Fe0.7V0.3BO3 0.51 501 0.41 0.41 1.49 0.15 6Fe 0.50 471 0.49 0.45 1.40 0.14 5Fe1V 0.49 430 0.49 0.57 1.94 0.12 4Fe2V 0.50 374 0.45 0.56 1.06 0.13 3Fe3V

0.49 273 0.30 1.77 1.06 0.22 “Fe3BO6” 0.49 0.43 0.51 0.23 “Fe3BO6” Fe0.05V0.95BO3 0.50 0.44 0.48 1 “FeBO3” δ ± ± ∆ ± Γ ± Γ Γ ± ± Note: , 0.02 mm/s; Hhf, 5 kOe; EQ, 0.04 mm/s; 34, 0.02 mm/s; 16/ 34, 0.04 mm/s; and S, 0.05.

Table 4. Hyperﬁne interaction parameters for a Fe0.05V0.95BO3 crystal obtained at different temperatures δ ∆ Γ T, K Hhf EQ 34 S Assignment 4.2 0.28 ± 0.005 507.2 ± 0.03 0.22 ± 0.01 0.40 ± 0.02 70 0.23 ± 0.01 0.42 ± 0.01 0.29 ± 0.023 0.75 6V 0.29 ± 0.013 0.48 ± 0.016 0.16 ± 0.02 0.25 1Fe5V 130 0.23 ± 0.003 0.42 ± 0.004 0.33 ± 0.007 0.76 6V 0.29 ± 0.002 0.47 ± 0.003 0.14 ± 0.004 0.24 1Fe5V 300 0.27 ± 0.002 0.43 ± 0.003 0.26 ± 0.005 δ ± ± ∆ ± Γ ± ± Note: , 0.01 mm/s; Hhf, 0.03 kOe; EQ, 0.02 mm/s; 34, 0.02 mm/s; and S, 0.02.

2Fe4V positions in the Fe0.87V0.13BO3 crystal at room bauer spectra of the (Fe1 Ð xVx)2O3 system, according to temperature and in the Fe0.82V0.18BO3 and Fe0.7V0.3BO3 which low vanadium contents are conducive to the for- crystals at a temperature of 130 K. The hyperfine fields mation of Fe2+ÐV4+ ion pairs, did not find support in our at iron nucleus sites decrease monotonically in the studies. This implies that the electronic state of iron ∆ series of these configurations in discrete steps of Hhf = does not depend on substitution by vanadium. The iron 20Ð30 kOe, in full agreement with the usual concepts ion resides in an octahedral environment, and the elec- concerning a dilute magnetic insulator. The chemical tric field gradient at iron nucleus sites is directed paral- isomer shifts for these configurations are the same to lel to the [111] threefold axis. The quadrupole splitting within experimental error. The isomer shifts at room in the Fe1 Ð xVxBO3 solid solutions is close in magnitude ∆ temperature, δ = 0.38Ð0.40 mm/s, and at 130 K, δ = to EQ = 0.46 mm/s, the value for the unsubstituted 0.51Ð0.52 mm/s, indicate that iron in the Fe1 Ð xVxBO3 FeBO3 crystal. The small change in the hyperfine field system is trivalent. Note that the idea of substitution of for the Fe0.05V0.95BO3 sample at 4.2 K compared to that an average-sized ion used in [28] to interpret Möss- in FeBO3 indicates that the Hhf field in these crystals is

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1094 BAYUKOV et al. produced primarily by the iron ion itself and depends 13. L. V. Velikov, E. G. Rudashevskiœ, and V. N. Seleznev, only weakly on the environment. Izv. Akad. Nauk SSSR, Ser. Fiz. 36, 1531 (1972). 14. M. Eibschutz, L. Pfeiffer, and J. W. Nielsen, J. Appl. Phys. 41, 1276 (1970). ACKNOWLEDGMENTS 15. M. Kopcewicz, H. Engelmann, S. Stenger, G. V. Smir- One of the authors (M.M. A.-E.) would like to thank nov, U. Gonser, and H. G. Wagner, Appl. Phys. A 44, 131 the Deutsche Forschungsgemeinschaft for ﬁnancial (1987). support (SFB 608). 16. H. Schmid, Acta Crystallogr. 17, 1080 (1964). This study was supported by the Russian Founda- 17. T. A. Bither, Carol G. Frederick, T. E. Gier, J. F. Weiher, tion for Basic Research (project no. 03-02-16286) and and H. S. Young, Solid State Commun. 8, 109 (1970). the federal program “Integration” (project no. B0017). 18. Xu Ziguang, Matam Mahesh Kumar, and Ye Zuo Guang, in Proceedings of Annual March Meeting (Am. Phys. REFERENCES Soc., 2001). 19. De Lacklison, J. Chadwick, and J. L. Page, J. Phys. D: 1. I. Bernal, C. W. Struck, and J. G. White, Acta Crystal- Appl. Phys. 5, 810 (1972). logr. 16, 849 (1963). 20. M. W. Ruckman, R. A. Levy, and R. Chennette, J. Appl. 2. J. Haisma, H. J. Prins, and K. L. L. van Mierlo, J. Phys. Phys. 53, 1694 (1982). D: Appl. Phys. 7, 162 (1974). 3. B. Andlauer, J. Schneider, and W. Wettling, Appl. Phys. 21. O. A. Bayukov, V. P. Ikonnikov, M. I. Petrov, 10, 189 (1976). V. V. Rudenko, V. N. Seleznev, and V. V. Uskov, Tr. MKM-73b 3, 313 (1974). 4. R. Wolfe, A. J. Kurtzig, and R. C. LeCraw, J. Appl. Phys. 41, 1218 (1970). 22. N. B. Ivanova, V. V. Rudenko, A. D. Balaev, N. V. Kazak, 5. I. S. Édel’man and A. V. Malakhovskiœ, Opt. Spektrosk. V. V. Markov, S. G. Ovchinnikov, I. S. Édel’man, 35, 959 (1973). A. S. Fedorov, and P. V. Avramov, Zh. Éksp. Teor. Fiz. 121, 354 (2002) [JETP 94, 299 (2002)]. 6. A. J. Kurtzig, R. Wolfe, R. C. LeCraw, and J. W. Nielsen, Appl. Phys. Lett. 14, 350 (1969). 23. J. M. D. Coey and G. A. Sawatzky, Phys. Status Solidi B 44, 673 (1971). 7. R. Diehl, Solid State Commun. 17, 743 (1975). 8. M. Pernet, D. Elmaleh, and J.-C. Joubert, Solid State 24. A. S. Kamzin and L. A. Grigor’ev, Zh. Éksp. Teor. Fiz. Commun. 8, 1583 (1970). 105, 377 (1994) [JETP 78, 200 (1994)]. 9. J. C. Joubert, T. Shirk, W. B. White, and R. Roy, Mater. 25. J. G. White, A. Miller, and R. E. Nielsen, Acta Crystal- Res. Bull. 3, 671 (1968). logr. 19, 1060 (1965). 10. A. S. Kamzin, B. Shtahl, R. Gellert, M. Muller, E. Kan- 26. M. Hirano, T. Okuda, T. Tsushima, S. Umemura, keleit, and D. B. Vcherashniœ, Pis’ma Zh. Éksp. Teor. K. Kohn, and S. Nakamura, Solid State Commun. 15, Fiz. 71, 197 (2000) [JETP Lett. 71, 134 (2000)]. 1129 (1974). 11. A. M. Kodomtseva, R. Z. Levitin, Yu. F. Popov, 27. V. I. Goldanskii, E. F. Markov, and V. V. Karpov, Phys. V. N. Seleznev, and V. V. Uskov, Fiz. Tverd. Tela (Lenin- Lett. 3, 344 (1963). grad) 14, 214 (1972) [Sov. Phys. Solid State 14, 172 28. G. Shirane, D. E. Cox, and S. L. Ruby, Phys. Rev. 125, (1972)]. 1158 (1962). 12. M. P. Petrov, G. A. Smolenskiœ, A. P. Paugurt, S. A. Ki- zhaev, and M. K. Chizhov, Fiz. Tverd. Tela (Leningrad) 14, 109 (1972) [Sov. Phys. Solid State 14, 87 (1972)]. Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

LATTICE DYNAMICS AND PHASE TRANSITIONS

Pretransition State and Structural Transition in a Deformed Crystal E. E. Slyadnikov Institute of Strength Physics and Materials Science, Siberian Division, Russian Academy of Sciences, Akademicheskiœ pr. 8, Tomsk, 634021 Russia e-mail: [email protected] Received June 30, 2003; in ﬁnal form, August 26, 2003

Abstract—It is shown theoretically that a deformed crystal exhibiting structural instability can be treated as a quantum system of pseudospins. The quantum behavior of atoms becomes signiﬁcant when the characteristic distance between sites of the original crystal lattice and the corresponding sites of the ﬁnal phase is less than the amplitude of zero-point oscillations of atoms (0.1 Å) and when the area under the barrier separating the two × Ð10 minima of the double-well potential is less than aV2 = 5 10 eV cm. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION [4] and the γ α γ structural transition in auste- nite steels observed in the region of a stress concentra- Certain phenomena that occur during deformation tor [5]. Another prominent example of systems in of solids cannot be described in terms of the classical which an atom moves in a double-well potential is the microscopic models of crystals [1]. Among these phe- surface of a crystal. When a crystal is subjected to nomena are nonlinear elastic effects, pretransition phe- external fields, its surface undergoes structural phase nomena, and structural transitions that are accompa- transformations. Since the crystal surface exhibits nied by excitation of additional degrees of freedom of structural instability, the surface layer is characterized the atomic lattice under deformation [2]. Therefore, by two different types of atomic configurations, e.g., new microscopic models of crystals are needed to the bcc and fcc structures [6]. describe such phenomena. In this paper, we propose a microscopic model of a deformed crystal in which an These specific features of the behavior of a atomic lattice is treated as a quantum system of pseu- deformed crystal suggest that the one-particle potential dospins. Within this model, it is shown that the one-par- in which an atom moves can be represented as a sum of ticle potential acting on an atom can be approximated one-particle potentials characterizing the original and by a double-well potential and that the atoms of the lat- final phases (i.e., as an asymmetric double-well poten- tice of a deformed crystal obey quantum-mechanical tial). In the absence of external fields, the deeper poten- laws. The pseudospin model of a crystal allows one to tial well corresponds to the original structure and the avoid the problem of strong anharmonicity, which shallower well, to the final phase. The configuration in always arises near a structural transition. which an atom is located in the deeper well corresponds to the ground state, and the configuration with the atom The conclusion that the one-particle potential acting located in the shallower well corresponds to an excited on an atom can be approximated by a double-well (virtual) state, because the transfer of the atoms to the potential can be drawn from the calculated coefficients virtual state can occur only via a structural (coherent) of expansion of the interatomic pair interaction poten- transition. In this paper, we discuss the possible physi- tial in powers of atomic displacements. According to cal cause of the coherent transition and the physical calculations [3], the anharmonic terms in the pair mechanism through which a pretransition state occurs potential increase sharply near structural transitions in a deformed crystal as a superposition of several and become comparable in magnitude to the harmonic structures with the formation of new allowed states in terms. The potential profile changes; more specifically, the space of interstitial sites [2]. an infinitely high potential barrier observed far from the transition point is replaced by a barrier of finite height, which is indicative of the existence of another mini- 2. THE POTENTIAL IN WHICH AN ATOM MOVES mum in the interatomic pair interaction potential in the IN A DEFORMED CRYSTAL pretransition state [3]. In order to find the potential acting on an atom, we Experimental evidence for the formation of a one- introduce the atomic density in the crystal: particle double-well potential acting on an atom is the ρ(), δ(), () existence of a structural pretransition state to which r t = ∑v i rrÐ i t , (1) titanium nickelide transfers with varying temperature i

1096 SLYADNIKOV where vi is the volume of the ith atom. The energy of respectively, are signiﬁcantly smaller than the distance the atomic system characterized by the atomic density between the wells, then the potential 〈U(r)〉 in which an ρ(r, t) can be represented in the form of a functional atom moves can be written in the form series, 〈〉U()r = ÐV[]δ()δra+ /2 + ()raÐ /2 . (8) () ()ρ, (), Et = E0 + ∫V 1 r t r t dr It follows from Eq. (8) that the one-particle potential in (2) which an atom moves in a deformed crystal is a double- () ()ρ, ()ρ, (), … + 1/2 ∫V 2 rr' r t r' t drrd ' + . well potential.

Here, Vk is the k-particle interaction potential between atoms. The one-particle potential acting on an atom at 3. ATOM IN AN ASYMMETRIC time t can be found as DOUBLE-WELL POTENTIAL Let us consider a crystal undergoing a structural u()r, t = δEt()/δρ()r, t . (3) transition from the original phase to a final phase stim- Substituting Eq. (2) into Eq. (3) gives ulated by external influences (temperature variations, external forces). In the absence of external influences, (), (), ()ρ, (), … u r t = V 1 r t ++∫V 2 rr' r' t dr' . (4) the crystal is in the original phase and a lattice atom moves in an asymmetric potential with two local min- According to Eq. (4), the potential profile u(r, t) ima differing in depth. The deeper potential well corre- depends on ρ(r, t). Therefore, the problem should be sponds to the initial structure, and the shallower well solved self-consistently, which can be performed using corresponds to the final phase. We assume that the min- the pseudopotential theory [7], the electron density ima are located on the x axis coinciding with the crys- functional [8], or computer simulation methods [9]. tallographic axis of the crystal along which the struc- In considering the macroscopic properties, we tural transition occurs. In this case, the potential Ua(x) should perform averaging over microscopic fluctua- in which an atom moves can be written as ρ tions in atomic density (r, t), i.e., over time t. Accord- () δ()δ() ing to the ergodic hypothesis [10], we can average over Ua x = ÐV 1b xa+ /2 Ð V 2b xaÐ /2 . (9) an ensemble of effective (smoothened in time) potential Here, δ(x) is a Dirac function; a is the distance between profiles {U(r)} rather than average the single potential the minima of the potential; V1 and V2 are the depths of profile u(r, t). In this case, the average potential profile the left-hand (deeper) and the right-hand (shallower) takes the form of a functional integral, well, respectively (V1 > V2); and b is the width of these 〈〉U()r = U()r PU{}()r DU()r , (5) local wells. We assume that the distance between the ∫ local wells is much larger than their width (a ӷ b). It is where P{U(r)} is the probability that the specific pro- convenient to separate the asymmetric potential Ua(x) file U(r) belonging to the ensemble {U(r)} will occur into a symmetric part Us(x) with local wells of equal ∆ in the crystal. Let us assume that, when the deformed depth and a correction Ua(x) taking into account the crystal undergoes a structural transition, the potential difference in depth (asymmetry) between the local profiles corresponding to the original and final phases, wells: UM(r) and UA(r), respectively, are the most probable. () () ∆ () Ua x = Us x + Ua x , For the sake of simplicity, we also assume that, in the (10) () δ()δ() vicinity of the structural transition, the probability Us x = ÐV 2b xa+ /2 Ð V 2b xaÐ /2 , P{U(r)} is the sum of δ functions ∆U ()x = ()V Ð V bδ()xa+ /2 . (11) {}() ()δ[]() () a 2 1 PUr = 1/2 U r Ð U A r (6) First, we consider the motion of a lattice atom in the ()δ[]() () + 1/2 U r Ð UM r . symmetric potential and then take into account the Substituting Eq. (6) into Eq. (5) gives asymmetry of the potential by using perturbation theory under the assumption of smallness of the asymmet- 〈〉() ()() ()() Ӷ U r = 1/2 U A r + 1/2 UM r . (7) ric correction, (V1 Ð V2)/V1 1. The motion of an atom in the potential well U (x) is described by the Since the averaged one-particle potential profile is a s periodic function of the space coordinates, it will suf- Schrödinger equation [11] fice to consider this profile for the unit cell centered at []ΨÐ()∂ប/2m 2/∂x2 + U ()x ()x = Ð εΨ()x , (12) the origin of coordinates. The distance between the s potential wells corresponding to the site of the original where Ψ(x) is the wave function of the atom and Ðε is structure and to the site of the finite phase in the unit its energy (ε > 0). An atom moving in the symmetric ≈ Ð9 cell is equal to a 10 cm [5]. If the widths of the local potential Us(x) has two equilibrium positions. If the wells of the original structure, UM(r Ð a/2) and the final tunneling effect can be ignored, the ground state of the ε phase, UA(r + a/2), centered at r = Ða/2 and r = a/2, atom in each local well is doubly degenerate; i.e., + =

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 PRETRANSITION STATE AND STRUCTURAL TRANSITION 1097

ε Ψ Ψ Ð9 × Ð for the even and odd wave functions, +(x) and Ð a = 10 cm, the area under the barrier is V2a = 4 (x), respectively. Allowance for the quantum tunneling 10−10 eV cm, quantum tunneling of an atom is signifi- of the atom through the potential barrier between the cant (i.e., the quantum properties of atoms of the crystal ε ε local wells lifts the degeneracy ( + > Ð). To find the lattice should be taken into account), and the asymme- ε ε Ψ energy eigenvalues + and Ð and the eigenfunctions + try of the potential becomes weaker. Ψ and Ð from Eq. (12), we represent the wave functions κ Ӷ Ψ Ψ In the limit of intensive tunneling, ±a 1, it fol- + and Ð in the form (for x < Ða/2) lows from Eqs. (14) and (15) that the level splitting បω Ψ± = A±{}exp[]κ±()xaÐ /2 ± exp[]κ±()xa+ /2 , is large and the tunneling effect is pronounced. For V = 1.6 × 10Ð1 eV, b = 10Ð10 cm, a = 4 × 10Ð10 cm, m = 2 (13) 2 K = ()ប /m /()V ba . Ð22 Ð4 q 2 10 g, and V1 Ð V2 = 10 eV, Eqs. (14) and (15) give Ð1 Ð1 κ × 9 Ð1 κ × 8 Ð1 Here, κ± are the characteristic localization lengths for Kq = 1.03, + = 3.6 10 cm , Ð = 1.5 10 cm , បω × Ð2 ប∆ × Ð6 the even (+) and odd (Ð) wave functions and A± are the = 4 10 eV, and = 0.05(V2 Ð V1) = Ð5 10 eV. normalization constants. Wave functions (13) must sat- Thus, for the distance a = 4 × 10Ð10 cm, the area under × Ð11 isfy the corresponding boundary and normalization the barrier is V2a = 6.4 10 eV cm, quantum tunnel- conditions, from which the quantities κ± and A± can be ing plays a decisive role in the motion of an atom, the determined. atom is highly delocalized in the double-well potential, Solving the Schrödinger equation (12) with the use and the asymmetry of the potential is very weak. of Eq. (13), we find ()κប2 []± ()κ /m ± = V 2b 1 exp Ð ±a , 4. PSEUDOSPIN FORMALISM AND THE HAMILTONIAN OF A SYSTEM 2 ()2A± /κ± []11± ()κ+ κ±a exp()Ð ±a = 1, (14) OF PSEUDOSPINS 2 The estimates made above show that quantum tun- ε± = ()κប /2m ±. neling of an atom between the related sites of the orig- We calculate the correction ប∆ to the energy eigen- inal and final phases is of importance in a structurally values associated with the asymmetry of the potential unstable crystal. In other words, in addition to small- ∆ Ua(x) to first order in perturbation theory using the amplitude thermal vibrations, an atom in the deeper even and odd wave functions (13). The result is (left-hand) well of the double-well potential undergoes ប∆Ψ〈|∆|〉Ψ quantum zero-point oscillations (tunneling) of a certain = Ð Ua + amplitude along a certain direction, which corresponds ()[ ()κ = V 2 Ð V 1 bA+ AÐ 1 + exp Ð +a (15) to additional discrete degrees of freedom. Therefore, the wave function of the atom depends not only on the ()κ ()κ κ ] Ð expÐ Ða Ð exp Ð +a Ð Ða . continuous space coordinate x but also on a discrete variable related to the projection of the pseudospin onto In the “weak”-tunneling limit κ±a ӷ 1, it follows from Eqs. (14) and (15) that the splitting of the energy the z axis. For the two-level system at hand, the wave Ψ z levels of the even and odd states បω = ε Ð ε tends to function of an atom has the form of a spinor (x, S ) + Ð consisting of two components, namely, the even zero and the tunneling effect is practically absent. Let Ψ Ψ Ψ Ψ us estimate the quantities in Eqs. (14) and (15) for a (x, +1/2) = +(x) and odd (x, Ð1/2) = Ð(x) coordi- Ð10 nate functions corresponding to different values of the typical transition metal. Putting V2 = 4 eV, b = 10 cm, Ð8 Ð22 Ð4 z component of the pseudospin. a = 10 cm, m = 10 g, and V1 Ð V2 = 10 eV, we In terms of the model considered above, the energy Ð1 × 2 κ ≅ κ × 10 Ð1 obtain Kq = 6.4 10 , + Ð = 6.4 10 cm , of the system of pseudospins subjected to a mechanical បω ប∆ × Ð4 0, and = 1.6(V2 Ð V1) = Ð1.6 10 eV. Thus, force field is described by the Hamiltonian for the distance a = 10Ð8 cm, the area under the barrier separating the minima of the double-well potential is 1 int × Ð8 HH= ∑ l + ∑Hl . (16) V2a = 4 10 eV cm, quantum tunneling is practically absent, and the potential is highly asymmetric. l l In the intermediate case of moderate tunneling, In the basis formed by the even and odd wave functions κ±a ≈ 1, the level splitting បω is nonzero, but the tun- 1 Ψ±(x), the Hamiltonian H has the form × Ð1 Ð10 l neling effect is weak. For V2 = 4 10 eV, b = 10 cm, Ð9 Ð22 Ð4 1 z x a = 10 cm, m = 10 g, and V1 Ð V2 = 10 eV, បω ប∆ Hl = Sl + Sl , Ð1 κ × 9 Ð1 Eqs. (14) and (15) give Kq = 6.4, + = 6.40 10 cm , (17) κ × 9 Ð1 បω × Ð4 ប∆ ∆ () x () x x − = 6.41 10 cm , = 8.8 10 eV, and = = Ð 1/2 ∑ JlkSk Ð 1/3 ∑IlkmSk Sm , × Ð5 0.16(V2 Ð V1) = Ð1.6 10 eV. Thus, for the distance k km,

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1098 SLYADNIKOV where the asymmetry of the double-well potential is  associated with two- and three-particle interactions ()បω បΩ 2  បΩ ប 〈〉z ប ប hi ==hi 0 + t + Ð a + ∑ Jij S j between pseudospins and Jlk and Ilkm are the two- and  j three-particle interaction constants of pseudospins, (23) respectively.  2 1/2 + ∑បI 〈〉S z 〈〉S z  . Assuming that the external mechanical force stimu- ijm j m  lates quantum tunneling and decreases the asymmetry jm, of the double-well potential, we describe the interaction of the pseudospins with the mechanical force field 5. STATIONARY STATES Ω a Ω t Wl(t) = (l (t), 0, l (t)) by the Hamiltonian OF THE PSEUDOSPIN SYSTEM Let us consider the case where there is no external int បΩ a() x បΩ t() z Ω Ω Hl = l t Sl + l t Sl . (18) mechanical force, t = a = 0. We assume that the structural transition from the initial to the final phase is Here, Ω t (t) = Gtµ xy and Ω a (t) = Gaµ yx are the z and associated with the “longitudinal” interaction of pseu- l l l l dospins oriented along the z axis and is characterized by x components of the mechanical force field, respec- z z yx the order parameter S = 〈〉S . Expanding the self-con- tively; Gt and Ga are positive constants; and µ is the i l sistency equation (22) in powers of the order parameter shear component of the stress tensor stimulating the z structural transition. S near the structural transition temperature TMA, we obtain In the basis of the wave functions ϕ and ϕ local- L R z z 2 z 3 ized in the left- and right-hand wells, respectively, the αS ++δ()S γ ()S = 0, (24) Hamiltonian of the pseudospin system in the mechani- α ប []()βω ()បω α [] cal force field has the form = J0 1 Ð J0/2 0 tanh 0/2 = 0 TTÐ c , δδ 1/2 () = 0 T MA Ð T sgn T MA Ð T , x z z បω ()ប 3 3 H = ∑ Sl Ð 1/2 ∑ JlkSl Sk γ ប ()βω [ ()បω = J0 J0 /4 0 tanh 0/2 (25) l lk, ()ββបω Ð2 ()បω ] Ð 0/2 cosh 0/2 , () ប z z z Ð 1/3 ∑ IlkmSl Sk Sm (19) ()បω []()ω Ð1 kBT c = 0/2 arctanh 2 0/J0 , lkm,, α where J0 = ∑ Jij , I0 = ∑ , Iijm , 0 = + ∑()បΩ a()t S z + បΩ t()t S x . i im l l l l ប2 2 2 Ð2 បω δ ប Ð1/2 l ()/2J0 /4kBT c cosh ( 0 kBTc), and 0 = 2 I0 T MA . The coefficients α and δ can change sign, and γ is pos- In the molecular-field approximation [11], the itive. The corresponding Landau expansion of the free Hamiltonian of the pseudospin system (19) is replaced energy near the temperature TMA has the form by the effective Hamiltonian 2 4 ∆F = α()Sz /2 ++δ()Sz /3 γ ()Sz /4. (26) M []∂ 〈〉∂ 〈〉 The solution to the self-consistency equation (24) is H ==Ð∑hiSi ∑ H / Si Si, (20) i i Sz = []Ðδδ± 2 Ð 4αγ /2γ for δ2 ≥ 4αγ , (27)

 z 2  បω បΩ ,,បΩ S = 0 for δ ≥ 4αγ . (28) hi = Ð 0 Ð0t Ð a ∫  It follows from Eq. (27) that there is a temperature (21) range (T+, TÐ) over which the crystal lattice can be in a  z z z z pretransition state S = 0. The temperature at which the + ∑បJ 〈〉S + ∑បI 〈〉S 〈〉S  . ij j ijm j m  transition from the original phase to the pretransition j jm, + α γ δ 2 α γ Ð1∆ state occurs is T = TMA + 4 0 [0 Ð 4 0 ] T, and the In this case, the average value of the pseudopotential temperature at which the crystal undergoes the transi- in the ith unit cell is deﬁned as tion from the pretransition state to the ﬁnal phase is T − = T Ð 4α γ[δ 2 + 4α γ]Ð1∆T, where ∆T = T Ð T > 0. 〈〉S = Tr[]S exp()ÐβH M /Tr[]exp()ÐβH M MA 0 0 0 MA c i i i i (22) Let us consider the case of strong interaction ∂()∂β()()()() β ប = lnZi / hi = 1/2 hi/hi tanh hi/2 , between pseudospins ( J0 = 0.4 eV) and weak tunnel-

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 PRETRANSITION STATE AND STRUCTURAL TRANSITION 1099

បω ≈ α z ing, 0 0.0001 eV. From the condition = 0, the The order parameter S is determined from the thermo- temperature of the transition to the pretransition state dynamic equilibrium condition ≈ ≈ can be estimated to be kBTc 0.1 eV and Tc 1000 K. ω Ӷ ӷ ∂∆Φ ∂ z α z δ()z 2 γ ()z 3 បΩ Therefore, we have 2 0/J0 1 and Tc TMA. In the / S ==S +++S S a 0.(30) region T > T , Eq. (24) has a nonzero (positive) solu- MA It can be seen from Eq. (30) that the solution Sz = 0 tion corresponding to the thermodynamically stable exists if បΩ = 0. This condition is satisfied when there original phase. Below the temperature T , Eq. (24) has a MA is no external force, σ = 0, and the stress field in the another nonzero (negative) thermodynamically stable crystal is equal to zero. Another condition under which solution. Therefore, in the case of strong interaction z បΩ ≈ z between pseudospins and weak tunneling, the crystal the solution S = 0 exists is the relation a S . This relation takes place in the vicinity of the critical value bypasses the pretransition state and directly undergoes σ σ Ω the transition from the initial to the final phase. of the external force = c, where the component a of the mechanical force field is of a relaxation character Now, we consider the case of a moderate strength of λ λ σ σ ប due to softening of the elastic constant = 0( Ð c). interaction between pseudospins ( J0 = 0.04 eV) and of បω ≈ In this case, the thermodynamic equilibrium equation moderate tunneling ( 0 0.001 eV). From the condi- (30) should be supplemented by the equilibrium equation α = 0, the temperature of the transition to the pre- ≈ ≈ tion transition state is found to be kBTc 0.01 eV and Tc ω z 100 K. Therefore, 2 0/J0 < 1 and Tc < TMA. Above the ∂∆Φ/∂Ω ==បS Ð0.λ ()Ωσ Ð σ (31) ӷ a 0 c a transition temperature, in the region Tc TMA, Eq. (24) Ω has a single nonzero (positive) thermodynamically sta- This equation relates the component a of the mechan- ble solution, which corresponds to the initial phase. ical force field to the order parameter, and the corresponding self-consistency equation takes the form When the temperature TMA is approached in the region T ≥ T > T , there is a unique thermodynamically sta- z z 2 z 3 MA c α˜ S ++δ˜ ()S γ˜ ()S = 0, (32) ble solution to Eq. (24), Sz = 0, which corresponds to Ð2 Ð2 the pretransition state of the crystal. Below Tc (in the α αλ ()σ σ ប δ˜ δλ ()σ σ ប ˜ ==0 c Ð + 1, 0 c Ð , region T < Tc), Eq. (24) has a nonzero (negative) ther- (33) γ γλ ()σ σ បÐ2 modynamically stable solution. Therefore, in the case ˜ = 0 c Ð . of a moderate strength of interaction between pseudospins and of a moderate tunneling intensity, the trans- Self-consistency equation (32) has a solution formation of the crystal from the initial to the final 2 2 phase goes through the pretransition state. Sz = []Ðδ˜ ± δ˜ Ð 4α˜ γ˜ /2γ˜ for δ˜ ≥ 4α˜ γ˜ , (34) Finally, let us consider the case of a moderate ប z δ2 ≤ αγ strength of interaction between pseudospins ( J0 = S = 0 for 4 . (35) បω ≈ 0.04 eV) and of intensive tunneling ( 0 0.04 eV). It can be seen from Eq. (34) that there is a range (σÐ, σ+) From the self-consistency equation (24), it follows that for the external force over which the crystal lattice is in the structural transition from the pretransition state to the pretransition state. The value of the external force the initial (or final) phase can occur if the “transverse” ω causing the transition from the original phase to the pre- tunneling field 2 0 is weaker than the “longitudinal” σÐ σ γ δ2λ Ð1 transition state is = c Ð 4 [ 0] , and the value of interaction field of pseudospins J0. Indeed, the temper- the external force causing the transition from the pre- ature Tc is found by linearizing Eq. (24). The linearized σ+ σ ω ≤ transition state to the final phase is = c + equation has a real solution for Tc only if 2 0/J0 1. ω ≥ (4µប6 λ Ð5 δÐ2)1/5. For the sake of definiteness, we Therefore, in the case of 2 0/J0 1, there is no solution 0 z assumed that the coefficient δ is negative in the original for Tc and S = 0 is a unique thermodynamically stable σ σ z solution to Eq. (24) at any temperature. Thus, in the phase; therefore, when < c, we have S > 0 and the σ σ case of moderate interaction between pseudospins and crystal is in the original phase, whereas for > c we intensive tunneling, the crystal can be only in the pre- have Sz < 0 and the crystal is in the final phase. transition state. Thus, when the external force is within the range Let the crystal be subjected to a mechanical force (σ−, σ+), the thermodynamically stable solution to self- Ω z field with a nonzero component a decreasing the consistency equation (32) is S = 0 and, hence, the crys- asymmetry of the double-well potential. In this case, tal is in the pretransition state. the Landau expansion of the thermodynamic potential near the structural transition point has the form [12] 6. DISCUSSION OF THE RESULTS z 2 z 3 ∆Φ= α()S /2 + δ()S /3 Thus, the proposed model describes a deformed (29) crystal as a quantum system of pseudospins. The quan- γ ()z 4 បΩ z ()λΩ2 + S /4 + aS Ð 1/2 a . tum behavior of lattice atoms becomes significant when

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1100 SLYADNIKOV the characteristic distance between the sites of the orig- the pretransition state is associated with the fact that the inal structure and the corresponding sites of the final three-particle interatomic interaction responsible for phase is less than the zero-point oscillation amplitude the formation of the original and final phases weakens of atoms (0.1 Å) and when the area under the barrier significantly near the temperature TMA of the transition separating the minima of the double-well potential is from the initial to the final phase, with the consequence × Ð10 less than aV2 = 5 10 eV cm. that the two-particle interatomic interaction becomes Near the structural transition from the initial to the important and takes part in the formation of the pretran- final phase stimulated by external influences (tempera- sition state. ture variations, external forces), the area under the bar- It is reasonable to suggest that quantum tunneling of rier between the minima of the double-well potential atoms is the physical basis of the mechanism for the decreases under an external influence. As a result, the coherent structural transformations occurring in a crys- quantum-tunneling effects become significant, the tal under loading [4, 5]. Since the wave function of an asymmetry of the double-well potential decreases, and atom is significantly delocalized because of quantum the initial state of the lattice with the asymmetric dou- tunneling, the pretransition state can be considered a ble-well potential becomes unstable with respect to the superposition of the initial and final structural states of occurrence of the pretransition state of the lattice with the crystal [2]. a symmetric double-well potential. The pretransition state is taken to mean a condensed state of the crystal in REFERENCES which a lattice atom is completely delocalized in the symmetric double-well potential because of quantum 1. V. E. Panin, Fiz. Mezomekh. 1 (1), 5 (1998). tunneling; i.e., the atom occupies the site of the initial 2. V. E. Egorushkin, E. V. Savushkin, V. E. Panin, and phase and the corresponding site of the final phase with Yu. A. Khon, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 1, 9 equal probability. (1987). 3. É. V. Kozlov, L. L. Meœsner, A. A. Klopotov, and Thus, on the one hand, a crystal exhibiting structural A. S. Taœlashev, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 5, instability transforms into the pretransition state as the 118 (1985). temperature is varied near the structural transition tem- 4. V. G. Pushin, V. V. Kondrat’ev, and V. N. Khachin, Izv. perature TMA; on the other hand, the deformed crystal Vyssh. Uchebn. Zaved., Fiz., No. 5, 5 (1985). transforms into the pretransition state as an external 5. A. N. Tyumentsev, I. Yu. Litovchenko, Yu. P. Pinzhin, σ force varies in the vicinity of the critical value c. A. D. Korotaev, and N. S. Surikova, Fiz. Met. Metall- Let us compare the quantum-tunneling frequency oved. 95 (2), 86 (2003). and the frequency of thermally assisted hopping of an 6. V. E. Panin, Fiz. Mezomekh. 2 (6), 5 (1999). electron in a double-well potential in the case where the 7. V. Heine, M. L. Cohen, and D. Weaire, Solid State Phys- ≈ ics, Ed. by H. Ehrenreich, F. Seitz, and D. Turnbull (Aca- pretransition state occurs at a temperature Tn 300 K. ≈ demic, New York, 1970), Vol. 24 [Theory of Pseudopo- The average thermal energy of an atom is kBTn tential (Mir, Moscow, 1973)]. 0.02 eV, and the thermally assisted hopping frequency 8. E. H. Lieb, Int. J. Quantum Chem. 34, 243 (1983). ω ប ≈ 13 Ð1 is n = kBTn/ 10 s . In the case where the barrier 9. Monte Carlo Methods in Statistical Physics, Ed. by height between the minima of the double-well potential K. Binder (Springer, Berlin, 1979; Mir, Moscow, 1982). is V2 = kBTn = 0.02 eV and the distance between a site 10. P. Resibois and M. De Leener, Classical Kinetic Theory of the original lattice and the corresponding site of the of Fluids (Wiley, New York, 1977; Mir, Moscow, 1980). final lattice is a = 10Ð9 cm, the quantum-tunneling fre- 11. L. D. Landau and E. M. Lifshitz, Course of Theoretical ω ≈ 14 Ð1 quency of an atom is 0 10 s . Therefore, in a crys- Physics, Vol. 3: Quantum Mechanics: Non-Relativistic tal exhibiting structural instability at a temperature Theory, 4th ed. (Nauka, Moscow, 1989; Pergamon, New much lower than room temperature, the quantum-tun- York, 1977). neling mechanism is more efficient than the thermally 12. L. D. Landau and E. M. Lifshitz, Course of Theoretical assisted hopping mechanism. Furthermore, if V ӷ Physics, Vol. 5: Statistical Physics, 3rd ed. (Nauka, Mos- 2 cow, 1976; Pergamon, Oxford, 1980), Part 1. 0.02 eV at room temperature, the thermally assisted hopping mechanism is also of minor importance in determining the structural transition. The occurrence of Translated by Yu. Epifanov

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

Physics of the Solid State, Vol. 46, No. 6, 2004, pp. 1101Ð1106. Translated from Fizika Tverdogo Tela, Vol. 46, No. 6, 2004, pp. 1071Ð1075. Original Russian Text Copyright © 2004 by Poklonski, Vyrko, Zabrodskiœ. LATTICE DYNAMICS AND PHASE TRANSITIONS

Electrostatic Models of InsulatorÐMetal and MetalÐInsulator Concentration Phase Transitions in Ge and Si Crystals Doped by Hydrogen-Like Impurities N. A. Poklonski*, S. A. Vyrko*, and A. G. Zabrodskiœ** *Belarussian State University, pr. F. Skoriny 4, Minsk, 220050 Belarus e-mail: [email protected] **Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia Received October 29, 2003

Abstract—Two electrostatic models have been developed that allow calculation of the critical concentration of hydrogen-like impurities in three-dimensional crystalline semiconductors corresponding to the insulatorÐmetal and metalÐinsulator transition in the zero temperature limit. The insulatorÐmetal transition manifests itself as a divergence of the static permittivity observed in lightly compensated semiconductors as the concentration of polarizable impurities increases to the critical level. The metalÐinsulator transition is signaled by the divergence of the dc electrical resistivity in heavily doped semiconductors as the compensation of the majority impurity increases (or its concentration decreases). The critical impurity concentration corresponds to the coincidence of the percolation level for the majority carriers with the Fermi level. The results of the calculations made with these models ﬁt the experimental data obtained for n- and p-type silicon and germanium within a broad range of their doping levels and impurity compensation. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION pensation ratios (K 1), in terms of percolation theory [7], according to which the transition can be consid- The critical concentration Nc of the majority dopant for the insulatorÐmetal (IÐM) and metalÐinsulator (MÐ ered a consequence of the Fermi level EF becoming I) phase transitions is derived in the zero temperature equalized with the percolation level Eµ in the potential limit (T 0) from low-frequency measurements of relief of large-scale (non-tunneling) fluctuations. For the macroscopic permittivity and electrical resistivity, small compensation ratios (the value K = 0 corresponds respectively. It is believed that the critical concentra- to the Mott transition), the fluctuations are not large tions of the transition determined by these two methods scale, so the critical concentration nc can be described coincide [1]. A relevant review and a comparison of the 1/3 ≈ microscopic models describing the difference between by Mott’s relation [8Ð12] nc aB 0.25, where aB = an insulator and a conductor under dc bias can be found 4πεប2/me2 is the Bohr radius for the electron (hole) in [2]. Performing calculations and making predictions with an effective mass m in a lattice with static permit- ε ε ε ε from these models meets, however, with difficulties. tivity = r 0 ( 0 is the free-space permittivity). The On the insulator side of the IÐM transition, increas- case of intermediate compensations (K ≈ 0.5) has been ing the dopant concentration N up to the critical level Nc explored to a lesser extent, thus making description of brings about a divergence of the relative permittivity the dependence of the critical carrier concentration n ε c r(N) of a crystalline sample [3Ð5]. Quantitative on the compensation of the majority impurity K at the ε description of the behavior of r(N) in the vicinity of the MÐI transition an important problem. IÐM transition is, however, lacking (see, e.g., [5]). On the metallic side, the MÐI transition manifests The present communication reports on an analytical itself experimentally for T 0 as a divergence of the description of the IÐM and MÐI transitions in crystal- dc electrical resistivity as the concentration of the line semiconductors doped by hydrogen-like impuri- dopant decreases or as the degree of its compensation ties. The variation of the macroscopic static permittiv- increases. On the metallic side of the transition, the ity at the IÐM transition is treated in terms of the Clau- majority carrier concentration is nc = (1 Ð K)Nc, where siusÐMossottiÐLorentzÐLorenz (CMLL) model. The K is the semiconductor compensation. The critical car- MÐI transition is considered with due account of spatial rier concentration nc for the MÐI Anderson transition fluctuations of electrostatic energy in the crystal. Only driven by the introduction of compensating impurities pure Coulomb interaction among the nearest neighbor into a heavily doped semiconductor has been experi- charges (the majority carriers and impurity ions) is mentally shown [6] to be well described, for high com- included.

1102 POKLONSKI et al. 2. THE INSULATORÐMETAL TRANSITION 0), Eqs. (1) and (2) yield the relative permittivity of the crystal lattice (matrix): For the sake of definiteness, we consider an n-type crystalline semiconductor with a concentration of α N ε ()N 0 ==1 + ------m m - ε . (3) donors N = N0 + N+1 in charge states (0) and (+1) and r 0 1 Ð α N /β r with acceptors in the charge state (Ð1) with concentra- m m m tion NÐ1 = KN, where K is the donor compensation ratio. Using Eqs. (1)Ð(3), we obtain the macroscopic relative The charge neutrality condition can be written as N+1 = permittivity of the donor-doped semiconductor: KN = NÐ1. On the insulator side, at a temperature T 0, 3()βε Ð 1 + 3()αε Ð 1 + β N the criterion for the onset of the IÐM transition is diver- ε ()N = 1 + ------r m r m 0 0; (4) ε r 0 3β Ð ()αε Ð 1 + β N gence of the relative permittivity r(N) of a crystalline m r m 0 0 sample with the hydrogen-like donor concentration N in the absence of neutral donors (N0 0), we come increasing to the critical value Nc. ε ε to r(N0 0) = r. The permittivity of the material governs the connec- ε ∞ The criterion r(N0) for the onset of the IÐM tion between the local electric field El acting on each transition for N Nc or N0 N0c = (1 Ð K)Nc can (native or impurity) polarizable atom and the average be used to derive from Eq. (4) the following relation: macroscopic field Eav [13Ð15]. This connection is β ε β α β 3 m Ð ( r Ð 1 + m)(1 Ð K) 0Nc = 0. For m = 3, we obtain approximated by the CMLL relation. It is usually the critical donor concentration assumed that the CMLL relation is applicable only to systems made up of single point dipoles. However, in 9 reviews [13, 14], it was shown convincingly that the Nc = ()ε------()α-, 1 Ð K r + 2 0 CMLL relation is also valid for insulators and semicon- (5) ductors of a fairly general class, including those con- 1/3 0.542 Nc aH = ------, sisting of atoms or molecules with overlapping electron []()ε1 Ð K ()+ 2 1/3 shells. r α π 3 In terms of our model [11], with allowance for the where 0 = 18 aH is the polarizability of the dopant in superposition of the electric fields of the crystal matrix charge state (0) and a = e2/8πε ε I is the electron atoms with a concentration N and polarizability α of H r 0 d(a) m m (hole) orbit radius in a single hydrogen-like donor a matrix atom and the electric fields of electrically neu- (acceptor) localized in a crystal lattice with relative per- tral donors with a concentration N0 = (1 Ð K)N and ε α mittivity r. polarizability 0 of a donor atom, we can write (cf. [16, 17]) The Nc(aH) relation calculated from Eq. (5) for K 0 is plotted in Fig. 1 by solid lines [(1) for Si α α ε ε m Nm 0 N0 with r = 11.5 and (2) for Ge with r = 15.4]. Note that El = Eav + ------β + ------El, (1) the critical concentration N calculated with models m 3 c [16, 17] for ε (N ) was found to be ε (ε + 2)/3 and ε β r 0 r r r where m is a fitting parameter taking into account the times larger, respectively, than the value obtained using β point symmetry of the lattice ( m = 3 corresponds, in Eq. (5) for the same polarizability α = 18πa 3 per accordance with the CMLL model, to the case of the 0 H matrix atoms making up a cubic lattice or distributed at donor (acceptor). ӷ random) and Nm N0. Equations (4) and (5) yield the dependence of the macroscopic permittivity on the doping level of a crys- The polarizability of a single neutral hydrogen-like talline semiconductor (the ratio of the dopant concen- α π 3 2 πε donor [18] is 0 = 18 aH , where aH = e /8 Id is the tration N on the insulator side to its critical concentra- Bohr orbit radius of the optical electron of a donor with tion Nc): ionization energy Id. For N < Nc, the average radius of ε ε () r + 2N/Nc the spherical crystal region per neutral donor, which is r N = ------, (6) Ð1/3 1 Ð N/Nc 0.62[(1 Ð K)N] , exceeds aH by more than a factor of 2.5 even for K 0 [1, 3, 4]. Therefore, the use of the where N/N = N /N , with N = (1 Ð K)N being the con- ε ε ε c 0 0c 0 crystal lattice permittivity = r 0 in the relation for the centration of the majority dopant in charge state (0). Bohr radius aH is valid. ε The r(N) relation calculated from Eq. (6) for crys- The total macroscopic polarization of crystal matrix talline silicon is plotted in Fig. 2 together with experi- atoms and neutral donors can be written as [15] mental data. The experimental data [3, 5, 19, 20] were ()α α []ε () processed using average values of the critical dopant m Nm + 0 N0 El = r N0 Ð 1 Eav, (2) concentrations for K 0 (see the data in Fig. 1): ε × 18 Ð3 × where r(N0) is the relative permittivity of the doped Nc(Si : As) = 7.8 10 cm , Nc(Si : P) = 3.81 18 Ð3 × 18 Ð3 semiconductor. For a crystal without impurities (N 10 cm , and Nc(Si : B) = 4.1 10 cm .

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 ELECTROSTATIC MODELS 1103

Note that the use of Eqs. (5) and (6) is more appro- –3 logNc [cm ] priate for low compensation ratios of the majority 19.0 dopant (K Ӷ 1). a d 18.8 1 b e 3. THE METALÐINSULATOR TRANSITION c f 18.6 Taking into account spatial fluctuations of the electrostatic potential, the average electron concentration n in the conduction band of a degenerate n-type semicon- 18.4 ductor can be described by the relation [7, 21, 22] ~ 17.6 +∞ E ν()3/2 g m 2m ()1/2 () 17.4 n = ------2 3 - ∫ fEd ∫ EÐ U PUdU h n 2π ប i o Ð∞ Ð∞ 2 (7) 17.2 +∞ ∞ j p ν k q = ----- PU()dUfk2dk, 2 ∫ ∫ 17.0 l r π ~ ~ Ð∞ 0 0 0.1 0.2 0.5 0.6 0.7 where ν is the number of valleys, i.e., the energy min- logaH [nm] ima in the band allowed for the majority carriers (ν = 4 ν ν for n-Ge, = 6 for n-Si, = 1 for p-Ge and p-Si); m is Fig. 1. Critical concentration Nc of hydrogen-like impuri- the effective electron mass in one valley; f = [1 + ties for the onset of the IÐM phase transition in Ge and Si Ð1 plotted vs. Bohr radius a for K ≤ 0.01. Points are experi- exp((E Ð EF)/kBT)] is the FermiÐDirac function; H mental data: (a) n-Si : As [23], (b) n-Si : P [24], (c) n-Si : P P(U) = (W 2π )Ð1exp(ÐU2/2W2) is the Gaussian distri- [25], (d) n-Si : P [26], (e) p-Si : B [27], (f) n-Si : Sb [28], bution of fluctuations of the electrostatic electron (g) n-Ge : As [29], (h) n-Ge : As [30], (i) n-Ge : As [1, 31], (j) n-Ge : As [32], (k) n-Ge : P [29], (l) n-Ge : P [1, 31], potential energy U created by point charges in the crys- (m) p-Ge : Ga [29], (n) p-Ge : Ga [33], (o) p-Ge : Ga [34], tal (impurity ions and conduction electrons); W is the (p) n-Ge : Sb [29], (q) n-Ge : Sb [35], and (r) n-Ge : Sb [36]. 2 rms fluctuation; and the total energy E = (បk) /2m + U Line 1 is Nc(aH) for Si calculated from Eq. (5), and line 2 is of a conduction electron with quasi-momentum បk is the same for Ge. reckoned from the conduction band bottom of the undoped crystal (in the band gap, the electron energy E is negative). ε ε On the metallic side of the MÐI transition, where all r(N)/ r impurities are ionized and n = (1 Ð K)N, Eq. (7) for tem- 20 perature T 0 can be recast to [11, 22]

() a EF kF U ν 2 15 b n = ----- PU()dUkdk c π2 ∫ ∫ ∞ d Ð 0 (8) E ν F 10 []()3 () = ------∫ kF U PUdU, 3π2 Ð∞ 5 បÐ1 1/2 where kF(U) = (2m(EF Ð U)) is the Fermi quasi- wave vector of an electron with kinetic energy EF Ð U > 0 in the conduction band. The charge neutrality condition of the crystal can be written as n + KN = N, where 0 0.5 1.0 N is the concentration of donors and KN is that of 1 – N/Nc acceptors. Based on Eq. (8) and the condition of equality of the ε Fig. 2. Macroscopic relative permittivity r(N) of lightly percolation level (mobility edge) Eµ under a dc bias and compensated silicon plotted vs. majority dopant concentra- the Fermi level EF, the dependence of the critical major- tion N. Points are experimental data: (a) n-Si : As [3], (b) n- ity dopant concentration for the onset of the MÐI tran- Si : P [19], (c) n-Si : P [20], and (d) p-Si : B [5]. The curve ε ε presents r(N) calculated from Eq. (6) for r = 11.5. sition, Nc = nc/(1 Ð K), on the compensation ratio K for

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1104 POKLONSKI et al.

1/3 ν2/3 Nc aB/ critical concentration of the impurity, Nc, on its compensation K in the form a g 0.5 1/3 b h N a 0.1 c B ------c i ------2/3 = 2/3. (10) d j ν ()1 Ð K e 0.4 k The function N (K) calculated from Eq. (10) is plotted f c in Fig. 3 with a solid line. 2 πε 0.3 The quantity W = Ws = [(e /4 )/2 ](1 + 2/3 1/3 K) [Nc/(1 Ð K)] in model [7] originates from the screening of large-scale dopant-ion concentration ﬂuc- 1 0.2 tuations by electrons. Assuming the critical fractional volume of the semiconductor that contains electrons and corresponds to their percolation level to be 0.17, we

0.1 obtain Eµ/Ws = Ð0.6752 . The function Nc(K) calculated within model [7] using Eq. (9) for W = Ws for a compensated semiconductor is shown in Fig. 3 with a 0 0.2 0.4 0.6 0.8 1.0 dashed line. K

Fig. 3. Critical concentration Nc of donors (acceptors) in Ge 4. RESULTS AND DISCUSSION and Si plotted vs. their compensation ratio K. Points are exper- The dependences of the critical concentration N for imental data: (a) p-Ge : Ga [6, 38, 39], (b) p-Ge : Ga [33], c (c) p-Ge : Ga [40], (d) n-Ge : As [6, 29, 41], (e) n-Ge : As the Mott transition at K = 0 on the Bohr radius aH, cal- ε ∞ [30], (f) n-Ge : As [35], (g) n-Ge : Sb [35], (h) n-Ge : Sb culated using the criterion r(N0) [see Eq. (5)], [29], (i) n-Ge : Sb [42], (j) n-Ge : P [6, 29], and (k) n-Si : P are displayed graphically in Fig. 1 for Si and Ge [24]. Solid line is calculated using Eq. (10), and dashed line is calculated within model [7]. Dotted line at K = 0 identifies (curves 1 and 2, respectively). The points in Fig. 1 are the range of experimental data on Nc covered in [1, 23Ð36] experimental data for n- and p-type germanium and silicon [1, 23Ð36]. 1/3 ν2/3 and plotted in Fig. 1 with an average value Nc aB/ = 0.1. The critical impurity concentration Nc on the metallic side of the MÐI transition is given by Eq. (9) and is a function of the ratio of the Fermi level (equal to the this dopant can be presented, according to [11, 22], in percolation level Eµ) to the impurity band width W. The the form (cf. [7]) parameter Eµ/W was found by extrapolating Eq. (9) to the limit K = 0 and by subsequent comparison with 1/2 1/3 W experimental data [1, 23Ð36] available for Nc for the Nc aB = 0.238------case of K ≤ 0.01 (Fig. 1). EB The dependence of the critical concentration Nc of Eµ/W 1/3 (9) 3/2 2 the majority dopant for the MÐI transition on the com- ν Eµ Ðx × ------Ð x exp ------dx , pensation ratio K on the metallic side of the transition, 1 Ð K ∫ W 2 Ð∞ calculated using Eq. (10), is plotted by a solid line in Fig. 3, which also displays the available experimental πε ε ប2 2 2 πε ε data we are aware of on n- and p-Ge [6, 29, 30, 33Ð35, where aB = 4 r 0 /me and EB = e /8 r 0aB are the Bohr radius and energy of an electron (hole), respec- 38Ð42] and n-Si : P [24], with the dashed line showing ε ε the prediction made in [7] for W = Ws. We see that the tively, and r 0 is the static permittivity of the undoped crystal. proposed model fits experimental data well practically throughout the total range of compensations. In the model proposed in [11, 22], the quantity W = A comparison of the experimental data presented in ≈ 2 πε π 1/3 Wnn 1.64(e /4 )(8 Nc/3) originates from the Cou- Figs. 1 and 3 with Eq. (9) for W = Wnn shows that the lomb interaction of the nearest neighbor charges (impu- percolation level for a Gaussian distribution of electro- rity ions, electrons) only. The value Eµ/W = Ð1.15 is static potential fluctuations lies within the interval nn − found by fitting equality (9) to experimental data [1, 1.5W < Eµ < ÐW (here, the minus sign means that the percolation level lies in the band gap). This result cor- 23Ð36] for uncompensated semiconductors [Nc(K 1/3 ν2/3 relates well with estimates made by other researchers 0)] aB/ = 0.1 (Fig. 1). In what follows, we assume that the Eµ/Wnn ratio is Ð1.15 for all compensations (0 < 1 The temperature dependence of the large-scale potential relief K < 1). In this case, Eq. (9) yields the dependence of the W = Ws is treated in [37].

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 ELECTROSTATIC MODELS 1105

[7, 43, 44]. Note that model [7] is valid only for large 6. A. G. Zabrodskiœ, Fiz. Tekh. Poluprovodn. (Leningrad) compensation ratios K. As seen from Fig. 3, for K < 0.5, 14 (8), 1492 (1980) [Sov. Phys. Semicond. 14, 886 (1980)]. the calculation made with model [7] yields values of Nc less than the experimental ﬁgures. 7. B. I. Shklovskiœ and A. L. Éfros, Electronic Properties of Doped Semiconductors (Nauka, Moscow, 1979; Springer, New York, 1984). 5. CONCLUSIONS 8. V. F. Gantmakher, Usp. Fiz. Nauk 172 (11), 1283 (2002) Thus, we have developed models for the IÐM and [Phys. Usp. 45, 1165 (2002)]. MÐI transitions in crystalline covalent semiconductors. 9. A. A. Likal’ter, Usp. Fiz. Nauk 170 (8), 831 (2000) [Phys. Usp. 43, 777 (2000)]. On the insulator side, the transition manifests itself ε 10. N. Mott, Usp. Fiz. Nauk 105 (2), 321 (1971). as a divergence of the relative permittivity r(N0) of lightly compensated semiconductors, which is due to 11. N. A. Poklonskiœ and A. I. Syaglo, Fiz. Tverd. Tela polarization of neutral impurity atoms as their concen- (St. Petersburg) 40 (1), 147 (1998) [Phys. Solid State 40, tration increases from N to N = (1 Ð K)N . The critical 132 (1998)]. 0 0c c 12. A. G. Zabrodskii, Philos. Mag. B 81 (9), 1131 (2001). concentration N0c is expressed through the Bohr radius 2 πε 13. A. P. Vinogradov, Radiotekh. Élektron. (Moscow) 45 (8), aH = e /8 Id(a) for an electron (hole) on a neutral donor 901 (2000). (acceptor) with an ionization energy Id(a) in a crystal ε ε ε ε 14. V. L. Ginzburg and E. G. Maksimov, Sverkhprovodi- with lattice permittivity = r 0. The r(N0) dependence most: Fiz., Khim., Tekh. 5 (9), 1543 (1992). is quantitatively described for Si crystals near the IÐM 15. I. E. Tamm, The Principles of Electricity Theory, 10th transition (N0 N0c). ed. (Nauka, Moscow, 1989). The MÐI transition on the metallic side is observed 16. J. Kastellan and F. Seitz, in Semi-Conducting Materials as a collapse of metallic conduction as the concentra- (Butterworths, London, 1951; Inostrannaya Literatura, tion of the dopant atoms is decreased or their compen- Moscow, 1954). sation ratio is increased. Use is made of the Bohr radius 17. S. Dhar and A. H. Marshak, Solid-State Electron. 28 (8), πεប2 2 aB = 4 /me for the majority carriers, which depends 763 (1985). on the effective mass m of the electron (hole) and the 18. L. D. Landau and E. M. Lifshitz, Course of Theoretical ε ε 2 static permittivity r 0 of the undoped crystal matrix. It Physics, Vol. 3: Quantum Mechanics: Non-Relativistic is assumed that the amplitude of electrostatic potential Theory, 3rd ed. (Nauka, Moscow, 1974; Pergamon, New ﬂuctuations is determined by the interaction of the York, 1977). nearest neighbor charges (impurity ions and majority 19. M. Capizzi, G. A. Thomas, F. DeRosa, R. N. Bhatt, and carriers). The dependence of the critical majority T. M. Rice, Phys. Rev. Lett. 44 (15), 1019 (1980). 20. H. F. Hess, K. DeConde, T. F. Rosenbaum, and dopant concentration Nc for the onset of the MÐI transition on the compensation ratio K for this dopant was G. A. Thomas, Phys. Rev. B 25 (8), 5578 (1982). found and compared with the prediction from model [7] 21. E. O. Kane, Solid-State Electron. 28 (1/2), 3 (1985). and experimental data available for Ge and Si within a 22. N. A. Poklonskiœ and S. A. Vyrko, Zh. Prikl. Spektrosk. broad range of variation of the compensation ratio. 69 (3), 375 (2002) [J. Appl. Spectr. (Kluwer) 69 (3), 434 (2002)]. 23. P. F. Newman and D. F. Holcomb, Phys. Rev. B 28 (2), ACKNOWLEDGMENTS 638 (1983). This study was supported by the Belarussian Foun- 24. U. Thomanschefsky and D. F. Holcomb, Phys. Rev. B 45 dation for Basic Research (project no. F01-199) and the (23), 13356 (1992). Russian Foundation for Basic Research (project no. 01- 25. P. Dai, Y. Zhang, and M. P. Sarachik, Phys. Rev. B 49 02-17813). (19), 14039 (1994). 26. G. A. Thomas, M. Paalanen, and T. F. Rosenbaum, Phys. Rev. B 27 (6), 3897 (1983). REFERENCES 27. P. Dai, S. Bogdanovich, Y. Zhang, and M. P. Sarachik, 1. P. P. Edwards and M. J. Sienko, Phys. Rev. B 17 (6), Phys. Rev. B 52 (16), 12439 (1995). 2575 (1978). 28. A. P. Long, H. V. Myron, and M. Pepper, J. Phys. C 17 2. R. Resta, J. Phys.: Condens. Matter 14 (20), R625 (17), L425 (1984). (2002). 29. A. G. Zabrodskiœ, M. V. Alekseenko, A. G. Andreev, and 3. T. G. Castner, Philos. Mag. B 42 (6), 873 (1980). M. P. Timofeev, in Abstracts of 25th All-Union Meeting 4. T. G. Castner, N. K. Lee, H. S. Tan, L. Moberly, and on Low-Temperature Physics (Leningrad, 1988), Part 3, O. Symko, J. Low Temp. Phys. 38 (3Ð4), 447 (1980). p. 60. 5. M. Lee, J. G. Massey, V. L. Nguyen, and B. I. Shklovskii, 30. R. Rentzsch, M. Müller, Ch. Reich, B. Sandow, Phys. Rev. B 60 (3), 1582 (1999). A. N. Ionov, P. Fozooni, M. J. Lea, V. Ginodman, and I. Shlimak, Phys. Status Solidi B 218 (1), 233 (2000). 2 For Ge and Si crystals, both n and p type, a ≠ a Clearly enough, 31. M. N. Alexander and D. F. Holcomb, Rev. Mod. Phys. 40 ε H B for the hydrogen atom, r = 1, m = m0, and aH = aB. (4), 815 (1968).

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32. A. T. Lonchakov, G. A. Matveev, and I. M. Tsi- 41. A. G. Zabrodskiœ and K. N. Zinov’eva, Zh. Éksp. Teor. dil’kovskiœ, Fiz. Tekh. Poluprovodn. (Leningrad) 22 (8), Fiz. 86 (2), 727 (1984) [Sov. Phys. JETP 59, 425 1396 (1988) [Sov. Phys. Semicond. 22, 884 (1988)]. (1984)]. 33. H. Fritzsche, Philos. Mag. B 42 (6), 835 (1980). 42. F. R. Allen, R. H. Wallis, and C. J. Adkins, in Proceed- 34. K. M. Itoh, Phys. Status Solidi B 218 (1), 211 (2000). ings of 5th International Conference on Amorphous and 35. W. Sasaki and C. Yamanouchi, J. Non-Cryst. Solids 4, Liquid Semiconductors, Garmisch-Partenkirchen, 1973, 183 (1970). Ed. by J. Stuke and W. Brening (Taylor and Francis, Lon- 36. S. B. Field and T. F. Rosenbaum, Phys. Rev. Lett. 55 (5), don, 1974), Vol. 2, p. 895. 522 (1985). 43. J. M. Ziman, Models of Disorder: the Theoretical Phys- 37. V. G. Karpov, Fiz. Tekh. Poluprovodn. (Leningrad) 15 ics of Homogeneously Disordered Systems (Cambridge (2), 217 (1981) [Sov. Phys. Semicond. 15, 127 (1981)]. Univ. Press, London, 1979; Mir, Moscow, 1982). 38. A. G. Zabrodskii and A. G. Andreev, Int. J. Mod. Phys. B 8 (7), 883 (1994). 44. V. L. Bonch-Bruevich, Usp. Fiz. Nauk 140 (4), 583 39. A. G. Zabrodskii, A. G. Andreev, and S. V. Egorov, Phys. (1983) [Sov. Phys. Usp. 26, 664 (1983)]. Status Solidi B 205 (1), 61 (1998). 40. R. Rentzsch, O. Chiatti, M. Müller, and A. N. Ionov, Phys. Status Solidi B 230 (1), 237 (2002). Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

Physics of the Solid State, Vol. 46, No. 6, 2004, pp. 1107Ð1109. Translated from Fizika Tverdogo Tela, Vol. 46, No. 6, 2004, pp. 1076Ð1077. Original Russian Text Copyright © 2004 by Meœlanov, Abramova, Musaev, Gadzhialiev. LOW-DIMENSIONAL SYSTEMS AND SURFACE PHYSICS

Chemisorption on a Quantum-Well Wire R. P. Meœlanov*, B. A. Abramova**, G. M. Musaev**, and M. M. Gadzhialiev*** * Institute of Geothermal Problems, Dagestan Scientiﬁc Center, Russian Academy of Sciences, pr. Kalinina 39a, Makhachkala, 367030 Russia e-mail: [email protected] ** Dagestan State University, ul. M. Gadzhieva 43a, Makhachkala, 367025 Russia *** Institute of Physics, Dagestan Scientiﬁc Center, Russian Academy of Sciences, ul. 26 Bakinskikh Komissarov 94, Makhachkala, 367003 Russia Received September 5, 2003

Abstract—The speciﬁc features of chemisorption on a quantum-well wire are studied. It is shown that the energy of an electron of an adatom chemisorbed on the quantum-well wire changes jumpwise with variations in the wire radius. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION tion functions of electrons of an adatom and a substrate Since particular research interest has been expressed are presented in our previous work [5]. in the properties of quantum wires, it is important to In our case, the electron energy spectrum of a quan- investigate the chemisorption on a quantum-well wire. tum-well wire can be described within the model of an The specific features of chemisorption of an atom are infinitely deep two-dimensional potential well in the governed by the density of states of the electronic sub- following form [11]: system of the substrate. For a quantum-confined film, it ប2α 2 2 has been demonstrated that the energy of the electronic n px Enp = ------+ ------. (1) subsystem of a chemisorbed atom is an oscillating 2mR2 2m function of the film thickness and external magnetic α field [1–5]. This is associated with the fact that the den- Here, R is the radius of the quantum-well wire, n are α sity of states of a quantum-confined metallic film the roots of the Bessel function Jl( kl) = 0 in ascending depends on the film thickness and external quantizing order, m is the electron mass, and px is the electron magnetic field in a stepwise manner. momentum along the quantum-well wire. In particular, In this work, we investigated the case where a quan- for the first four roots, we have the following values: α α α α α α tum-well wire serves as a substrate. 1 = 10 = 2.42, 2 = 11 = 3.835, 3 = 21 = 5.14, and α α 4 = 12 = 5.5. 2. THEORETICAL BACKGROUND The density of states of a quantum-well wire takes the form Theoretical treatment will be performed in the framework of the AndersonÐNewns model [6Ð10] gen- ∞ dp dx Γ eralized to the case of chemisorption on low-dimen- ρω()= ∑ ------x ------np ∫ 2πប ()ω 2 Γ 2 sional systems [1Ð5]. According to the AndersonÐ Ð Enp + np Newns model [6Ð10], the energy of an electron of the n Ð∞ x (2) adatom can be represented by the relationship E = Θω() ε a, s Lx 2m Ð n ε + U〈n〉, where ε is the energy of an electron of an = ()Γ 0 = ------∑------. a a πប ()ωεÐ isolated atom, U is the potential of the intra-atomic n n Coulomb repulsion, and 〈n〉 is the perturbation of the Here, Lx is the linear size of the quantum-well wire and electron density of the adatom due to the interaction Θ(x) is the Heaviside step function. The electron con- with the substrate. For the perturbation of the electron centration can be represented as follows: density of the adatom 〈n〉, we can write the relationship nF ∞ 2m dω < N = ------∑ µεÐ, (3) 〈〉n = ∫ ------g()ω , π2ប 2 n π R n = 1 Ð∞ where n is the number of filled discrete energy levels < F where g is the correlation function of the electron of of the quantum-well wire. Expression (3) relates the the adatom. Basic relationships for equilibrium correla- chemical potential µ, the electron concentration N, and

1108 MEŒLANOV et al.

20 1.0

0.8 15 0 ε 〉

/ 0.6 n µ 〈 10 0.4 0.2 5 0.25 0.75 1.25 1.75 2.25 2.75 R/R0 1.0 1.5 2.0 2.5 3.0 R/R0 Fig. 1. Dependence of the dimensionless chemical potential µ/ε on the dimensionless radius R/R of the quantum-well Fig. 2. Dependence of the perturbed density of electrons of 0 0 〈 〉 wire. the adatom n on the dimensionless radius R/R0 of the quantum-well wire. the radius R of the quantum-well wire: µ = µ(R, N). It follows from expression (3) that the chemical potential where the spectral function of the electron of the quan- is an oscillating function of the wire radius. The wire tum-well wire is given by the expression radius at which the next n + l energy level is filled is Γ ()ω determined from the condition µ(R , N) = ε . No gen- A ()ω = ------ip -. (7) n n + 1 ip ()ω 2 Γ 2 ()ω eral expression describing the radius of a quantum-well Ð Eip + ip wire R has been derived to date. For example, for the n By using the expressions (4)Ð(7) and taking into quantum-wire radii R1 and R2, we have the following account the electron energy spectrum of the quantum- α 2 α 2 1/6 α 2 expressions: R1 = R0(2 Ð 1 ) and R2 = R0[23 Ð well wire, which is described by relationship (1), we obtain the final expression for the perturbed density of α 2 α 2 α 4 α 2α 2 α 2()α 2 α 2 1/6 1 Ð 2 + 23 + 1 2 Ð 3 1 + 2 ] . Here, R0 = electrons of the adatom: 2 Ð1/3 (π N) is the characteristic scale. Figure 1 shows the n µ ρ F dependence of the dimensionless chemical potential on 〈〉n = ------0 ∑ dω the dimensionless radius of the quantum-well wire 2π ∫ µ/ε n = 0 ε 0(R/R0) calculated with the use of expression (3). As n µ/ε can be seen from Fig. 1, the dependence 0(R/R0) ()ωεÐ (8) exhibits an oscillatory behavior. × ------n , nF 2 When calculating the energy of the electron of the []ωε 〈〉2 ρ Θω()ωε ε Ð a Ð Un + 0 ∑ Ð n Ð n adatom Ea, s, we used the expression n = 1 ∞ ρ | |2 ប 1 < where 0 = V Lx 2m / . Figure 2 shows the depen- 〈〉n = ------dωg()ω . (4) 2π ∫ dence of the perturbed density of electrons of the ada- Ð∞ tom on the dimensionless radius of the quantum-well 〈 〉 wire n (R/R0) calculated according to expression (8). The correlation function g< (ω) can be written in the fol- As follows from the calculations, the perturbed density lowing form [5]: of electrons of the adatom depends on the film thickness. Specifically, upon filling of the next quantum < < g()ω ==gR()σω ()ω gA()ω V 2 f ()ω a()ω . (5) level of the quantum-well wire, the perturbed density of a electrons of the adatom 〈n〉 jumpwise increases, which, Here, f(ω) is the FermiÐDirac distribution function. in turn, leads to a jumpwise change in the chemisorp- ω tion energy of the adatom. The calculations were per- The spectral function of the adatom a( ) is determined ρ ε by the relationship formed for the following parameters: 0 = 5, U/ 0 = 7, ε ε ε ប2 2 and a/ 0 = 0.1. Here, 0 = /2mR0 is the characteristic energy. ∑ A ()ω ip Experimental observations of the aforementioned ()ω ip a = ------2, (6) effects will make it possible to determine the specific []ωε 〈〉2 4 ()ω features of the energy characteristics of an adatom and Ð a Ð Un + V ∑ Aip the electronic subsystem of the quantum-well wire. The ip effect of oscillations in the energy of interaction

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 CHEMISORPTION ON A QUANTUM-WELL WIRE 1109 between the adatom and the quantum-well wire is of 5. R. P. Meœlanov, B. A. Abramova, M. M. Gadzhialiev, and interest from the viewpoint of controlled changes in the V. V. Dzhabrailov, Fiz. Tverd. Tela (St. Petersburg) 44 electronic properties of quantum-well wires. The con- (11), 2097 (2002) [Phys. Solid State 44, 2196 (2002)]. dition of observation of oscillations in the interaction 6. P. W. Anderson, Phys. Rev. 124, 41 (1961). ∆ε ӷ ប τ energy in a magnetic ﬁeld has the form n T, / 7. D. M. Newns, Phys. Rev. 178, 3 (1969). (where T is the absolute temperature and τ is the relax- 8. T. Einstein, J. Hertz, and J. Schrieffer, in Theory of ation time). Chemisorption, Ed. by J. R. Smith (Springer, Berlin, 1980; Mir, Moscow, 1983). 9. S. Yu. Davydov, Fiz. Tverd. Tela (St. Petersburg) 42 (7), REFERENCES 1331 (2000) [Phys. Solid State 42, 1370 (2000)]. 1. R. P. Meœlanov, Fiz. Tverd. Tela (Leningrad) 31 (7), 270 10. S. Yu. Davydov, Fiz. Tverd. Tela (St. Petersburg) 41 (9), (1989) [Sov. Phys. Solid State 31, 1254 (1989)]. 1543 (1999) [Phys. Solid State 41, 1413 (1999)]. 2. R. P. Meœlanov, Fiz. Tverd. Tela (Leningrad) 32 (9), 2839 11. V. M. Galitskiœ, B. M. Karnakov, and V. I. Kogan, Prob- (1990) [Sov. Phys. Solid State 32, 1649 (1990)]. lems in Quantum Mechanics (Nauka, Moscow, 1981). 3. R. P. Meœlanov, Poverkhnost, No. 6, 37 (1994). 4. R. P. Meœlanov, Poverkhnost, No. 3, 52 (1999). Translated by I. Volkov

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

Physics of the Solid State, Vol. 46, No. 6, 2004, pp. 1110Ð1112. Translated from Fizika Tverdogo Tela, Vol. 46, No. 6, 2004, pp. 1078Ð1080. Original Russian Text Copyright © 2004 by Kompan, Kuznetsov, Rozanov, Yakubovich.

LOW-DIMENSIONAL SYSTEMS AND SURFACE PHYSICS

Hysteretic Galvano-Mechanical Effect on Charging and Discharging Ionistor Structures M. E. Kompan*, V. P. Kuznetsov**, V. V. Rozanov***, and A. V. Yakubovich**** * Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia ** Girikond Research Institute Open Society, St. Petersburg, 190000 Russia *** Institute of Analytical Instrumentation, Russian Academy of Sciences, Rizhskiœ pr. 26, St. Petersburg, 198103 Russia **** Lyceum Physicotechnical School, Physicotechnical Institute, Russian Academy of Sciences, St. Petersburg, 194021 Russia Received June 30, 2003; in ﬁnal form, September 23, 2003

Abstract—The size of an ionistor capacitor structure is found to change during its charging and discharging. The specific feature of this effect that distinguishes it from other known effects of changing sizes under electric and magnetic fields (the piezoelectric effect and striction) is the complete hysteresis; i.e., the size reached under the exposure remains unchanged after the field is turned off. © 2004 MAIK “Nauka/Interperiodica”.

Submicrometer- and nanometer-scale engineering composite anode size is stabilized by the coal skeleton. and technology dictate, among other things, the need As a result, the charging (discharging) process should for precisely controlled methods and devices for mov- cause an increase (decrease) in the total size. In fact, the ing instruments and objects in the corresponding spatial mechanical action of the ionistor chargingÐdischarging scale. Currently, these functions are almost exclusively is caused by the fact that the electric current flowing implemented using piezoelectric movers. In recent through the superionics is accompanied by a galvanic decades, their design has been perfected due to the mass transfer. After the control current is switched off, development of scanning probe microscopy techniques the newly formed metal interlayer on the cathode (see, e.g., [1]). Alternative techniques for implementing remains unchanged and the structure size remains the controlled submicrometer-scale displacements can same as it was at the instant the current was turned off. present additional technological possibilities. As follows from the model of the process, the change in In this paper, we report on the detection of the effect the structure size is formally determined by the charge of a change in size arising during charging and dis- accumulated by the structure; namely, the change in charging of capacitor structures based on superionic size is proportional to the integral of the control current. conductors (ionistors). The effect has a clear physical It is this feature that qualitatively distinguishes the ion- nature; this effect is observed as low voltages are istor structure as a mover from the known devices based applied to the ionistor structure and significantly differs on other effects. Furthermore, in comparison with other from the commonly used piezoelectric effect by the fol- possible devices based on the galvanic mass transfer, a lowing feature. The size reached under the applied volt- device based on solid materials is more convenient and age remains unchanged when the power supply (control reliable and can operate in a wide temperature range. signal) is removed from the ionistor structures. In practical implementation of the idea, it was nec- An ionistor is a device based on a superionic mate- essary to take into account the fact that the superionic rial, intended for reversible charge accumulation. An material most used in commercial industry, RbAg4I5, is ionistor consists of three layers: a composite anode (an unstable in air under light. For this reason, experiments activated carbonÐsuperionic interpenetrating compos- were performed with ionistor structures encapsulated in ite); a superionic interlayer (RbAg4I5), suppressing the a thin metal case similar to that of a watch battery. The electron current component flowing through the struc- height and diameter of the encapsulated element used ture; and a metal cathode. During charging, as a voltage in the experiment were 3.5 and 23 mm, respectively. of corresponding polarity is applied, silver cations go The experiments were carried out at room temperature. away from the coalÐsuperionic conductor heterointer- The change in the size was preliminarily detected by face and are reduced to metal silver at the negatively interference measurements. Testing the ability of the biased cathode [2]. Material deposited on the cathode ionistor structures to controllably and reversibly pro- forms an additional metal interlayer. The anode, from vide very small changes in size, as well as the possibil- which the cations leave, changes in size only slightly, ity to retain sizes when the power supply is off, was of since, first, the relative change in the cation content in particular interest. For this reason, most of the experi- the superionic material remains minor and, second, the ments were carried out at small control influences. The

HYSTERETIC GALVANO-MECHANICAL EFFECT 1111

(a) Test current source

nm 40000 1000 30000 20000 Cantilever 40000 10000 30000 20000 10000 0 nm Ionistor (b)

Fig. 1. Experimental layout. changes in size were measured using an atomic-force microscope (AFM). The experimental layout is shown in Fig. 1. The Fig. 2. (a) Surface area scanned by an atomic-force micro- SMENA AFM operated in the tapping mode. Since the scope as the control action was applied to the ionistor structure and (b) the actual profile of the same area recorded in atomic-force microscope does not operate immediately the absence of action (demagnified). in the size measurement mode, a chargingÐdischarging current (control influence) was applied during scanning of the top surface of the encapsulated structure. As a result, the microscope recorded a seemingly complex +10 mA (a) surface profile, which was a reflection of the dynamic change in size. Figure 2 (bottom) shows one of such profiles; the top of the figure shows the surface profile recorded in the absence of control influence. The observed longitudinal depressions in the top profile are 0 methodical distortions caused by the preceding cantilever passage above sizable local hills. Figure 3 (top) shows a control current series; the bottom of the figure shows the surface section (similar –10 mA to that shown in Fig. 2) measured during this current action. It is clearly seen that the ionistor structure size (b) is indeed a function of the integral of the control current. It is also seen from Figs. 2 and 3 that the structure size remains unchanged in the absence of a control signal (e.g., portions A, B). Comparison of Figs. 2 and 3 demonstrates that a certain slope observed in these 1000 regions is caused by the initial surface profile. , nm

The samples under study were characterized by a Z size/charge conversion factor of 200Ð100 nm/C. We B note that the charge/size conversion factor was appre- 500 ciably higher for small control currents. The measurement accuracy was limited by the SMENA model char- A acteristics and surface roughness. In the first experiments, the capsule material roughness was found not to allow realization of the potential of the AFM. There- 0 10000 20000 30000 40000 fore, further experiments were conducted over the sur- Y, nm face of an additional glass coating deposited onto the Fig. 3. (a) Test current series used to obtain the relief shown capsule. We can see from Fig. 2 that surface defects are in Fig. 2 and (b) a section of the surface profile shown in nevertheless observed; however, sufficiently accurate Fig. 2; A and B are the portions corresponding to the measurements of changes in size become possible. absence of control current.

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

1112 KOMPAN et al. Although the nature of the effect is in general clear, in the ionistor structure size, one should consider the the numerical estimate of the expected effect and the contribution from the decrease in the anode size and the experimentally determined value differ. The chemical inﬂuence of the shell. Details of the effect are of interest type of the released material is beyond question; this for further investigation. material can only be silver, whose ions are charge car- The result of this study is the detection of a change riers in RbAg4I5. Let us estimate the thickness d of the in the ionistor structure size during recharging. A char- newly formed interlayer under the assumption that this acteristic difference is the strong hysteretic property of layer is formed by ordinary metal silver. Elementary the effect, i.e., retention of the structure size when the formulas of electrolysis yield control voltage is turned off. This hysteretic galvano- d = µIt/ρN eS, (1) mechanical effect offers new functionalities for devel- a oping nanometer mechanical devices and systems. where µ is the atomic weight of silver, It is the charge accumulated in the ionistor, ρ is the density of metal sil- ACKNOWLEDGMENTS ver, Na is Avogadro’s number, e is the elementary charge, and S is the cathode area. Substituting numeri- The authors are grateful to the NTÐMDT enterprise cal values into expression (1) for a cathode of 20 mm in for the scanning atomic-force microscope put at the dis- diameter yields the value ~340 nm/C. As noted, the posal of the laboratory of the Physicotechnical School experimentally determined numerical value of the Lyceum. charge/size conversion factor is 200Ð100 nm/C. This disagreement can be explained as follows. The REFERENCES disagreement means that the mass transfer process in a solid-state composite structure is more complex than in 1. A. Achuthan, A. K. Keng, and W. C. Ming, Smart Mater. Struct. 10 (5), 914 (2001). the case described by an elementary model of electrolysis. In the case under consideration, the cause of the 2. Yu. Ya. Gurevich and Yu. I. Kharkats, Superionic Con- ductors (Nauka, Moscow, 1982). slightly smaller effect in comparison with the estimate could be a partial release of metal into micro- and nan- 3. Yu. M. Gerbshteœn, V. P. Kuznetsov, and S. E. Nikitin, Fiz. Tverd. Tela (Leningrad) 27 (10), 2996 (1985) [Sov. opores during its deposition onto the cathode. This Phys. Solid State 27, 1799 (1985)]. behavior of a deposited material in systems similar to that under study is known and has been studied previously [3]. Moreover, to accurately estimate the change Translated by A. Kazantsev

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

Physics of the Solid State, Vol. 46, No. 6, 2004, pp. 1113Ð1124. Translated from Fizika Tverdogo Tela, Vol. 46, No. 6, 2004, pp. 1081Ð1091. Original Russian Text Copyright © 2004 by Antsygina, Poltavskaya, Chishko.

LOW-DIMENSIONAL SYSTEMS AND SURFACE PHYSICS

TranslationalÐRotational Interaction in Dynamics and Thermodynamics of a Two-Dimensional Atomic Crystal with Molecular Impurities T. N. Antsygina, M. I. Poltavskaya, and K. A. Chishko Verkin Institute of Low-Temperature Physics and Engineering, National Academy of Sciences of Ukraine, Kharkov, 61103 Ukraine e-mail: [email protected] Received May 13, 2003; in ﬁnal form, October 28, 2003

Abstract—The interaction between the rotational degrees of freedom of a diatomic impurity molecule and phonon excitations of a two-dimensional atomic matrix commensurate to the substrate is investigated theoretically. It is shown that the translational–rotational interaction leads to renormalization of the crystal ﬁeld constants and a change in the form of the operator for the rotational kinetic energy as compared to the corresponding expression for a free rotator. The contribution from the rotational degrees of freedom of impurities to the low-temperature heat capacity of a diluted solution of diatomic molecules in the two-dimensional atomic matrix is calculated. The possibility of experimentally observing the predicted effects is discussed. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION of the molecule and the translations of its center of inertia. However, even such a simplified model provides Investigation of the effects associated with interac- deeper insight into the basic features in the dynamics of tion between translational and rotational degrees of an impurity molecule with a mobile center of inertia in freedom in solids is an important problem in the phys- an external crystal field with low symmetry that is char- ics of condensed matter [1]. In particular, the inclusion acteristic of two-dimensional systems. of these effects has made it possible to offer an adequate quantitative interpretation of the low-temperature The inclusion of phonon degrees of freedom is not anomalies in heat capacity due to rotational degrees of only a further step toward generalizing the results freedom of molecular substitutional impurities in three- obtained previously [9] but is necessary for correct dimensional atomic cryomatrices [2Ð7]. description of the low-temperature thermodynamics of two-dimensional atomic matrices with molecular In two-dimensional atomicÐmolecular cryosolu- impurities. It should be noted that, unlike the problem tions, the dynamics of an impurity molecule appears to examined in [9], the problem under consideration be substantially more complex than that in a three- becomes many-particle and there arises a nontrivial dimensional system. At present, these objects have not problem of adequate description of the interaction been adequately studied both experimentally and theo- between the rotational degrees of freedom and local retically. Additional factors (such as the type of sub- (quasi-local) levels in the phonon spectrum of the strate, coverage, etc.) affecting the physical properties impure crystal. In the present work, we theoretically of two-dimensional crystals provide a means for pro- analyzed how translationalÐrotational interaction ducing (under different conditions) systems with radi- affects both the rotational dynamics of diatomic impu- cally different characteristics [8]. As a consequence, rity molecules in a two-dimensional close-packed these systems can exhibit a wide variety of effects asso- atomic matrix and the impurity heat capacity of the ciated with translationalÐrotational interactions. system. In the general case, theoretical analysis of the dynamics of a molecular impurity in an atomic matrix 2. FORMULATION OF THE PROBLEM with due regard for the mutual influence of phonon modes of the crystal and rotational modes of the molec- Let us consider a diatomic homonuclear molecule ular impurity is a sufficiently complex problem [1]. In with mass M and internuclear distance 2d, which is a this respect, in our earlier work [9], we investigated a substitutional impurity in a two-dimensional close- model in which a diatomic impurity molecule moves in packed monoatomic matrix (with the coordination a crystal field induced by the immobile environment number in the layer z1 = 6) located on a substrate. We and took into account only the effects induced upon examine the case where the matrix and substrate struc- interaction between the rotational degrees of freedom tures are commensurate, so that each atom has z2 near-

1114 ANTSYGINA et al. est neighbors in the substrate. The substrate has either a number of particles in the layer. In this case, Hamilto- triangular (z2 = 3) or hexagonal (z2 = 6) lattice. For def- nian (3) takes the form initeness, we assume that the impurity is located at the origin of the coordinates. The OZ axis is perpendicular π 2 1 ν Ω 2 ξ 2 to the layer and is directed away from the substrate. The Hph = ---∑ ------+ m ν ν 2 m OX and OY axes are oriented along the matrix plane. ν We will restrict our consideration to the case of an iso- (5) topic impurity. The complete Hamiltonian of the sys- ε + ------∑()πeν, eν νπν*, tem has the form 2MN ' ' νν, '

HH= rot ++Hph H1. (1) where πν = Ðiប∂/∂ξν and Ων are the frequencies of phonon excitations of the impurity-free crystal. For the Here, Hrot is the kinetic energy of the impurity mole- two-dimensional commensurate structure under con- cule, which is deﬁned by the relationship sideration, the spectrum of the ideal triangular lattice can be calculated exactly. The appropriate expressions 2 Ω 1 ∂ ∂ 1 ∂ for the three spectral branches ν can be represented in H = ÐB ------sinϑ------+,------(2) rot ϑ∂ϑ∂ϑ 2 2 the following form [10]: sin sin ϑ∂ϕ 2 2 4 2 ប2 Ω , ()k = ∆+ ---()Ω + D where B = /(2I) is the rotational constant of the impu- lt 3 rity molecule, I = Md2 is the moment of inertia of the molecule, and ϑ and ϕ are the azimuthal and polar × []12Ð cos χχcos()χ+ v cos()Ð v (6) angles specifying the orientation of the molecular axis, Ω2 respectively. The Hamiltonian of the phonon subsystem ± ------[()cos4χ Ð cos 2χcos2v 2 3 Hph can be written as 2 χ 2 v ]1/2 2 +3sin 2 sin 2 , () p H = H 0 + ε------0 , (3) ph ph Ω 2() ∆2 2M z k = z 4 (7) where + ---D[]12Ð cos χχcos()χ+ v cos()Ð v , 3 2 ()0 pf 1 ij i j where H = ∑------+ --- ∑ Λ , u u (4) ph 2m 2 ff' f f ' ,,, f ff' ij k R 3k R χ ==------x -1, v ------y 1, k =()k , k . is the phonon Hamiltonian of the impurity-free atomic 4 4 x y layer, m is the mass of the matrix atom, ε = (m Ð M)/m ∆ ∆ Ω is the mass defect, uf is the displacement of the atom Here, , z, , and D are the constants expressed located at the site f from the equilibrium position, pf = through the mass of the lattice atom and parameters of ប∂ ∂ Λ ij the potentials of the interaction of layer atoms with Ði / uf, ff, ' is the matrix of force constants, and each other and with the substrate [10] and R1 is the dis- i, j = x, y, z. tance from the matrix atom to the nearest neighbor in Now, we expand the displacement vectors u into a the layer. As can be seen from relationships (6) and (7), f all branches of the phonon spectrum of the impurity- series in terms of the polarization unit vectors eν of free commensurate structure are characterized by gaps; phonon excitations of the impurity-free layer; that is, ∆ ∆ Ӷ Ω2 in this case, z > and, as a rule, D . Therefore, the z mode is virtually dispersionless. In the presence of Ð1/2 ξ () an isotopic impurity, the phonon spectrum involves uf = N ∑ νeν exp ikRf , local and quasi-local frequencies [11Ð13]. When the ν ε ω impurity is light ( > 0), the local levels loc are located ν α above the upper edge of the continuous spectrum. If where = (k, ); k is the two-dimensional wave vector; the impurity is heavy (ε < 0), the frequencies ω lie α = l, t, z are indices characterizing the polarization of loc the phonon branch (l and t correspond to the longitudi- below the lower edge of the continuous spectrum, i.e., nal and transverse modes, respectively, polarized in the in the gap. layer plane, and z refers to the mode polarized perpen- The Hamiltonian H1 involved in expression (1) dicularly to the layer); Rf is the radius vector of the site describes the interaction between the impurity and the f; ξν are the expansion coefﬁcients; and N is the total environment. This interaction will be taken into

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 TRANSLATIONALÐROTATIONAL INTERACTION 1115 account in the atomÐatom potential approximation symmetry, in which the crystal ﬁeld Hc arises only in [14]; that is, the fourth order in d/R [15], in the system under investigation, these terms appear even in the second order. Nonetheless, we will retain the fourth-order terms in H = {}V ()r Ð r + V ()r Ð r Ð U ()r Ð u 1 ∑ 1 d a 1 d b 1 d 0 the relationship for the crystal ﬁeld, because, as will be d (8) shown below, the interaction of phonons with the impurity molecule substantially affects these terms. As a {} + ∑ V 2()rD Ð ra + V 2()rD Ð rb Ð U2()rD Ð u0 . result, we have D G H = Ð------0w 2 Here, V1 is the potential of the interaction between the c 2 z impurity atom and the matrix atom; V2 is the potential (9) 1 4 2 2 of the interaction between the impurity atom and the + ---[]G w + ∆ , G w w ()w Ð 3w . 1 z z2 3 2 y z y x substrate atom; U1 is the potential of the interaction 2 between the matrix atoms; U2 is the potential of the Here, interaction between the matrix atom and the substrate ± atom; ra, b = u0 wd are the coordinates of the atoms in  2 ᏹ the impurity molecule; w = (sinϑcosϕ, sinϑsinϕ, G0 = d  ∑ zi i ϑ  cos ) is the unit vector specifying the spatial orienta- i = 12, tion of the impurity molecule; rd = R1d + ud is the coor-  dinate of the layer atom of the nearest environment; ()2 []ᏹ 2()2 ᏼ -Ð z2 1 Ð b 3 2 + d 5b Ð 2 2 , rD = R2D is the coordinate of the substrate atom of the  nearest environment; R2 is the distance from the matrix atom to the nearest neighbor atom in the substrate; d 4 and D are the unit vectors specifying the direction to the d 5 2 2 G = ----- ∑ z ᏼ + ---z ()1 Ð b ()17Ð b ᏼ , nearest neighbors in the layer and the substrate, respec- 1 2 i i 3 2 2 i = 12, tively; u0 is the displacement of the center of inertia of the molecule from the equilibrium position; and ud is 4 4d 3 2 the displacement of the lattice atom from the equilib- G = ------z b 1 Ð b ᏼ , (10) rium position. 2 3 2 2

When writing expression (8), we took into account 2 that the impurity is isotopic (and, hence, the equilib- ᏹ d d 1 dAi i = Ai + -----Ri------------, rium position of the center of inertia of the impurity lies 8 dRi Ri dRi in the layer plane) and, moreover, restricted our treatment to the rigid-substrate approximation. Further- 1 d 3 d A ᏼ = ---R ------R ------i , more, it was assumed that the potentials V (r) and U (r) i i i 4 i i 8 dRi dRi R are short-range potentials (which is the case in real sys- i tems). Consequently, only the interaction with nearest d 1 dVi neighbors was included in Hamiltonian (8). Note also Ai = Ri------------, that the theory proposed below is assumed to be valid, dRi Ri dRi in principle, for potentials Vi(r) and Ui(r) of arbitrary and the parameters b for the substrates with the triangu- type. Hence, all further calculations will be carried out without specifying the form of these potentials, except lar and hexagonal lattices are equal to R1/( 3 R2) and for numerical evaluations in which, for definiteness, the R1/(3R2), respectively. It can be seen from expres- interatomic interaction will be described using the Len- sions (10) that the interactions of the impurity with the nard-Jones potential [14]. layer and the substrate make additive contributions to By assuming that the displacement of the center of the coefficients G0, 1. An analysis of these coefficients inertia of the impurity molecule from the equilibrium demonstrates that the substrate and matrix fields tend to position is small compared to the distances Ri and orient the impurity molecule along the layer plane and perpendicular to the substrate, respectively. As a conse- allowing for the smallness of the ratios d/Ri, we expand Hamiltonian (5) into a series in terms of these small quence, the equilibrium position of the impurity is parameters [9]. In subsequent calculations, insignifi- governed by the competition between the above two cant constants that lead only to a change in the energy factors. From expression (9), it also follows that the origin will be omitted. relationships for the crystal field are different for z2 = 6 The first-order terms do not depend on the orienta- and 3. For z2 = 3, the crystal field has symmetry group tion of the molecular axis and determine the equilib- S6 (sixfold inversion-rotational axis) [16, 17]. The term 4 rium position of the layer with respect to the substrate. determining this symmetry is of the order of (d/Ri) , In contrast to three-dimensional crystals with cubic and the coefficient G2 of this term depends only on the

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1116 ANTSYGINA et al. parameters of the substrate field [see relationships (9), βប (10)]. For z2 = 6, the symmetry of the crystal field τ ()τ S ==∫ d L , L Lph +++Lrot Lint Lc, (17) should correspond to symmetry group C6. However, the corresponding terms arise only in the sixth order in 0 d/Ri, which is not considered in our work. As a result, the symmetry of the crystal field H at z = 6 corre- m ú 2 2 2 ε ú ú c 2 L = Ð----∑ ξν +,Ων ξν Ð ----∑()ξeν, eν νξ*ν' sponds to the continuous two-dimensional rotation ph 2 N ' ν ν' group C∞. 2 I ∂w I 2 2 2 The terms describing the interaction of the phonon L ==Ð------Ð---()ϑú + ϕú sin ϑ , subsystem with the rotational degrees of freedom of the rot 2∂τ 2 impurity molecule appear only beginning with the () third-order terms [9, 18]. In this approximation, the Lint + Lc = Ð Hint + Hc , interaction Hamiltonian under consideration can be written in the form where β = 1/T (the Boltzmann constant is taken equal to unity) and τ is the imaginary time. The dot denotes 2 differentiation with respect to τ. Expression (17) can be d ()*ξ Hint = Ð------∑ f ν ν + c.c. , represented as the sum of two terms. The first term S 2 N (11) 0 ν depends only on the rotational degrees of freedom of αβ the impurity molecule, and the second term S includes f ν = Γν Qαβ + Cν. 1 both rotational and phonon variables: Here, we introduced the following designations: βប βប αβ αβγ γ α β Γν = s ()k eν + 2h ()k eν , τ() τ() S0 ==∫ d Lrot + Lc , S1 ∫ d Lph + Lint . (18) 1 ααγ γ γ (12) 0 0 Cν = ---[]s ()k + 5h ()k eν , 3 Let us now expand the functions ξν(τ) and fν(τ) (entering into the second term S1) into a series in terms αβγ() ᏷ δαδβδγ () of the Matsubara frequencies: s k = 1∑ exp ikdR1 d ∞ (13) 1 ˜ 2πn ξν()τ ==------∑ ξν()ω exp()Ðiω τ , ω ------. ᏷ ∆α∆β∆γ βប n n n βប + 2∑ , n = Ð∞ D As a result, the integrand in the partition function (16) transforms into a form that is bilinear in the Fou- α() κ δα ()κ ∆α h k = 1∑ exp ikdR1 + 2∑ , (14) ˜ rier amplitudes ξν and ˜f ν . The bilinear form can be d D rearranged into diagonal form by changing the vari- 2 d A A ables ᏷ ==R ------i , κ -----i, (15) i i dR 2 i R i Ri i ξ˜ ()ω ζ()Φω ()ω ν n = ν n + ν n , where Qαβ = wαwβ Ð ∆αβ/3 is the dimensionless quadru- 2 εω2 (19) pole moment of the impurity molecule. Therefore, the d ˜ n Φν ω ------ν ω ν (n ) = 2 2 f ()n +,------2 -()e g complete Hamiltonian of the system is the sum of rela- m()ω + Ων 1 Ð εω σα tionships (2), (5), (9), and (11). n n where 3. DYNAMICS OF THE IMPURITY ˜ ()ω MOLECULE 1 f ν n eν g = ----∑------, N ω 2 Ω 2 In this section, we analyze how the interaction ν n + ν between the rotational degrees of freedom of the impurity molecule and the matrix phonons affects the rota- σ ≡ σ 1 1 tional motion of the molecule. The magnitudes of the lt, ⊥ = ------∑ ------, corresponding effects are calculated by functional inte- 2N ω 2 Ω 2() k, γ = lt, n + γ k gration [19]. Within an insignificant normalizing factor, the partition function Z has the form 1 1 σ = ----∑------. ZD= []ξτ()D[]w()τ exp()S/ប , (16) z N ω 2 Ω 2() ∫ k n + z k

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 TRANSLATIONALÐROTATIONAL INTERACTION 1117

Integration over the phonon variables gives the follow- that, since ε < 1 and any function Fν satisfies the ine- ing expression for the partition function: quality ∆ 2 []()τ S0 + S ZZ==phZ1, Z1 ∫D w exp ------, (20) 1 2 1 ប ----∑ Fν ≥ ----∑Fν , (24) N N ν ν ∞ 2 4 ˜ d  1 f ν()ω the addition ∆H is a nonnegative quantity. ∆S = ------∑ ----∑------n - rot 2mβប N ω 2 Ω 2 Now, we analyze relationships (22) and (23) with ∞  ν + ν n = Ð n the aim of elucidating the physical effects caused by the 2 2 ∆ ∆  (21) additions Hc and Hrot. First and foremost, it should εω2 gz g⊥ + n ------+ ------, be noted that the expression for fν [see formulas (11)Ð εω2σ εω2σ  1 Ð n z 1 Ð n ⊥ (14)] involves summation over the nearest neighbors in (),, the layer plane. For lattices with sufficiently large coor- g⊥ = gx gy 0 . dination numbers, an efficient method for calculating the corresponding sums consists in replacing the sum- In expression (20), the factor Zph represents the phonon partition function of the two-dimensional crys- mation over the nearest neighbors by integration over tal and the quantity Z accounts for the rotational the unit circle [10, 21]. This approximation is justified, 1 because corrections (22) and (23) are integrated charac- motion of the molecule with due regard for the contri- teristics that are weakly sensitive to the dependence of bution of the phonon subsystem. As follows from the integrands on the direction of the wave vector. As a expressions (20) and (21), this interaction leads to the result, the sums entering into relationships (12)Ð(14) appearance of the additional term ∆S. In principle, take the form expression (21) is the solution of the posed problem. This solution will be analyzed with allowance made 1 δαδβ()() for the fact that the rotational states for the majority of ----∑ eνd exp ikdR1 molecular impurities in two-dimensional atomic lat- z1 tices are characterized by low energies (the spacing d ()(), α β between the ground and first excited rotational levels, ÐiJ3 kR1 eν nk nk nk ∆ as a rule, is less than [17]) and, hence, the specific () (25) effects due to the rotational excitations manifest them- iJ2 kR1 []α β α β ()∆, + ------eν nk ++nk eν eν nk αβ, , selves only at low temperatures. For this reason, our kR1 interest here is in the temperature range T ≤ ប∆ or បω α α loc. On this basis, by analogy with the transforma- 1 δ () () ----∑ exp ikdR1 iJ1 kR1 nk , tions performed in [9, 18, 20], we found that, in the z1 ប∆ បω d basic approximation in T/( ) or T/( loc), the addition ∆S is separated into terms of two types. The terms where Jn(x) is the nth-order Bessel function. Moreover, of the first type renormalize the constants of the crystal for convenience of calculation, exact expressions (6) field; that is, and (7) can be replaced by the approximate relationships for the branches of the phonon spectrum [10, 21], 4 2 ˜ d f ν which were derived in the aforementioned approxi- Hc ==H + ∆H , ∆H Ð------∑------. (22) c c c 2mN Ω 2 mation: ν ν 2 2 2 2 Ω , (kR ) = ∆ + ()Ω + D []Ω1 Ð J ()kR ± J (),kR The terms of the second type lead to a change in the lt 1 0 1 2 1 form of the kinetic energy: Ω 2()∆2 []() z kR1 = z + D 1 Ð J0 kR1 . ˜ ∆ Hrot = Hrot + Hrot, By substituting relationships (11)Ð(14) into expres- 2 4 2 sion (22) and using formula (25), we find that the inclu- ú úν (23) ∆ d 1 f ν ε 1 f eν sion of the interaction between the rotational degrees of Hrot = ----------∑------Ð ----∑------. 2mN Ω 4 N Ω 2 freedom of the impurity molecule and the phonons does ν ν ν ν not lead to a change in the general form of expression (9) Unlike correction (22) to the crystal field, addition for the crystal field but results in the renormalization of (23) to the kinetic energy depends on the mass defect ε. all the coefficients entering into it; that is, This is obvious from the general relationship (21). G˜ 2 ε ω 2 H H˜ = Ð------0w Indeed, the term dependent on contains the factor n ; c c 2 z i.e., it is determined by the derivatives of the dimen- (26) 1 ˜ 4 ˜ 2 2 sionless quadrupole moment and, hence, makes a con- + ---[]G1w + ∆ , G2w w ()w Ð 3w . tribution to the kinetic energy of the system. Note also 2 z z2 3 y z y x

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1118 ANTSYGINA et al.

Here, ()m 1 1 ()m 1 1 s⊥ ==------∑ ------, sz ----∑------. (31) L S 2m 2m ˜ ∆ ∆ ∆ ∆ ,, 2N Ωα N Ω Gi ===Gi + Gi, Gi Gi +Gi , i 012, k, α = lt, k z where the superscripts L and S indicate that the contri- Note that, for short-range potentials (of Lennard- | | butions are predominantly made by the interactions of Jones type), the inequalities K0 > 0, K1 < 0, and K2 < the impurity molecule with the matrix and substrate ∆ S atoms, respectively. From symmetry considerations, it K0 are satisfied and, hence, G0 > 0. is a priori evident that allowance made for the interaction of the impurity molecule with nearest neighbors in The crystal field (9) contains additive contributions the layer leads only to renormalization of the coeffi- associated with the interaction of the impurity molecule cients G and G and does not affect the last term in for- with the layer and substrate atoms. An analysis shows 0 1 that these contributions each decrease in the amplitude ∆ L mula (26). The corresponding corrections G01, can be owing to additions (27) and (29), respectively, which written as appear upon the translationalÐrotational interaction. This result is physically quite obvious. Actually, the z 2d4 rotator moves in the environment of mobile atoms of ∆G L = Ð------1 -()Ꮽ ++3Ꮾ Ꮿ , 0 6m 1 1 1 the matrix and is most strongly affected by the high- (27) frequency phonons that generate the highest strains in z 2d4 the first coordination sphere around the impurity mol- ∆G L = Ð------1 -()Ꮽ + Ꮾ , 1 4m 1 1 ecule. In this sense, the problem under consideration is similar to the known problem regarding the motion of where a particle in a rapidly oscillating field [22], when the averaging over oscillations results in an effective 2 1 α decrease in the depth of the initial potential well. Ꮽ = ----∑------1-, m N Ω 2m k l A different situation is observed for the additions to the kinetic energy. It follows from relationship (23) that 2 2 1 ()α Ð α ()α + α the interaction between the rotational degrees of free- Ꮾ = ------2 3 +,------2 3 - m ∑ 2m 2m dom of the impurity molecule and the phonons leads to 8N Ω Ω (28) k l t a substantial change in the form of the Hamiltonian α α α Hrot. Indeed, after a number of manipulations, the addi- 1 1()3 2 Ð 1 ∆ Ꮿ = ----∑------, α = ()᏷ + 2κ J (),kR tion Hrot can be rewritten in the form m N Ω 2m 1 1 1 1 1 k l 1 2 2 α ==()᏷ + 4κ J ()kR , α ᏷ J ()kR . ∆H = ---()∆I⊥wú ⊥ ++∆I wú ∆ , ∆I wú wú . (32) 2 1 1 1 1 3 1 3 1 rot 2 z z z2 3 ij i j From expressions (27) and (28), it follows that the Here, the corrections ∆I⊥ to the moment of inertia of ∆ L ∆ L , z corrections G0 and G1 are negative in sign. The the impurity molecule have the following form: additions to the crystal field due to the interaction with the immobile substrate can be represented in the form 2 4 L S L z1 d 2 ∆I⊥, ==∆I⊥, + ∆I⊥, , ∆I⊥ ------Ꮾ w⊥ , 2 4 z z z m 2 ∆ S z2 d []()2 2∆ ()1 ()1 G0 = ------2K0 Ð K2 z , 3 s⊥ Ð K0K1sz , 2m 2 z 2d4 ∆I L = ------1 -Ꮽ w 2, 2 4 z m 2 z S z2 d 2 2 ()1 2 ()1 ∆G = ------[]()4K Ð K ∆ , s⊥ Ð K s , (29) (33) 1 4m 0 2 z2 3 1 z 2 4 S z2 d 2 2 2 2 ∆I⊥ = ------ζ⊥()K w + ∆ , K w⊥ , 2 4 m 0 z z2 3 2 S z2 d ()1 ∆G = Ð------K K s⊥ , 2 m 0 2 2 4 S z2 d 2 2 2 2 ∆I = ------[]K ζ⊥()13Ð w + K ζ w , where z m 0 z 1 z z

2()2᏷ κ where K0 = 1 Ð b b 2 + 2 2 , 2[]()2 ᏷ κ ζ ()2 ε()()1 2 K1 = 1 Ð3b b Ð 2 2 Ð 4 2 , (30) ⊥, z = s⊥, z Ð s⊥, z . 3 b ᏷ Furthermore, for substrates with a triangular lattice K2 = ----- 2, 2 (z2 = 3), there arises a term containing off-diagonal ele-

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 TRANSLATIONALÐROTATIONAL INTERACTION 1119 ments (with respect to wú iwú j ); that is, parameters corresponding to the interatomic interaction in the gaseous phase [15]. It was found that the maximum relative change in the moment of inertia is 2w w 2w w 2w w 2 4 y z x z x y approximately equal to 30% and the renormalization of z d ∆ 2 ζ 2 2 the crystal field amplitudes can be as large as 50Ð60%. Iij = Ð------⊥K0K2 2w w Ð2w w w Ð w . (34) m x z y z x y Undeniably, the above estimates are rather crude, 2w w w 2 Ð0w 2 because the Lennard-Jones potential used is extremely x y x y sensitive to the choice of parameters, whose real values Therefore, the interaction of the impurity molecule for two-dimensional systems can differ substantially with phonons results in a radical change in its inertial from those for gaseous phases [21]. Nonetheless, we properties; more precisely, the effective kinetic energy can argue that the renormalization effect associated of the rotator takes the generalized positively definite with the translational–rotational interaction is signifi- quadratic form [see expression (23)] of the components cant and should be accounted for when discussing of the angular velocity, which corresponds to the sym- physical phenomena in the systems under investigation. metry of the system under investigation (C∞ or S6). By Finally, we should note that, in the limit of the ∆ virtue of the positive definiteness of the addition Hrot immobile environment, the expressions for the renor- ∆ (32), we can write the condition I⊥, z > 0. Actually, malization of the crystal field constants and the addi- L tions to the rotational kinetic energy of the impurity since ζ⊥ ≥ 0 [see inequality (24)], the quantities ∆I⊥, , z z molecule transform into the corresponding expressions S and ∆I⊥ are obviously positive. As regards the addition obtained in our earlier work [9]. ∆ S Iz , this quantity can appear to be negative at specific ∆ molecular orientations; however, the total addition Iz 4. IMPURITY HEAT CAPACITY is always positive. Consequently, the interaction Now, we calculate the contribution made to the low- between the rotational degrees of freedom of the impu- temperature thermodynamic properties of a two- rity molecule and the phonons leads to an increase in dimensional atomic crystal by the rotational degrees of the principal moments of inertia of the impurity mole- freedom of a system formed by noninteracting diatomic cule, i.e., to an increase in its weight. impurity molecules with due regard for the transla- It should be noted that the separation of the renor- tionalÐrotational interaction. Our treatment will be car- malizations of the crystal field and the moments of iner- ried out in the tight-binding limit when a molecule exe- tia into the layer and substrate contributions is conven- cutes small librations with respect to the equilibrium tional in a sense. Although each contribution explicitly position coinciding with the normal to the layer plane. depends on the parameters of the potential associated In this case, the Hamiltonian of the system can be with one component of the system (only the substrate or derived under the assumption [20] only the matrix), all the contributions, according to expressions (27)Ð(29) and (33), are functions of the fre- Qαβ ==〈〉Qαβ + qαβ, ξν 〈〉ηξν + ν, (35) quencies Ων of phonon excitations, which characterize the system as a whole. where the angle brackets denote the thermodynamic average of the corresponding functions and qαβ is a All the corrections to the crystal field and the 〈 〉 4 small addition. Only the diagonal elements Qxx = moment of inertia are of the order of (d/R) . In explicit 〈Q 〉 = Ð1/3 and 〈Q 〉 = 2/3 are nonzero. The quantities form, the corrections expressed through the parameters yy zz 〈ξν〉, which are nonzero due to the translationalÐrota- of the system in the general case are rather complicated [see formulas (27)Ð(31)]. However, the situation can be tional interaction, satisfy the relationship [20] simplified in some cases, for example, in the tight-bind- 2 〈〉 ing limit, when the crystal field is sufficiently strong as 〈〉ξ d f ν ν = ------2 . (36) compared to the rotational constant of the impurity NmΩν molecule. If the condition G0 Ð 2G1 > 0 is satisfied, the molecule executes small librations with respect to the The deviation of the rotator axis from the equilib- equilibrium position at which the molecular axis is per- rium position is conveniently described in terms of the pendicular to the layer. In this case, the off-diagonal variables u = sinϑcosϕ and v = sinϑsinϕ, which are elements arising in the inertia tensor due to the transla- small quantities for the system under consideration. tionalÐrotational interaction prove to be negligible and Then, the vector w takes the form only the corrections to the diagonal elements make significant contributions. We performed numerical calcu- u2 + v 2 w = u,,v 1 Ð ------. (37) lations of the renormalizations for a number of systems 2 under the assumption that the interaction of the impurity molecule with atoms of the matrix and the substrate By substituting formulas (35)Ð(37) into relation- is described by the Lennard-Jones potential with the ships (2), (5), (9), and (11) and retaining the terms up to

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1120 ANTSYGINA et al. the second order, we obtain the Hamiltonian of the One more specific feature of the two-dimensional sys- system: tem (as compared to the three-dimensional case) is that ᏴᏴᏴ Ᏼ Ᏼ Ᏼ it has no threshold mass defect for localized vibrations; ==0 + 1, 0 ph + 2. (38) i.e., localized vibrations occur at any ε < 1. Note that to Here, Ᏼ is the Hamiltonian of the phonon subsystem, every local level there necessarily corresponds a conju- ph gated quasi-local state. These results are similar to 2 those obtained for localized states in the vicinity of a 1 πν 2 2 Ᏼ = ---∑ ------+ mΩν ην rectilinear dislocation [23]. The translationalÐrota- ph 2 m ν tional interaction leads not only to renormalization of (39) the rotational parameters of the impurity molecule but ε also to a modification of the vibrational spectrum of the + ------∑()πeν, eν νπν*, 2MN ' ' system. νν, ' The thermodynamic characteristics of the system Ᏼ and 2 is the Hamiltonian related to the rotational described by Hamiltonian (38) will be calculated using degrees of freedom of the impurity molecule, the following procedure. The auxiliary numeral factor λ is introduced so that Ᏼ B ()κ2 2 ()2 2 2 = ----- Pu + Pv + u + v . (40) Ᏼλ = Ᏼ + λᏴ . (45) ប2 0 1 Ᏼ It is evident that Hamiltonian (45) coincides with The Hamiltonian 1 describing the interaction Hamiltonian (38) at λ = 1. The free energy correspond- between the rotational degrees of freedom of the mole- ing to Hamiltonian (45) obeys the relationship cule and phonon excitations has the form ∂Ᏺ()λ Sp[]Ᏼ exp()ÐβᏴλ ------〈〉Ᏼ 〈〉Ᏼ ------1 ∂λ = 1 λ, 1 λ = []()βᏴ . (46) Ᏼ 1 ()η()η Sp exp Ð λ 1 = ------∑ bνuc+ νv ν + ν* . (41) 2 N ν Then, the free energy of the system under consider- Here, we introduced the following designations: ation can be represented in the form 1 π ប∂ ∂η ប∂ ∂ ()v ν ==Ði / ν, Pu()v Ði / u , ᏲᏲ∆Ᏺ ∆Ᏺ 〈〉Ᏼ λ ==0 + , ∫ 1 λd , (47) B G κ = --- + ------0 Ð G 0 4 2 1 (42) where ∆Ᏺ is the addition associated with the transla- 4 2 Ᏺ d 2 ()2 z tionalÐrotational interaction and 0 is the free energy + ------z K ()K Ð K s +,----1 ()2Ꮽ Ð Ꮿ 4m 2 1 1 0 z 3 2 2 in the absence of this interaction. The latter quantity includes the contributions from the impurity-free layer, 2 x 2 y local and quasi-local frequencies, and the rotational bν ==z d K eν , cν z d K eν . (43) 2 0 2 0 degrees of freedom of impurities with nonrenormalized Hamiltonian (39) describes the phonon subsystem parameters. of a two-dimensional crystal that is located on the sub- Therefore, the problem is reduced to calculating the strate and contains an isotopic impurity. It is known 〈Ᏼ 〉 quantity 1 λ. This can be accomplished in terms of that, in the presence of such an impurity, the vibrational two-time Green’s functions. The two-time retarded spectrum is characterized by local and quasi-local fre- commutator Green’s function is defined by the expres- quencies [11]. Note that, in the spectrum of a three- sion [24] dimensional cubic crystal, all local frequencies are tri- i ply degenerate [11Ð13]. By contrast, the spectrum of 〈〉〈|Aˆ ()t Bˆ ()t' 〉= Ð---Θ()ttÐ ' 〈〉[]Aˆ ()t , Bˆ ()t' . the two-dimensional atomic crystal on the substrate ប involves a doubly degenerate vibration polarized in the layer plane and a nondegenerate vibration polarized The equations for the Fourier components of the perpendicularly to the layer plane. The corresponding Green’s function for the system described by Hamilto- equations for determining the local and quasi-local fre- nian (45) are written as quencies have the form [10] ε ωη〈〉〈|〉 1 〈〉〈|〉π ()π, 〈〉〈|〉 i ν u ω = ---- ν u ω + ------∑ eν eν' ν' u ω, 2 m MN 1 1 ν 1 Ð0,εω S⊥()ω ==S⊥()ω ------∑ ------, ' 2N ω2 Ω 2 k, γ = lt, Ð γ ()k 2 iωπ〈〉〈|〉ν u = ÐmΩν 〈〉〈|〉ην u Ð λbν 〈〉〈|〉uu ω, (44) ω ω 2 1 1 (48) εω ---- 2B 1 Ð0.∑------= iω 〈〉〈|〉uu ω = ------〈〉〈|〉P u , N ω2 Ω 2() 2 u ω k Ð z k ប

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 TRANSLATIONALÐROTATIONAL INTERACTION 1121

is the part of the impurity contribution that is modiﬁed ω 〈〉〈|〉 κ 〈〉〈|〉 λ 〈〉〈|〉η through the translationalÐrotational interaction. This i Pu u ω = Ð2 uu ω Ð ∑bν ν u ω Ð 1. ν last term, which involves the contributions from the rotational and translational degrees of freedom of the The equations for the Green’s functions with the impurity, can be written in the form variable v have a similar form. After eliminating the functions 〈〈πν|u〉〉ω and 〈〈P |u〉〉ω, Eqs. (48) are reduced ∞ u ប βបω to a system of integral equations with degenerate ker- ∆Ᏺ˜ = --- lim ∫dωcoth------φωδ(), , nels whose solution can be found in an explicit form. As πδ → 0 2 a result, for the Green’s function under consideration 0 (54) ()ωδ, φωδ(), P 1 = arctan------()ω . Ᏻλ()ω = ------∑()bν 〈〉〈|〉ην u ω + cν 〈〉〈|〉ην v ω , (49) R N ν Here, we obtain 2λΦ P()ωδ, = 2ωδ[]1 Ð ερ() ω ()2ω2 Ð ω 2 Ᏻλ()ω = ------. (50) 0 λ2Φ (55) 1 Ð µω()εω[]2()ω2 ω 2 Ð Ð 0 + a , Here, 2 2 2 aS⊥()ω R()ω = () ωÐ ω []1 Ð εω ρω()Ð aρω(), Φ = ------, (51) 0 (56) ()ω2 ω 2 []εω2 ()ω Ð 0 1 Ð S⊥ S⊥()ρωω + iδ = ()+ iµω(). ω κ ប where 0 = 2B / is the frequency of rotator libra- The procedure for calculating integral (54) depends ω tions in the absence of the translationalÐrotational on the location of the librational frequency 0 with 2 4 2 ∆ interaction and a = 2z d K B /(mប2) is the parameter respect to the lower edge of the continuous spectrum 2 0 (hereinafter, the continuous spectrum will be consid- characterizing this interaction. It follows from expres- ered to mean only excitations belonging to the l and t sions (44) and (51) that, within our approximation, dis- modes of the impurity-free monolayer). placements of the center of inertia of the impurity molecule along the OZ axis do not affect the rotational For the majority of real systems, the librational fre- ω motion of the molecule. This is obvious even from the quency 0 of the nonrenormalized rotator is low com- form of relationship (41), which contains only the in- Ω pared to the upper edge max of the continuous spec- plane components of the displacement of the center of trum of the impurity-free two-dimensional crystal. This inertia of the rotator in the monolayer. ω implies that the librational frequency 0 lies either in By using the spectral representation for the Green’s ω ∆ the gap ( 0 < ) or in the range of the continuous spec- function [24] trum in the vicinity of the lower edge. It is these low fre- ∞ quencies ω that are of interest from the standpoint of ប Ᏻ ()ω δ Ᏻ ()ω δ 0 〈〉Ᏼ i λ + i Ð λ Ð i ω the possibility of observing effects due to the rotational 1 λ = ------lim ∫ ------d (52) 2πδ → 0 exp()βបω Ð 1 heat capacity in experiments. Indeed, in the impurity Ð∞ system, apart from the contribution of the rotational and expression (47), after integration over λ, we derive degrees of freedom, there are contributions from trans- the following relationship for the addition to the free lational excitations (associated with both the continu- energy: ous spectrum and the local and quasi-local states). ω ∞ Therefore, the lower the frequency 0 and the lower the iប dω temperature, the simpler the separation of the rotational ∆Ᏺ = Ð------lim ∫ ------2πδ → 0 exp()βបω Ð 1 (53) contribution to the heat capacity against the back- Ð∞ ground of the translational contribution. × {}[]Φω()δ []Φω()δ ln1 Ð + i Ð ln 1 Ð Ð i . Initially, we consider the systems for which the ω After substituting expression (51) into relationship librational frequency 0 lies in the gap of the phonon ω ∆ ε (53) and a number of transformations, the addition ∆Ᏺ spectrum ( 0 < ). For a light impurity ( > 0), local can be represented as the sum of three terms. Two terms and quasi-local translational levels of the impurity mol- cancel out both the contribution made to the free energy ecule are located in the vicinity of the upper edge Ω Ᏺ max 0 by the rotational degrees of freedom of the rotator and their inﬂuence on the low-temperature thermody- with the nonrenormalized parameters and the contribu- namic characteristics of the system can be ignored. tions from the local and quasi-local states (polarized in According to relationships (54)Ð(56), the contributions the layer plane) of the phonon spectrum. The third term made to the free energy and the heat capacity (per par-

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1122 ANTSYGINA et al. ticle) by the rotational degrees of freedom have the As follows from relationships (61), irrespective of form the mutual arrangement of the nonrenormalized librational and local levels, the translationalÐrotational បω ˜ 0 interaction leads to a decrease in the free energy and, ∆Ᏺ˜ = 2T ln 2sinh------, rot 2T hence, to an increase in the heat capacity of the system (57) at low temperatures. It is easy to see that, at ω ӷ ω , បω˜ បω˜ 2 loc 0 0 0 ω˜ ∆C = 2 ------arcsinh------, relationship (61) for 0 coincides with formula (59). 2T 2T If the parameter a of the translationalÐrotational where ω˜ is the smallest root of the equation interaction is comparable to or exceeds the spacing 0 between the rotational and local levels, i.e., at a ≥ R()ω = 0. (58) ω 2 ω 2 2 (loc Ð 0 ) , the frequencies are mixed. As a result, This root is given by the formula molecular librations and local vibrations cease to be well-defined eigenstates of the system. ω2 ω 2[]ζ ()1 ˜ 0 = 0 1 Ð a ⊥ Ð as⊥ . (59) A more complex situation is observed in the case when the librational frequency is located in the range of In the absence of the translationalÐrotational inter- the continuous spectrum of phonon excitations in the ω˜ ω ∆ action, the frequency 0 coincides with the frequency vicinity of the lower edge ( 0 > ) [12, 20]. For a heavy ω ω ∆ 0 of unperturbed librations. Expression (57) describes impurity ( loc < ), the contribution from the rotational the heat capacity of a two-dimensional Einstein oscilla- degrees of freedom to the thermodynamic functions of ω tor with the renormalized librational frequency ˜ 0 . It is the system is small compared to that from the local fre- easy to verify that formula (59) coincides with the cor- quencies. Therefore, we dwell on the more interesting responding formula derived from relationships (27), case of a light impurity. In order to derive the contribu- (29), (33), and (34) in the tight-binding limit. As could tion from the rotational degrees of freedom to the free be expected, the renormalization of the parameters of energy, we rewrite relationship (54) in the following the rotational motion leads to an effective decrease in form: the frequency ω and, hence, to an increase in the rela- ∞ 0 បω ∂φ() ω, δ tive fraction of the rotational degrees of freedom in the ∆Ᏺ˜ 2T ω  = Ð------lim ∫d ln2sinh------. (62) low-temperature heat capacity of the system. π δ → 0 2T ∂ω ∆ In the case of a heavy impurity (ε < 0), apart from ω the frequency 0, there is a local vibrational level at The analytical and numerical treatment demon- ω ∆ ∂φ ∂ω loc < in the gap. The contribution from this level to strated that the function / exhibits a sharp maxi- ω the thermodynamic functions can be comparable to that mum at the frequency ˜ 0 and rapidly decays away from the rotational degrees of freedom. Note that the from this frequency. The frequency ω˜ coincides to main contribution made by the impurity to the free 0 energy and the heat capacity consists of two terms high accuracy with the frequency determined from rela- defined by formulas similar to expression (57) with fre- tionship (59). This agreement is explained by the fact that the density of phonon states in the system under quencies determined as the roots of Eq. (58), which can ω Ω be conveniently represented in the form consideration drastically increases at max. ω ∆ Ӷ ∆ Therefore, at 0 Ð , the main contribution to the 2 2 2 2 2 ()1 2 ()2 ()ωω Ð ω ()Ð ω Ð0.aω ()s⊥ + ω s⊥ = (60) renormalization of the rotational frequency of the 0 loc loc impurity molecule is made by high-frequency phonons. In the case when the difference between the unper- ∂φ ∂ω ω ω By approximating the function / in the vicinity turbed frequencies loc and 0 is large compared to the of the maximum, we obtain the contribution from the parameter a of the translationalÐrotational interaction, rotational motion of the molecule to the free energy: Ӷ ω 2 ω 2 2 i.e., when the inequality a (loc Ð 0 ) is satisfied, Ω max the excitations under consideration, as before, can be 2T បω γ ∆Ᏺ˜ = ------∫ dωln2sinh ------0 , (63) treated as librational and local excitations with the π 2T ()ωω˜ 2 γ 2 renormalized frequencies ∆ Ð 0 + 0 ω2 ω 2[]()ω , ω ()1 where γ = Ðaµ(ω˜ )/(2ω˜ ) is the half-width of the ˜ 0 = 0 1 Ð af 0 loc Ð as⊥ , 0 0 0 Lorentzian peak of the integrand in the vicinity of the ω2 ω 2[]()ω , ω ˜ loc = 0 1 + af 0 loc , ω˜ (61) frequency 0 . It should be emphasized that approxi- ()1 2 ()2 ω ω mation (63) holds true if the frequency 0 is not too ()ω , ω s⊥ + loc s⊥ f 0 loc = ------. close to the lower edge of the continuous spectrum, i.e., ω 2 ω 2 ω ∆ ӷ γ loc Ð 0 if the inequality 0 Ð 0 is satisfied.

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 TRANSLATIONALÐROTATIONAL INTERACTION 1123

The integrand in expression (63) has the form of a rity molecule in the two-dimensional matrix located on γ standard dispersion relation in which the half-width 0 the substrate represents a parametric rotator whose is proportional to the interaction parameter a. Physi- dynamics signiﬁcantly differs from the behavior of cally, this result can be explained as follows: the both a free rotator and a rotator in a three-dimensional phonon subsystem is excited by the rotator and, in turn, cubic lattice. has an effect on the rotator state. This effect, as in any Upon introduction of molecular impurities into the distributed system, is characterized by a retardation two-dimensional atomic lattice, the low-temperature determined by the response function of the system. heat capacity of the system is characterized by an addi- γ Actually, the quantity 0 includes the interaction param- tional contribution that is associated with the rotational eter a and the characteristics of the phonon subsystem. degrees of freedom of the impurity molecule and can be γ observed experimentally. For commensurate structures, Since the half-width 0 is small, the contribution from the rotational degrees of freedom of the impurity two-dimensional solutions of light impurities are molecule to the heat capacity can be approximately rep- experimentally simpler, because the contribution made resented by expression (57). Therefore, the rotational to the heat capacity by local translational frequencies part of the heat capacity at low temperatures has an does not affect the contribution from the rotational exponential form, degrees of freedom. The case of a heavy impurity is theoretically more interesting and, at the same time, is បω˜ 2 បω˜ more complex for experimental investigations. This is ∆C = 2 ------0 exp Ð------0 , (64) T T explained by the difﬁculties encountered in correct separation of the contributions from the localized and rota- whereas the corresponding dependence for three- tional states. dimensional crystals is described by a power function Thus, the results obtained in the present work dem- [20]. The result obtained is explained by the gap char- onstrated that the inclusion of the translationalÐrota- acter of the phonon spectrum of the two-dimensional tional interactions is of considerable importance in ana- atomic crystal commensurate to the substrate [8, 21]. In lyzing and interpreting the dynamics and thermody- this respect, we recall that the heat capacity of the namics of molecular impurities in two-dimensional matrix also has an exponential form [8, 21], crystals. ∆ ∆ C ∼ --- exp Ð--- . ph T T REFERENCES 1. R. M. Lynden-Bell and K. H. Michel, Rev. Mod. Phys. The above dependence of the heat capacity on the tem- 66 (3), 721 (1994). perature differs from that represented by relationship 2. A. I. Krivchikov, M. I. Bagatskiœ, V. G. Manzheliœ, et al., (64) in terms of the preexponential factor. This circum- Fiz. Nizk. Temp. 14 (11), 1208 (1988) [Sov. J. Low stance can appear to be useful in separating the rota- Temp. Phys. 14, 667 (1988)]. tional contribution from the total experimentally mea- 3. P. I. Muromtsev, M. I. Bagatskiœ, V. G. Manzheliœ, et al., sured heat capacity of the system, especially in the case Fiz. Nizk. Temp. 16 (8), 1058 (1990) [Sov. J. Low Temp. ω ∆ when 0 and are close in magnitude. Phys. 16, 616 (1990)]. 4. P. I. Muromtsev, M. I. Bagatskiœ, V. G. Manzheliœ, and I. Ya. Minchina, Fiz. Nizk. Temp. 20 (3), 247 (1994) 5. CONCLUSIONS [Low Temp. Phys. 20, 195 (1994)]. The behavior of a diatomic impurity molecule in a 5. V. G. Manzheliœ, E. A. Kosobutskaya, V. V. Sumarokov, two-dimensional atomic matrix located on a substrate et al., Fiz. Nizk. Temp. 12 (2), 151 (1986) [Sov. J. Low turns out to be considerably more complex than that in Temp. Phys. 12, 86 (1986)]. a three-dimensional atomic matrix with cubic symme- 6. T. N. Antsygina and V. A. Slyusarev, Fiz. Nizk. Temp. 19 try [18, 20]. In a three-dimensional system, owing to (1), 102 (1993) [Low Temp. Phys. 19, 73 (1993)]. the high symmetry of the environment, the interaction 7. T. N. Antsygina and V. A. Slyusarev, Fiz. Nizk. Temp. 20 of the rotator with phonons leads only to an increase in (3), 255 (1994) [Low Temp. Phys. 20, 202 (1994)]. its moment of inertia without changing the character of 8. J. G. Dash, Fiz. Nizk. Temp. 1 (7), 839 (1975) [Sov. J. the rotational motion. In a two-dimensional system, Low Temp. Phys. 1, 401 (1975)]. which is characterized by the anisotropy due to a pref- 9. T. N. Antsygina, M. I. Poltavskaya, and K. A. Chishko, erential direction (perpendicular to the layer plane), the Fiz. Tverd. Tela (St. Petersburg) 44 (7), 1215 (2002) interaction of the rotator with phonons brings about [Phys. Solid State 44, 1268 (2002)]. substantial changes: the inertia tensor components 10. T. N. Antsygina, K. A. Chishko, and I. I. Poltavsky, J. become dependent on the orientation of the axis of the Low Temp. Phys. 126 (1/2), 15 (2002). impurity molecule. In this case, the inertia tensor 11. I. M. Lifshits and A. M. Kosevich, Rep. Prog. Phys. 29, remains diagonal when the substrate atoms form a hex- 217 (1966). agonal structure and has off-diagonal components for 12. Yu. Kagan and Ya. Iosilevskiœ, Zh. Éksp. Teor. Fiz. 45 substrates with triangular lattices. Therefore, the impu- (3), 819 (1963) [Sov. Phys. JETP 18, 562 (1963)].

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13. H. Bottger, Principles of the Theory of Lattice Dynamics 20. T. N. Antsygina and V. A. Slyusarev, Teor. Mat. Fiz. 77 (Physik, Weinheim, 1983; Mir, Moscow, 1986). (2), 234 (1988). 14. A. I. Kitaœgorodskiœ, Molecular Crystals (Nauka, Mos- 21. T. N. Antsygina, I. I. Poltavskiœ, M. I. Poltavskaya, and cow, 1972). K. A. Chishko, Fiz. Nizk. Temp. 28 (6), 621 (2002) [Low 15. Cryocrystals, Ed. by B. I. Verkin and A. F. Prikhot’ko Temp. Phys. 28, 442 (2002)]. (Naukova Dumka, Kiev, 1983). 22. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 1: Mechanics, 2nd ed. (Nauka, Moscow, 16. M. I. Poltavskaya and K. A. Chishko, Fiz. Nizk. Temp. 1963; Pergamon, Oxford, 1965). 26 (4), 394 (2000) [Low Temp. Phys. 26, 289 (2000)]. 23. A. M. Kosevich, The Theory of Crystal Lattice (Vishcha 17. M. I. Poltavskaya and K. A. Chishko, Fiz. Nizk. Temp. Shkola, Kharkov, 1988). 26 (8), 837 (2000) [Low Temp. Phys. 26, 615 (2000)]. 24. D. N. Zubarev, Nonequilibrium Statistical Thermody- 18. N. Antsygina, K. A. Chishko, and V. A. Slusarev, Phys. namics (Nauka, Moscow, 1971; Consultants Bureau, Rev. B 55 (6), 3548 (1997). New York, 1974). 19. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965; Mir, Moscow, 1968). Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

LOW-DIMENSIONAL SYSTEMS AND SURFACE PHYSICS

Photoinduced Ion Transfer in Heterojunctions Based on Solid Electrolytes with a Mixed IonicÐElectronic (Hole) Conductivity A. I. Stetsun Institute of Semiconductor Physics, National Academy of Sciences of Ukraine, Kiev, 03028 Ukraine Received May 19, 2003; in ﬁnal form, August 26, 2003

Abstract—The concept of photoinduced ion transfer in heterojunctions based on solid electrolytes with a mixed ionicÐelectronic (hole) conductivity is justiﬁed. This concept is conﬁrmed by the experiments on photodeposition of a metal in a heterojunction based on a solid electrolyte with a mixed conductivity. The possibility of using heterojunctions based on a solid electrolyte layer with a mixed ionicÐelectronic (hole) conductivity for optical recording of information is considered. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION The electric field induced by separated electrons and holes is opposed to the electric field generated in the Since the advent of heterojunctions and, especially, space-charge regions of the heterojunction. The with the advance of their technology and theory, hetero- strength of the photoinduced field can be defined as junctions have been widely used in fabricating diodes, V transistors, photodiodes, solar cells, light-emitting E = ------, (2) diodes, semiconductor lasers, etc. [1Ð3]. Despite the ∆x extensive application in many fields, the possibility of ∆ using heterojunctions for optical recording of informa- where x is the distance between electrons and holes tion has not been adequately studied. In this respect, the separated by the heterojunction. Under exposure of the present work is devoted to the design and investigation heterojunction to light, the electric field is induced both of specific heterojunctions intended for optical record- in the space-charge regions of the heterojunction and in ing of information. the quasi-neutral region. Let us now assume that charged particles (differing It is known that one of the applications of hetero- from electrons and holes), such as ions with a particular junctions is based on the generation of a photovoltage charge and mobility, are placed in either of two parts of under illumination [4, 5]. The generation of a photo- the photoelectric heterojunction. In this case, the elec- voltage is associated with the fact that electronÐhole tric field arising upon generating a photovoltage should pairs excited by light in both parts of the heterojunction initiate ion migration across one of the parts of the het- are separated by an electric field in space-charge erojunction. Therefore, illumination of the heterojunc- regions of the heterojunction. The photovoltage can be tion, which induces the photovoltage, should also stim- determined from the relationship [4] ulate an ion mass transfer. Correspondingly, the difference in the concentration distribution of ions over the ∆σ ∆σ ∆σ n + p n dχ thickness in one of the parts of the heterojunction V = ------ξ dx Ð ------dx ∫σ 0 ∫σ dx before and after exposure to light can be used for optical recording of information. ∆σ p dχ dE Ð ------+ ------g dx (1) Since mobile charges of two types, namely, ions and ∫σ dx dx electrons (or holes), reside in solid electrolytes with a mixed ionicÐelectronic (hole) conductivity, it is possi- ∆σ d ∆σ d + kT ------p ----- ()ln N Ð ------n ----- ()ln N dx, ble to fabricate a specific heterojunction between a con- ∫ σ dx v σ dx c ventional semiconductor with electronic (or hole) conductivity and a solid electrolyte (whose illumination where σ is the conductivity induced under illumination, generates a photovoltage) with a mixed conductivity. ∆σ is the photoconductivity, χ is the electron affinity, This heterojunction is characterized by the photoin- Nc is the density of states in the conduction band, Nv is duced ion transfer across one of the two parts of the het- ξ the density of states in the valence band, 0 is the erojunction. strength of the electric field in the heterojunction, and The interface between two materials with different Eg is the band gap. bond lengths contains a large number of defects in the

1126 STETSUN

Table 1. Characteristics of electrical conductivity of As2S3 lamp at different illuminances. In the course of illumi- films photodoped with silver at different contents (tempera- nation, the sample was cooled with an air stream pro- ture, 300 K) [7] duced by a fan. σ , 10Ð5 After exposure to light, the samples were examined C , at. % σ , ΩÐ1 cmÐ1 p η Ag i ΩÐ1 cmÐ1 i under a Biolam optical microscope. The concentration distribution of the metal over the thickness of a solid Ð8 20 7.6 × 10 0.17 0.04 electrolyte layer was investigated on a JAMP-10S 25 8.2 × 10Ð6 2.8 0.23 Auger analyzer. 29 3.5 × 10Ð5 4.7 0.43 Heterojunctions were prepared through thermal 32 3.0 × 10Ð4 6.3 0.83 evaporation under vacuum with the use of a VUP-5 vac- σ σ uum setup. During evaporation, the residual pressure Note: C is the silver content, is the ionic conductivity, is Ð3 Ag ηi p was no less than 10 Pa. the hole conductivity, and i is the transport number. Photodoping of As2S3 layers with silver was carried out according to the procedure described in [10, 11]. form of dangling bonds [3]. Dangling bonds can serve After deposition of a silver-photodoped As2S3 layer on both as electron trapping centers and as recombination a quartz or silicon substrate, this layer was coated with centers. The recombination of electrons and ions at the a CdSe layer through thermal evaporation. interface formed by two parts of the heterojunction The optical reflectance and transmittance spectra of brings about the photodeposition of a chemical element the heterojunctions before and after illumination were responsible for ionic conductivity in the solid electro- recorded on a KSVU-23 spectrophotometer in the lyte. wavelength range 200Ð1200 nm. For the purpose of experimentally verifying the pos- The photovoltage was measured with the use of a sibility of photoinducing mass transfer of ions across V7-45 voltmeter. The internal resistance of the voltme- the layer of a solid electrolyte with mixed conductivity 16 Ω in a heterojunction, we investigated how illumination ter was equal to 10 . affects the properties of heterojunctions between AIIBVI semiconductors and vitreous As2S3 layers photodoped 3. EXPERIMENTAL RESULTS with silver at a content of 2Ð32 at. %. The heterojunctions studied in this work contained II VI The choice of the A B semiconductor material as 400- to 500-nm-thick As2S3 layers photodoped with sil- a constituent of a heterojunction was made for the rea- ver and 200- to 500-nm-thick CdSe layers. son that these materials are characterized by a high pho- At the first stage of our investigation, we examined tovoltage [6]. heterojunctions between a CdSe layer and a silver- Silver-photodoped As S layers are solid electro- 2 3 photodoped As2S3 layer without exposure to light. Ini- lytes with a mixed (ionic and hole) conductivity [7Ð9]. tially, these unexposed heterojunctions were studied Photodoping allows one to vary the metal content in a using an optical microscope. No inhomogeneities, pre- semiconductor layer over a wide range and, thus, to cipitates, or clusters were found in the heterojunction control the ionic and hole conductivities (Table 1). In structure. turn, this makes it possible to vary substantially the conditions of experimental investigations into the prop- The unexposed heterojunctions were subjected to erties of heterojunctions with solid electrolytes. chemical treatment. The CdSe layer was removed by chemical etching in a mixture of HNO and HCl acids. The photoinduced transfer of silver ions and the 3 The remaining As2S3 layer photodoped with silver was photodeposition of silver in a heterojunction were examined with the use of an electron microscope from found experimentally in CdSeÐAs2S3Agx (x = 0.9Ð2.4) the Auger analyzer. This examination also did not heterojunctions. The experimental results obtained in reveal precipitates or clusters in the silver-photodoped this work proved that heterojunctions with mixed layer. The results of Auger spectroscopy demonstrated ionicÐelectronic (hole) conductors can be used for opti- that the silver content in the surface region of the As2S3 cal recording of information. layer photodoped with silver at a content of 25 at. % The main objective of the present work was to jus- differs from this value by no more than 1 at. %. tify the concept of photoinduced ion transfer and to reveal experimentally the photoinduced transfer of sil- At the second stage of the investigation, the hetero- ver ions across one of the parts of the heterojunction junctions were exposed to light. with a solid electrolyte under illumination. For heterojunctions between a CdSe layer and an As2S3 layer photodoped with silver at a content of 15Ð 32 at. % (the chemical composition corresponds to the 2. EXPERIMENTAL TECHNIQUE formula As2S3Agx at x = 0.9Ð2.4), exposure from the Photoinduced changes in heterojunctions were initi- side of the photodoped layer to light of a KG-220-500 ated under exposure to light of a KG-220-500 halogen halogen lamp at an illuminance of 8 × 104 lx for 2 h

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 PHOTOINDUCED ION TRANSFER 1127 leads to the formation of silver precipitates of two R types, namely, small-sized and large-sized precipitates, 0.4 in the heterojunction. The precipitates are formed in the (a) 1 photodoped layer in the vicinity of the interface. The 0.3 characteristic sizes of the precipitates are equal to 0.2Ð 0.5 (smaller precipitates) and 2Ð5 µm (larger precipitates). 0.2 2 According to the results of Auger spectroscopy, the exposure of the heterojunctions to light results in an 0.1 increase in the content of silver ions in the vicinity of 0 the interface between the layers. These experiments 1200 1000 800 600 400 200 were performed as follows. A CdSeÐAs S heterojunc- λ 2 3 T , nm tion photodoped with silver at a content of 25 at. % (the 1.0 chemical composition corresponds to the formula 1 (b) As S Ag ) was exposed to light of the halogen lamp 2 3 1.67 0.8 at an illuminance of 8 × 104 lx for 20 min in the cham- ber of the Auger analyzer. Then, under exposure of the 2 sample at the same illuminance for 20 min, the CdSe 0.6 layer was etched away with an argon ion beam. Expo- sure under these conditions brought about the forma- 0.4 tion of precipitates 0.2Ð0.3 µm in size in the surface region of the photodoped layer. In the regions free of 0.2 precipitates, the silver content in the surface region of the photodoped layer increased by 5 at. % as compared 0 to that in the bulk of the photodoped layer. 1200 1000 800 600 400 200 λ, nm Moreover, the CdSeÐAs2S3Ag1.67 heterojunction was exposed and etched under the same conditions but Fig. 1. (a) Reflectance (R) and (b) transmittance (T) spectra 4 at an illuminance of 2 × 10 lx. In this case, no forma- of the CdSeÐAs2S3Ag1.67 heterojunction (1) before and tion of photoprecipitates was observed and the silver (2) after exposure to light. content in the surface region of the As2S3Ag1.67 layer increased by 3 at. %. directed such that it initiates the migration of positive The size of the photoprecipitates formed under silver ions toward the interface. exposure of the CdSeÐAs2S3Ag1.67 heterojunction to light depends on the illuminance and exposure time. The reflectance and transmittance spectra of the ini- The exposure of the heterojunction to light at an illumi- tial and exposed CdSeÐAs2S3Ag1.67 heterojunctions are 4 nance of 8 × 10 lx for 2 h leads to saturation of the pho- depicted in Fig. 1. The exposure of the As2S3Ag1.67 het- todeposition of silver, i.e., to the formation of large- erojunction to light results in a decrease in the ampli- sized silver photoprecipitates whose height is equal to tude of interference oscillations, which indicates an the thickness of the photodoped layer. increase in the absorption by the photoinduced layer. Similar results were obtained for the CdSeÐ As was shown in our earlier work [12], an increase in the silver content in the photodoped As2S3 layer leads As2S3Agx (x = 0.9Ð2.4) heterojunctions exposed from the side of the CdSe layer. to a shift in the absorption edge toward the long-wavelength range and to an increase in the absorption in the In the heterojunctions under investigation, we mea- layer. Therefore, the observed decrease in the ampli- sured a dark voltage with a positive potential at the tude of interference oscillations can be associated with CdSe layer. However, under exposure of the hetero- an increase in the silver content in photodoped layer junction to light, we revealed a photovoltage with a regions of different thicknesses. negative potential at the CdSe layer. The photosensitivity of the CdSeÐAs S Ag het- For example, in the CdSeÐAs2S3Ag1.67 heterojunc- 2 3 1.67 tion between a CdSe layer 200 nm thick and an erojunction was calculated from the spectral dependence of the reflectance. The photosensitivity S was As2S3Ag1.67 layer 430 nm thick, the dark voltage with a positive potential at the CdSe layer is equal to 105 mV. defined as the reciprocal of the radiant exposure The exposure of this heterojunction to light at an illu- responsible for the relative change in the reflectance R, minance of 8 × 104 lx induces a photovoltage of 90 mV that is, with a negative potential at the CdSe layer. Upon gen- A eration of the photovoltage with this sign, the electric S = ----- , (3) field induced in the layer of the solid electrolyte is Et

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1128 STETSUN

Table 2. Photosensitivities of the CdSeÐAs2S3Ag1.67 heterojunction leads to photoinduced mass transfer of silver junction at different wavelengths ions to the interface between the layers. This mass λ Ð9 Ð1 Ð1 transfer is induced by an electric field arising upon gen- , nm S, 10 lx s eration of a photovoltage. Thus, the experimental 1200 0.30 results obtained confirm the concept of photoinduced 1180 0.35 ion transfer in a heterojunction with a solid electrolyte, which was proposed in formulating the problem to be 1140 0.20 solved in this work. 1100 1.10 The photoinduced processes occurring in a hetero- 1060 1.20 junction that contains a solid electrolyte with a mixed 1020 0.30 ionicÐelectronic (hole) conductivity can be explained 980 0.10 in terms of the energy band diagram of the heterojunction. For this purpose, we will consider the formation of 940 0.14 a heterojunction between a CdSe layer and an As2S3 900 0.20 layer doped with silver at a content of 25 at. %. 860 0.21 The electron affinity of silver-doped As2S3 exceeds 820 0.70 the electron affinity of CdSe by 0.3 eV [13]. As a con- 780 0.10 sequence, when CdSe is thermally evaporated onto the As S Ag layer, electrons diffuse from CdSe into the 740 0.30 2 3 1.67 As2S3Ag1.67 layer and recombine with holes. These pro- 700 0.20 cesses bring about the formation of a positively charged 660 0.70 region in the CdSe layer and a negatively charged 620 0.40 region in the As2S3Ag1.67 layer. 580 0.50 The formation of the space-charge regions in the 540 0.01 heterojunction that contains the solid electrolyte with mixed conductivity results in two competing processes of ion migration. where E is the illuminance, t is the exposure time, and First, the ions involved in the space-charge region of the solid electrolyte are expelled from this region under ∆R A = ------. (4) the action of an electric field induced by the space- R charge regions. The photosensitivities of the CdSeÐAs S Ag Second, the ion migration can be initiated by the 2 3 1.67 interaction between the charges that form the positively bilayer structure at different wavelengths are presented and negatively charged regions in the heterojunction in Table 2. with positive ions located in the bulk of the solid electrolyte layer. If the charges distributed in the positively 4. DISCUSSION and negatively charged regions in the heterojunction induce a dipole field at distances considerably larger The observed increase in the content of silver ions in than their sizes, the positive ions should be attracted to the vicinity of the interface formed by the heterojunc- the negatively charged region of depletion of the het- tion layers and the formation of photoprecipitates at the erojunction. interface under exposure of the heterojunction to light These two processes of ion migration result in the have demonstrated that the illumination of the hetero- formation of a close-packed layer of ions after the space-charge region of the solid electrolyte. The existence of such close-packed layers of ions is characteristic of interfaces between solid electrolytes and other materials, in particular, in double electrical layers [14]. In the heterojunction with a solid electrolyte, the ion migration occurs until the sum of forces acting on each

1.75 eV ion from the space-charge regions and other particles 0.50 eV becomes equal to zero. The band diagram thus obtained for the CdSeÐ 0.20 eV 1.70 eV As2S3Ag1.67 heterojunction is shown in Fig. 2. This dia- CdSe As2S3Ag1.67 gram was constructed without regard for the migration of ions from deep within the solid electrolyte layer to Fig. 2. Band diagram of the CdSeÐAs2S3Ag1.67 heterojunc- the space-charge region, because the strength of the tion. The parameters of the material are taken from [6, 10]. electric ﬁeld of the space dipole is substantially weaker

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 PHOTOINDUCED ION TRANSFER 1129

+ than that inside the space-charge region of the solid ++++ ––+ + electrolyte. ++++ ––+ + –– + + + – –– ++++ ––+ + + The above band diagram is similar to the band dia- p – + + +

, –

e + + – ++++ ––+ gram of a graded-gap heterojunction [15]. C – + + – ++++ ––+ + – + + The illumination of the CdSeÐAs S Ag hetero- ++++ ––+ + 2 3 1.67 – + – ++++ ––+ + junction leads to the generation of electronÐhole pairs – + + that are separated by the electric field in the space- – + + charge regions of the heterojunction. The potential bar- CdSe As2S3Ag1.67 rier on the side of the As2S3Ag1.67 layer can be overcome by electrons through injection, injection with pre- Fig. 3. Schematic drawing of the space-charge regions in the CdSeÐAs2S3Ag1.67 heterojunction and the concentra- liminary thermalization, and tunneling. As follows tion distributions of photoexcited electrons and holes sepa- from the results of investigations of the CdSeÐ rated by the electric field of the heterojunction over its As2S3Ag2.4 heterojunction [16], the injection is the thickness. most important mechanism of overcoming the potential barrier by electrons. Owing to these processes, electrons are accumulated in quasi-neutral regions of the heterojunction toward the CdSe layer (the electron CdSe layer, whereas holes are concentrated in quasi- wind effect [17]). neutral regions of the As2S3Ag1.67 photodoped layer. The interface between the CdSe and As2S3Ag1.67 The schematic drawing of the space-charge regions layers is a boundary between polycrystalline and amorphous materials with different bond lengths. Therefore, in the CdSeÐAs2S3Ag1.67 heterojunction is presented in Fig. 3. The concentration distributions of photoexcited this interface contains a large number of defects in the electrons and holes separated by the electric field of the form of dangling bonds. It is known that dangling heterojunction are also schematically shown in this fig- bonds can serve both as electron trapping centers and as ure. The distributions of photoexcited electrons and recombination centers [2, 3]. As a consequence, the holes are given according to the data taken from [3]. photodeposition of silver occurs through the recombination of an electron and an ion at the interface. The Under exposure of the heterojunction to light, the precipitates can grow either through the same process electric field induced in the region between photoex- (the capture of an electron by a dangling bond and cited electrons and holes separated by the electric field recombination of this electron with another ion, fol- of the heterojunction (Fig. 3) acts on positive Ag+ ions lowed by binding of both ions) or through the capture and thus induces mass transfer of silver ions toward the of an electron upon the photodeposition of silver and interface. The former field affects both the quasi-neu- recombination of the next ion with this electron. When tral region of the solid electrolyte layer and the space- multiply repeated, these processes provide the growth charge region. of precipitates. Owing to the spatial distribution of the electric field The theoretical analysis of the photoinduced pro- induced by photoexcited carriers and the electric field cesses observed in heterojunctions based on mixed of the heterojunction, the most intensive mass transfer ionicÐelectronic (hole) conductors and the analysis of of ions occurs from the quasi-neutral region to the the experimental results obtained have demonstrated depletion region of the solid electrolyte. This can be that, depending on the conditions of illumination of the explained by the fact that the strength of the electric heterojunction with a solid electrolyte, there are two field in the quasi-neutral region is relatively high and is variants of photoinduced ion transfer: (1) photoinduced directed toward the interface between the layers, ion transfer with subsequent photodeposition of the whereas the field strength of the heterojunction in the chemical element responsible for the ionic conductivity space-charge region is opposite in direction (i.e., it is and (2) photoinduced ion transfer without photodeposi- opposed to the electric field of separated electron–hole tion. pairs). The second variant is observed under exposure of After the ion transfer from the quasi-neutral region the heterojunction to light at lower illuminances when to the space-charge region, silver ions diffuse in the the electric field induced upon the generation of a pho- space-charge region of the solid electrolyte toward the tovoltage cannot ensure mass transfer of ions to the interface between the layers. This diffusion becomes space-charge region of the solid electrolyte toward the possible under exposure of the heterojunction to light, interface between the layers. because the electric field of separated electron–hole The analysis of the results obtained in the present pairs reduces the strength of the electric field of the het- work allowed us to propose a new method of designing erojunction and, in the limiting case at high illumi- a medium for optical recording of information. This nances, can provide the band alignment of the hetero- medium can be prepared in the form of a bilayer or mul- junction. Furthermore, silver ions are entrained by elec- tilayer thin-film structure in which heterojunctions trons that moves in response to the electric field of the capable of generating a photovoltage are formed

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1130 STETSUN between layers of a solid electrolyte with mixed con- The analysis of the photoinduced processes occur- ductivity and layers of other materials with electronic ring in heterojunctions based on solid electrolytes with or hole conductivity. Exposure of the heterostructure to mixed conductivity showed that, depending on the type light leads to photoinduced transfer of ions across the of band diagram of the heterojunction and the type of layer of the solid electrolyte. Moreover, depending on conductivity of the constituent materials, the photoin- the conditions of illumination of the heterojunction and duced ion transfer can occur in two directions. This pro- the properties of the constituent materials, there can cess can proceed from deep within the solid electrolyte occur photodeposition of the chemical element respon- layer to the interface between the layers or in the oppo- sible for the ionic conductivity in the solid electrolyte. site direction (from the interface between the layers to In this medium, the data recording is based on the dif- the surface of the solid electrolyte layer). ference between the reﬂectances (or transmittances) in exposed and unexposed regions. REFERENCES The proposed method makes it possible to prepare media for optical recording of information from numer- 1. A. G. Milnes and D. L. Feucht, Heterojunctions and MetalÐSemiconductor Junctions (Academic, New York, ous combinations of solid electrolytes (possessing a 1972; Mir, Moscow, 1975). mixed conductivity) with semiconductors, dielectrics, 2. B. L. Sharma and R. K. Purohit, Semiconductor Hetero- and metals. Consequently, heterostructures based on junctions (Pergamon, Oxford, 1974; Sovetskoe Radio, solid electrolytes with mixed conductivity can be Moscow, 1979). expected to differ substantially in terms of their physi- 3. A. L. Fahrenbruch and R. H. Bube, Fundamentals of cal properties. In this case, the possibility of producing Solar Cells (Academic, New York, 1987; Énergoatomiz- heterojunctions with band diagrams of different types dat, Moscow, 1987). has assumed a particular importance. Furthermore, 4. Current Topics in Photovoltaic, Ed. by T. G. Coutts and solid electrolytes differ in the type and magnitude of J. D. Meakin (Academic, London, 1972; Mir, Moscow, conductivity. For example, the conductivity of 1988), p. 25. As2S3Agx (x = 0.9Ð2.4) glasses falls in the range from 5. A. van der Ziel, Solid State Physical Electronics (Pren- 7.6 × 10Ð8 to 3.08 × 10Ð4 ΩÐ1 cmÐ1 [7], whereas the ionic tice Hall, Englewood Cliffs, 1957), p. 284. conductivity of the (Ag2GeS3)48(AgI)52 glass is equal to 6. K. L. Chopra and S. R. Das, Thin Film Solar Cells (Ple- 6 × 10Ð3 ΩÐ1 cmÐ1 [18]. num, New York, 1983; Mir, Moscow, 1986). It should be noted that the ionic conductivities of 7. V. A. Dan’ko, I. Z. Indutnyœ, and V. I. Min’ko, Fiz. Khim. Cu RbCl I and Ag RbI crystals are considerably Stekla 18, 128 (1992). 4 3 2 4 5 8. I. Z. Indutnyœ, Zh. Nauchn. Prikl. Fotogr. 39, 65 (1994). higher (0.5 and 0.3 ΩÐ1 cmÐ1, respectively) [19]. 9. K. Tanaka, J. Non-Cryst. Solids 137Ð138, 1021 (1991). If the electronic (hole) component of the conductiv- 10. I. Z. Indutnyœ, M. T. Kostyshin, O. P. Kosyarum, ity is provided in the aforementioned materials either V. I. Min’ko, E. V. Mikhaœlovskaya, and P. F. Roma- through doping with impurities or by other methods nenko, Photoinduced Interactions in MetalÐSemicon- and the materials thus modiﬁed are used to prepare het- ductor Structures (Naukova Dumka, Kiev, 1992). erojunctions capable of generating a photovoltage, it 11. A. I. Stetsun, I. Z. Indutnyi, and V. G. Kravets, J. Non- can be expected that the time of data recording with Cryst. Solids 202, 113 (1996). these heterojunctions will be several orders of magni- 12. I. Z. Indutnyœ, A. I. Stetsun, M. V. Sopinskiœ, and tude shorter than that for the CdSeÐAs2S3Agx (x = 0.9Ð V. D. Nechiporuk, Optoélektron. Poluprovodn. Tekh. 2.4) heterostructures. 30, 42 (1995). 13. V. S. Fomenko, Emission Properties of Materials (Nauk- ova Dumka, Kiev, 1981), p. 184. 5. CONCLUSIONS 14. V. M. Arutyunyan, Usp. Fiz. Nauk 158, 255 (1989) [Sov. Thus, the results obtained in this work demonstrated Phys. Usp. 32, 521 (1989)]. that a necessary condition for photoinduced mass trans- 15. W. G. Oldham and A. G. Milnes, Solid State Electron. 6, fer of ions in a heterojunction that contains a solid elec- 121 (1963). trolyte with a mixed ionicÐelectronic (hole) conductiv- 16. A. I. Stetsun, Fiz. Tekh. Poluprovodn. (St. Petersburg) ity is the generation of a photovoltage. 37, 1197 (2003) [Semiconductors 37, 1169 (2003)]. 17. V. B. Fiks, Ionic Conductivity in Metals and Semicon- Depending on the conditions of illumination of the ductors (Nauka, Moscow, 1983), pp. 108Ð123. heterojunction and the properties of the constituent 18. A. Feltz, J. Non-Cryst. Solids 90, 545 (1987). materials, there are two variants of photoinduced ion 19. Yu. Ya. Gurevich, Solid Electrolytes (Nauka, Moscow, transfer: (1) photoinduced ion transfer with subsequent 1986). photodeposition of the chemical element responsible for the ionic conductivity and (2) photoinduced ion transfer without photodeposition. Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

Physics of the Solid State, Vol. 46, No. 6, 2004, pp. 1131Ð1140. Translated from Fizika Tverdogo Tela, Vol. 46, No. 6, 2004, pp. 1098Ð1107. Original Russian Text Copyright © 2004 by Galchenkov, Ivanov, Pyataœkin. LOW-DIMENSIONAL SYSTEMS AND SURFACE PHYSICS

Electron Localization in Conducting LangmuirÐBlodgett Films L. A. Galchenkov, S. N. Ivanov, and I. I. Pyataœkin Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Mokhovaya ul. 11, Moscow, 125009 Russia e-mail: [email protected] Received October 14, 2003

Abstract—The temperature dependence of the intracrystallite conductivity of Langmuir–Blodgett films of the (C16H33ÐTCNQ)0.4(C17H35ÐDMTTF)0.6 charge-transfer complex (CTC) is studied by measuring the surface acoustic wave attenuation in a piezoelectric delay line coated with such a film. (C16H33ÐTCNQ)0.4(C17H35Ð DMTTF)0.6 is a surface-active CTC made from a 1.5 : 1 mixture of heptadecyl-dimethyltetrathiafulvalene (C17H35ÐDMTTF) and hexadecyl-tetracyanoquinodimethane (C16H33ÐTCNQ). The temperature dependence of the intracrystallite conductivity is found to have a maximum at TMD = 193.5 K. Above TMD, the conductivity of the films is metallic (∂σ/∂T < 0), while below this temperature it obeys a law that is close to the one-dimensional Mott law. The decrease in the conductivity with decreasing temperature at T < TMD is shown to be related to the localization of electron states in the quasi-one-dimensional system under study and to be caused by the presence of impurities and defects in the TCNQ chains, along which a charge is transferred. The detected variation in the conductivity with temperature below TMD is found to qualitatively and quantitatively agree with the model of localization in a weakly disordered quasi-one-dimensional system proposed earlier by Nakhmedov, Prigodin, and Samukhin. Fitting the experimental results to the theoretical dependences obtained in the framework of this model allows us to find the electron–phonon and electron–impurity scattering times. The structural parameters of the conducting layer are used to estimate the density of states at the Fermi level and the Fermi velocity in the films. With these values, the mean free path and the localization length in the films under study are determined. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION Two main causes of the decreased electrical conductivity of LB films are currently known: first, their poly- Conducting Langmuir–Blodgett (LB) films based crystalline structure and, as a result, the presence of on surface-active charge-transfer complexes (CTCs) intercrystallite potential barriers causing thermally are being intensively studied now because of their activated transport through them, and, second, the sig- potential application in molecular electronics (ME) [1]. nificant difference in the mechanism of intracrystallite The LB technology makes it possible to fabricate conductivity in the films, compared to that in the bulk homogeneous conducting films having a nanometer- parent crystals. It is obvious that a detailed study of scale thickness and stable properties from CTC mole- electron transport in crystallites might contribute to cules, which opens up broad fields for their application improving the conductivity of such films as a whole. in thin-film ME devices [2]. Moreover, such studies could reveal new physical phenomena, since the conductivity mechanism in LB sys- In the last decade, substantial progress has been tems can have unique features that are absent in bulk achieved in producing LB structures with increasing crystals because of the low dimensionality of these sys- conductivity. Among CTC-based LB films, the maxi- tems. mum conductivity has been reached in systems based on bis(ethylenedioxy)tetrathiafulvalene (BEDOÐTTF) Electron transport in individual crystallites of LB with various acceptors. For example, films made from films was first investigated in [6] using a microwave- a mixture of (BEDOÐTTF)0.71(C10H21ÐTCNQ)0.29 with cavity perturbation technique that had been success- arachidic acid have a conductivity σ of about 10 S/cm fully applied earlier for studying the conductivity of at room temperature [3], whereas LB films made from bulk crystals of organic conductors [7, 8]. However, the a mixture of BEDOÐTTF with behenic acid exhibit a application of this technique for examining LB systems record room-temperature value σ ≈ 40 S/cm [4]. It encounters difficulties, since it is impossible to separate should be noted that the conductivities of LB systems such films from the substrates. The microwave field are several times lower than those of the corresponding perturbation caused by a substrate with thickness of bulk organic-conductor crystals.1 This circumstance about 0.5 mm is many orders of magnitude higher than narrows the area of application of LB films in ME. the perturbation caused by an LB film with thickness of 300–500 Å. Therefore, to ensure the required measure- 1 σ ≈ For example, (BEDOÐTTF)ReO4 á H2O crystals have ment accuracy, it is necessary to provide a stable driv- 147 S/cm at room temperature [5]. ing-oscillator frequency and stable parameters of the

1132 GALCHENKOV et al. because of the localization of electron states in the 1 quasi-one-dimensional system under study and that the Film Ω metallic behavior of conductivity at T > TMD is caused by the suppression of the localization effects by inelas- 2 tic electron–phonon scattering, which becomes significant at high temperatures. It should be noted that the structure of the films under study and the conductivity a bc mechanism in them are typical of conducting LB systems based on quasi-one-dimensional CTCs. There- 3 fore, all the conductivity mechanism features caused by one-dimensional localization that are discussed here are common for this class of conducting LB films.

6 2. EXPERIMENTAL TECHNIQUE 4 Figure 1 shows the measurement circuit. As SAW delay lines, we used substrates of lithium niobate ° (LiNbO3, Y + 128 cut) with three groups of interdigital 5 7 transducers (IDTs) photolithographically patterned onto the substrate surface. A pulse-modulated signal with a high-frequency component was applied to the Fig. 1. Schematic diagram of the experimental setup: central IDT group from a high-frequency oscillator. (1) cryostat cold finger, (2) sapphire substrate with sput- Thus, this group was the source of SAWs and the two tered Au electrodes for measuring the conductivity by the other transducer groups were receivers. The film stud- two-probe method, (3) delay line, (4) high-frequency oscillator, (5) pulse oscillator, (6) precision attenuator, and (7) ied was deposited between IDT groups b and c (Fig. 1), oscilloscope. and the space between groups a and b remained clear. This measurement circuit allowed us to eliminate sys- tematic errors related to random spontaneous changes experimental device used during measurement. These in the output power of the driving oscillator. Signals circumstances determine the measurement accuracy from IDT groups a and c were fed to a two-channel and the accessible temperature range. Earlier [9], we high-frequency oscilloscope, and the readout of the sig- proposed another approach for studying the tempera- nal levels was performed visually. To increase the sen- ture dependence of the intracrystallite conductivity of sitivity, a precision attenuator was built in one of the films, which was based on measuring the attenuation of receiving paths. surface acoustic waves (SAWs) in piezoelectric delay Before the deposition of the LB film, the surface of lines coated with an LB film. In that work, we showed the delay line was carefully cleaned chemically and that the SAW attenuation is mainly determined by the treated in an oxygen plasma. Then, two layers of stearic film conductivity and is directly proportional to the acid were deposited in the gap between groups b and c acoustic-wave frequency. Moreover, the film-induced for hydrophobization of the surface. They were coated attenuation is higher by 1Ð1.5 order of magnitude with 18–20 layers of the film under study by the verti- (depending on the SAW frequency) than the attenuation cal-lifting LB technique. The deposition of the film induced by the losses due to the divergence of a SAW onto a sapphire substrate with contacts for two-probe beam and dissipative processes that are unrelated to dc conductivity measurements (Fig. 1) was performed conductivity (losses in the material delays lines and the simultaneously with the film deposition onto the delay attenuation caused by the film viscosity). Therefore, the line. Then, the delay line and the sample for two-probe studying of electron transport in crystallites with this measurement were glued to the cold finger of the insert technique is much simpler and not a less accurate pro- of an evacuated cryostat, which was used to measure cedure than using the microwave technique. the temperature dependence of the acoustic-wave atten- In this work, we measured the temperature depen- uation. dences of the intracrystallite conductivity of (C16H33Ð As shown in [9], the main cause of SAW attenuation TCNQ)0.4(C17H35ÐDMTTF)0.6 films using the acoustic is the interaction of the field of an acoustic wave with technique developed in [9]. The temperature depen- charge carriers in the film. The alternating electric field dence of the intracrystallite conductivity is found to with the SAW frequency that accompanies a strain have a maximum at TMD = 193.5 K. Above TMD, the wave in the piezoelectric substrate induces electric cur- conductivity of the films exhibits metallic behavior rents in the deposited film and, hence, Joule’s losses. As (∂σ/∂T < 0), whereas below this temperature it follows a result of this interaction, the wave power is absorbed. a close-to-Mott law. We show that the conductivity However, this source of attenuation is not the only one, decreases with decreasing temperature at T < TMD since the wave also loses its energy due to the viscosity

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

ELECTRON LOCALIZATION 1133 of the film and dissipative processes occurring in the 1 material of the delay line. To exclude the contributions from these sources, we performed the following check 2 –1 experiment. The insert with the glued samples was 10 (b) removed from the cryostat and placed inside a quartz (a) tube, which was then filled with nitrogen. Then, the H35 C17 SSCH samples were irradiated with light from a DRT 125-1 3 H mercury ultraviolet lamp located at a distance of 7Ð 10–2 S S CH 3 h 8 cm from the sample for 200 h (the radiation intensity C17H35–DMTTF C16H33 was about 100 µW/cm2); as a result, the film became Conductivity, S/cm NC CN Substrate nonconducting. Then, the insert was again placed in the NC CN cryostat and we measured the temperature dependence 10–3 C16H33–TCNQ of the attenuation caused by the nonconducting film. 468 The results of these measurements were subtracted 103/T, K–1 from the data obtained in the experiments performed on the conducting film. The thus-determined attenuation Fig. 2. Temperature dependences of the conductivity of the (per unit length) was then used to calculate the film LB film measured with (1) the acoustic technique and conductivity from the Adler formula [9], whose tem- (2) the dc two-probe method. The temperature dependence of the dc conductivity is fitted to the function σ ∝ perature dependence is discussed in this work. − ≈ exp( Ta/T), where Ta 1393 K. Inset (a) shows the struc- We measured several delay lines with IDTs having tural formulas of the molecules forming the film. Inset resonance frequencies in the range 200Ð400 MHz and (b) schematically shows the cross section of the LB system obtained similar results. Here, we present the data consisting of four monolayers. The conducting bilayers of thickness h ≈ 10 Å are formed by the –TCNQ and –DMTTF obtained on the typical sample with fundamental fre- head groups, and the dielectric bilayers of thickness H ≈ 25 Å quency f = 355.6 MHz. 0 are formed by the ÐC16H33 and ÐC17H35 side groups.

3. RESULTS AND DISCUSSION (a) (b) Figure 2 shows the temperature dependences of the Edc Eac conductivity of the film found using the acoustic and usual dc techniques (curves 1 and 2, respectively). The temperature dependence measured from the SAW attenuation is seen to have a characteristic feature at TMD = 193.5 K; above TMD, the conductivity of the film is metallic (∂σ/∂T < 0), whereas below this temperature it varies according to the law lnσ ∝ Ð1/Tγ (where 0 < γ Crystallites < 1). At the same time, the measured dc conductivity Barriers of the film decreases with decreasing temperature σ ∝ (c) according to the Arrhenius law exp(ÐTa/T), where σ ≈ ⊥ Ta 1393 K. This temperature dependence is mono- TCNQ stacks e tonic and has no characteristic features at TMD. Note σ|| that the activation energy kBTa of the dc conductivity measured in this work (0.12 eV) coincides with the value measured earlier in [10] with the four-probe method. Fig. 3. Two-dimensional polycrystalline structure of the conducting bilayer: (a) dc conductivity (determined by It is generally accepted [1] that the activation char- intercrystallite barriers) and (b) transport in the high-fre- acter of the dc conductivity of the LB film is caused by quency field of an acoustic wave. Within a half-period of its polycrystalline structure, because the film consists field oscillations, there is no time for charge to accumulate of randomly oriented two-dimensional (2D) crystallites at grain boundaries; therefore, the applied electric field is not shielded by the space charge. Thus, the effect of inter- (Fig. 3a); therefore, charge carriers need to overcome crystallite barriers on the conductivity is eliminated. (c) intercrystallite potential barriers, whose height deter- Transport at the microscopic level (schematic); electrons mines the activation energy kBTa. The regions adjacent move along the TCNQ stacks, jumping from one stack to to grain boundaries make a dominant contribution to the similar nearest neighbor stack from time to time. the resistivity of the film; the variation of the electric potential within crystallites is small as compared to its jump at the boundaries. Therefore, the temperature film turns out to be masked, and it is virtually impossi- dependence of the dc conductivity mainly characterizes ble to extract it from the results of such measurements. the properties of the intercrystallite barriers. Informa- Unlike the dc measurements, the acoustic technique tion on the electron transport within crystallites of the measures a quantity that is related closely to the intrac-

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1134 GALCHENKOV et al. σ ω 〈σ 〉 σ ω 4.76 Å (a) sented in the form ( , T) = (T) + ac( , T), where 〈σ(T)〉 is the dc conductivity of the 2D polycrystal with Side group σ ω short-circuited intercrystallite barriers and ac( , T) is its ac conductivity, which is an isotropic quantity. The Side group problem of finding 〈σ〉 was solved by Dykhne [11]: 〈σ〉 = σ σ , where σ and σ are the principal val- TCNQ stacks 11 22 11 22 ues of the intracrystallite conductivity tensor. As will be σ 〈σ〉 shown below, ac is several times smaller than at frequencies of several hundreds of megahertz. As a result, it is the intracrystallite conductivity averaged over all possible crystallite orientations that is mea- DMTTF stacks sured using the acoustic technique. Hence, the temperature dependences of the conductivity measured with this technique (Fig. 2, data set 1) reflect the features inherent in the electron transport within crystallites. 6.37 Å Below, we will consider the microscopic structure of the films, the mechanism of charge transfer within aLB (b) crystallites, and the related theoretical models. This consideration will allow us to choose expressions for σ σ σ 11, 22, and ac, which will be used for fitting the σ π experimental data to the function (2 f0, T) = σ ()σ() σ π LB 11 T 22 T + ac(2 f0, T) and finding the quanti- ֊ a h/2 ties that characterize the electron transport in the films.

3.1. The Structure of the Films and a Model of Electron Transport in Them Inset b in Fig. 2 schematically shows the cross section of the LB system consisting of four monolayers Fig. 4. (a) Schematic representation of the conducting formed by CTC molecules (Fig. 2, inset a). The thick- monolayer structure (according to data from [13]); for sim- nesses of the dielectric (H ≈ 25 Å) and conducting (h ≈ plicity, only the first two ÐCH2Ð links of the side groups − 10 Å) bilayers were determined by small-angle x-ray ( CnH2n + 1, n = 16, 17) are shown. (b) Structure of the conducting bilayer; the axes of the TCNQ and DMTTF stacks diffraction [12]. The structure of the conducting layer are normal to the sketch plane. aLB ≈ 7.65 Å [13], h/2 ≈ 5 Å, was schematically presented in [13] (Fig. 4a). Based on those data, we show the detailed structure of the con- LB 2 LB 2 and a⊥ = ()h/2 + ()a ≈ 9.2 Å. ducting bilayer in Fig. 4b. It is seen that the head groups (–DMTTF and –TCNQ) in the film are arranged in separate stacks (chains), similarly to the molecular packing rystallite conductivity. Indeed, as noted above, the in bulk TTFÐTCNQ crystals. However, the distance SAW attenuation in delay lines coated with a conduct- between the neighboring DMTTF and TCNQ stacks in ing Langmuir film is determined by the ohmic losses the layer is aLB ≈ 7.65 Å and the lattice parameter along caused by the coupling of the high-frequency (107Ð the TCNQ chains is bLB ≈ 4.76 Å, whereas the corre- 109 Hz) electric field of an acoustic wave with charge sponding distances in TTFÐTCNQ crystals are equal to carriers in the film. The non-Arrhenius temperature acryst/2 ≈ 6.8 Å and bcryst ≈ 3.8 Å, respectively [14]. behavior of the film conductivity measured with the Charge carriers in the bilayer move mainly along the SAW technique suggests that charge carriers do not DMTTF and TCNQ stacks, just as in the case of bulk accumulate at grain boundaries at such high frequen- crystals. From time to time, the carriers jump between cies; therefore, the shielding of the applied electric field similar nearest neighbor chains. In other words, the sys- of the acoustic wave by the space charge is insignifi- tem is quasi-one-dimensional. The negative sign of the cant. Thus, electron transport in a field of such a high Hall coefficient detected in our films in [15] indicates frequency is insensitive to intercrystallite barriers and that, just as in the crystals, the films exhibit electronic the ohmic losses are completely determined by the con- conductivity and that a charge is mainly transferred ductivity mechanism within the crystallites. Actually, along the acceptor TCNQ stacks. Note that the distance with the SAW technique, we measure the conductivity between the nearest neighbor TCNQ chains in the σ(ω, T) of the polycrystal formed by randomly oriented LB ()2 ()LB 2 ≈ 2D crystallites with short-circuited intercrystallite bar- bilayer is a⊥ = h/2 + a 9.2 Å (Fig. 4b), riers (Fig. 3b). The conductivity σ(ω, T) can be repre- which virtually coincides with the distance between the

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 ELECTRON LOCALIZATION 1135 nearest neighbor TCNQ stacks in the bulk crystals In the range T1 < T < T0 (mode I), the transverse con- 2 cryst cryst ductivity is described by the Mott-type relation (a⊥ = c /2 ≈ 9.1 Å) [14]. σ 2 ε 2 ν τ ប 2 []()1/2 As noted above, the dc conductivity within the 2D ⊥ = 2e g()F a⊥ ph()t⊥ / exp Ð22T 0/Tz , (1) crystallite of the film is characterized by a 2 × 2 tensor and the longitudinal conductivity obeys the Arrhenius σ ij. One of the principal axes of this tensor is parallel to law the TCNQ stacks, and the other axis is orthogonal to σ 2 ()νε ξ 2 ()π () them (Fig. 3c). In the principal axes frame, we have || = 2e g F ph || /2 T 0/2T σ σ δ δ σ σ (2) ij = ik kj, where kj is the unit tensor and 11 = || and × []() σ σ exp ÐT 0/2T , 22 = ⊥ are the conductivities along the TCNQ stacks πប τ ε (longitudinal conductivity) and along the orthogonal where T0 = /(4kB ), e is the electron charge, g( F) = direction to them (transverse conductivity), respec- 2 1/(πបv a⊥ ) is the density of states (per spin) at the tively. Impurities and defects in the TCNQ chains, F along which a charge moves, cause electron-wave scat- Fermi level, vF is the Fermi velocity, a⊥ is the distance κ κ tering, whose intensity is characterized by the electronÐ between nearest chains, T1 = 3T0/(4 ), = π impurity scattering time τ. The quantum interference of ln[4kBT0/( t⊥)], z is the number of nearest neighbor scattered electron waves leads to the localization of chains (in the two-dimensional case in question, z = 2; Ð1 electronic states, thereby changing the conductivity in Fig. 4b), ξ|| = 4v τ, and ν = τ . character and decreasing its magnitude. On the con- F ph in κ2 trary, the interaction between individual TCNQ chains, In the range T2 < T < T1, where T2 = T0 /(2 ) which is characterized by the interchain transfer inte- (mode II), the transverse conductivity is still described gral t⊥, favors delocalization. The presence of these two by Eq. (1), but the longitudinal conductivity is deter- 3 Ӷ opposite tendencies results in the possibility of the mined by Eqs. (68)Ð(71) from [18]. Finally, at T T2 quasi-one-dimensional disordered system being in (mode III), the temperature dependences of the longitu- either metallic and dielectric state, depending on the dinal and transverse conductivities are determined by 4 σ relation between the values of t⊥ and ប/τ. Note that the Eqs. (77)Ð(80) from [18]. As shown in [18], ac mani- ω ≥ ω ν inelastic scattering of electrons by phonons with suffi- fests itself at frequencies 1 = phexp[Ð3T0/(2T)]; σ ω ω ω σ ω ∝ ciently high energy dispersion (the intensity of this pro- more specifically, ac( , T) = 0 at < 1 and ac( , T) ωs(ω, T) ω ≤ ω ≤ ω ν 1/2 cess is characterized by the electronÐphonon scattering at 1 2 = phexp[Ð2(2T0/T) ], where τ τ Ӷ τ ω ω ω time in) also weakens the localization and, at in , s( , T) 1 at 2. At higher frequencies ω ω completely suppresses it [16]. It was found in [17] that, ( > 2), the ac conductivity should be calculated at t⊥τ/ប = 0.3, the quasi-one-dimensional disordered within the pair approximation. Therefore, we have σ ω σ ω system at hand undergoes a metalÐinsulator transition ac( , T) = p( , T), where (the system is in the insulator state at t⊥ < 0.3ប/τ and in ()τ 3 2 ()ε 2 the metallic state at t⊥ > 0.3ប/τ). The frequency and 4v F e N F 2 2 σ = ------a⊥ k Tωνln ()/ω , (3) temperature dependences of the conductivity of this p 96 B ph ប τ system at various relations between t⊥ and / were ε ε ≅ ប τ in accordance with [19], where N( F) = 2g( F). The analyzed in [18], where the cases t⊥ tc = 0.3 / (the Ӷ expressions describing the frequency dependence of system is near the metalÐinsulator transition) and t⊥ tc the conductivity in the different ω ranges transform (the system is characterized by a weak transverse over- continuously into one another. Below, we will find lapping and is far from the transition) were considered. which of the modes of the dc and ac conductivities In what follows, the expressions derived in that work describes our experimental data at T < T most ade- σ σ σ ω MD for ||(T), ⊥(T), and ac( , T) will be used for fitting the quately. experimental temperature dependence of the conductivity to the function σ(2πf , T) = σ||()σT ⊥()T + 0 3.2. Determination of g(ε ), t⊥, and the Form σ π ν F ac(2 f0, T) in order to determine the parameters char- of ph(T) acterizing the electron transport within the crystallites. σ σ σ ω The relations given for ||(T), ⊥(T), and ac( , T) ν ε The experimental temperature dependence (Fig. 2, are seen to depend on four parameters: ph, T0, g( F), ε data set 1) shows that our films belong to systems with and t⊥. The density of states g( F) and the interchain Ӷ weak interchain overlapping (t⊥ tc), where, according transfer integral t⊥ can be estimated from the structural to [18], the conductivity is nonmetallic in the tempera- 2 τ τ Equations (48) and (63) in [18], describing σ|| and σ⊥ in the range ture range corresponding to the inequality < in(T). In T1 < T < T0, contain misprints and, hence, somewhat differ from this range, three characteristic regions (modes I, II, III) Eqs. (1) and (2) in this work. in the temperature dependence of the conductivity can 3 Equation (68) in [18] contains an incorrect sign in the exponent. 4 be distinguished (see [18, Fig. 2]). In Eqs. (77)Ð(79) from [18], T2 should be replaced by T0.

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1136 GALCHENKOV et al. data given in Fig. 4. In turn, the fitting of the measured As noted in [16, 18], the ν (T) = τ Ð1 (T) dependence temperature dependences (Fig. 2, data set 1) to a func- ph in σ π ν is different in temperature ranges above the Debye tem- tion (2 f0, T) allows us to determine ph and T0 and, Θ hence, to find the electron–phonon and electron–impu- perature D of the system under study and below it. At ε Θ τ Ð1 ∝ Θ τ Ð1 ∝ 3 rity scattering times. Using these values and g( F), we T > D, we have in T, and, at T < D, in T . The will find the mean free path and the localization length Debye temperature of DMTTFÐTCNQ crystals was in the films. Θ cryst ≈ found to be D 85 K [25]. Let us show that, in LB The interchain transfer integral t⊥ characterizes the Θ LB ≤ Θ cryst intensity of electron tunneling between neighboring films, D D . Indeed, as noted above, the dis- TCNQ chains and decreases exponentially with tance between the neighboring DMTTF and TCNQ increasing interchain distance a⊥. As noted above, this chains in the film and the distance between the nearest LB neighbor molecules in the chain are greater than the distance in the film bilayer, a⊥ , virtually coincides respective distances in the crystals. Therefore, the with the distance between TCNQ chains in the bulk bonding forces between individual molecules in the cryst LB crystals, a⊥ ; therefore, t⊥ in the films should coin- chain and between chains in the monolayer are weaker cryst than the corresponding forces in the crystals. Moreover, cide with t⊥ . The latter quantity is known to be the insulator layers (Fig. 2, inset b) formed by the cryst cryst hydrocarbon side groups (ÐC H , n = 16, 17) t⊥ ≈ 5 meV [20]. This value of t⊥ is confirmed by n 2n + 1 weaken the interaction between the conducting bilayers modern quantum-chemical HartreeÐFock calculations of the film. As a result, the LB structures turn out to be LB [21, 22]. Thus, we have t⊥ ≈ 5 meV. “softer” systems than the crystals, which results in the Θ LB ≤ Θ cryst Based on the structural parameters of the bilayer inequality D D . Hence, we should assume that shown in Fig. 4, we can find the value of vF in the film ν τ Ð1 ∝ ph = in T over the whole temperature range under and determine the value of g(ε ). Indeed, since v ∝ bt|| F F study (125 < T < 300 K). (where t|| is the intrachain transfer integral and b is the To quantitatively describe the temperature depen- translation period along a chain), we obtain v cryst /v LB = F F τ Ð1 cryst cryst LB LB cryst dence of in in TTFÐTCNQ crystals, Bright et al. [26] b t|| /()b t|| . The value of t|| is known to be τ Ð1 cryst LB proposed using the Hopfield relation in (T) = 135 meV [20], and the values of b and b are given π λ ប λ LB 2 kBT / , where is the dimensionless electronÐ above; therefore, we only have to calculate t|| . Simi- phonon interaction constant (according to [27], λ ≈ larly to t⊥, t|| also decreases exponentially with increas- 0.23 in TTFÐTCNQ). When fitting our experimental σ π ing distance u between the neighboring TCNQ mole- data to the dependence (2 f0, T), we also assume that ν π λ ប cules in the stack and can be represented in the form t|| = ph(T) = 2 kBT / . Since the electronÐphonon interac- Auexp(Ðu/r0) [19]. The constants A and r0 can be deter- tion constant in the LB films under study is unknown, cryst the parameter λ is taken to be free and adjusted during mined from the condition t|| cryst = t|| and uu= fitting. ∂ ∂ cryst t||/ u cryst = –0.20 eV/Å [23] (here, u = Finally, the quantity T , which enters into Eqs. (1) uu= 0 bcrystcosθ = 3.17 Å [14], where θ is the angle between and (2) and is related to the electronÐimpurity scatter- the stack axis and the normal to the plane of the TCNQ ing time, is also taken to be a free parameter and is molecule). Assuming that θ in our films has the same determined by fitting. value as in TTFÐTCNQ crystals, we obtain uLB = bLBcosθ = 3.95 Å. After simple calculations, we arrive 3.3. Fitting the Experimental Data cryst LB LB at T < TMD at t|| /t|| ≈ 3.18, whence it follows that t|| ≈ cryst LB Before processing the experimental data, let us con- 42.5 meV and v /v ≈ 2.55. The Fermi velocity in σ π F F sider the second term in the expression (2 f0, T) = cryst ≈ × σ ()σ() σ π TTFÐTCNQ crystals is known to be v F 1.2 || T ⊥ T + ac(2 f0, T). As seen from Eq. (3), LB σ π 7 ≈ × 6 ac(2 f0, T) decreases almost linearly with decreasing 10 cm/s [23]; therefore, we have v F 4.7 10 cm/s. LB cryst temperature, whereas at T < TMD the experimental Taking into account a⊥ ≈ a⊥ , we calculate the ratio dependence (Fig. 2, data set 1) most likely decrease of the density of states at the Fermi level in the films and exponentially with decreasing temperature. Therefore, LB ε cryst ε ≈ v cryst v LB ≈ we can conclude that, in the temperature range under the crystals: g ( F)/g ( F) F /F 2.55. The study, the ac contribution to the conductivity at the cryst ε × 21 Ð1 Ð3 value of g ( F) is known to be 4.7 10 eV cm given frequency f0 = 355.6 MHz is much lower than the LB ε ≈ × 22 Ð1 Ð3 [24], whence it follows that g ( F) 1.2 10 eV cm . dc contribution. Hence, as a fitting function, we can

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 ELECTRON LOCALIZATION 1137

σ ()σ() σ π simply use || T ⊥ T instead of (2 f0, T). After 1 λ T0 and have been determined, we can calculate 2 σ π ac(2 f0, T) and check the validity of neglecting this σ π TMD term in (2 f0, T). 10–1 ε × 22 Ð1 Ð3 Assuming t⊥ = 5 meV and g( F) = 1.2 10 eV cm 3 λ and varying T0 and , we fit the experimental data σ ()σ() (Fig. 2, data set 1, T < TMD) to the ⊥ T || T Conductivity, S/cm dependence, whose components are determined from –2 Eqs. (1) and (2), respectively.5 As a result, we find that 10 4 ± λ ± T0 = 1344 4 K and = 0.26 0.01. As noted above, 468 103/T, K–1 the expressions for σ⊥(T) and σ||(T) in mode I are valid in the temperature range T < T < T however, the cal- 1 0 Fig. 5. Temperature dependences of the conductivity culated value of T0 leads to T1 = 298 K > TMD. Whence obtained using different approximations: (1) the experimen- it follows that Eqs. (1) and (2) cannot be applied for tal data determined with the acoustic technique (Fig. 2); describing our results at T < T . (2) the temperature dependence of the conductivity of the MD σ ∝ Ð1 type T constructed according to [18] at T > TMD; and The expressions for the longitudinal and transverse (3) the best fit to the experimental results at T < T with conductivities in mode III also cannot be used to MD σ ()σ() σ σ explain our data. Indeed, these expressions are valid at the dependence || T ⊥ T , where ||(T) and ⊥(T) are T Ӷ T . Therefore, for them to describe the experimen- the mode II longitudinal and transverse conductivities, 2 respectively. The best fit to the experimental points is tal data over the whole range T < TMD, at least the con- ± λ ± achieved at T0 = 1317 6 K and = 0.11 0.01. (4) The dition T > T must be satisfied. At T = 193.5 K, the σ π 2 MD MD temperature dependence of p(2 f0, T) calculated by λ latter inequality results in T0 > 12 500 K. However, the Eq. (3) using the determined values of T0 and ; f0 = calculation by Eqs. (77)Ð(80) from [18] at such a high 355.6 MHz. λ λ value of T0 and reasonable values of ( < 3 [28]) gives values of the conductivity that are eight orders of mag- σ π σ π nitude smaller than the experimental values. we find that, at T < 141 K, ac(2 f0, T) = p(2 f0, T) (Fig. 5). At temperatures T > 217 K, σ (2πf , T) = 0, Our experimental data can adequately be described ac 0 and the range 141 < T < 217 K corresponds to the tran- by the expressions for conductivities in mode II. Fitting σ sition range where ac increases monotonically from 0 the data in Fig. 2 (data set 1, T < TMD) to the σ to p. Therefore, in the range 141 < T < 300 K, curve 4 σ⊥()σT ||()T dependence, where σ⊥ is described by σ π in Fig. 5 is the upper estimate for ac(2 f0, T). The ac Eq. (1) and σ|| by Eqs. (68)Ð(71) from [18], we obtain component of the conductivity is seen to be far less than ± λ ± 6 σ π Ӷ 〈σ 〉 T0 = 1317 6 K and = 0.11 0.01. The fitting results the dc component, ac(2 f0, T) (T) , over the are shown in Fig. 5, curve 3. Using the known value of whole temperature range under study, which supports σ π T0, the temperatures T1 and T2 (which determine the our assumption that the contribution of ac(2 f0, T) to σ π limits of applicability of the mode II formulas) are (2 f0, T) can be neglected. found to be T1 = 294 K and T2 = 58 K. It is seen that the temperature range under study (125 K < T < T ) falls Using the calculated value of T0, we determine the MD τ πប ≈ in this range. electronÐimpurity scattering time = /(4kBT0) λ × Ð15 LB Now, having the values of T0 and , let us return to 4.6 10 s. With this time and the value of v F cal- σ π the problem of estimating ac(2 f0, T). Curve 4 in culated above, we can determine the mean free path l = σ π LB Fig. 5 shows the p(2 f0, T) dependence calculated by v τ and the localization length l = 4l [16] in the λ F loc Eq. (3) using the values of T0 and obtained. Compar- π ω ω ≈ LB LB τ ≈ ing 2 f0 with the values of 1 and 2 introduced above, film; they are l 0.45b and lloc = 4v F 8.6 Å. The value of lloc is seen to be smaller than the double lattice 5 Fitting was performed using the nonlinear LevenbergÐMarquardt parameter along the TCNQ chains. least squares method, which is available in the Origin software package intended for numerical data processing and visualiza- τ λ Let us compare the obtained values of and l with tion. The final values of T0 and thus obtained are insensitive to the choice of their initial values. analogous parameters in other disordered quasi-one- dimensional systems in which the effects of electron 6 Recent electronic band structure calculations for TTFÐTCNQ crystals using a modification of the pseudopotential method [29] localization are also important. The best known exam- give values of t⊥ that are, on average, three times greater than the ples of such systems are conducting polymers and dis- generally accepted value t⊥ = 5 meV [20]. Fitting our data at t⊥ = ordered 1 : 2 complexes of quinolinium (Qn) and acri- 22 Ð1 Ð3 15 meV and g(ε ) = 1.2 × 10 eV cm gives T = 1609 ± 7 K dinium (Ad) with TCNQ [Qn(TCNQ)2 and F 0 τ and λ = 0.11 ± 0.01. Ad(TCNQ)2, respectively]. In [30], the values of and

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1138 GALCHENKOV et al. l in polypyrrole (PPy) doped with hexafluorophosphate inelastic electronÐphonon interaction, which becomes (PF ) were found to be 5.5 × 10Ð16 s and 0.85a, respec- substantial at sufficiently high temperatures (when the 6 τ Ӷ τ tively (a is the length of the repeated polymer structural inequality in is satisfied). In this case, the longitu- unit). The corresponding parameters of Qn(TCNQ)2 dinal and transverse conductivities are given by were found to be 1.3 × 10Ð14 s and 2.5b, respectively Eqs. (28) in [18]. According to those equations, the [31] (b is the lattice parameter along the TCNQ stacks). temperature dependences of the conductivities are fully τ Ð1 It is seen that our LB films are intermediate in degree of determined by the inelastic-relaxation rate in (T), electron localization between the conducting polymers which reflects the specific features of the phonon sub- and TCNQ salts with asymmetric cations. system of quasi-one-dimensional materials. As noted In the literature, the degree of electron localization Θ τ Ð1 ∝ is estimated by comparing the value of k l with unity above, at high temperatures (T > D), we have in T; F σ (k is the Fermi wave vector). If k l ≤ 1, the localization therefore, according to [18], we can expect ⊥(T), F F Ð1 ӷ σ||(T) ∝ T at T > T . The conductivity 〈σ(T)〉 mea- is strong (Anderson localization); otherwise (kFl 1), MD sured in our experiments should also obey this law, the localization is weak. In our films, we have kFl = LB Ð1 πρl/b , where ρ is the degree of charge transfer from namely, 〈σ(T)〉 = σ⊥()σT ||()T ∝ T . However, Fig. 5 donor to acceptor. Assuming that the value of ρ in our shows that the temperature variation of the curve is films is identical to that in TTFÐTCNQ bulk crystals much weaker. This discrepancy can be due to the sim- ρ ≈ ≈ ( 0.6 [20]), we obtain kFl 0.85, which indicates τ Ð1 strong electron localization in them. plified approach used to describe the in (T) dependence. Indeed, the properties of the phonon subsystem As noted above, LB films based on a mixture of of low-dimensional organic materials are specific: at BEDOÐTTF with fatty acids have the highest conduc- high temperatures, one has to take into account tivity among CTC films. Recently, negative magnetore- intramolecular phonon excitations in addition to lattice sistance was detected in films prepared from a mixture phonons. Gor’kov et al. [33] tried to describe the tem- of BEDOÐTTF with stearic acid (SA) [32]; this fact τ Ð1 was explained in that work in terms of weak electron perature dependence of in in such systems; however, localization. Note that the localization in our films is they considered only the case of low temperatures (T Ӷ Θ much stronger than in the BEDO–TTF + SA films, D). At high temperatures, a large number of phonon whose conductivity is metallic down to 120 K and modes is excited and the situation becomes very com- decreases weakly (logarithmically) with decreasing τ Ð1 temperature in the range 15 < T < 120 K. This fact is plex. Therefore, the description of in (T) with the explained by a strong interaction between the chains of Hopfield relation, which is justified at not very high BEDOÐTTF molecules in the conducting bilayers of temperatures above the Debye temperature, can be too such films, which results in the formation of a quasi- rough an approximation at T > 200 K. The situation is two-dimensional electron system. In other words, also complicated by the fact that, at sufficiently high charge carriers in the films studied in [32] move along temperatures, thermally excited phonons contribute to conducting planes rather than along quasi-one-dimen- the random potential [31], which leads to additional sional chains, as in our case. It is known that, at the electron localization. same defect concentration, localization effects in a Note that the crystals of the Qn(TCNQ)2 salts with a quasi-one-dimensional system are much more pro- weak intrinsic structural disorder caused by a random nounced than in a quasi-two-dimensional system. orientation of asymmetric Qn cations also exhibit a Thus, the temperature dependence of the intracrys- change in the character of conductivity from semicon- ∂σ ∂ tallite conductivity of the films under study at T < TMD ducting ( / T > 0) to metallic at 240 K, which is agrees with the model of electron-state localization in a caused by suppression of the localization by inelastic disordered quasi-one-dimensional system [18]. Fitting electronÐphonon scattering. The mechanisms of elec- the experimental data to the relations from that work tron transport and localization in this compound can be leads to reasonable values of the electronÐphonon and explained in the same way as in the case described electronÐimpurity scattering times, as well as the mean above for films. However, the temperature dependence free path and the localization length. of the conductivity of these crystals also deviates from the law σ ∝ TÐ1; namely, at T > 240 K, σ ∝ TÐn, where n = 1.5Ð2 [7]. 3.4. Metallic Conductivity at T > TMD However, the most serious discrepancy between the We now consider the temperature range T > TMD, predictions made in [18] and the experimental data ∂σ ∂ Ð1 where the conductivity has metallic behavior ( / T < [which indicates that the approximation used for τ (T) 0). In terms of the electron localization described in above, the change in the character of conductivity at becomes rough at high temperatures] manifests itself when the temperature of the conductivity maximum T > TMD is explained by the fact that the quantum interference of scattered electron waves is suppressed by the T MD* is estimated. In terms of the model developed in

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 ELECTRON LOCALIZATION 1139

[18], T MD* can be found from the condition that, at this and is caused by the presence of impurities, defects, and temperature, the inelastic electronÐphonon scattering distortions in the linear conducting chains of TCNQ time becomes equal to the electronÐimpurity scattering molecules, along which charges are transferred. The τ π * λ ប Ð1 observed temperature dependence of the conductivity time, i.e., = (2 kB T MD / ) . Whence it follows that agrees qualitatively and quantitatively with the model of ប π λτ T MD* = /(2 kB ) = 2426 K, which exceeds the exper- electron localization in a weakly disordered quasi-one- imental value of T by an order of magnitude. Similar dimensional system [18]. Fitting the experimental results MD to the theoretical dependences taken from [18] allowed estimations for Qn(TCNQ) lead to T * = 406 K.7 τ × 2 MD us to evaluate the electronÐphonon [ in(193.5 K) = 6 This value also differs from the experimental tempera- 10Ð14 s] and electronÐimpurity (τ = 4.6 × 10Ð15 s) scat- ture of the conductivity maximum (240 K), although tering times. Based on the structural parameters of the this discrepancy is not as high as in the case of the films. conducting layer, we estimated the density of states at LB ε ≈ × 22 Ð1 Ð3 The fact that using the Hopfield relation for describ- the Fermi level (g ( F) 1.2 10 eV cm ) and the τ Ð1 LB ≈ × 6 ing in (T) at high temperatures causes problems was Fermi velocity (v F 4.7 10 cm/s) in the films noted long ago. For example, Bright et al. [26] estab- under study, which allowed us to determine the mean lished that the experimental room-temperature values free path l and the localization length lloc = 4l = τ Ð1 λ LB τ ≈ of in in TTFÐTCNQ crystals correspond to equal to 4v F 8.6 Å in the films. Our estimations show that 8 ≤ approximately 1.3 in the Hopfield relation. However, a strong (Anderson) localization mode (kFl 1) is real- as mentioned in review [20], this high value of λ is ized in the films. about an order of magnitude greater than the value required for correct description of the whole spectrum The metallic temperature behavior of the conductiv- of the magnetic and structural properties of these crys- ity at T > TMD is caused by suppression of the localiza- tals. Due to the reasons given above, the problem of tion effects by inelastic electronÐphonon scattering, Ð1 which becomes substantial at high temperatures, when finding the form of τ (T) in quasi-one-dimensional τ Ӷ τ in the inequality in is satisfied. The experimental tem- organic materials at high temperatures has not yet been perature dependence of the film conductivity at T > TMD solved, and this circumstance hinders comparing the differs from that predicted in [18]. Moreover, the esti- results from [18] with the experimental data at T > T . MD mates of TMD based on the results of that work are an At the same time, as shown above, the Hopfield relation order of magnitude higher than the experimental val- τ Ð1 ues. We believe that these discrepancies can be is a valid approximation for in in the temperature Θ explained by the imperfection of the approximation range D < T < 200 K and its using leads to reasonable τ Ð1 values for λ in the films. used for in (T), whose roughness becomes more pronounced at high temperatures. In concluding, it should τ Ð1 4. CONCLUSIONS be noted that the problem of finding the form of in (T) in quasi-one-dimensional organic materials at high We have presented the results of studying the tem- temperatures has remained unsolved for more than perature dependence of the intracrystallite conductivity thirty years [20, 26]. of the LB films of (C16H33ÐTCNQ)0.4(C17H35Ð DMTTF)0.6 measured with the acoustic technique proposed by us in [9]. We have found that the temperature ACKNOWLEDGMENTS dependence of the conductivity has a characteristic fea- This work was supported by the Russian Foundation ture at TMD = 193.5 K; more specifically, the conductiv- ∂σ ∂ for Basic Research (project nos. 01-02-16068, 03-02- ity of the films above TMD is metallic ( / T < 0), σ 22001) and the federal program of support for leading whereas below this temperature it obeys the law (T) = scientific schools (project no. NSh 1391.2003.2). σ 1/2 σ 0(T)exp[Ð(8T0/Tz) ], where 0(T) varies with temperature more slowly than the exponential factor. We believe that the nonmetallic behavior of the conductiv- REFERENCES ity at T < TMD is related to the localization of electron states in the quasi-one-dimensional system under study 1. P. Delhaés and V. M. Yartsev, in Spectroscopy of New Materials, Ed. by R. J. H. Clark and R. E. Hester (Wiley, Chichester, 1993), Vol. 22, p. 199. 7 When calculating TMD* , we assumed that the electronÐphonon λ 2. S. N. Ivanov, L. A. Galchenkov, and F. Ya. Nad’, interaction constant in Qn(TCNQ)2 is the same as that in TTFÐ Radiotekh. Élektron. (Moscow) 38 (12), 2249 (1993). TCNQ crystals, λ ≈ 0.23 [27]. 8 Note that calculating T * in the films at λ = 1.3 gives T * = 3. T. Nakamura, G. Yunome, R. Azumi, M. Tanaka, MD MD H. Tachibana, M. Matsumoto, S. Horiuchi, H. Yamochi, 205 K, which is close to the experimental value TMD = 193.5 K. and G. Saito, J. Phys. Chem. 98 (7), 1882 (1994).

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1140 GALCHENKOV et al.

4. H. Ohnuki, T. Noda, M. Izumi, T. Imakubo, and R. Kato, 19. Z. H. Wang, A. Ray, A. G. MacDiarmid, and A. J. Epstein, Phys. Rev. B 55 (16), R10225 (1997). Phys. Rev. B 43 (5), 4373 (1991). 5. S. Kahlich, D. Schweitzer, I. Heinen, S. E. Lan, 20. D. Jérome and H. J. Schulz, Adv. Phys. 31 (4), 299 B. Nuber, H. J. Keller, K. Winzer, and H. W. Helberg, (1982). Solid State Commun. 80 (3), 191 (1991). 21. E. B. Starikov, Int. J. Quantum Chem. 66 (1), 47 (1998). 6. J. Richard, M. Vandevyver, P. Lesieur, A. Barraud, and 22. E. B. Starikov, private communication. K. Holczer, J. Phys. D: Appl. Phys. 19 (12), 2421 (1986). 23. E. M. Conwell, Phys. Rev. B 22 (4), 1761 (1980). 7. I. F. Shchegolev, Phys. Status Solidi A 12 (1), 9 (1972). 24. S. K. Khanna, E. Ehrenfreund, A. F. Garito, and 8. H. W. Helberg and M. Dressel, J. Phys. I (France) 6 (12), A. J. Heeger, Phys. Rev. B 10 (6), 2205 (1974). 1683 (1996). 25. T. Wei, P. S. Kalyanaraman, K. D. Singer, and A. F. Garito, 9. V. Chernov, L. Galchenkov, S. Ivanov, P. Monceau, Phys. Rev. B 20 (12), 5090 (1979). I. Pyataikin, and M. Saint-Paul, Solid State Commun. 97 26. A. A. Bright, A. F. Garito, and A. J. Heeger, Phys. Rev. B (1), 49 (1996). 10 (4), 1328 (1974). 10. L. A. Galchenkov, S. N. Ivanov, F. Ya. Nad’, V. P. Cher- 27. A. J. Berlinsky, Contemp. Phys. 17 (4), 331 (1976). nov, T. S. Berzina, and V. I. Troitskiœ, Biol. Membr. 7 28. L. N. Bulaevskiœ, V. L. Ginzburg, G. F. Zharkov, (10), 1105 (1990). D. A. Kirzhnits, Yu. V. Kopaev, E. G. Maksimov, and D. I. Khomskiœ, in Problem of High-Temperature Super- 11. A. M. Dykhne, Zh. Éksp. Teor. Fiz. 59 (1), 110 (1970) conductivity, Ed. by V. L. Ginzburg and D. A. Kirzhnits [Sov. Phys. JETP 32, 63 (1971)]. (Nauka, Moscow, 1977). 12. O. V. Konovalov and I. I. Pyataœkin, unpublished. 29. S. Ishibashi and M. Kohyama, Phys. Rev. B 62 (12), 13. F. Rustichelli, S. Dante, P. Mariani, I. V. Myagkov, and 7839 (2000). V. I. Troitsky, Thin Solid Films 242 (1Ð2), 267 (1994). 30. K. Lee, R. Menon, C. O. Yoon, and A. J. Heeger, Phys. 14. T. J. Kistenmacher, T. E. Phillips, and D. O. Cowan, Acta Rev. B 52 (7), 4779 (1995). Crystallogr. B 30 (3), 763 (1974). 31. A. A. Gogolin, S. P. Zolotukhin, V. I. Mel’nikov, 15. L. A. Galchenkov, S. N. Ivanov, F. Ya. Nad’, V. P. Cher- É. I. Rashba, and I. F. Shchegolev, Pis’ma Zh. Éksp. nov, T. S. Berzina, and V. I. Troitsky, Synth. Met. 42 (1Ð Teor. Fiz. 22 (11), 564 (1975) [JETP Lett. 22, 278 2), 1471 (1991). (1975)]. 16. A. A. Gogolin, V. I. Mel’nikov, and É. I. Rashba, Zh. 32. Y. Ishizaki, M. Izumi, H. Ohnuki, K. Kalita-Lipinska, Éksp. Teor. Fiz. 69 (1), 327 (1975) [Sov. Phys. JETP 42, T. Imakubo, and K. Kobayashi, Phys. Rev. B 63 (13), 168 (1975)]. 134201 (2001). 33. L. P. Gor’kov, O. N. Dorokhov, and F. V. Prigara, Zh. 17. V. N. Prigodin and Yu. A. Firsov, Pis’ma Zh. Éksp. Teor. Éksp. Teor. Fiz. 85 (4), 1470 (1983) [Sov. Phys. JETP 58, Fiz. 38 (5), 241 (1983) [JETP Lett. 38, 284 (1983)]. 852 (1983)]. 18. É. P. Nakhmedov, V. N. Prigodin, and A. N. Samukhin, Fiz. Tverd. Tela (Leningrad) 31 (3), 31 (1989) [Sov. Phys. Solid State 31, 368 (1989)]. Translated by K. Shakhlevich

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

LOW-DIMENSIONAL SYSTEMS AND SURFACE PHYSICS

Adsorption of Barium and Rare-Earth Metals on Silicon S. Yu. Davydov Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia e-mail: [email protected] Received September 8, 2003

Abstract—The change in the work function of the Si(111) surface caused by deposition of a submonolayer ﬁlm of barium has been calculated with due account of dipoleÐdipole adatom repulsion and metallization effects. The results of the calculations agree satisfactorily with experimental data. The character of the variation in the model parameters in the BaÐSmÐEuÐYb series is discussed. The electron tunneling probability from an adatom quasi-level to the substrate conduction band and the energy barrier for interatomic transitions of electrons in an adlayer were estimated. © 2004 MAIK “Nauka/Interperiodica”.

1. Submonolayer films of alkali (AM) and alkaline- the former case, the adsorbed atom has one electron in earth metal (AEM) atoms have been intensely used in the outer s shell, while in the latter, the s shell contains emission electronics to lower the work function of two electrons. Because the barium 6s shell also con- high-melting thermionic cathodes; as a result, the AM tains two electrons, it appears only natural to use here and AEM interaction with metallic substrates has been the results obtained in [5]. studied fairly well [1, 2]. AM films on semiconductor In accordance with [5], the adatom charge Z is substrates have been explored to a much lesser extent, because such adsorption systems undergo structural ΩξΘ3/2 ()Θ ()Θ 2 Ð Z rearrangement and feature metalÐsemiconductor elec- Z = π--- arctan------ΓΘ() -, tronic transitions (see, e.g., [3] and references therein). (1) As for AEM films on semiconductors, their investiga- ξ ==2e2λ2 N 3/2 A, ΓΓ()1 + γΘ . tion is only at the earliest stage. ML 0 Here, ξ is the adatom dipoleÐdipole repulsion constant, A scanning tunneling microscopy study [4] λ explored the effect of a barium film deposited on a e is the positronic charge, 2 is the adsorption bond Si(111)-(7 × 7) surface on the variation of the work length (which is equal to the thickness of the double ∆φ Θ Θ electric layer), A = 10 is a dimensionless coefficient that function ( ) of the adsystem, where = N/NML is Ω the coverage (N is the concentration of adatoms and depends weakly on the adatom array geometry, = (I Ð φ + ∆) is the position of the adatom quasi-level relative NML is the concentration of adatoms in a monolayer). It Θ to the substrate Fermi level, I is the s-shell ionization was established that an increase in brings about a φ ∆ 2 λ lowering of the work function of the system and rear- energy, is the substrate work function, = e /4 is the rangement of the silicon (111) surface from the (7 × 7) Coulomb shift of the level of the adatom caused by the × × × interaction of its electron with the substrate electrons, structure to (3 1) (5 1) (2 8). This com- Γ ∆φ Θ 0 is the quasi-level half-width of a single adatom, and munication reports on a calculation of ( ) in the γ Ba/Si(111)-(7 × 7) system carried out within the model is a dimensionless parameter taking into account the developed in [5] for describing the adsorption of rare- broadening of the adatom quasi-level into a two-dimen- earth metals (REM) on semiconductors. sional band, i.e., the metallization effects. ∆φ Θ 2. To describe AM adsorption on semiconductor The change in the work function ( ) resulting substrates, a model was proposed in [6, 7] based on the from adsorption is given by AndersonÐNewns Hamiltonian [2]. This model takes ∆φ() Θ = Ð ΦΘZ()Θ , into account the dipoleÐdipole adatom repulsion and (2) Φ π 2 λ the metallization effects associated with electron = 4 e NML . exchange among atoms in the adlayer. The model disregarded the specific adlayer structure, which made it 3. Prior to continuing our calculations, we have first possible to describe the ∆φ(Θ) relation for AM adsorp- to specify the model parameters. The value of NML can tion on silicon [6] and rutile [7] fairly simply (and sat- be derived from the distance between the nearest neigh- isfactorily). This model was subsequently extended to bors in bulk barium samples, d = 4.35 Å [8]. As in [5], λ cover REM adsorption on Si(111) [5]. Considered in we set = 0.7ra, where ra = 2.24 Å is the atomic radius terms of the AndersonÐNewns Hamiltonian, the differ- of barium [9]. As in [4], we assume the silicon work ence between the AM and REM adsorption is that, in function φ to be 4.8 eV and the barium ionization

1142 DAVYDOV

Table 1. Model parameters Ω Γ Φ ξ γ 14 2 λ Film , eV 0, eV , eV , eV NML, 10 atoms/cm , Å Z0 ZML Ba 1.88 1.88 15.1 8.7 0.575 5.3 1.6 0.50 0.14 Sm 1.725 2.55 17.6 10.0 1.87 7.5 1.3 0.38 0.08 Eu 1.50 1.87 15.5 8.6 0.45 6.0 1.4 0.43 0.12 Yb 1.12 1.37 15.3 8.1 0.10 6.5 1.3 0.44 0.11

Γ γ energy to be I = 5.21 eV. To find 0 and , we make use adatom depolarization through the dipoleÐdipole and of the scheme proposed in [6, 7] based on fitting the val- exchange interactions, i.e., through the same effects ∆φ Θ ∆φ ≡ ∆φ Θ ues of (d /d )Θ → 0 and ML ( = 1) to experi- that are significant in adsorption on metals [2]. This mental data. Note that, for the work function of the does not come as a surprise. Indeed, the electron affin- χ monolayer coverage, we took the value corresponding ity for silicon is = 3.99 eV and the band gap is Eg = to the intensity of the Auger signal equal to unity [4, 1.11 eV [10]; therefore, the adatom quasi-level, which Fig. 2]. The model parameters found for Ba are pre- for zero coverage is at an energy Ω = 1.88 eV, overlaps sented in Table 1, which also lists the parameters for the silicon conduction band, a typical “metallic” situa- 1 Ω Ω ξ REMs for comparison. tion. For monolayer coverage, where ML = Ð ZML = The calculated ∆φ(Θ) data are compared with the 0.66 eV, the situation for the silicon substrate possess- experimental data in the figure. The agreement between ing intrinsic or electronic conduction (the material used the calculations and experiment is seen to be quite sat- in [4] is n-Si) does not change. Thus, we believe that isfactory. Note that not all of the available experimental structural rearrangement of the silicon surface is not the primary reason for the decrease in the work function; data are displayed in the figure. The reason for this is φ that the values of ∆φ for the coverage region Θ ≈ 0.15 rather, both the decrease in and the reconstruction of quoted in [4] have a large scatter and vary from 0.2 to the silicon surface are a consequence of a change in the 0.8 eV [4, Fig. 2]. The figure shows ∆φ(0.15) = 0.8 eV, electronic structure of the adlayer (see, e.g., [11] and the value closest to the calculation. (Note that, at larger references therein). coverages, the experimental scatter is also quite large.) 4. Consider the variation of the model fitting param- Γ γ The agreement between the calculated and experi- eters 0 and in the adsorbate series BaÐSmÐEuÐYb mental data permits us to interpret the change in the (Table 1). Such a consideration is appropriate for two work function differently from what was accepted in reasons. First, all these atoms have a filled outer 6s [4], where the reduction of the work function was shell, and second, barium is a model atom for calculat- believed to be a consequence of a rearrangement of the ing the REM cohesion energy [12] and the REM silicon surface caused by barium adsorption. Our adsorption energy on d metals [13]. model, disregarding reconstruction of both the sub- ∆φ Θ In the AndersonÐNewns model, the quasi-level half- strate and the adlayer, assigns the ( ) dependence to Γ πρ 2 ρ width is 0 = sV , where s is the substrate density of states (in our particular case, the density of states in the 0 conduction band), which for the sake of simplicity is × assumed to be constant, and V is the matrix element of Ba/Si(111)-(7 7) the adatomÐsubstrate interaction representing a two- –0.5 Theory center tunneling integral between the states of the ada- Experiment tom |a〉 and of the substrate |s〉 [14]. According to [15], the matrix element V squared can be presented in the –1.0 form

2 ∼ ρ Ð1ρ Ð1 –1.5 V s a Ds, (3) ρ where s(a) is the density of states of the substrate (ada- –2.0 Work function change, eV tom) and Ds is the probability of electron tunneling through the energy barrier separating the adatom from –2.5 the substrate. The probability of an electron having an 0 0.2 0.4 0.6 0.8 1.0 energy E = χ Ð I + ∆ (the energy is reckoned from the Coverage 1 The metallic situation is identiﬁed in this case with the possibility Variation of the work function plotted vs. Si(111) coverage of direct electron tunneling from the adatom quasi-level to free by barium atoms. substrate states.

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 ADSORPTION OF BARIUM AND RARE-EARTH METALS ON SILICON 1143

Table 2. Probability of adatom electron tunneling Ds into the substrate Parameter Ba Sm Eu Yb

Ds 0.13 0.22 0.145 0.09 ρ Ds/ a, eV 1.52 2.57 1.40 0.65 Ua, eV 3.78 1.69 4.33 6.41 ρ Γ Note: The ratio Ds/ a is proportional to the half-width of the single adatom quasi-level 0; Ua is the barrier height for electron tunneling from the adatom quasi-level into the adlayer.

χ conduction band bottom) to cross a barrier of height data in Tables 1 and 2, we obtain the values of Ua listed and width b is [16] in Table 2. The values of Ua for Ba, Sm, and Eu appear 2κ 2 quite reasonable, whereas for ytterbium the barrier 4k1 2 height is obviously a gross overestimate, which is a Ds = ------2 -, (4) ()k 2 + κ 2 sinh2 ()κ b + 4k 2κ 2 consequence of the extreme smallness of the corre- 1 2 2 1 2 sponding parameter γ = 0.1. (In this connection, it where បk = 2mE and បκ = 2m()χ Ð E . should be stressed that the magnitude of the fitting 1 2 parameter γ derived from [6, Eq. (6)] or [7, Eq. (4)], is We set the barrier width b equal to the half-sum of extremely sensitive to variation of the adatom charge in the atomic radii of silicon and of the adsorbed atom. the monolayer, ZML, which is small. Indeed, a decrease Because Γ = πρ V2, we have, in accordance with 0 s in ZML for Yb by one one-hundredth brings about a two- Eq. (3), fold increase in the parameter γ.2 Γ ∼ ρ 0 Ds/ a, (5) To sum up, we have shown that, first, the model pro- where the density of states for the adatom is ρ = posed originally for describing the adsorption of alkali a metal atoms on semiconductors has a broader area of Γ π Ω2 Γ 2 0/ ( + 0 ). The calculated tunneling probability Ds application. Second, we have demonstrated that the fit- ρ and the Ds/ a ratio are listed in Table 2, from which it ting parameters of the proposed model are quite reason- ρ is seen that the value of the ratio Ds/ a correlates well able. From this it follows that the major factor govern- Γ ing structural transitions in the substrate, which were with the value of 0 for Ba, Sm, and Eu, whereas for Yb the agreement is not as good. pointed out in [4], is the depolarization of the adatoms, γ which grows in significance as their concentration Consider now the parameter . If one takes into increases. account only direct electron exchange between the nearest neighbor adatoms, γ can be shown to have the form [17] REFERENCES γ 2 ()Ω2 Γ 2 2 ≡ 2 = T / + 0 , T zn.n.tML , (6) 1. L. A. Bol’shov, A. P. Napartovich, A. G. Naumovets, and where z is the number of nearest neighbors in a A. G. Fedorus, Usp. Fiz. Nauk 122 (1), 125 (1977) [Sov. n.n. Phys. Usp. 20, 432 (1977)]. monolayer and tML is the electron transfer integral between the nearest neighbor adatoms in a monolayer. 2. O. M. Braun and V. K. Medvedev, Usp. Fiz. Nauk 157 Equation (6) can be recast as (4), 631 (1989) [Sov. Phys. Usp. 32, 328 (1989)]. γΓΓ Γ πρ 2 3. S. Yu. Davydov, Fiz. Tverd. Tela (St. Petersburg) 42 (6), ==a/ 0, a aT . (7) 1129 (2000) [Phys. Solid State 42, 1164 (2000)]. We have introduced here the half-width of the adatom 4. M. Komani, M. Sasaki, R. Ozawa, and S. Yamamoto, Γ quasi-level a related to the possibility of electron tun- Appl. Surf. Sci. 146 (1), 158 (1999). neling in the adlayer. We can thus set 5. S. Yu. Davydov and A. V. Pavlyk, Fiz. Tverd. Tela γ = Da/Ds, (8) (St. Petersburg) 45 (7), 1325 (2003) [Phys. Solid State 45, 1388 (2003)]. where Da is the probability of an electron tunneling into 6. S. Yu. Davydov and A. V. Pavlyk, Fiz. Tekh. Polupro- the adlayer by crossing a barrier of height Ua and width γ vodn. (St. Petersburg) 35 (7), 796 (2001) [Semiconduc- d/2. Hence, parameter determines the probability ratio tors 35, 796 (2001)]. for an adatom electron to tunnel into the adlayer and into the substrate. 7. S. Yu. Davydov and I. V. Noskov, Pis’ma Zh. Tekh. Fiz. 27 (20), 1 (2001) [Tech. Phys. Lett. 27, 844 (2001)]. The probability Da is calculated from Eq. (4), with χ the electron affinity replaced by Ua and a replaced by 2 γ ∆φ Θ γ For such a small variation in the parameter , the ( ) function d/2. Equating Da to the product Ds calculated from the changes only insignificantly.

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1144 DAVYDOV

8. C. Kittel, Introduction to Solid State Physics, 5th ed. 14. C. Kittel, Quantum Theory of Solids (Wiley, New York, (Wiley, New York, 1976; Nauka, Moscow, 1978). 1963; Nauka, Moscow, 1967), Chap. 18. 9. Physical Quantities. Handbook, Ed. by I. S. Grigor’ev 15. J. R. Schrieffer, Theory of Superconductivity (Benjamin, and E. Z. Meœlikhov (Énergoatomizdat, Moscow, 1991). New York, 1964; Nauka, Moscow, 1970), Chap. 3. 10. T. Bechshtedt and R. Enderlein, Semiconductor Surfaces and Interfaces. Their Atomic and Electronic Structures 16. L. D. Landau and E. M. Lifshitz, Course of Theoretical (Academie, Berlin, 1988; Mir, Moscow, 1990). Physics, Vol. 3: Quantum Mechanics: Non-Relativistic 11. S. Yu. Davydov, Fiz. Tverd. Tela (Leningrad) 21 (8), Theory, 3rd ed. (Nauka, Moscow, 1974; Pergamon, New 2283 (1979) [Sov. Phys. Solid State 21, 1314 (1979)]. York, 1977). 12. J. M. Wills and W. A. Harrison, Phys. Rev. B 29 (10), 17. S. Yu. Davydov, Surf. Sci. 411 (1Ð3), L878 (1998). 5486 (1984). 13. S. Yu. Davydov and S. K. Tikhonov, Surf. Sci. 371 (1Ð3), 157 (1997). Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

Physics of the Solid State, Vol. 46, No. 6, 2004, pp. 1145Ð1148. Translated from Fizika Tverdogo Tela, Vol. 46, No. 6, 2004, pp. 1111Ð1114. Original Russian Text Copyright © 2004 by Shulakov, Braœko, Bukin, Drozd. LOW-DIMENSIONAL SYSTEMS AND SURFACE PHYSICS

X-ray Spectral Analysis of the Interface of a Thin Al2O3 Film Prepared on Silicon by Atomic Layer Deposition A. S. Shulakov, A. P. Braœko, S. V. Bukin, and V. E. Drozd Fock Institute of Physics, St. Petersburg State University, ul. Ul’yanovskaya 1, Petrodvorets, St. Petersburg, 198504 Russia e-mail: [email protected] Received October 23, 2003

Abstract—The extent and phase chemical composition of the interface forming under atomic layer deposition (ALD) of a 6-nm-thick Al2O3 ﬁlm on the surface of crystalline silicon (c-Si) has been studied by depth- resolved, ultrasoft x-ray emission spectroscopy. ALD is shown to produce a layer of mixed Al2O3 and SiO2 oxides about 6Ð8 nm thick, in which silicon dioxide is present even on the sample surface and its concentration increases as one approaches the interface with the substrate. It is assumed that such a complex structure of the layer is the result of interdiffusion of oxygen into the layer and of silicon from the substrate to the surface over grain boundaries of polycrystalline Al2O3, followed by silicon oxidation. Neither the formation of clusters of metallic aluminum near the boundary with c-Si nor aluminum diffusion into the substrate was revealed. It was established that ALD-deposited Al2O3 layers with a thickness up to 60 nm have similar structure. © 2004 MAIK “Nauka/Interperiodica”.

The method of atomic layer deposition (ALD) has remain contradictory in many respects. This is appar- recently been attracting increasing interest as a promis- ently associated with the application of destructive ing means of forming heterostructures for the develop- methods to a depth-resolved analysis of the structure ment of modern micro- and nanoelectronics devices. In and chemical composition of films. These methods also view of the extremely small size of the nanostructures, do not permit one to study thick coatings. their formation requires the control of the growth pro- The present communication reports on a study of cess at the monatomic layer. To solve this problem, one interfaces using a nondestructive method of probing the has to apply to the surface of one compound very thin phase chemical composition, namely, depth-resolved perfect layers of other compounds in a given order and ultrasoft x-ray emission spectroscopy [9, 10]. This of a predetermined thickness. The principles underly- method makes use of the dependence of the depth of ing the ALD method are outlined in [1]. escape of characteristic x-ray radiation on the primary In this study, we used the ALD process to deposit the electron energy. The characteristic x-ray emission bands are created by valence band electrons spontane- oxide Al2O3 on the Si(100) surface layers. Because aluminum oxide has a considerably higher value of dielec- ously filling the vacancies produced by electron impact tric permittivity k than silicon oxide, this structure is in the inner atomic shells. The dependence of the shape considered a promising material for microelectronics of the x-ray emission bands on the chemical state of the (the so-called high-k material). Obviously enough, the emitting atom permits reliable determination of the electrophysical properties of the heterostructure are phase chemical composition of a sample. determined not only by the characteristics of the thin Silicon plates were preliminarily exposed to hydrof- Al O layer but also by the characteristics of the inter- luoric acid vapor and a glow discharge plasma in the 2 3 ° face with silicon. The present study addresses this presence of water vapor at a temperature of 200Ð300 C issue. We are interested here in the phase chemical under reduced pressure. The actual atomic layer deposition process consisted of the following cycles. The composition of the Al2O3/Si interface and its extent (thickness). reactor was filled by vapors of trimethyl aluminum (TMA) for a few tenths of a second, with subsequent Considerable attention has been focused recently on evacuation of the reagent and admission of water vapor the analysis of the properties of Al2O3 films fabricated to the reactor, which prepared the surface for the next by ALD [2Ð8]. The formation of the structure of thin ALD cycle. These cycles were repeated the required ALD-deposited Al2O3 films and the underlying pro- number of times. Samples with Al2O3 layer thicknesses cesses have been studied primarily by electron (x-ray varying from a few nanometers to a few hundreds of photoelectron and Auger electron) spectroscopy, sec- nanometers were prepared. X-ray spectroscopy showed ondary ion mass spectrometry, recording the elastic the interfaces of thin films, with thicknesses of up to recoil of light atoms upon scattering of heavy atoms, about 60 nm, to have similar properties, which vary etc. However, the results obtained in such studies with increasing coating thickness (i.e., interface depth).

1146 SHULAKOV et al. with a wavelength resolution of 0.2 nm. The primary Soft x-ray emission bands

µ electron current did not exceed 1 mA, and the reproduc- D A E d ibility of the results and the constancy of the electro- µ b SiO physical parameters of the structures were monitored (Al2O3) c 2 periodically. e B Al2O3 , arb. units Figure 1 shows L2, 3 x-ray emission bands of pure a 3 Al O crystalline aluminum and silicon (c-Si) and the bands 2 of the same atoms in stoichiometric oxides. These Si Intensity, arb. units F C of Al bands will be used as references when analyzing the

Absorption coefficient emission spectra of an ALD-deposited Al O /Si film. Si L 2 3 Al L2,3 2,3 Figure 1 also shows the spectral response of the absorp- 50 60 70 80 90 100 110 tion coefficient of x-ray radiation by an Al2O3 film. The Photon energy, eV fine structure of the absorption spectrum near the L2, 3 absorption edge may distort the spectral shape of the Fig. 1. Ultrasoft x-ray L2, 3 emission bands of crystalline silicon radiation passing through the film, and this aluminum and silicon (solid lines) and the emission bands effect should be taken into account in analyzing the of these atoms in stoichiometric oxides (dashed lines); µ(Al O ) is the Al O absorption coefficient (dotted line). shape of the spectra. This effect (self-absorption of 2 3 2 3 escaping radiation) was found to be small for the thin coatings studied here and could not produce noticeable distortions in the spectral shape. Al2O3(6 nm)/Si Figure 2 displays x-ray emission spectra of a sample with the silicon surface coated by a 6-nm-thick Al2O3 E D A B layer (Al2O3(6 nm)/Si) that demonstrate the dependence of the spectral shape on E0 in the range including C Al and Si L2, 3 bands. The labeling of the maxima of the spectra is the same as that in Fig. 1, and the spectra are normalized against the intensity of the Si L2, 3 band. b c The spectra obtained at the minimum primary electron energy (0.3 keV) are seen to contain L2, 3 bands of the Al and Si atoms entering into Al2O3 and SiO2, respectively (maxima A, B, C, D, E). After E0 has been , a 0 increased to 0.5 keV, a maximum b appears in the sili- 0.3 E con spectrum. As E is increased still further, maxima a

0.4 0 Intensity, arb. units arb. Intensity, and c become evident and the silicon spectrum practi- 0.5 cally coincides with the c-Si spectrum (Fig. 1). The 0.6 Al L shape of the Al L2, 3 band is independent of E0, with only 50 2,3 Si L2,3 0.8 60 keV 70 its relative intensity changing. At E0 = 1.5 keV, the con- 1.0 Photon energy,80 90 eV tribution from the aluminum band intensity to the total 100 110 1.5 intensity of recorded radiation becomes negligible.

Primary electron energy The explanation for the observed variations is Fig. 2. Variation in the shape of x-ray emission spectra with straightforward. Indeed, at the lowest energy E0 = primary electron energy for a sample consisting of a 6-nm- 0.3 keV, the sample layer responsible for the x-ray thick Al2O3 layer prepared by ALD on the Si(100) surface. emission bands does not exceed the Al2O3 coating in The silicon L2, 3 bands are normalized in intensity. thickness and contains, in addition to the aluminum oxide, oxidized silicon atoms in the coordination characteristic of SiO2. As E0 increases, the lower boundary Because the amount of the data obtained is too large to of the emitting layer shifts away from the surface to be presented in one publication, we describe only the cross the substrate interface and its contribution to the results of a study of thin coatings using the example of spectrum increases (the c-SiL2, 3 band in pure silicon, a 6-nm-thick layer. The studies of deeper lying Al2O3/Si maxima a, b, and c) and quickly becomes dominant. interfaces will be reported later. Analysis of the band shape evolution with increas- We studied the dependence of the shape of the alu- ing E0 suggests the obvious conclusion that ALD of minum and silicon L2, 3 x-ray emission bands on the pri- Al2O3 on an oxide-free surface of a silicon single crys- mary electron energy E0. The measurements were con- tal produces an interface containing, nevertheless, a ducted on an RSL-1500 ultrasoft x-ray spectrometer considerable amount of the silicon oxide SiO2. This [11]. The spectra in the range 50Ð110 eV were obtained conclusion can be supported in a more revealing way.

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 X-RAY SPECTRAL ANALYSIS OF THE INTERFACE 1147

Figure 3 shows deconvolution of an experimental silicon spectrum into constituents for two values of E . For Si L2,3 emission bands 0 b Al2O3 (6 nm)/Si B + c constituents, we chose only the SiL2, 3 bands in c-Si and A + a SiO2 (Fig. 1). The fit to the experimental spectrum Experiment E = 0.6 keV unfolded into constituents by the standard least squares SiO 0 2 Si/SiO = 3.6 procedure turned out to be very good. This shows that, Si 2 in the region of the interface, silicon atoms are coordi- Sum nated either as in c-Si (the substrate) or as in SiO2 (the coating). Figure 3 identifies the fine structure in the C experimental spectra in the notation used in Fig. 1. Intensity, arb. units While the above analysis provides evidence for the E0 = 0.3 keV presence of not only aluminum oxide but also a sub- C Si/SiO2 = 0.43 stantial amount of SiO2 (in a concentration comparable to that of Al2O3) in the surface layer, it does not offer insight into the silicon oxide distribution in depth. This 70 75 80 85 90 95 100 105 110 information can be extracted from data on the depen- Photon energy, eV dence of the relative band intensities on E . Figure 4 0 Fig. 3. Deconvolution of experimental silicon L2, 3 spectra displays the behavior of the integrated intensities of the into the constituent L2, 3 bands of Si in SiO2 (solid lines) SiL2, 3 bands in SiO2 and c-Si obtained from unfolded and in c-Si (dashed lines) plotted for E0 = 0.3 and 0.6 keV. spectra and normalized against the AlL2, 3 band inten- The notation of the maxima is the same as that in Fig. 1. sity in Al2O3 (Fig. 2). The relative intensity of the c-Si spectrum grows rapidly, which reflects the shift of the center of gravity of the excitation into the substrate with 10 increasing E0. Obviously enough, excitation of the c-Si band starts when the energy E0 reaches a value close to 0.3 keV. This means that the c-Si interface is overlaid c-Si and, in order for the electrons crossing the layer to be able to ionize the Si 2p shell, their energy has to exceed the binding energy of the electrons in this shell 5 (99.8 eV) by approximately 200 eV. σ σ Si(E0)/ Al(E0) The E0 dependence for the SiO2 band is more complex; its relative intensity increases and saturates at E0 = 0.8 keV. Extrapolating this graph to zero through the first three points shows that the SiO2 spectrum SiO2

± Relative integrated intensity appears when E0 enters the region of 100.0 0.5 eV, which includes the binding energy of the Si 2p elec- 00.1 0.3 0.5 0.7 0.9 1.1 trons in SiO2 (104.1 eV). The matching (to within E0, primary electron energy, keV experimental error) of the extrapolation point to the binding energy of the scanning 2p level means that sil- Fig. 4. Relative integrated intensities of Si L2, 3 bands in icon oxides are not only present in the synthesized SiO2 (solid line) and in Si (dash-dotted line) plotted vs. primary electron energy E l. The spectra are normalized Al2O3 ﬁlm but emerge onto its surface as well. Let us 0 estimate the pattern of the silicon dioxide distribution against the Al L2, 3 band intensity in Al2O3. Dashed line is across the coating thickness. In our case, where the extrapolation of the dependence for SiO2 to zero. Circles atomic numbers of the elements and the ionization are the ionization cross-section ratios of the Si and Al 2p energies of the scanning 2p shells are similar and the levels calculated from the Grysinsky relation [12]. ﬁlm thickness is small, it can be shown that the dependence of the intensity ratio of the Si and Al L2, 3 spectra observed difference in shape between the experimental on E0 should be determined primarily by the ratio of the and theoretical curves can be accounted for as follows. 2p-shell electron-impact ionization cross sections, σ σ The theoretical graph is constructed under the assump- Si(E0)/ Al(E0). This ratio was calculated using the tion that the Al and Si atoms are present in equal con- Grysinsky formula for the cross sections of np shell centrations (the cross sections are given per atom); ionization by electron impact [12] and is shown in σ σ Fig. 4 by circles (the experimental and theoretical therefore, by varying the scale of the Si(E0)/ Al(E0) curves are matched in magnitude at the 0.8-keV point). curve, we vary, as it were, the relative concentration of The cross-section ratio also grows with E0, but substan- the Si and Al atoms. Far from the thresholds, the cross tially faster than the experimental intensity ratio. For sections follow the same course and, therefore, their ≥ E0 0.8 keV, the graphs become identical in shape. The ratio saturates. This corresponds obviously to about

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1148 SHULAKOV et al.

0.8 keV (Fig. 4). The relative intensity of the SiL2, 3 sion of the oxygen and silicon, followed by silicon oxi- band should also reach saturation with increasing E0, dation. One could agree with the suggestion from [8] because at high enough values of E0 the conditions of that oxygen diffuses most probably over grain bound- excitation of Si and Al spectra in a thin near-surface aries of the Al2O3 polycrystalline layer, although this film become practically the same, irrespective of the requires supporting evidence. The distributions of ele- relative concentration distributions. The same reason ments in depth presented in [8] indicate that Al atoms accounts for the saturation in the theoretical relation; penetrate deep into the substrate. Our results indicate thus, there is nothing strange in the fact that the satura- that the boundary between the c-Si substrate and the tion starts in both cases at the same point (0.8 keV). variable-composition Al2O3ÐSiO2 layer is sharp, with Thus, by normalizing the theoretical and experimental no Al diffusion into the c-Si observed. relations in this region, we reduce the relative, layer- Our study has demonstrated a high efficiency of averaged atomic concentrations in the theoretical curve ultrasoft x-ray emission spectroscopy in investigating to the experimental value. After this procedure, the the- thin surface layers, and our data offer the possibility of oretical curve provides an estimate of the relative Si examining the results obtained by traditional methods radiation intensity for a certain constant average silicon from a different standpoint. concentration. As seen from Fig. 4, for any E0, the ordi- nates of the experimental curve lie below the theoretical values, but the difference between them decreases with ACKNOWLEDGMENTS increasing E0. This fact is a direct indication of the Si This study was supported by the Russian Founda- atom concentration (in the SiO2 coordination) increas- tion for Basic Research, project no. 01-03-32771. ing as one approaches the interface. Let us estimate the thickness of the surface layer REFERENCES coating the silicon substrate. For this purpose, we can use the empirical relation describing the behavior of the 1. V. B. Aleskovskii and V. E. Drozd, Acta Polytech. maximum escape depth from which the Si L radiation Scand., Chem. Technol. Ser. 195, 155 (1990). 2, 3 2. Y. K. Kim, S. H. Lee, S. J. Choi, H. B. Park, Y. D. Seo, emerges from SiO2 [9] by properly changing the layer K. H. Chin, D. Kim, J. S. Lim, W. D. Kim, K. J. Nam, density. The coating density should apparently be an M.-H. Cho, K. H. Hwang, and Y. S. Kim, in Proceedings average between the densities of SiO2 and Al2O3, i.e., of the International Electron Devices Meeting (San approximately 3Ð4 g/cm3, in which case the layer thick- Francisco, CA, 2000). ness is 6Ð8 nm. If we take into account that ALD is capa- 3. W. S. Yang, U. K. Kim, S.-Y. Yang, J. H. Choi, ble of depositing Al2O3 layers 6 nm thick with a high H. B. Park, S. I. Lee, and J.-B. Yoo, Surf. Coat. Technol. accuracy, this estimate appears reasonable. Indeed, the 131, 79 (2000). presence of SiO2 in noticeable amounts, irrespective of 4. J. Kim, D. Kwong, K. Chakrabarti, C. Lee, K. Oh, and J. its location, should bring about film swelling. Lee, J. Appl. Phys. 92 (11), 6739 (2002). As follows from an analysis of our results, ALD of 5. V. Consier, H. Bender, M. Caumax, J. Chen, T. Conard, H. Nohira, O. Richard, W. Tsai, W. Vandervorst, a thin Al2O3 layer on a c-Si surface freed of its native E. Young, C. Zhao, S. D. Gendt, M. Heyns, J. W. H. Maes, oxide is a complex process. It gives rise to the forma- M. Tuominen, C. Rochart, M. Oliver, and A. Chabli, in tion of a 6Ð8-nm-thick layer of mixed Al2O3 and SiO2 Proceedings of the International Workshop on Gate oxides, in which silicon dioxide is present even on the Insulator (IWGI) (Japan, 2001). surface of the structure and its concentration increases 6. M. Kadoshima, T. Nabatame, T. Yasuda, M. Nishizawa, as one approaches the c-Si substrate interface. Strictly M. Ikeda, T. Horikawa, and A. Toriumi, in Proceedings speaking, the Al2O3/Si interface does not form, with a of Atomic Layer Deposition Conference (Seoul, 2002). more complex Al2O3ÐSiO2/Si boundary appearing in its 7. G. Parson, D. Niu, T. Gougoushi, M. J. Kelly, and place. Clusters of metallic aluminum were reported to T. Abatemarco, in Proceedings of Atomic Layer Deposi- have been observed to form in the vicinity of the inter- tion Conference (Seoul, 2002). face [2–4]. No such clusters at the thin film interface 8. S. Jakschik, U. Schroeder, T. Hecht, D. Krueger, G. Doll- were found by us within the limits of sensitivity of our inger, A. Bergmaier, C. Luhmann, and J. W. Bartha, measurements (less than 0.5 wt %) for a thickness of Appl. Surf. Sci. 211, 352 (2003). the emitting layer of about 6Ð8 nm. 9. A. S. Shulakov, Cryst. Res. Technol. 23, 835 (1988). Noticeable oxygen diffusion was found to occur 10. A. Zimina, A. S. Shulakov, S. Eizebitt, and W. Eberhardt, Surf. Sci. Lett. 9 (1), 461 (2002). through the Al O layer in the course of ALD [6Ð8], and 2 3 11. A. P. Lukirskiœ, V. A. Fomichev, and A. V. Rudnev, Appar. silicon was observed to diffuse from the substrate to the Metody Rentgen. Anal. 9, 89 (1970). surface during the ALD and annealing [8]. Our results corroborate the findings reported in [6–8]. Indeed, the 12. M. Grysinsky, Phys. Rev. 138, A305 (1965); Phys. Rev. 138, A322 (1965). most probable mechanism for the formation, in the Al2O3 film, of the SiO2 oxide, whose concentration increases as one approaches the substrate, is interdiffu- Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

Physics of the Solid State, Vol. 46, No. 6, 2004, pp. 1149Ð1157. Translated from Fizika Tverdogo Tela, Vol. 46, No. 6, 2004, pp. 1115Ð1122. Original Russian Text Copyright © 2004 by Laœus, Slutsker, Gofman, Gilyarov. POLYMERS AND LIQUID CRYSTALS

Correlation between Characteristics of Thermal and Stress Reversible Deformations in Solids with Different Structures L. A. Laœus†*, A. I. Slutsker**, I. V. Gofman*, and V. L. Gilyarov** * Institute of Macromolecular Compounds, Russian Academy of Sciences, Bol’shoœ pr. 31, St. Petersburg, 199004 Russia ** Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia e-mail: [email protected] Received August 25, 2003

Abstract—The dependences of the coefficients of reversible thermal expansion on the tensile stress and the temperature dependences of the elastic modulus are measured for solids with different structures, such as a metal, rigid-chain oriented polymers, and a flexible-chain oriented polymer—poly(ethylene) in the devitrified state. For these materials, the elastic moduli differ by several orders of magnitude, whereas the thermal expansion coefficients can differ not only in magnitude but also in sign. It is found that, for a solid, the derivative of the thermal expansion coefficient with respect to the stress is close to the derivative of the reciprocal of the elastic modulus with respect to the temperature. The inference is made that these parameters do not depend on the specific features of the solids under investigation. Calculations are performed for several mechanisms of thermal and stress deformations of solids with different structures. The results of these calculations are in reasonable agreement with experimental data. © 2004 MAIK “Nauka/Interperiodica”.

† 1. INTRODUCTION temperature-dependent elastic compliance EÐ1(T), we can write the following relationship: Thermal and stress reversible deformations of solids have different origins. Ð1 2 ∂α ∂()E ∂ ε ------==------. (1) Reversible thermal deformation (thermal expan- ∂σ ∂T ∂σ∂T sion) is caused by a change in the temperature T and is characterized by the thermal expansion coefficient α = Therefore, the derivative of the thermal expansion ∂ε ∂ coefficient with respect to the stress and the derivative ∂ε l ∂------, where = ---- is the relative change in the size of of the reciprocal of the elastic modulus with respect to T l Ð1 Ð1 a solid. the temperature have identical dimensions (Pa K ) and coincide in magnitude (i.e., there is a peculiar Reversible stress deformation (elastic deformation) invariant). Relationship (1) is general and, hence, is caused by a change in the mechanical stress σ and is should be satisfied for any solid. ∂σ characterized by the elastic modulus E = ------or the Moreover, solids with different structures exhibit a ∂ε great variety of mechanisms of thermal expansion and reciprocal of this quantity, i.e., the elastic compliance elastic deformation and differ substantially in terms of ∂ε Ð1 the thermal expansion coefficients (and even in their E = ∂σ------. sign) and elastic moduli. A considerable amount of data is available in the lit- The quantities α and EÐ1 have different dimensions Ð1 Ð1 erature on the temperature dependence of the elastic (K and Pa , respectively) and, therefore, cannot be modulus for low-molecular crystalline materials (pre- quantitatively compared for a particular solid. dominantly, metals) [1] and polymers [2]. There are At the same time, the thermal expansion coefficient only a few works concerned with the stress dependence and the elastic compliance EÐ1 for a solid are not con- of the thermal expansion coefficient for metals [3, 4] stants and depend on the stress and the temperature. For that provide a means for verifying relationships (1). the stress-dependent thermal expansion coefficient However, there is virtually no data on the stress depen- α(σ) and the temperature-dependent elastic modulus or dence of the thermal expansion coefficient for polymers (it is these polymers that can have negative coefficients † Deceased. of thermal expansion).

1150 LAŒUS et al.

3 The stress dependence of the thermal expansion 2 (a) coefﬁcient was measured as follows: the sample at the 1 initial temperature was subjected to a number of speci- 1.0 ﬁed stresses; under each stress, the sample was heated at different temperatures; and the thermal strain of the sample upon heating at these temperatures was mea-

3 sured.

10 The temperature dependence of the elastic modulus ×

ε 0.5 was measured as follows: the sample was thermostated at several speciﬁed temperatures; at each temperature, the sample was subjected to a number of different stresses; and the stress strain of the sample under these stresses was measured. During measurements, the temperature and stress were varied over sufﬁciently narrow ranges; as a result, 300 340 380 the thermal and stress strains were virtually recoverable T, K (i.e., plastic and shrinkage deformations were absent). 4 3 6 The experiments were performed with a metal sam- (b) 2 1 ple and several polymer samples.

2.1. Metal 4 3 The high-elasticity steel wire used as a metal sample

10 was 0.4 mm in diameter and possessed a positive coef- ×

ε ficient of thermal expansion. 2 2.2. Polymers Highly oriented fibers (10–12 µm in diameter) from rigid-chain polymers, namely, poly(4,4'-oxydiphe- 400 800 nylene pyromellitic imide) (PM), poly(p-phenylene σ, MPa terephthalamide) (Kevlar-49, K-49), and poly(amido- benzimidazole) (synthetic high-polymer material, Fig. 1. (a) Temperature and (b) stress dependences of the SHPM), had a negative coefficient of thermal expan- recoverable strain of the steel wire. (a) Tensile stress σ: sion along the axis. (1) 303, (2) 730, and (3) 1090 MPa. (b) Temperature T: (1) 293, (2) 323, (3) 353, and (4) 381 K. Oriented rods (~1 mm in diameter) from a flexible- chain polymer, namely, poly(ethylene) (PE), had a large negative coefficient of thermal expansion along In the present work, we measured the thermal defor- the axis. mation with a variation in the stress and the stress deformation with a variation in the temperature for metals and polymers. The stress derivatives of the ther- 3. RESULTS AND DISCUSSION mal expansion coefficient were compared with the tem- 3.1. Metal perature derivatives of the elastic modulus. The ratios The temperature dependences of the recoverable of the experimentally obtained derivatives of these strain of the steel wire under different tensile stresses quantities were interpreted in terms of several mecha- are shown in Fig. 1a. The stress dependences of the nisms of thermal expansion and elastic deformation of recoverable strain of the steel wire at different temper- solids with different structures. atures are depicted in Fig. 1b. Since the thermal strains are positive, we deal with a conventional thermal 2. EXPERIMENTAL TECHNIQUE AND OBJECTS expansion. As can be seen from these figures, the ther- OF INVESTIGATION mal expansion of the sample increases under tensile stresses (Fig. 1a), whereas an increase in the tempera- The thermal and stress deformations were investi- ture leads to an increase in the elastic elongation gated using rodlike samples with an operating length of (Fig. 1b). 100 mm. The deformation of samples was measured It can be seen from Fig. 1 that the temperature and with an accuracy of 3 µm. The temperature was kept stress dependences of the recoverable strain exhibit an constant to within 0.5 K. The tensile stress was con- almost linear behavior. From the slopes of these depen- trolled to within 2%. dences, we can determine the thermal expansion coef-

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

CORRELATION BETWEEN CHARACTERISTICS 1151

ﬁcients α = ∆ε/∆T at different stresses σ and the values of the reciprocal of the elastic modulus EÐ1 = ∆ε/∆σ at (a) different temperatures T. The stress dependence of the 1.22 thermal expansion coefﬁcient α(σ) and the temperature dependence of the reciprocal of the elastic modulus − E 1(T) for the steel wire are plotted in Figs. 2a and 2b, –1 , K respectively. After linear approximation of these depen- 5 1.19 ∆α ∆σ ∆ Ð

dences, we determined the quantities / and (E 10 × 1)/∆T, which are given in the table. The thermal expansion coefﬁcients α at T = 293 K and σ = 0 and the elastic α moduli E at T = 293 K are also listed in the table. More- over, the table presents the characteristics of thermal 1.16 and stress deformations for metallic copper [4, 5]. As follows from the table, the quantities ∆α/∆σ and ∆ Ð1 ∆ 300 700 1100 (E )/ T are close to each other (especially for cop- σ per). The small difference between these quantities for , MPa the steel can be associated with the errors in the mea- 0.55 (b) surements, because, as can be seen from Fig. 1, the effects of the stress σ on ε(T) and the temperature T on ε(σ) are insignificant. –1 Let us consider the factors responsible for the close 0.54 ∆α ∆σ ∆ Ð1 ∆ , Pa values of / and (E )/ T and determine the mag- 11

nitudes of these derivatives for metals. It should be 10 × noted that, in this case, the dominant role is played by 0.53 –1

the anharmonicity of the interatomic interaction. It is E this factor that governs not only the thermal expansion but also the stress dependence of the thermal expansion coefficient and the temperature dependence of the elas- 0.52 tic modulus. 300 340 380 In the simplest model of a solid (a diatomic mole- T, K cule with an anharmonic bond), the potential of the interatomic interaction can be represented by an Fig. 2. (a) Stress dependence of the thermal expansion coef- approximate expression, that is, ficient and (b) temperature dependence of the reciprocal of the elastic modulus for the steel wire. 1 2 1 3 Ur()≈ ÐD + --- fr()Ð a Ð ---gr()Ð a , (2) 2 3 Then, from relationships (3) and (4) for small ∆l, we where D is the dissociation energy, f is the coefficient of obtain linear elasticity, g is the anharmonicity coefficient, and 2 a is the equilibrium interatomic distance. α E a αα≈ 14+ ------0 0 ∆l , (6) According to [6], from expression (2), we derive an 0k approximate relationship for the elastic modulus, 2 Ð1 Ð1α E a f E ≈ E 12+ ------0 0 ∆l , E ≈ ---, (3) 0 k a (7) α gk f 0 gk 0 ==------, E0 ----- . α ≈ ------, (4) f 2a a f 2a 0 An analysis of expressions (6) and (7) shows that the where K is the Boltzmann constant. thermal expansion coefficient and the elastic modulus The elongation of the anharmonic bond by ∆l leads depend on the bond elongation, which, in turn, can be to a change (a decrease) in the bond rigidity, i.e., the caused by both the stress σ and heating (thermal expan- coefficient f, which can be represented as follows: sion). Therefore, the thermal expansion coefficient α and the elastic modulus E depend on the stress σ and ≈ 2g∆ the temperature T. Consequently, upon heating of the ff01 Ð ------l , (5) f 0 loaded sample (when measuring the dependence of the thermal expansion coefficient on the stress), the strain ∆ where f0 is the initial value of f (at l = 0). is determined not only by the thermal expansion but

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1152 LAŒUS et al.

Characteristics of thermal and stress deformations Calculation α, KÐ1 (at 293 K, ∆α/∆σ, ∆(EÐ1)/∆T, Material E, Pa (at 293 K) ∆α ∆σ σ = 0) PaÐ1 KÐ1 PaÐ1 KÐ1 / = ∆(EÐ1)/∆T, PaÐ1 KÐ1 Metals Steel 1.12 × 10Ð5 1.9 × 1011 0.9 × 10Ð15 2.7 × 10Ð15 (0.6Ð1.3) × 10Ð15 Copper [4, 5] 1.65 × 10Ð5 1.2 × 1011 2.9 × 10Ð15 3.1 × 10Ð15 Polymers PM Ð0.66 × 10Ð5 1.1 × 1011 6 × 10Ð15 7 × 10Ð15 K-49 Ð0.53 × 10Ð5 1.2 × 1011 3 × 10Ð15 6 × 10Ð15 ~5 × 10Ð15 SHPM Ð0.75 × 10Ð5 1.2 × 1011 4 × 10Ð15 4 × 10Ð15 PE Ð34 × 10Ð5 0.7 × 109 0.6 × 10Ð11 2.2 × 10Ð11 ~1 × 10Ð11 also has a stress component due to the change in the invariant is explained in terms of the anharmonicity of elastic modulus with variations in temperature. Corre- the interatomic interaction. spondingly, upon loading of the heated sample (when measuring the dependence of the elastic modulus on the temperature), the strain is governed not only by the 3.2. Rigid-Chain Oriented Polymers elastic elongation but also has a thermal expansion Rigid-chain polymers are polymers whose skeletons component associated with the change in the thermal contain ring groups that prevent conformational transi- expansion coefficient under a stress. Thus, the thermal tions and, thus, provide a straightened form of chain expansion coefficient and the elastic modulus, which molecules in the polymer material. In oriented poly- are defined as derivatives of the strain with respect to mers, the axes of chain molecules are predominantly the temperature and stress, respectively, turn out to be aligned along the axis of the orientation of the polymer more complex functions. (fiber axes). Making allowance for the combined character of the In this work, the thermal expansion coefficients and bond elongation involving the stress component ∆lσ = elastic moduli were measured in the longitudinal direc- Ð1 ∆σ ∆ α∆ tion under tensile stresses applied along the fiber axis. a(E ) and the thermal component lT = a T, we obtain the expression Figure 3 shows the temperature dependences of the recoverable longitudinal strain at different tensile Ð1 2 ∆α ∆()E α 3 stresses (Fig. 3a) and the stress dependences of the ------= ------≈ 6-----0-a . (8) ∆σ ∆T k recoverable longitudinal strain at different temperatures (Fig. 3b) for a rigid-chain oriented polymer— Therefore, the analysis of the diatomic model con- polyimide PM. firms the validity of the general relationship (1), i.e., the First and foremost, it can be seen from Fig. 3a that equality of the cross derivatives. the longitudinal thermal expansion of the fiber is nega- Now, we quantitatively estimate the magnitude of tive. The tensile stress leads to a decrease in the magni- the “invariant.” According to the data presented in the tude of the negative expansion (contraction). table, the thermal expansion coefficient can be esti- The stress strain of the fiber is positive (Fig. 3b). An α ≈ × Ð5 Ð1 mated as 0 (1.1Ð1.6) 10 K . The atomic volume increase in the temperature results in an increase in the stress strain. 3 ≈ ≈ A can be taken as a Va or Va ------ρ- , where A is the Consequently, the stress strain and the temperature NA ρ dependence of the stress strain for the fiber are similar atomic mass, is the density, and NA is the Avogadro ≈ × to those for the metals, whereas the thermal strain and number. For iron and copper, we have Va 1.2 − the stress dependence of the magnitude of the thermal 10 2 nm3. As a result, from expression (8), we obtain strain for the fiber are opposite in sign to those for the metals (cf. Figs. 3, 1). As can be seen from Fig. 3, the α 2 6-----0-a3 ≈ ()0.6Ð1.3 × 10Ð15 PaÐ1 KÐ1. temperature and stress dependences of the recoverable k longitudinal strain exhibit an almost linear behavior. From the slopes of these dependences, we can deter- We can assert that the calculations in the framework mine the thermal expansion coefficients α = ∆ε/∆T at of a rather crude model of a solid lead to a quantitative different stresses σ (from Fig. 3a) and the elastic mod- result that is in close agreement with the experimental uli E = ∆σ/∆ε at different temperatures T (from data. Fig. 3b). The thermal expansion coefficients α at σ = 0 Thus, for low-molecular materials with a conven- and the elastic moduli E at T = 293 K are presented in tional positive coefficient of thermal expansion, the the table. As can be seen, the elastic modulus E for

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 CORRELATION BETWEEN CHARACTERISTICS 1153

T, K σ, MPa 300 350 400 450 100 200

(a) –5.0 (a)

–0.4 –1 –5.5 3 , K 6 10 × 10

ε ×

α –6.0

–0.8 3 –6.5 2 1 (b) 4 3 (b) 2 1.02 0.8 1 –1 , Pa 11 0.98 3 10 ×

10 × –1

0.4 ε E

0.94 300 340 380 T, K

50 100 Fig. 4. (a) Stress dependence of the longitudinal thermal σ, MPa expansion coefficient and (b) temperature dependence of the reciprocal of the longitudinal elastic modulus for polyimide PM. Fig. 3. (a) Temperature and (b) stress dependences of the recoverable longitudinal strain for a rigid-chain oriented polymer—polyimide PM. (a) Tensile stress σ: (1) 2, (2) 126, and (3) 253 MPa. (b) Temperature T: (1) 293, tent with the general relationship (1), as is the case with (2) 323, (3) 353, and (4) 381 K. negative coefficients of thermal expansion. Let us now analyze the mechanism ensuring the polyimide PM is close in magnitude but opposite in validity of this relationship for rigid-chain polymers. sign to the elastic moduli E for steel and copper. This mechanism is associated with the specific features of the molecular dynamics in these polymers, which are The stress dependence of the longitudinal thermal primarily responsible for the negative sign of the longi- α σ expansion coefficient ( ) and the temperature depen- tudinal thermal expansion coefficient. The skeleton of dence of the reciprocal of the longitudinal elastic mod- chain polymer molecules has a high longitudinal rigid- ulus EÐ1(T) for polyimide PM are depicted in Figs. 4a ity due to the rigidity of covalent interatomic bonds. As and 4b, respectively. The table presents the quantities a consequence, the characteristic temperature of longi- ∆α/∆σ and ∆(EÐ1)/∆T, which were determined from the tudinal vibrations of polymer molecules is relatively slopes of the linear approximations of the dependences high (1000Ð1500 K [2]); hence, virtually no longitudi- α(σ) and EÐ1(T) shown in Figs. 4a and 4b. nal vibrations are excited at room temperature. The The results of similar measurements for other rigid- characteristic temperature of transverse vibrations is chain oriented polymers (K-49, SHPM) are also given considerably lower (200Ð400 K [2]). It is these trans- in the table. It can be seen from the table that the values verse vibrations that bring about the so-called membrane effect [7] or, more specifically, a decrease in the of each parameter (α, E, ∆α/∆σ, or ∆(EÐ1)/∆T) for dif- projection of the molecular skeleton contour onto the ferent rigid-chain polymers are close to each other. molecular axis, i.e., a decrease in the axial length of the We call the reader’s attention to the main result of molecule with an increase in the temperature (negative these investigations: the close values of ∆α/∆σ and coefficient of longitudinal thermal expansion). The the- ∆(EÐ1)/∆T for different rigid-chain polymers are consis- ory of negative longitudinal expansion in crystals and

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1154 LAŒUS et al. vitrified oriented polymers was developed in [8, 9]. axis a|| ≈ 0.13 nm [11], and the cross-sectional area of 2 This analysis did not include the effect of stresses on the molecule Sm ~ 0.3 nm [11]. the thermal expansion coefficient. Upon substituting the above parameters into rela- In our calculations, we used the model allowing for tionships (10)Ð(12), we obtain the longitudinal thermal the transverse vibrations, which was proposed earlier in expansion coefficient α ≈ Ð1 × 10Ð5 KÐ1 at σ = 0, the lon- [9], and additionally took into account the effect of the gitudinal elastic modulus E ≅ 1.5 × 1011 Pa at T = longitudinal stress σ on the sample. As a result, we 293 K, and the derivatives dα/dσ = d(EÐ1)/dT ≈ 5 × derived the following expression for the negative longi- −15 Ð1 Ð1 tudinal strain in a sample due to the combined effect of 10 Pa K . temperature and longitudinal stress (to the first order in The results of these calculations are in reasonable T and σ): agreement with the experimental data for rigid-chain polymers (see table). S S k ε(), σ ≈ 1 k m σ m σ Thus, for rigid-chain oriented polymers with spe- T Ð------T ++------q------2 -3 T, (9) 3 f ⊥a|| f ||a|| f ⊥ a|| cific negative coefficients of thermal expansion, the invariant is explained in terms of the anisotropy of the where f|| and f⊥ are the longitudinal and transverse rigid- molecular dynamics. ities of the polymer molecule, respectively; Sm is the cross-sectional area of the molecule; a|| is the length of the interatomic bond in the molecular skeleton; and q is 3.3. A Flexible-Chain Oriented a coefficient of the order of unity. Polymer—Poly(ethylene) From formula (9), we obtain the relationships for Flexible-chain polymers are polymers with simple the longitudinal thermal expansion coefficient and the “one-dimensional” molecular skeletons of the ÐCÐCÐCÐ longitudinal elastic modulus: type, which can twist upon devitrification due to conformational transitions [2]. ∂ε 1 k S k α ==------Ð ------+ q------m -σ, (10) The temperature dependences of the recoverable ∂ 2 3 T 3 f ⊥a|| f ⊥ a|| longitudinal strain at different tensile stresses for a flexible-chain oriented polymer—poly(ethylene) are Ð1 ∂ε Sm Smk shown in Fig. 5a. The stress dependences of the recov- E ==------+ q------2 -3T. (11) ∂σ f ||a|| erable longitudinal strain at different temperatures for f ⊥ a|| the flexible-chain oriented poly(ethylene) are depicted According to relationship (10), the thermal expan- in Fig. 5b. sion coefficient α is negative at σ = 0 and decreases in As can be seen from Fig. 5, the behavior of the flex- magnitude with an increase in the longitudinal stress σ, ible-chain oriented poly(ethylene) is qualitatively simi- which is actually observed in the experiment. lar to that of the rigid-chain oriented polymers. Indeed, As follows from expression (11), the quantity EÐ1 the longitudinal thermal expansion of poly(ethylene) is should increase with an increase in the temperature. also negative and the tensile stress leads to a decrease This behavior is also in agreement with the experimen- in the magnitude of the thermal expansion (Fig. 5a). tal data. The stress strain of poly(ethylene) also increases with an increase in the temperature (Fig. 5b). However, An analysis of relationships (10) and (11) shows quantitatively, the thermal expansion coefficients and that the aforementioned properties of the thermal elastic moduli of the flexible-chain oriented poly(ethyl- expansion coefficient and the elastic modulus are asso- ene) differ substantially from those of the rigid-chain ciated with the transverse vibrations. This can be oriented polymers (see below). judged from the coefficient f⊥ (the rigidity of transverse vibrations) involved in these formulas. It is worth noting noted that the temperature dependences of the recoverable longitudinal strain ε(T) The main result following from relationships (10) (Fig. 5a) are nonlinear (this inconstancy of the thermal and (11) can be represented as expansion coefficient with a variation in the tempera- Ð1 2 ture is in agreement with the data obtained for oriented ∂α ∂()E ∂ ε Smk ------===------q------. (12) poly(ethylene) in [12]). In order to elucidate the charac- ∂σ ∂ ∂σ∂ 2 3 T T f ⊥ a|| ter of the stress dependence of the thermal expansion coefficient, we analyzed the values of α(σ) = ∂ε/∂T in This means that the processes under consideration are the vicinity of only one temperature (300 K). The ther- consistent with the general relationship (1) (invariant). mal expansion coefficient α for the flexible-chain ori- The quantitative estimates were made for the follow- ented poly(ethylene) at σ = 0 is given in the table, and ing parameters: the longitudinal rigidity of the carbon- the stress dependence of the longitudinal thermal Ð1 chain molecular skeleton f|| ≈ 400 N m [10], the trans- expansion coefficient α(σ) is plotted in Fig. 6a. Ð1 verse rigidity of the molecular skeleton f⊥ ≈ 20 N m The stress dependences of the recoverable longitu- [10], the projection of the CÐC bond onto the molecular dinal strain ε(σ) are approximately linear (Fig. 5b). The

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 CORRELATION BETWEEN CHARACTERISTICS 1155

T, K σ, MPa 290 300 310 5 10 15 20 –2.0 (a) (a) –1

, K –2.5 4 –2 10 ×

3 –3.0 10 ×

ε –4 (b)

4 –1 1.7 , Pa

3 9 –6 10 1.5 2 ×

1 –1 E 3 1.3 (b) 2 290 300 310 10 T, K

1 Fig. 6. (a) Stress dependence of the longitudinal thermal 8 expansion coefﬁcient and (b) temperature dependence of the reciprocal of the longitudinal elastic modulus for

3 poly(ethylene).

10 6 ×

ε that the general relationship (1) is satisfied for the flex- 4 ible-chain oriented poly(ethylene), as is the case with metals and rigid-chain oriented polymers. However, quantitatively, the characteristics of 2 poly(ethylene) and rigid-chain polymers differ significantly. For poly(ethylene), the negative coefficient of longitudinal thermal expansion is approximately one 2 4 6 and a half orders of magnitude larger, the longitudinal σ, MPa elastic modulus is nearly two orders of magnitude smaller, and the temperatureÐstress invariant [∆α/∆σ = Fig. 5. (a) Temperature and (b) stress dependences of the ∆(EÐ1)/∆T] is almost four orders of magnitude greater recoverable longitudinal strain for a flexible-chain oriented than those for the rigid-chain polymers. polymer—poly(ethylene). (a) Tensile stress σ: (1) 5, (2) 10, (3) 15, and (4) 20 MPa. (b) Temperature T: (1) 293, (2) 305, These differences suggest that the data obtained for and (3) 311 K. poly(ethylene) cannot be interpreted within the model proposed for rigid-chain polymers. Actually, the ∂σ ∂ε extremely small elastic modulus of poly(ethylene) indi- elastic modulus E = / for poly(ethylene) at T = 293 K cates that the stress strain is inconsistent with the elon- is presented in the table, and the temperature depen- gation of covalent bonds in the polymer skeleton. Cor- dence of the reciprocal of the longitudinal elastic mod- Ð1 respondingly, the large negative coefficient of thermal ulus E (T) is shown in Fig. 6b. expansion has defied description in the framework of The table also presents the quantities ∆α/∆σ and the membrane mechanism. It is evident that, in the flex- ∆(E−1)/∆T, which were determined from the slopes of ible-chain oriented poly(ethylene), the thermal and the linear approximations of the dependences α(σ) and stress deformations occur through a different mecha- EÐ1(T) depicted in Figs. 6a and 6b. nism. As follows from the table, the quantities ∆α/∆σ and In poly(ethylene), the thermal and stress deforma- ∆(EÐ1)/∆T are in satisfactory agreement. This implies tion processes proceed through the conformational

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1156 LAŒUS et al. mechanism. This mechanism involves local conforma- conformers, Sm is the cross-sectional area of the mole- tional transitions of fragments in the chain molecule cule, and χ is the degree of crystallinity (volume frac- that lead to a change in the axial length of the molecule tion of crystallites). (projection of the molecular skeleton contour onto the Thus, the general relationship (1), i.e., the equality molecular axis) [13]. of the derivatives, is satisfied for poly(ethylene). The polymer molecule can be considered as a Now, we perform a number for quantitative estima- sequence of trans conformers (straightened fragments) tions. and gauche conformers (bent fragments). The conformers can transform from one type into the other type, Poly(ethylene) is an amorphousÐcrystalline poly- which results in a change in the axial length of the mol- mer. The glass transition temperature of poly(ethylene) ecule. These transitions are accompanied by overcom- is approximately equal to 240 K. At higher tempera- ing the potential barrier due to local fluctuations of the tures (T > 240 K), poly(ethylene) is in the devitrified state. For oriented poly(ethylene) samples, the degree thermal energy [13]. The difference between the poten- χ ≈ tial energies of the conformers leads to an increase in of crystallinity is estimated as 0.4. An elementary the equilibrium concentration of gauche conformers conformer in poly(ethylene) involves four CH2 groups. with an increase in the temperature [13] and, hence, to The length of the elementary trans conformer is a negative longitudinal expansion. Under longitudinal approximately equal to 0.5 nm, whereas the length of tensile stresses, the difference between the energies of the elementary gauche conformer is approximately the gauche and trans conformers increases. This leads equal to 0.3 nm. The difference between the potential to a decrease in the equilibrium concentration (at a energies of the trans and gauche conformers is esti- ∆ ≈ × Ð2 given temperature) of gauche conformers and, corre- mated as Ue 2.3 10 eV [2]. spondingly, to an increase in the axial length of the mol- A single conformational transition of an elementary ecule (i.e., to an elongation of the oriented polymer). conformer in a long-chain molecule is virtually impos- Therefore, in amorphous (noncrystalline) regions of sible. Such transitions occur with the participation of a flexible-chain devitrified oriented polymer, the ther- several (three or four) conformers [2, 13]. Hence, the mal and stress deformations occur through the mecha- calculations were performed for the following parame- ≈ δ ≈ ∆ ≈ × nism of conformational transitions. The approximate ters: lt (1.5Ð2.0) nm, l (0.6Ð0.8) nm, U (7Ð10) Ð2 ≈ 2 calculations of the longitudinal strain of the devitrified 10 eV, and Sm 0.3 nm . (i.e., at temperatures above the glass transition point) Then, from relationships (13)Ð(15), we found α ≈ oriented polymer due to conformational transitions −20 × 10Ð5 KÐ1 at σ = 0, E ≈ 1 × 109 Pa at T = 293 K, were carried out in our previous works [9, 14]. and ∂α/∂σ = ∂(EÐ1)/∂T ≈ 0.5 × 10Ð11 PaÐ1 KÐ1. The expressions derived for the quantities under With due regard for the roughness of the calcula- investigation are as follows. tions and the errors in the measurements, the calculated The longitudinal thermal expansion coefficient is and experimental data (see table) are in reasonable determined by the formula agreement. ∆ε δl 1 ∆U α = ------≈ Ð()1 Ð χ ------exp Ð------∆ 2  T lt kT kT 4. CONCLUSIONS (13) ∆ The thermal expansion of solids proceeds through ×∆ δ σU U Ð.lSm ------Ð 1 different mechanisms. These are the vibrationalÐanhar- kT monic mechanism (due to the anharmonic interatomic The reciprocal of the elastic modulus has the form interaction) in low-molecular solids, the vibrationalÐ anisotropic (membrane) mechanism in polymer crys- Ð1 ∆ε δl 1 ∆U tals and oriented polymers in the vitreous state (without E = ------≈ ()1 Ð χ ------exp Ð------δlS . (14) ∆σ l kT kT m participation of the anharmonic interatomic interac- t tion), and the conformational mechanism (associated From relationships (13) and (14), we obtain with the jumpwise fluctuation dynamics) in polymers in the devitrified state. As a result, the thermal expan- ∂α ∂()EÐ1 sion coefficients of solids can differ not only by several ------= ------∂σ ∂T orders of magnitude but also in sign. (15) It should be noted that solids undergo elastic defor- δ 2 ∆ ∆ ≈ ()χ l Sm 1 U U mation also through different mechanisms, such as 1 Ð ------2 expÐ------------Ð 1 , ltk T kT kT stress elongation (contraction) of interatomic bonds in low-molecular materials and polymers in the vitreous δ where lt is the axial length of the trans conformer, l is state and conformational transitions in polymers in the the difference between the axial lengths of the trans and devitrified state. In these cases, the elastic moduli of gauche conformers, ∆U is the initial difference different solids are always positive but can differ by between the potential energies of the gauche and trans several orders of magnitude.

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 CORRELATION BETWEEN CHARACTERISTICS 1157

All the above mechanisms were revealed in the REFERENCES materials studied in this work (see table). 1. Physical Quantities: A Handbook, Ed. by I. S. Grigor’ev The results obtained in the above investigation dem- and E. Z. Meœlikhov (Énergoatomizdat, Moscow, 1991). onstrated that, although solids involved in the processes 2. Yu. K. Godovskiœ, Thermal Physics of Polymers under consideration can exhibit quite different thermal (Khimiya, Moscow, 1982). and stress behaviors, all materials share a number of 3. S. I. Novikova, Thermal Expansion of Solids (Nauka, traits. In particular, the thermal expansion coefficient Moscow, 1974). depends on the mechanical stress; however, tensile stress can lead to an increase in the thermal expansion 4. N. F. Kunin and V. N. Kunin, Fiz. Met. Metalloved. 5 (1), coefficient for metals and a decrease in the (negative!) 173 (1957). coefficient of thermal expansion for polymers. The 5. A. Nadai, Theory of Flow and Fracture of Solids, 2nd ed. elastic modulus for all materials depends on the tem- (McGraw-Hill, New York, 1963; Mir, Moscow, 1969), perature in a similar manner: i.e., it decreases with an Vol. 2. increase in the temperature. The equality between the 6. Ya. I. Frenkel, Introduction to the Theory of Metals temperature and stress derivatives ∂α/∂σ = ∂(EÐ1)/∂T (GITTL, Moscow, 1948). holds good for all solids (with allowance made for inad- 7. I. M. Lifshitz, Zh. Éksp. Teor. Fiz. 22 (4), 475 (1952). vertent errors in the measurements). This implies that 8. F. C. Chen, C. L. Choy, and K. Young, J. Polym. Sci., the general relationship (1) (thermodynamically inde- Polym. Phys. Ed. 19 (12), 2313 (1980). pendent of the specific properties of different solids) is 9. A. I. Slutsker, V. L. Gilyarov, G. Dadobaev, L. A. Laœus, valid for all solids. I. V. Gofman, and Yu. I. Polikarpov, Fiz. Tverd. Tela The results of the approximate calculations per- (St. Petersburg) 44 (5), 923 (2002) [Phys. Solid State 44, formed in this work are in reasonable agreement with 964 (2002)]. the experimental data. These results confirm and justify 10. L. R. G. Treloar, Polymer, No. 1, 95 (1960). the concept regarding different mechanisms of thermal 11. K. Bunn, Trans. Faraday Soc., No. 35, 482 (1939). and stress deformations of solids with different structures. 12. F. C. Chen, C. L. Choy, and K. Young, J. Polym. Sci., Polym. Phys. Ed. 19 (2), 335 (1981). Moreover, the values of the invariant ∂α/∂σ = ∂ −1 ∂ 13. M. V. Vol’kenshtein, Configurational Statistics of Poly- (E )/ T, which can differ by several orders of magni- meric Chains (Akad. Nauk SSSR, Moscow, 1959; Inter- tude for solids (see table), correspond to different science, New York, 1963). mechanisms of thermal and stress deformations of sol- 14. A. I. Slutsker, V. L. Gilyarov, L. A. Laœus, I. V. Gofman, ids with different structures. Yu. I. Polikarpov, and B. A. Averkin, in Nonlinear Prob- lems in Mechanics and Physics of Strained Solids ACKNOWLEDGMENTS (S.-Peterb. Gos. Univ., St. Petersburg, 2002), No. 5. This work was supported by the Russian Foundation for Basic Research, project no. 00-03-33064. Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

POLYMERS AND LIQUID CRYSTALS

Thermal Fluctuations in Smectic-A Films on the Surface of Solid Substrates L. V. Mirantsev Institute for Problems of Mechanical Engineering, Russian Academy of Sciences, Bol’shoœ pr. 61, Vasil’evskiœ Ostrov, St. Petersburg, 199178 Russia e-mail: [email protected] Received September 18, 2003; in ﬁnal form, November 10, 2003

Abstract—The static and dynamic characteristics of layer displacement fluctuations in smectic-A films supported on the surface of a solid substrate are calculated with due regard for the profiles of the flexural and tensile (compressive) moduli of smectic layers. The difference in the surfaces bounding the film and the asymmetry of the profiles of the elastic moduli with respect to the central layer of the film are taken into account. The profiles of fluctuations of smectic-layer displacements and the correlations between these fluctuations are determined for the films formed by liquid-crystal compounds that can undergo a bulk smectic-AÐnematic phase transition. The dynamic correlation functions derived for these fluctuations are used for calculating the correlations between the intensities of x-ray scattering by a film at different instants of time. It is demonstrated that, in smectic-A films supported on the surface of a solid substrate, unlike free-standing smectic-A films, the effect of temperature on the dynamics of layer displacement fluctuations can be observed in experiments on dynamic x-ray scattering from films that are not very thick (the number of layers N ~ 20) and at considerably smaller recoil- momentum components in the film plane. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION lying the substrate [10Ð12, 15]. In particular, de Boer Smectic liquid crystals have a one-dimensional lay- [16] showed that, for these films, the correlation func- ered structure [1] and, consequently, can form free- tions of layer displacement fluctuations, which deter- 2 mine the intensity of diffuse x-ray scattering, can be standing films with a macroscopic area (~cm ) [2] and separated into two components. One component is gov- a thickness that can be varied from several hundreds to erned by the thermal fluctuations of smectic-layer dis- two smectic layers or even to one layer [3, 4]. In this placements, and the other component is determined by respect, smectic liquid crystals are the most appropriate the defects on the substrate surface. Therefore, the ade- model objects for use in studying the physics of two- quate theoretical description of thermal fluctuations of dimensional systems. Moreover, in thin free-standing layer displacements in a smectic liquid-crystal film smectic films, the effects associated with their micro- offers possibilities for using x-ray scattering as an effi- scopic thickness are combined with surface effects. As cient tool for obtaining information on both thermal a result, these films exhibit effects and phenomena that fluctuations in the film and defects on the surface of a are not observed in the bulk phase of liquid crystals [5Ð substrate. 9]. For this reason, over the last 20Ð25 years, free- standing smectic films have been extensively studied The thermal fluctuations of layer displacements in both experimentally and theoretically. smectic-A (SmA) films supported on the surface of a In recent years, considerable interest has been solid substrate were theoretically described by de Boer expressed by researchers in smectic films supported on [16] and Romanov and Ul’yanov [17]. In these works, the surface of a solid substrate [10Ð15]. These systems, the thermal fluctuations of layer displacements were like free-standing smectic films, have been predomi- treated in the framework of the well-known discrete nantly studied using specular and diffuse (nonspecular) model (Holyst model) [18Ð20] proposed for describing small-angle x-ray scattering, as well as neutron scatter- the layer displacement fluctuations in free-standing ing. The aforementioned scattering methods provide a smectic-A films. Within this model, the smectic-A film means for determining the equilibrium characteristics is considered to be spatially homogeneous and charac- of smectic liquid-crystal films (such as the number of terized by the following parameters: the number N of smectic layers, the layer thickness, and the type of smectic layers, the surface tension coefficient γ of the molecular packing in a layer) and reliable data on ther- free surface (note that, in [16], the surface tension coef- mal fluctuations in films. Over the last decade, experi- ficient of the film–substrate interface was treated as ments on diffuse x-ray scattering have revealed useful infinitely large, whereas in [17], the equivalent assump- information about defects on the surface of a solid sub- tion was made that the smectic layer at the boundary strate that induce distortions in layers of the film over- with the substrate is fixed), and the flexural modulus K

THERMAL FLUCTUATIONS IN SMECTIC-A FILMS 1159 and tensile (compressive) modulus B of the smectic lay- peratures. This is associated with the fact that the sur- ers. The last two quantities are considered to be identi- faces bounding the film (i.e., the free surface and the cal for all layers in the film irrespective of their location film–substrate interface) are not equivalent and can and are taken equal to the corresponding elastic modu- have different orienting effects on molecules in the lus in the bulk of the smectic-A phase. In my previous film. As a result, the film has no symmetry with respect works [21Ð23], it was noted that, even for free-standing to the central layer, which is inconsistent with the smectic-A films, the above assumption is physically assumption of spatial homogeneity. This indicates that justified only at temperatures substantially lower than the theoretical approach used in [16, 17] for describing the temperatures of the SmAÐnematic (SmAÐN) or the thermal fluctuations of layer displacements is inad- SmAÐisotropic (SmAÐI) phase transitions in the bulk of equate for the films under consideration. liquid crystals. In this case, the film has a pronounced smectic-A structure throughout the volume and the ori- Although the static properties of thermal fluctua- entational and translational orders of molecules and, tions of layer displacements in free-standing smectic hence, the elastic characteristics of the inner and sur- films have already long been investigated, their dynam- face layers of the film only slightly differ from each ics has come under the scrutiny of science only in other. However, thin smectic layers of particular liquid recent years. The dynamics of thermal fluctuations of crystals can exist at temperatures considerably higher layer displacements has been experimentally investi- than the temperatures at which the smectic order in the gated using coherent soft [29] and hard [30] x-ray bulk of mesogens disappears [5Ð9]. The microscopic dynamic scattering. The experimental results obtained model proposed in [24Ð27] for free-standing smectic-A were theoretically interpreted in [31Ð34]. The theoreti- films describes the behavior of these films upon heating cal interpretation was based on the linearized hydrody- above the temperatures of the SmAÐnematic and SmAÐ namic equations for the smectic-A phase and the afore- isotropic phase transitions rather well. According to the mentioned Holyst model [18Ð20] for the energy of predictions of this model, both the orientational and layer displacement fluctuations in free-standing smec- translational orders in inner layers of the film at these tic-A films. The models proposed in [31Ð34] make it temperatures can be substantially less pronounced than possible to determine the dynamic correlation functions those in the vicinity of the free bounding surfaces. This for layer displacement fluctuations in films and then to theoretical result was experimentally confirmed in use these functions for calculating the correlations studies of the optical transmission spectra of free- between the intensities of x-ray scattering by films at standing smectic-A films [28] and the specular x-ray different instants of time. The predictions of these mod- reflectivity of smectic-A films above the temperature of els are in qualitative agreement with the experimental the SmAÐnematic phase transition in the bulk of liquid data on coherent x-ray dynamic scattering in free- crystals [9]. It is well known [1] that, in the bulk of liq- standing smectic-A films with different thicknesses [29, uid crystals, the flexural modulus K is proportional to 30]. The dynamics of thermal fluctuations of layer dis- the square of the orientational order parameter p and the placements in smectic-A films supported on the surface tensile (compressive) modulus B of smectic layers is of a solid substrate has not been experimentally studied proportional to the square of the translational order to date. However, this dynamics was theoretically ana- parameter s. Under the assumption that these relation- lyzed in [17] within the Holyst model [18Ð20], accord- ships are also valid for liquid-crystal films, the elastic ing to which the spatial inhomogeneity of the smectic- moduli of the smectic layers at temperatures higher A films is disregarded. As a consequence, the results than the temperatures of the SmAÐnematic or SmAÐiso- obtained in [17] turned out to be independent of the tropic phase transitions in the bulk of liquid crystals temperature, because the above assumption leads to should decrease with an increase in the distance to the complete disregard of the temperature dependence of free surface and reach minimum values at the center of the viscoelastic parameters for layers of the film. the film. Since this profile of the elastic moduli is completely ignored in the model proposed in [18Ð20], the The influence of the profiles of the flexural modulus model at the aforementioned temperatures should K and the tensile (compressive) modulus B of smectic incorrectly predict the fluctuations of smectic-layer dis- layers and, hence, the temperature on the dynamics of placements and correlations between these fluctuations. thermal fluctuations of layer displacements in free- The influence of spatial inhomogeneity of the free- standing smectic-A films was theoretically treated in standing smectic-A films on the thermal fluctuations of my recent work [35]. It was demonstrated that this layer displacements was theoretically analyzed in [21Ð effect can be revealed in experiments on x-ray dynamic 23]. It turned out that the results obtained differ signif- scattering in sufficiently thick films (N ≥ 100) and at icantly from the data predicted by the Holyst model at large recoil-momentum components in the film plane the highest temperatures of existence of free-standing (≥106 cmÐ1). In films with a smaller thickness and at smectic-A films with a specified thickness. However, it smaller recoil-momentum components in the film should be noted that the assumption regarding the spa- plane, the dominant role in the dynamics of thermal tial homogeneity of the smectic film supported on the fluctuations of layer displacements in free-standing surface of a substrate can also be invalid at lower tem- smectic-A films is played by the so-called acoustic

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

1160 MIRANTSEV mode [31, 32] with a virtually temperature-independent especially when the orienting effect of the solid sub- relaxation time, strate on liquid-crystal molecules is appreciably weaker than that of the free surface of the film. Moreover, the τ()1 η γ = Ndin 3/2 , (1) dynamic correlation functions derived for the layer displacement fluctuations were used to calculate the corre- η where din is the interlayer spacing and 3 is the viscos- lations between the intensities of x-ray scattering by the ity coefficient of interlayer sliding. This mode corre- film at different instants of time. It was demonstrated sponds to the motion of the film during which the inter- that, in smectic-A films supported on the surface of a layer spacing remains constant, and, hence, the motion solid substrate, unlike free-standing smectic-A films, does not depend on the profiles of the elastic moduli K the effect of temperature on the dynamics of layer dis- and B. However, the acoustic mode is absent in smectic placement fluctuations can be observed in experiments films supported on the surface of a solid substrate [17]. on dynamic x-ray scattering from not very thick films Therefore, thermal fluctuations of layer displacements (N ~ 20) and at considerably smaller recoil-momentum in the smectic film are always accompanied by a change components (≤105 cmÐ1) in the film plane. in the interlayer spacing and the dynamic characteristics of thermal fluctuations should depend on the profiles of the elastic moduli K and B and, consequently, on 2. FLUCTUATIONS OF LAYER DISPLACEMENTS the temperature even for not very thick films and at not IN A SMECTIC-A FILM ON THE SURFACE very large recoil-momentum components in the film OF A SOLID SUBSTRATE AND FLUCTUATION plane. To put it differently, the thermal fluctuations of CORRELATIONS layer displacements in the smectic films supported on In [16, 17], it was shown that thermal fluctuations of the substrate surface should be affected by the spatial smectic-layer displacements from equilibrium posi- inhomogeneity to a greater extent than those in free- tions in a smectic-A film supported on the surface of a standing smectic-A films. solid substrate can be described in the framework of the In this work, the static and dynamic characteristics discrete model proposed in [18Ð20] for calculating of layer displacement fluctuations in smectic-A films thermal fluctuations of layer displacements in free- supported on the surface of a solid substrate were cal- standing smectic-A films. We will consider a smectic-A culated with due regard for the dependences of the flex- film that is supported on the surface of a substrate and ural modulus K and the tensile (compressive) modulus consists of N discrete layers. Let un(x, y) be fluctuation B of smectic layers on the distance to the surfaces displacements of layers from the equilibrium positions bounding the film. The computational procedure was zn = (n Ð 1)d along the z axis perpendicular to the layer similar to that used in [21Ð23, 35] for calculating the plane, where d is the thickness of the smectic layer and thermal fluctuations in free-standing smectic-A films n is the number of the smectic layer. These displace- but also took into account the difference in the surfaces ments are characterized by the excess free energy F, bounding the film (the surface tension coefficient γ was which involves the bulk contribution FB described by taken to be finite for the free surface and infinitely large Eqs. (2) and (3) from [21] and the surface contribution for the film–substrate interface) and the asymmetry of FS. According to [16], the surface contribution FS can the profiles of the elastic moduli K and B with respect be written in the form to the central layer of the film. The elastic modulus profiles were determined in terms of the microscopic 1 []γ ∇ ()2 γ ∇ ()2 FS = ---∫ 1 ⊥u1 R + N ⊥uN R dR. (2) model proposed in [24Ð27] with allowance made for 2 different orienting effects of the bounding surfaces on γ Here, 1 is the surface tension coefficient of the free sur- molecules in the film. The profiles of smectic-layer dis- face of the film, γ is the surface tension coefficient of placement fluctuations and correlations between these N the film–substrate interface, R is the radius vector in the fluctuations were calculated for the films formed by liq- 2 2 2 ∇ uid-crystal compounds that can undergo a bulk SmAÐ plane of smectic layers (R = x + y ), and ⊥ is the pro- nematic phase transition. The calculations were per- jection of the operator — onto the (x, y) plane. formed both at temperatures considerably below the With the use of the continuous Fourier transform temperature of the phase transition in the bulk of liquid () ()π Ð2 () () crystals and at the highest temperatures of existence of un R = 2 ∫un q⊥ exp iq⊥R dq⊥ , (3) films with a specified thickness. It was shown that, at temperatures below the temperature at which the smec- we obtain a simple expression for the excess free tic order disappears in the bulk of liquid crystals, the energy F: inclusion of the profiles of the elastic moduli K and B N does not lead to disagreement with the data obtained 1 F = --- ∑ u ()q⊥ M u ()Ðq⊥ dq⊥, (4) within the Holyst model [16, 17]. However, at temper- 2 ∫ k kn n atures considerably above the temperature of the bulk kn, = 1 phase transition, this inclusion can result in substantial where Mkn are the elements of the banded matrix. In this deviations from the predictions of the Holyst model, matrix, only the diagonal and first adjacent elements

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 THERMAL FLUCTUATIONS IN SMECTIC-A FILMS 1161 are nonzero. The latter elements are defined by the fol- 27] for a thin liquid-crystal layer with two bounding lowing formulas: surfaces, the above situation can be described by the model parameter α = 2exp[Ð(πr /d)2] ≤ 0.98, where γ 2 4 () 0 M11 ==1q⊥ ++K1dq⊥ B1 + B2 /2d b1, (5) r0 is the characteristic radius of interaction for the γ 2 4 () model pair intermolecular potential used in the MNN ==Nq⊥ ++K Ndq⊥ BN Ð 1 + BN /2d bN, (6) McMillan theory. As in [21Ð23], the calculations were 4 performed at α = 0.871. The thermal fluctuations of M ==K dq⊥ + ()B ++2B B /2d b , nn n n Ð 1 n n + 1 n (7) layer displacements in the film and the correlations n = 2, N Ð 1, between these fluctuations were calculated at two temperatures: the first temperature T1 was considerably M ==M Ð()B + B /2d =c , n + 1n nn + 1 n n + 1 n (8) lower than the temperature of the SmAÐnematic phase n = 1, N Ð 1, transition in the bulk of the liquid crystal, whereas the second temperature T2 was only slightly below the lim- where Kn is the flexural modulus of the nth layer of the iting temperature at which the film with a specified film and Bn is the tensile (compressive) modulus of the thickness can exist. According to [24Ð27], the limiting nth layer. When the elastic moduli K and B in the bulk temperature is the temperature at which the smectic of the smectic-A phase of a liquid crystal are known at order in the bulk of the film disappears almost com- ≡ ≡ a temperature T0 [K(T0) K0, B(T0) B0], which is pletely (s 0). As in [21–23], the first temperature lower than the temperature of the SmAÐisotropic or was taken as T1 = 0.204(V0/kB) [in [36], the temperature SmAÐnematic phase transition, the elastic moduli Kn of the SmAÐnematic phase transition in the bulk of the α and Bn for any liquid-crystal film with a specified thick- liquid crystal at = 0.871 was determined to be TAN = ness at arbitrary temperature T (at which the film exists) 0.2091(V0/kB)], where V0 is the intermolecular interac- can be determined using the microscopic model pro- tion constant in the McMillan theory. In the framework posed in [24Ð27] for a thin liquid-crystal layer with two of the model developed in [24Ð27], the second temper- bounding surfaces. The algorithm for calculating these ature T2 is governed by the orienting effect of the sur- moduli was described in detail in my previous works faces bounding the film on liquid-crystal molecules. [21, 23]. This effect is simulated by the effective external fields, Knowing the elements Mkn of the direct matrix, we whose strengths are proportional to the interaction con- Ð1 can determine the elements (M )kn of the inverse stants W1 (the free surface of the film) and WN (the film– matrix. Then, the latter elements can be used to calcu- substrate interface). As in my previous works [21Ð23], late the mean fluctuations of layer displacements in the the ratio of the constant W1 to the intermolecular inter- σ 〈 2 〉1/2 action constant V was taken equal to 1.6. As regards film n = un (0) and the correlations between the 0 〈 〉 σ σ the constant WN, we considered three cases. In the first fluctuations gk, n(R) = uk(R)un(0) /( k n). According to [18, 19], we have case, the constant WN was equal to W1 (the orienting effects of bounding surfaces on molecules in the film 2 2 k T Ð1 σ 〈〉() B () are identical). In the second case, we set WN = 5W1 (the n ==un 0 ------∫ M nndq⊥, (9) ()2π 2 profound orienting effect of the film–substrate interface). In the third case, the calculations were carried out k T 〈〉() B ()Ð1 () at WN = W1/5 (the orienting effect of the film–substrate uk R un0 = ------∫ M knexp iq⊥R dq⊥, (10) ()2π 2 interface is weaker than that of the free surface). As a result, the second temperatures were as follows: T2 = where k is the Boltzmann constant. On the right-hand B 0.21035(V0/kB) in the first case (as in [21–23]), T2 = side of relationships (9) and (10), the lower limits of 0.21039(V0/kB) in the second case, and T2 = integration are determined by the transverse sizes L of 0.2093(V /k ) in the third case. As in [21–23], the flex- the film and the upper limits of integration are deter- 0 B ural modulus K0 and the tensile (compressive) modulus mined by the molecular diameter a (2π/L ≤ |q⊥| ≤ 2π/a). B of smectic layers for the bulk liquid-crystal phase at If the transverse sizes of the film are macroscopic (L ~ 0 cm), the lower limits of integration in relationships (9) the temperature T1 lower than the phase transition tem- Ð6 and (10) can be set equal to zero. perature TAN were assumed to be K0 = 10 dyn and B0 = σ 108 dyn/cm2 (typical values for the majority of liquid The fluctuations of smectic layer displacements n crystals [1]). The surface tension coefficient γ for the and correlations gk, n(R) were numerically calculated 1 for the smectic-A film supported on the surface of a free surface was equal to 25 dyn/cm (taken from [9]), γ solid substrate and consisting of N = 24 layers. It was and the surface tension coefficient N (by analogy with γ γ assumed that the film is formed by a liquid crystal that [16]) was assumed to be infinitely large ( N = 1000 1). can undergo a weak first-order SmAÐnematic phase The thickness of the smectic layers in the film did not transition. According to the McMillan model [36] for a depend on the temperature and was taken equal to the bulk smectic-A phase and the model developed in [24Ð molecular length l = 30 Å, and the molecular diameter

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1162 MIRANTSEV

K/K0 tuations of smectic layer displacements were consid- 1.6 ered in free-standing smectic-A films (see Figs. 1, 2 in [21, 23]). The profiles determined in the second case (WN = 5W1) are also almost identical to the profiles calculated in the first case, even though the values of K/K 1.2 0 and B/B0 for the Nth layer of the film are slightly larger 1 than those in the first case. However, in the third case 2 (WN = W1/5), the profiles for K/K0 (Fig. 1) and B/B0 0.8 (Fig. 2) differ substantially from the corresponding profiles obtained in [21, 23]. First, as would be expected, these profiles are not symmetric with respect to the cen- 0.4 ter of the film. Second, the minimum elastic moduli 0 5 10 15 20 25 (this is especially true for the moduli B) are observed n not in the central part of the film but in the layer adjacent to the surface of the solid substrate. It should also Fig. 1. Profiles of the flexural modulus K in a smectic-A film supported on the surface of a solid substrate: (1) at temper- be noted that, at the low temperature T1, the profiles of ature T1 below the temperature of the SmAÐnematic phase the elastic moduli K and B (curves 1 in Figs. 1, 2; transition and (2) at the highest temperature T2 of existence Figs. 1, 2 in [21, 23]) in all three cases are characterized α of the film. N = 24, = 0.871, W1/V0 = 1.6, WN = W1/5, by a rather extended plateau in the central part of the T1 = 0.204(V0/kB), and T2 = 0.2093(V0/kB). film, in which the moduli are nearly constant and close in magnitude to the bulk values K0 and B0, respectively. At the same time, it can be seen that this plateau is B/B0 absent at the highest temperature T2 of existence of the 1.2 film (curves 2 in Figs. 1, 2; Figs. 1, 2 in [21, 23]). Con- sequently, at the temperature T1, the major part of the 1 smectic-A film on the surface of the solid substrate is actually spatially homogeneous (or almost homogeneous) and the Holyst model [18Ð20] used in [16, 17] 0.8 should lead to results that differ very little from our data. However, in the vicinity of the limiting tempera- 2 ture at which the film with a specified thickness can exist (T = T2), the film in all three cases under consideration is spatially inhomogeneous (curves 2 in Figs. 1, 2; 0.4 Figs. 1, 2 in [21, 23]). Moreover, in the third case, the film is not symmetric with respect to its center and the elastic moduli K and B in the layers adjacent to the substrate surface are considerably smaller than those in the layers near the free surface. It is in this case that the results obtained in [16, 17] within the Holyst model 0 5 10152025 should differ most significantly from our data. n Then, the determined profiles of the elastic moduli Fig. 2. Profiles of the tensile (compressive) modulus B in a K and B were used to calculate the profiles of the layer smectic-A film supported on the surface of a solid substrate: σ displacement fluctuations n in the smectic-A film (con- (1) at temperature T1 below the temperature of the SmAÐ nematic phase transition and (2) at the highest temperature sisting of 24 layers) supported on the surface of a solid T2 of existence of the film. The parameters are the same as substrate at the low temperature T1 and the highest tem- in Fig. 1. perature T2 of existence of the film. The results of these calculations for the first (WN = W1) and third (WN = W1/5) cases are presented in Fig. 3 (the profiles for the a was equal to 4 Å (which are typical sizes for liquid- second case at W = 5W differ only slightly from the crystal molecules [1]). N 1 profiles at WN = W1). The profile of the displacement σ First and foremost, the profiles of the elastic moduli fluctuations n calculated within the Holyst model [18, K and B for the aforementioned ratios between the ori- 20] is also shown by the dashed line in Fig. 3. As could enting effects of the surfaces bounding the film were be expected, at T = T1, our results do not differ greatly calculated with the use of the model developed in [24Ð from those obtained in terms of this model. However, in 27]. As could be expected, in the first case, the calcu- the vicinity of the limiting temperature of existence of lated profiles completely coincide with similar profiles the film (T = T2), the inclusion of the profiles of the elas- obtained in my earlier works, in which the thermal fluc- tic moduli K and B leads to a substantial deviation from

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 THERMAL FLUCTUATIONS IN SMECTIC-A FILMS 1163 δ the predictions of the Holyst model [18, 20]. Indeed, in n, Å the first case, the maximum value of layer displacement 6 σ fluctuations n calculated with due regard for these pro- 1 files at the temperature T2 = 0.21035(V0/kB) is approxi- 2 σ mately half the maximum value of n at the temperature 3 σ T1. Furthermore, the maximum of the n profile is 4 4 shifted to the center of the film, where the moduli K and B have minimum values. At the same time, according to the calculations carried out in terms of the Holyst model, this increase in the temperature weakly affects the profile of these fluctuations. In the third case, an 2 increase in the temperature from T1 to T2 = 0.2093(V0/kB) leads to a more radical change in the pro- σ σ file of the fluctuations n. At T = T1, the fluctuations n only slightly depend on the number of the layer (except σ 0 5 10 15 20 25 for a drastic decrease in n to zero at n = 24). On the other hand, at the limiting temperature of existence of n σ the film, T = T2, the fluctuations n considerably Fig. 3. Calculated profiles of layer displacement fluctua- increase (by a factor of approximately two) with an tions in a smectic-A film supported on the surface of a solid substrate. The calculations were performed with due regard increase in n to the penultimate layer of the film and σ for the profiles of the moduli K and B for parameters (1) sharply decrease to zero at n = 24. Therefore, the n WN = W1 and T = T1 = 0.204(V0/kB), (2) WN = W1 and T = profile exhibits a maximum in the vicinity of the sur- T2 = 0.21035(V0/kB), (3) WN = W1/5 and T = T1 = face of the solid substrate, i.e., in the region where the 0.204(V0/kB), and (4) WN = W1/5 and T = T2 = Ð6 8 2 elastic moduli K and B have minimum values (Figs. 1, 0.2093(V0/kB). W1/V0 = 1.6, K0 = 10 dyn, B0 = 10 dyn/cm , γ 2). In other words, in all cases under consideration, the and 1 = 25 dyn/cm. The dashed line represents the results thermal fluctuations of layer displacements are maxi- obtained in the framework of the Holyst model [18Ð20]. mum in the region where the smectic layer structure is less pronounced. Note that the profile of the fluctuations is determined by the orienting effect of the bound- g (0) ing surfaces on molecules in the film. 12, n 1.0

We also calculated the correlations gk, n(R) between 1 displacement fluctuations of different layers in the 0.8 2 smectic-A films on the surface of the solid substrate. The results of these calculations for the correlations between the displacement fluctuations of the central 0.6 layer (k = 12) and the other layers (n = 1Ð24) at R = 0 and WN = W1 are presented in Fig. 4. At temperatures 0.4 (T = T1) below the temperature at which the smectic order disappears in the bulk of the liquid crystal, the 0.2 results of the calculations are very close to those obtained within the Holyst model [18Ð20] (the dashed line in Fig. 4). The correlation g12, n(0) substantially 0 5 10 15 20 25 decreases with an increase in the magnitude of the dif- n ference between the number n and 12. However, as would be expected, the dependences obtained are Fig. 4. Correlations g12, n(0) between the displacement fluc- asymmetric with respect to the center of the film. The tuations of the twelfth layer and the other layers (n = 1Ð24) decay of the correlation g12, n(0) toward the free surface in a smectic-A film supported on the surface of a solid sub- of the film occurs more slowly than that toward the sur- strate at temperatures (1) T = T1 = 0.204(V0/kB) and (2) T = α T2 = 0.21035(V0/kB). WN = W1. The parameters N, , W1, face of the solid substrate. The correlation g12, n(0) in γ the vicinity of the free surface (n = 1) is larger than 0.2, K0, B0, and 1 are the same as in Fig. 3. The dashed line represents the results obtained in the framework of the Holyst whereas the value of g12, n(0) for the penultimate film model. layer (n = 23) near the substrate is close to zero. This difference between the correlations is caused by the fact that the layers located near the free surface of the film on the surface of the solid substrate and cannot be can be displaced (from their equilibrium positions) syn- involved in a similar process due to high energy expen- chronously with the central layers. At the same time, ditures to tensile (compressive) deformation of these the film layers adjacent to the last layer are rigidly fixed layers.

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1164 MIRANTSEV

Our calculations also demonstrate that, upon heat- the smectic-A film is described by the equations of ing of the smectic-A films supported on the surface of a motion solid substrate to the limiting temperature of their exist- ∂2 ∂ ence T , the change in the correlations g (0) between un un 1 2 k, n ρ ------= Ðη ∆⊥------Ð ---δF/δu . (11) 0 2 3 ∂ n the displacement fluctuations of film layers is less pro- ∂t t d nounced than that in the profiles of these fluctuations ρ (Fig. 4). It should be noted that these changes are com- Here, 0 is the mean density of the liquid crystal form- pletely determined by the orienting effect of the bounding the film, t is the time, and F is the excess free energy ing surfaces on molecules in the film. When the orient- associated with these fluctuation displacements in the ing effect of the film–substrate interface is equal to the film. By taking the continuous Fourier transform (3) of Eqs. (11) and introducing the dimensionless variables orienting effect of the free surface of the film (WN = W1) 1/2 2 t' = [(K B ) /(dη )]t and q' = q⊥/q (where q = or exceeds it (WN = 5W1), an increase in the temperature 0 0 3 0 0 2 1/2 leads to a weakening of the correlations g12, n(0) [B0/K0d ] ), we obtain a system of equations that can (Fig. 4), as is the case with free-standing smectic-A be written in the following compact matrix form: films. If the orienting effect of the film–substrate inter- ∂2u ∂u face is weaker than that of the free surface of the film ()ρ K /η 2 ------n = Ðq'2------n Ð M' u . (12) 0 0 3 2 ∂ nm m (WN = W1/5), an increase in the temperature results in ∂t' t' an enhancement of these correlations g12, n(0). This behavior can be easily explained from the physical Here, Mnm' are elements of the banded matrix similar to standpoint. In the first two cases (WN = W1, WN = 5W1), the matrix Mnm introduced above. For the smectic-A heating of the smectic-A film supported on the surface film supported on the surface of a solid substrate, the nonzero elements can be defined by formulas (11)–(15) of a solid substrate to the limiting temperature T2 of its existence leads to a substantial decrease in the elastic from my earlier work [35] (devoted to the dynamics of moduli K and B in the central part of the film, whereas layer displacement fluctuations in free-standing smectic-A films) if the dimensionless surface tension coeffi- these moduli in the surface layers remain virtually γ γ Ð1/2 unchanged. As was noted above, the profiles of the cient = (K0B0) in Eqs. (11) and (12) from [35] is γ γ Ð1/2 γ γ Ð1/2 moduli K and B in these cases are almost identical to the replaced by 1 = 1(K0B0) and N = N(K0B0) , analogous profiles for free-standing smectic-A films respectively.

(see Figs. 1, 2 in [21, 23]). Since the fluctuation dis- As in [31, 32, 35], the solution un(q', t') of the system placements of the central smectic layers of the film are of equations (12) is represented in the form of an imparted to other layers through the layers adjacent to expansion the central layers, a considerable decrease in their rigid- N ity should result in a weakening of the correlations (), ()k (), ()k () un q' t' = ∑ un q' t' v n q' (13) g12, n(0) between the fluctuation displacements of the central layers and the other layers of the film. In the k = 1 () third case (WN = W1/5), an increase in the temperature k in terms of the eigenvectors v n (q') of the matrix to the limiting temperature T2 leads not only to a decrease in the rigidity of the central part of the film but Mnm' (q'). The time dependence of the kth normal mode () also to a more substantial decrease in the rigidity of the u k (q', t') is described by the relationship smectic layers adjacent to the surface of the solid sub- n strate (Figs. 1, 2, curves 2), whose fixation hinders syn- ()k (), ()k ()[] β()k () un q' t' = un+ q' exp + q' t' chronous fluctuation displacements of the film layers. (14) ()k ()[] β()k (), Therefore, in this case, heating of the film, for the most + unÐ q' exp Ð q' t' , part, lifts a barrier to these synchronous fluctuation dis- ()k placements and, hence, should result in an enhance- in which the exponents β± (q') are expressed through ment of the correlations g (0). () 12, n k (k) the relaxation times τ± (q') and the frequencies ω (q') with the use of relationships (18)Ð(21) from [35]. 3. DYNAMICS OF FLUCTUATIONS OF LAYER Knowing the time dependences of the normal modes ()k DISPLACEMENTS IN A SMECTIC-A FILM un (q', t'), we can calculate the dynamic correlation ON THE SURFACE OF A SOLID SUBSTRATE functions of layer displacements in the film. In the Fou- rier representation, these functions are defined as fol- According to the model proposed in [31, 32], the lows [31, 32]: time dependence of the fluctuation displacements (), 〈〉(), (), un(x, y) of layers from their equilibrium positions in Cmn, q' t' = um q' t' un Ð0q' . (15)

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 THERMAL FLUCTUATIONS IN SMECTIC-A FILMS 1165

C12, 12, arb. units C1, 1, arb. units 30 8

20 4 2 1 2 0 10 –4 1

01234 –8 0 0.1 0.2 0.3 0.4 0.5 t, 10–9 s t, 10–7 s Fig. 5. Time dependences of the dynamic correlation function C12, 12 at temperatures (1) T = T1 = 0.204(V0/kB) and Fig. 6. The same as in Fig. 5, but for the dynamic correlation 5 Ð1 × 4 Ð1 (2) T = T2 = 0.21035(V0/kB). WN = W1 and q⊥ = 10 cm . function C1, 1 at q⊥ = 2 10 cm .

With knowledge of the dynamic correlation func- and T (Fig. 5, curve 2). It can be seen from this figure 〈 2 tions Cm, n(q', t'), we can calculate the correlation I(q, that, at t = 0, the magnitude of the correlation function 〉 t)I(q, 0) between the intensities of x-ray scattering by C12, 12 at the limiting temperature T2 of existence of the the film at different instants of time (at t = 0 and t). The film is approximately two times larger than that at the 〈 〉 2 quantity I(q, t)I(q, 0) /I (q, 0) is expressed in terms of lower temperature T1. Furthermore, this correlation the correlation functions Cm, n(q', t') with the use of rela- function at T = T2 decays considerably more slowly tionships (27) and (28) taken from [35]. with time as compared to that at T = T1. The dependences of the other dynamic correlation functions (C1, 1, The dynamic correlation functions Cm, n(q⊥, t) and the correlations 〈I(q, t)I(q, 0)〉 between the intensities of C1, 12, etc.) exhibit a similar behavior. These dynamic x-ray scattering at different instants of time from the correlation functions for free-standing smectic-A films smectic-A film supported on the surface of a solid sub- at the same values of N and q⊥ are entirely independent strate were numerically calculated for films consisting of the temperature. The difference in the behavior of the of 24 layers with the same model parameters used for smectic-A films supported on the surface of a solid sub- calculating the profiles of the layer displacement fluc- strate and free-standing smectic-A films is explained by tuations and the correlations between them. The viscos- the fact that, in not very thick (N ≤ 100) free-standing 6 Ð1 ity coefficient of interlayer sliding was taken equal to smectic-A films and at q⊥ ≤ 10 cm , the dominant role η 3 = 0.4 g/(cm s), which is typical of smectic liquid in the dynamics of thermal fluctuations of layer dis- crystals. The calculations were performed for the afore- placements is played by the so-called acoustic mode mentioned three ratios between the orienting effects of [31, 32]. This mode corresponds to motion of the film the bounding surfaces on liquid-crystal molecules in during which the interlayer spacing remains constant, the film at temperatures T1 and T2. and, hence, the relaxation time does not depend on the profiles of the elastic moduli K and B. However, the First, we analyze how the temperature affects the acoustic mode is absent in the smectic-A films sup- dynamic correlation functions. In [35], it was demon- ported on the surface of solid substrates [17] and ther- strated that this effect in free-standing smectic-A films mal fluctuations of layer displacements in the film are can be observed in sufficiently thick films (N ≥ 100) and always accompanied by a change in the interlayer spac- at large recoil-momentum components in the film plane ()k (≥106 cmÐ1) (see Figs. 3, 4 in [3, 5]). However, for the ing. Consequently, the relaxation times τ± (q'), which smectic-A films supported on the surface of a solid sub- determine the time dependences of the dynamic corre- strate under consideration, this effect is well pro- lation functions, should depend on the profiles of the nounced in films with a smaller thickness and at sub- elastic moduli K and B and, hence, on the temperature stantially smaller recoil-momentum components. This already for films that are not very thick (N ≤ 100) and is illustrated in Fig. 5, which shows the time depen- at values of q⊥ that are not large. It should be noted that, dences of the dynamic correlation function C12, 12 cal- for the smectic-A films supported on the surface of a culated for the film consisting of 24 layers at q⊥ = solid substrate, the temperature dependence of these 5 Ð1 10 cm , WN = W1, and temperatures T1 (Fig. 5, curve 1) correlation functions manifests itself for even smaller

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1166 MIRANTSEV

〈I(t)I(0)〉/I2(0) observed even for smaller recoil-momentum compo- × 4 Ð1 1.0 nents in the film plane (q⊥ = 3 10 cm ) at which the dependence exhibits a pronounced oscillatory behavior (Fig. 7, curves 4, 5). At similar values of q⊥, this corre- 0.8 lation function for free-standing smectic-A films is entirely independent of the temperature of the film. Thus, compared to free-standing smectic-A films, in the 0.6 4 smectic-A films supported on the surface of a solid substrate, the dynamics of layer displacement fluctuations 0.4 is considerably more sensitive to temperature-induced changes in the internal structure. Experimental investi- 3 5 gation into this dynamics can provide valuable struc- 2 0.2 tural information. 1 0246810 ACKNOWLEDGMENTS t, 10–9 s This work was supported by the Russian Foundation Fig. 7. Time dependences of the correlation function 〈I(q, for Basic Research, project no. 01-03-32084. 〉 5 t)I(q, 0) at WN = (1, 2) W1 and (3Ð5) W1/5; q⊥ = (1Ð3) 10 × 4 Ð1 and (4, 5) 3 10 cm ; and T = (1, 4) 0.204(V0/kB), REFERENCES (2) 0.21035(V /k ), and (3, 5) 0.2093(V /k ). 0 B 0 B 1. P. de Gennes, The Physics of Liquid Crystals (Claren- don, Oxford, 1974; Mir, Moscow, 1977). 2. P. Pieranski, L. Beliard, J. P. Tournellec, et al., Physica A recoil-momentum components in the film plane. This is (Amsterdam) 194 (1Ð4), 364 (1993). illustrated in Fig. 6, which depicts the time depen- 3. C. Rosenblatt, R. Pindak, N. A. Clark, and R. B. Meyer, dences of the dynamic correlation function C1, 1 at q⊥ = Phys. Rev. Lett. 42 (18), 1220 (1979). × 4 Ð1 2 10 cm and temperatures T1 (Fig. 6, curve 1) and 4. M. Veum, C. C. Huang, C. F. Chou, and V. Surendranath, T2 (Fig. 6, curve 2). At such a small value of q⊥, both Phys. Rev. E 56 (2), 2298 (1997). dependences exhibit an oscillatory behavior; however, 5. T. Stoebe, P. Mach, and C. C. Huang, Phys. Rev. Lett. 73 the oscillations substantially differ from each other. (10), 1384 (1994). Then, the correlation functions 〈I(q, t)I(q, 0)〉 were 6. E. I. Demikhov, V. K. Dolganov, and K. P. Meletov, Phys. Rev. E 52 (2), R1285 (1995). calculated for the smectic-A film comprised of 24 layers and supported on the surface of a solid substrate. 7. V. K. Dolganov, E. I. Demikhov, R. Fouret, and C. Gors, The results of the calculations for coherent diffuse Phys. Lett. A 220, 242 (1996). x-ray scattering in the vicinity of the main Bragg peak 8. P. Johnson, P. Mach, E. D. Wedell, et al., Phys. Rev. E 55 π (4), 4386 (1997). (qz = 2 /d) for the recoil-momentum component in the 5 Ð1 9. E. A. L. Mol, G. C. L. Wong, J. M. Petit, et al., Physica film plane q⊥ = 10 cm and WN = W1 are presented in B (Amsterdam) 248 (1Ð4), 191 (1998). Fig. 7. In this figure, curve 1 corresponds to the temper- 10. R. E. Geer, R. Shashidar, A. F. Thibodeaux, and ature T1 lower than the temperature of the SmAÐnematic R. S. Duran, Phys. Rev. Lett. 71 (9), 1391 (1993). phase transition in the bulk of the liquid crystal and 11. R. E. Geer and R. Shashidar, Phys. Rev. E 51 (1), R8 curve 2 represents the calculated data at the tempera- (1995). ture T2 close to the limiting temperature of existence of 12. R. E. Geer, S. B. Qadri, R. Shashidar, et al., Phys. Rev. E the film. It can be seen from Fig. 7 that, upon heating of 52 (1), 671 (1995). the film, the correlation function 〈I(q, t)I(q, 0)〉 decays 13. G. Henn, M. Stamm, H. Roths, et al., Physica B more slowly with time. This effect is most pronounced (Amsterdam) 221 (1Ð3), 174 (1996). at t ~ 10Ð9 s. It should also be noted that the results of 14. M. W. J. van der Wielen, M. A. Cohen Stuart, G. J. Fleer, the calculations carried out in the case of a profound et al., Langmuir 13, 4762 (1992). orienting effect of the film–substrate interface on mol- 15. T. Salditt, C. Munster, J. Lu, et al., Phys. Rev. E 60 (6), ecules in the film (WN = 5W1) do not differ greatly from 7285 (1999). the above data. However, when the orienting effect of 16. D. K. G. de Boer, Phys. Rev. E 59 (2), 1880 (1999). the interface is weaker than that of the free surface of 17. V. P. Romanov and S. V. Ul’yanov, Phys. Rev. E 66, the film (WN = W1/5), the correlation function 061701 (2002). 〈 〉 I(q, t)I(q, 0) at the limiting temperature T2 of exist- 18. R. Holyst and D. J. Tweet, Phys. Rev. Lett. 65 (17), 2153 ence of the film decays even more slowly with time (1990). (Fig. 7, curve 3). Moreover, the calculations performed 19. R. Holyst, Phys. Rev. A 44 (6), 3692 (1991). demonstrate that the effect of temperature on the time 20. A. Poniewerski and R. Holyst, Phys. Rev. B 47 (15), dependence of the correlation function 〈I(q, t)I(q, 0)〉 is 9840 (1993).

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21. L. V. Mirantsev, Fiz. Tverd. Tela (St. Petersburg) 41 (10), 30. A. Fera, I. P. Dolbnya, G. Grubel, et al., Phys. Rev. Lett. 1882 (1999) [Phys. Solid State 41, 1729 (1999)]. 85 (11), 2316 (2000). 22. L. V. Mirantsev, Fiz. Tverd. Tela (St. Petersburg) 42 (8), 31. A. Poniewierski, R. Holyst, A. C. Price, et al., Phys. Rev. 1511 (2000) [Phys. Solid State 42, 1554 (2000)]. E 58 (2), 2027 (1998). 23. L. V. Mirantsev, Phys. Rev. E 62 (1), 647 (2000). 32. A. Poniewierski, R. Holyst, A. C. Price, and L. B. Soren- sen, Phys. Rev. E 59 (3), 3048 (1999). 24. L. V. Mirantsev, Phys. Lett. A 205, 412 (1995). 33. A. N. Shalaginov and D. E. Sullivan, Phys. Rev. E 62 (1), 25. L. V. Mirantsev, Liq. Cryst. 20 (4), 417 (1996). 699 (2000). 26. L. V. Mirantsev, Phys. Rev. E 55 (4), 4816 (1997). 34. V. P. Romanov and S. V. Ul’yanov, Phys. Rev. E 65 (2), 021706 (2002). 27. L. V. Mirantsev, Liq. Cryst. 27, 491 (2000). 35. L. V. Mirantsev, Kristallograﬁya 48 (4), 745 (2003) 28. V. K. Dolganov, V. M. Zhilin, and K. P. Meletov, Zh. [Crystallogr. Rep. 48, 693 (2003)]. Éksp. Teor. Fiz. 115 (5), 1833 (1999) [JETP 88, 1005 (1999)]. 36. W. L. McMillan, Phys. Rev. A 4 (3), 1238 (1971). 29. A. C. Price, L. B. Sorensen, S. D. Kevan, et al., Phys. Rev. Lett. 82 (4), 755 (1999). Translated by O. Borovik-Romanova

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Physics of the Solid State, Vol. 46, No. 6, 2004, pp. 1168Ð1172. Translated from Fizika Tverdogo Tela, Vol. 46, No. 6, 2004, pp. 1132Ð1136. Original Russian Text Copyright © 2004 by Avramov, Yakobson, Scuseria.

FULLERENES AND ATOMIC CLUSTERS

Effect of Carbon Network Defects on the Electronic Structure of Semiconductor Single-Wall Carbon Nanotubes P. V. Avramov1, 2, B. I. Yakobson2, and G. E. Scuseria2 1 Kirensky Institute of Physics, Siberian Division, Russian Academy of Sciences, Akademgorodok, Krasnoyarsk, 660036 Russia 2 Center for Biological and Environmental Nanotechnology, Rice University, Houston, Texas, 77005-1892 USA Received September 26, 2003

Abstract—For a single-wall (14, 0) carbon nanotube, the total density of electronic states of the ideal structure and of some possible defect structures is calculated in the framework of the band theory approach using Gaus- sian-type orbitals and the approximation of the generalized density gradient. It is shown that allowance for defects of the atomic structure of a nanotube makes it possible to adequately describe the existing experimental data on nanotube electronic structure. In the framework of the same approach, the total density of electronic states is calculated for an intermolecular contact of (5, 5) and (10, 0) single-wall carbon nanotubes formed due to the creation of a 5Ð7 defect. It is shown that the electronic states related to the contact region and the 5Ð 7 defect lie in vicinity of the Fermi level. © 2004 MAIK “Nauka/Interperiodica”.

Atomic and electronic structures of a defect (13, 13) Another type of defects in semiconductor carbon carbon nanotube and of an intermolecular contact nanotubes caused by adsorption of O2 and NO2 mole- between the (21, Ð2) and (22, Ð5) carbon nanotubes cules by the carbon network of these objects was exper- formed by introducing a 5Ð7 defect were studied using imentally studied in [4, 5]. Both chemical agents scanning tunneling microscopy (STM) and scanning increased the density of electronic states at the Fermi tunneling spectroscopy (STS) in [1Ð3]. In those studies, level and converted semiconductor nanotubes into it was shown that a defect introduced into an ideal metal nanotubes. On the basis of nonempirical calcula- atomic network of a (13, 13) metal nanotube generates tions of the electronic structure [6] of a chemisorbed O2 a number of spectroscopic features located between the molecule, these experimental results have been inter- first Van Hove singularities [1, 3] and that the energy preted in terms of extra charge carriers (holes) at the Fermi level and closing of the band gap in semiconduc- position and intensity of the new spectroscopic features tor p-type nanotubes. depend on the distance from the defect for which the scanning tunneling spectrum is measured. In [1, 3], the In the theoretical study performed in [7], on the observed oscillations were interpreted in terms of reso- basis of calculations of the electronic structure of sin- nant backscattering of the incident electron plane wave gle-wall semiconductor nanotubes performed within the tight-binding model and the local density functional by quasibound electronic states of the defect. Experi- approximation, it was predicted that a 5Ð7 defect (a mental STS spectra of the (21, Ð2)/(22, Ð5) intermolec- StoneÐWales defect) should also result in closing of the ular contact measured in the vicinity of the structural band gap in semiconductor nanotubes. defect have shown that the electronic states of the metallic (22, Ð5) structure and the semiconductor (21, The tight-binding method was also used to calculate − a number of intermolecular contacts between nano- 2) structure are mixed. The relative weights of the tubes [8]. These theoretical results have qualitatively spectral features depend on the position of the measure- confirmed the experimental observations reported ment point with respect to the contact. It has also been in [3]. shown that the features caused by the contact decay over a distance of several nanometers. Local variations Nevertheless, correct interpretation of some experimental data even for nonchiral single-wall nanotubes in the densities of states for single-wall nanotubes were [9] is still lacking. The simplest example is sample no. experimentally measured in [2] and qualitatively inter- 7 investigated in the pioneering work [9], where, in par- preted as interference between the incident and ticular, the STM spectra of single-wall nanotubes were reflected electron waves. The dispersion law in single- measured for the first time. According to [9], this sam- wall nanotubes with defects has been interpreted in ple (of diameter d = 1.1 nm with chiral angle ϕ = 30°) terms of quantum interference of electrons scattered by was a (14, 0) nanotube, which, according to [10], had to defects [1]. have semiconductor properties. Nevertheless, in the

EFFECT OF CARBON NETWORK DEFECTS 1169 experiment in [9], this sample had metallic properties. Pristine CNT (14, 0) In our opinion, the presence of defects in the structure Semiconductor of the carbon (14, 0) nanotube would provide the most c reasonable interpretation of such fundamental differ- PBE 6–31G* ences between theory and experiment. In order to check this assumption, we calculated the electronic structure of an ideal (14, 0) carbon nanotube and a (14, 0) carbon nanotube with different defects [a StoneÐWales defect, double vacancy in the carbon net- b work (2V defect), a defect produced by embedding two Semiconductor additional carbon atoms into the carbon network (an ad dimmer defect), and two defects produced by saturating PBE 6–31G* one of the double carbon bonds by the OH and H groups (2OH and 2H defects)]. All calculations were DOS, arb. units performed using the approximation of a generalized Experiment density gradient (the PerdewÐBurkeÐErnzerhof (PBE) a potential) [11] and the band theory approach using Gaussian-type orbitals in the 3Ð21G, 6Ð31G, and 6Ð 30° 31G* bases [12]. To correctly describe the electronic structure of isolated defects, the length of the unit cell was taken to be 20 Å. When modeling the variation in 2.3 eV the defect density in the atomic network, the length of the unit cell (and, accordingly, the number of carbon –1 0 1 atoms in it) was varied. To study the electronic structure Energy, eV of an intermolecular contact, we chose the structure formed by the contact of a metal (5, 5) and semiconduc- Fig. 1. (a) Experimental STS spectra [9] and the theoretical tor (10, 0) nanotubes. The intermolecular contact con- (b) PBE and (c) PBE0 densities of states (DOS) for semiconductor zigzag (14, 0) carbon nanotubes. The inset shows sidered is formed by introducing one pentagon and one a photograph of the structure obtained with a scanning tun- heptagon into the (5, 5) or (10, 0) nanotube. To model neling microscope [9]. It is clearly seen that the nanotube noninteracting contacts, the length of the unit cell was has a zigzag structure. increased to 40 Å. Optimization of the geometry of defect structures was performed using the method of the analytical gra- ture is 40 Å. Optimization of the geometry was per- dient of the potential energy surface and the PBE func- formed using the semiempirical quantum-chemical tional [11] in the 3Ð21G basis. The number of atoms PM3 method (for a cluster containing three unit cells) per unit cell was varied from 226 for 2H and 2OH and the analytical gradient of the potential energy. To defects (carbonÐcarbon bonds saturated by two atoms calculate the electronic structure, 64 points were taken of hydrogen or two functional groups, 2020 functions in k space in the Brillouin zone. For interpretation of in the 6Ð31G basis) to 334 for a 2V defect (3006 func- the data on the electronic structure of the intermolecu- tions in the 6Ð31G basis). The electronic structure of lar contact, band calculations for ideal (10, 0) and (5, 5) all objects was calculated using 128 points of k space nanotubes were performed using the PBE potential and in the Brillouin zone. The electronic structure of the 128 points in k space. ideal (14, 0) nanotube was calculated using both the Figure 1 shows the experimental STS spectra PBE functional [11] and the hybrid PBE0 functional (Fig. 1, a [9]) and the theoretical densities of electronic [13], whose distinctive feature is the admixture of 25% states calculated using the PBE (Fig. 1b) and PBE0 of the exact HartreeÐFock exchange to the pure PBE (Fig. 1, c) potentials for the ideal semiconductor (14, 0) potential of the density functional theory; this nanotube. It is seen from Fig. 1 that the experimental approach makes it possible to take into account elec- spectra exhibit metallic conductivity, which is in dis- tronic correlations in the system more fully. It is agreement with the previous theoretical prediction [10]. believed that this approach [13] is equivalent to the In contrast to the experimental spectrum, the theoretical MellerÐPlesset fourth-order perturbation theory. The densities of states obtained using both the PBE and PBE0 functional was shown to provide much better PBE0 potentials agree with the results from [10] and agreement between the total theoretical electronic den- are characterized by the presence of a band gap with a sities of states (DOS) and the experimental STS spec- width of about 1 eV (Fig. 1). tra of semiconductor nanotubes [14]. Figure 2 shows the experimental spectrum [9] The unit cell of the intermolecular (5, 5)/(10, 0) con- (Fig. 2, a) and a set of theoretical spectra for the struc- tact contains 360 carbon atoms (3240 functions in the tures with a StoneÐWales defect (Fig. 2, b) and 2H 3Ð21G basis). The length of the unit cell for this struc- (Fig. 2, c), 2OH (Fig. 2d), ad dimmer (Fig. 2, e), and 2V

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1170 AVRAMOV et al.

Theoretical DOS of CNT (14, 0) quately. There is a gap in the density of states only for with defects the structure with a StoneÐWales defect (Fig. 2, b); all other defects produce metallic-type densities of states. Characteristic energy-dependent oscillations in the density of states were simulated in [15] by using 1D plane waves exp(ikx), where x is the space coordinate. 2V defect f The incident plane wave can be resonantly reflected from a quasibound defect state with reflection coeffi- 2 Ef cient |R| [R = |R|exp(Ði(kx + δ)), where δ is the phase shift]. The corresponding standing wave can be written as ψ(k, x) = exp(ikx) + |R|exp(Ði(kx + δ)), which corresponds to spatial oscillations in the densities of states ρ(k, x) = |ψ(k, x)|2 = 1 + |R|2 + 2|R|cos(2kx + δ) [15]. defect ad dimmer e In order to simulate the effect of the defect density E f on the electronic structure of nanotubes, we performed calculations of the band structure for a 2V defect with two different cells. The first cell contained one defect 2OH defect E d per unit cell (334 atoms, which corresponds to a defect f density of 0.3%), whereas the second cell had two DOS, arb. units defects (the unit cell was the same; both defects were placed on opposite ends and opposite faces of the unit 2H defect c cell; the defect density was 0.6%). The translation vec- Ef tor for both unit cells was 25 Å. Using this special choice of the unit cells, we can write k1x ~ k2x (k1 is the wave vector of the defect state Stone–Wales defect with the former density and k2 is that for the defect state Ef b with the latter density); therefore, we can obtain the energy positions of the density-of-states oscillations for a both unit cells. Experiment For systems with low defect densities, we can use the model of independent centers [16] (assuming that 2.3 eV defects do not interact with each other) and the models of rigid bands and impurity bands (a rigid band does not –1 0 1 depend on the occupation numbers, whereas the nature Energy, eV of an impurity band is fully determined by the type and density of the impurity). Application of these two mod- Fig. 2. (a) Experimental STS spectrum of a (14, 0) carbon els means that we can distinguish between the two inde- nanotube (CNT) [9], (b) theoretical PBE 6Ð31G density of pendent electronic subsystems of the system (naturally, states for a (14, 0) nanotube with a StoneÐWales defect (the in narrow intervals of defect densities), namely, the unit cell contains 226 carbon atoms), (c) theoretical PBE subsystem of defect states TD and the subsystem TR 6−31G densities of states for a (14, 0) nanotube with a 2H defect (the unit cell contains 226 carbon atoms), (d) theoret- formed by the other states with occupation numbers x ical PBE 6Ð31G density of states for a (14, 0) nanotube with and (1 Ð x), respectively. a 2OH defect (the unit cell contains 226 carbon atoms), In terms of these two models, we can write two lin- (e) theoretical PBE 6Ð31G density of states for a (14, 0) nanotube with an ad dimmer defect created by sorption of a ear equations C fragment at the carbon wall (the unit cell contains 2 R 282 carbon atoms), and (f) theoretical density of states T T T = x T D + ()1 Ð x T R, obtained for a rigid band of a 2V defect using the total PBE 1 1 1 6Ð31G densities of states calculated for 2V defect densities T D ()R of 0.3 and 0.6% per unit cell (the unit cell contains 334 and T 2 = x2T + 1 Ð x2 T , 332 carbon atoms, respectively). T T where T 1 and T 2 are the total densities of states for the systems with different defect densities. defects (Fig. 2, f). Introducing structural defects into For systems with a significant number of noninter- the ideal graphite network results in the appearance of acting defects uniformly distributed over the entire states in the band gap. In all cases, the main features of atomic network, we can extract the quantity TR, since the experimental densities of states at energies of different defects can mutually suppress oscillations in approximately Ð0.9 and 1.4 eV are described ade- the densities of states due to interference effects. Most

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 EFFECT OF CARBON NETWORK DEFECTS 1171

nanostructures; thus, we may hope for a qualitative description of the electronic structure of the (5, 5)/(10, Bent (5, 5)–(10, 0) 0) intermolecular contact (Fig. 3). The length of the unit cell (~40 Å) was chosen to be sufficiently large to avoid significant interaction between the defects. By applying the translational symmetry to the unit cell, we can construct an infinite zigzag structure in which the sections with (5, 5) and (10, 0) structures are separated by the * structures containing five- and seven-membered rings PBE 3-21G (the upper inset in Fig. 3). Figure 3 shows the densities of states for the (5, 5) and (10, 0) nanotubes and for the (5, 5)/(10, 0) intermolecular contact. According to our nonempirical calcula- * * * * tions, the intermolecular contact is metallic with a non- (5, 5) background zero density of electronic states at the Fermi level, whose energy position is Ð4.7 eV. The features near the Ef Ef(5, 5) Fermi level are produced by the electronic states corresponding to the 5Ð7 defect, whereas the spectral fea- (10, 0) tures below and above the Fermi level originate from either (5, 5) or (10, 0) structures. DOS, arb. units Comparison of the experimental STS spectra and PBE 6–31G theoretical electronic densities of states calculated using the band theory approach, Gaussian-type orbitals, and the PBE potential of the approximation of the generalized density gradient shows that the method (5, 5) described above can be successfully applied to calculate the electronic structure of semiconductor single- wall nanotubes with defects. Noticeable differences in the densities of states of ideal nanotubes and nano- PBE 6–31G* tubes with defects can serve as an elementary test of the imperfection of the atomic structure of this type of object. Ef

ACKNOWLEDGMENTS –6 –5 –4 –3 This study was supported by the Nanoscale Science Energy, eV and Engineering Initiative of the National Science Foundation, award number EEC-0118007 (Rice Fig. 3. The total densities of states for the (5, 5) and (10, 0) CBEN), and by the Welch Foundation. nanotubes and for the (5, 5)/(10, 0) intermolecular contact. Arrows indicate the features related to the (5, 5) structure, triangles indicate the features related to the (10, 0) structure, REFERENCES and asterisks denote the features related to the intermolecular contact itself. The (5, 5), (10, 0), and (5, 5)/(10, 0) struc- 1. M. Ouyang, J.-L. Huang, and C. M. Lieber, Phys. Rev. tures are shown in the insets. Lett. 88, 066804 (2002). 2. M. Ouyang, J.-L. Huang, C. L. Cheung, and C. M. Lie- ber, Science 291, 97 (2001). likely, TR can be seen in spectroscopic experiments for 3. M. Ouyang, J.-L. Huang, and C. M. Lieber, Annu. Rev. such systems. Phys. Chem. 53, 201 (2002). Figure 2, f shows TR obtained by using the theoreti- 4. P. G. Collins, K. Bradley, M. Ishigami, and A. Zettl, Sci- cal densities of states for unit cells with 2V-defect den- ence 287, 1801 (2000). sities of 0.3 and 0.6%. It is clearly seen that such an 5. J. Kong, N. R. Franklin, C. Zhou, M. G. Chapline, approach provides an opportunity to quantitatively S. Peng, K. Cho, and H. Dai, Science 287, 622 (2000). describe the experimental density of states obtained in 6. S.-H. Jhi, S. G. Louie, and M. L. Cohen, Phys. Rev. Lett. [9] for a (14, 0) nanotube. 85, 1710 (2000). The calculations described above show the high 7. V. H. Crespi and M. L. Cohen, Phys. Rev. Lett. 79, 2093 quality of the results obtained for defect 1D carbon (1997).

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8. S. G. Louie, Top. Appl. Phys. 80, 113 (2001). 13. M. Ernzerhof and G. E. Scuseria, J. Chem. Phys. 110, 5029 (1999); C. Adamo and V. Barone, J. Chem. Phys. 9. J. W. G. Wildöer, L. C. Venema, A. G. Rinzler, 110, 6158 (1999). R. E. Smalley, and C. Dekker, Nature 391, 59 (1998). 14. P. V. Avramov, K. N. Kudin, and G. E. Scuseria, Chem. 10. J. W. Mintmire, B. I. Dunlop, and C. T. White, Phys. Rev. Phys. Lett. 370, 597 (2003). Lett. 68, 631 (1992). 15. M. Ouyang, J.-L. Huang, and C. M. Lieber, Annu. Rev. 11. J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Phys. Chem. 53, 201 (2002). Lett. 77, 3865 (1996); Phys. Rev. Lett. 78, 1396(E) 16. P. V. Avramov and S. G. Ovchinnikov, Fiz. Tverd. Tela (1997). (St. Petersburg) 37, 2559 (1995) [Phys. Solid State 37, 1405 (1995)]. 12. M. J. Frisch, G. W. Trucks, H. B. Schlegel, et al., GAUS- SIAN 01. Development Version (Gaussian, Pittsburgh, PA, 2001). Translated by I. Zvyagin

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Physics of the Solid State, Vol. 46, No. 6, 2004, pp. 1173Ð1178. Translated from Fizika Tverdogo Tela, Vol. 46, No. 6, 2004, pp. 1137Ð1142. Original Russian Text Copyright © 2004 by Zaporotskova, Lebedev, Chernozatonskiœ. FULLERENES AND ATOMIC CLUSTERS

Electronic Structure of Carbon Nanotubes Modified by Alkali Metal Atoms I. V. Zaporotskova*, N. G. Lebedev*, and L. A. Chernozatonskiœ** *Volgograd State University, Volgograd, 400062 Russia **Institute of Biochemical Physics, Russian Academy of Sciences, Moscow, 117334 Russia e-mail: [email protected] Received July 1, 2003; in ﬁnal form, November 3, 2003

Abstract—The electronic structure and parameters of the energy band structure of (n, 0)-type nanotubes modified by alkali metal atoms (Li, Na) and intercalated by potassium atoms are studied. The quantum-chemical semiempirical MNDO method and a model of the covalent cyclic cluster built in via ionic bonding are used to model infinitely long nanotubes. The electronic density of states of modified nanotubes is found. It is shown that semiconductorÐmetal transitions can occur in semiconductor nanotubes and that semimetal nanotubes can undergo metalÐmetal transitions. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION the synthesis mechanisms and to develop new experimental methods of NT fabrication. In this study, we A lot of attention is currently being paid to the present the results of our investigations into the elec- experimental and theoretical study of a recently discov- tronic structure and energy spectrum parameters of ered form of open-surface carbon called nanotubes small-diameter carbon NTs modified and intercalated by (NTs) [1Ð10]. In the course of partial thermal destruc- alkali metal atoms (Li, Na, K); these investigations were tion of graphite layers, not only fullerene molecules but started as early as in the middle of the 1990s [13Ð19]. also long tubes are formed, whose surface consists of regular hexagons. These tubes are up to several microns We discuss the results from calculating two possible long and several nanometers in diameter, can consist of alternatives for modifying zigzag-type (n, 0) carbon one or several layers (depending on the preparation NTs: (1) regular adsorption of alkali metal atoms on the conditions), and have open or closed ends. The methods external surface of an NT (surface modification) and of NT fabrication are explicitly described in reviews [6, (2) introduction of alkali metal atoms into the NT cav- 7]. The NT structures can be classified using the sym- ity (intercalation). bols (n, m), as suggested by Hamada et al. [4]. An (n, m) NT is obtained by rolling a graphite fragment so that one hexagon is located above another with the shift 2. SURFACE MODIFICATION mA1 + nA2, where A1 and A2 are primitive translation OF CARBON NANOTUBES vectors for the graphite fragment. Among the NTs A model of a covalent cyclic cluster built in via ionic (tubelenes) with cylindrical symmetry, (n, n) NTs are bonding [20], adapted to calculations for crystals and said to be of the “armchair” type and (n, 0) NTs are said polymers in the framework of the semiempirical quan- to be of the “zigzag” type. At present, it is possible to tum-chemical MNDO calculation procedure [21], was prepare and study the structure, elastic properties, and applied to calculate the electronic structure and energy conductivity of separate single-wall and multi-wall band structure parameters for (6, 0), (8, 0), and (12, 0) NTs and of bundles of single-wall tubes ~1 nm in diam- NTs modified by Li and Na atoms. The extended unit eter [6, 7, 10Ð12]. cells (EUCs) of the tubes under study contained three One of the interesting aspects of the study of carbon layers with 6, 8, and 12 carbon hexagons in each layer NTs is the possibility of fabricating structures having and 6, 8, and 12 alkali metal atoms, respectively, new mechanical, electronic, or optical properties located above the surface hexagons. The distance through doping or modification by other atoms. Imme- between the nearest neighbor carbon atoms was taken diately after the discovery of carbon NTs, the problem to be 1.4 Å. An example of EUCs for a (6, 0) NT mod- of filling and modifying NTs by various materials ified by Na atoms is shown in Fig. 1. Cyclic boundary arose. In particular, an NT filled by conductor, semi- conditions for molecular orbitals (MOs) of a cylindrical conductor, or superconductor material can be used in EUC were imposed in the direction of the NT axis. Two modern microelectronics as a miniature device [6Ð8]. possible positions of metal atoms above the tube sur- Theoretical study of the mechanisms of formation of face were considered: (I) metal atoms were located filled and modified NTs makes it possible to understand above the hexagon centers along the tubelene circle

1174 ZAPOROTSKOVA et al. metal atoms for (6, 0), (8, 0), and (12, 0) NTs, respectively. The geometrical parameters of the rings of alkali metal atoms were optimized in quantum-chemical calculations; for NTs, the rigid lattice approximation was used. The calculated energy structure parameters for modifications I and II of NTs are given in Table 1. Analysis of these results shows that modification I is energetically more favorable and, therefore, more stable in all cases. As the NT diameter increases, the difference in the specific energy (i.e., the energy per hexa- ∆ gon) between the two modifications = EII Ð EI decreases from 0.46 [for the (6, 0) tube] to 0.06 eV [for the (12, 0) tube] in the case of Li and from 0.42 to 0.04 eV for Na. This difference in specific energy approaches values of 0.05 and 0.02 eV, respectively, obtained in the case of the same modifications I and II with the same atoms for a single-layer C72 graphite cluster (this is actually the limiting case of an NT of infinite diameter; see Table 1). Analysis of the calculated electronic structure of modified NTs shows that the molecular orbitals (MOs) are grouped into energy bands. The 2s- and 2p-atomic orbitals (AO) of carbon atoms contribute to the molec- Fig. 1. Extended unit cell of a (6, 0) nanotube modified by ular orbitals corresponding to the states of the valence Na atoms. band. The MOs corresponding to the bottom of the conduction band are constructed from the 2p AOs of carbon atoms and 2s and 2p AOs of Li atoms or 3s and 3p AOs of Na atoms. Calculations reveal the presence of metal- 0.4 ∆ lic conductivity (the band gap Eg = 0) in the systems under study and the appearance of minibands of unoc- cupied levels in the energy spectrum corresponding to the AOs of alkali metal and C atoms. These features are 0.3 a consequence of the appearance of a superlattice of metal atoms and indirectly confirm the correctness of the above results. Due to the presence of defects (atoms of the superlattice), the band of the crystal splits into 0.2 minibands. Figure 2 shows the density of electronic N/EUC states in a pure (n, 0) carbon tubelene with n = 8. The density of states for the (8, 0) NT has a gap of about 0.1 2 eV near the Fermi level, thus indicating the semiconductor properties of the tube. Figure 3 shows the density of states for modification I with Li atoms. Compar- ison with the case of a pure carbon NT clearly shows an 0 increase in the peak amplitudes corresponding to mini- –15 –10 –5 0 5 10 15 bands near the Fermi level EF (the main contribution to E, eV the minibands comes from the alkali metal atoms). Generally, for the (n, 0) tubes, the modification by Fig. 2. Density of electronic states of a (8, 0) carbon nano- alkali metal atoms results in the appearance of mini- tube normalized to an extended unit cell (EUC) and calculated using the method of covalent cyclic cluster built in via bands in the band gap and, therefore, in the disappear- ionic bonding; the dashed line indicates the position of the ance of the energy gap, which indicates the onset of Fermi level (approximately Ð5eV). metallic conductivity in such NTs. We calculated the Li and Na atom adsorption energies for both surface modifications using the formula forming two “rings” not shifted with respect to each () Et + NEme Ð Emt other (a rectangular superlattice) (Fig. 1) and (II) the Ead = ------, (1) metal atoms of one ring were shifted with respect to the N atoms of the other ring by one hexagon (a rhombic where N is the number of metal atoms per EUC, Et is superlattice). One ring contained three, four, and six the total energy of a pure carbon (n, 0) NT, Eme is the

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ELECTRONIC STRUCTURE OF CARBON NANOTUBES 1175

Table 1. Energy structure parameters of modiﬁed (n, 0) nanotubes (n = 6, 8, 12) ∆ ∆ ∆ dt, Å Atom Modiﬁcation r, Å QE, eV Eg, eV Ev, eV , eV Ead, eV (6, 0) 4.6 Li I 1.6 0.68 Ð509.96 0.0 50.60 0.46 3.7 II 1.6 0.67 Ð509.50 0.0 49.52 2.3 Na I 1.7 0.65 Ð510.04 0.0 50.94 0.42 3.9 II 1.7 0.65 Ð509.62 0.0 49.74 2.7 (8, 0) 6.2 Li I 1.6 0.68 Ð511.54 0.0 50.10 0.24 3.4 II 1.6 0.66 Ð511.30 0.0 49.31 2.7 Na I 1.7 0.65 Ð511.66 0.0 50.28 0.22 3.7 II 1.7 0.65 Ð511.44 0.0 49.49 3.0 (12, 0) 9.3 Li I 1.7 0.68 Ð521.66 0.0 50.88 0.06 3.5 II 1.7 0.67 Ð521.60 0.0 49.90 2.5 Na I 1.8 0.65 Ð522.32 0.0 51.02 0.04 3.6 II 1.8 0.65 Ð522.28 0.0 50.10 2.8

Graphite C72 ∞ Li I 1.7 0.62 Ð504.21 0.0 50.51 0.05 Ð II 1.7 0.63 Ð504.16 0.0 50.40 Ð Na I 1.8 0.60 Ð504.16 0.0 50.50 0.02 Ð II 1.8 0.60 Ð504.14 0.0 50.45 Ð Note: I and II are the modiﬁcation types, r is the optimum distance between the metal atoms and the centers of carbon hexagons at the surface, and Q is the charge on a metal atom; E is the energy of the system per hexagon, ∆ = E Ð E , d is the tubelene diameter, ∆ ∆ II I t Eg is the band gap, and Ev is the valence band width.

Table 2. Adsorption energies (in electronvolts) of Li and Na atoms for the two modiﬁcations of (n, 0) nanotubes with n = 6, 8, and 12 Nanotube type (6, 0) (8, 0) (12, 0) Metal Li Na Li Na Li Na Modiﬁcation I II I II I II I II I II I II 3.7 2.3 3.9 2.7 3.4 2.7 3.7 3.0 3.5 2.5 3.6 2.8

metal atom energy (Li or Na), and Emt is the total energy respectively. A further increase in diameter does not of the modified NT. The results of the calculation are change these quantities appreciably. given in Table 2. The charge distribution over the metal atoms of the superlattices indicates that electron transfer to carbon It is seen from Table 2 that, for a rectangular super- NTs takes place, which increases the number of major- lattice, the adsorption energy is greater than for a rhom- ity charge carriers in the NTs; as a result, the semicon- bic superlattice. Thus, modification I actually turns out ductor tubes [e.g., (8, 0) NTs] begin to take on semi- to be energetically more favorable than modification II. metal properties and the conductivity of metal NTs [e.g., (6, 0) and (12, 0) tubes] increases. In our case, the The geometrical parameters of the rings of alkali superlattice parameter exceeds 4 Å for the chosen con- metals vary only slightly as the NT diameter is figuration of alkali metal atoms (Fig. 1). The lattice increased (Table 1). For (6, 0) and (8, 0) NTs, all parameters of bcc alkali metal crystals are 3.5 (Li) and adsorbed Li atoms are at a distance of 1.6 Å from the 4.2 Å (Na) [22] and are comparable to the respective tube surface for both superlattice modifications and the values for the superlattices under study. Moreover, as Na atoms are at a distance of 1.7 Å. Only for a substan- noted above, the distance between the metal atoms and tially increased NT diameter [in the case of a (12, 0) the tubes is 1.6–1.7 Å in most cases. All these facts NT] do the adsorption lengths change to 1.7 and 1.8 Å, indicate that, in an NT modified by alkali metals, there

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1176 ZAPOROTSKOVA et al.

0.25 (a)

0.20

0.15 N/EUC 0.10

0.05

0 –15 –10 –5 0 5 10 15 E, eV

Fig. 3. Density of electronic states of a (6, 0) nanotube modified by Li atoms (modification I); the density is normalized to an extended unit cell. The dashed line indicates the position of the Fermi level (~0 eV). (b) are two types of conducting channels, namely, the metal superlattice and the carbon NT itself. However, the alkali metal atoms are monovalent and have one electron in their valence electron shell; according to calculations, the electron is transferred to the NT. Therefore, in our opinion, the greater contribution to the conductivity of our NTs comes from the second channel, namely, the carbon NT itself. Thus, our calculations show once again that the surface modification of NTs by metal atoms results in the occurrence of a semiconductorÐmetal transition in semiconductor NTs and a metalÐsemiconductor transition in semimetal NTs. Therefore, by introducing, e.g., metal atoms between the layers of multilayer tubelenes, it is possible to create hollow alternating metal superlattices and NT conductors in a semiconducting carbon shell that have new conducting, magnetic, and electrical properties.

3. INTERCALATION OF CARBON NANOTUBES In addition to surface modiﬁcation of carbon NTs, we used the method of covalent-cyclic cluster built in via ionic bonding in the framework of the MNDO Fig. 4. Extended unit cell (a) of a (6, 0) nanotube with an approach to calculate the electronic structures of (n, 0) intercalated potassium atom and (b) of a (8, 0) nanotube NTs (n = 6, 8) with potassium atoms introduced into the with two intercalated potassium atoms. tube cavity. Two ways of intercalating carbon NTs were considered: one or two K atoms were placed at the center of an NT. The EUC of such a system consists of in graphite, i.e., 1.4 Å. In the case where two K atoms three layers containing six or eight carbon hexagons each and of one or two K atoms inside the NT that lie were intercalated, the distance between them was cho- on the symmetry axis joining the centers of the opposite sen to be ~4.5 Å (the minimum interatomic distance in hexagons (Fig. 4). The distances between the nearest the crystal). Application of the cyclic model (with the neighbor C atoms were chosen to be the same as those cyclic boundary conditions) makes it possible to cor-

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ELECTRONIC STRUCTURE OF CARBON NANOTUBES 1177

Table 3. Energy parameters of (n, 0) nanotubes (n = 6, 8) size gives rise to new magnetic and electric quantum- intercalated by potassium atoms confinement effects that can find numerous applications in modern microelectronics and nanoelectronics, Nanotube Number of Q ∆E , eV ∆E , eV type K atoms g v which have been rapidly developing in recent years. Our results agree well with previous quantum- (6, 0) 1 0.32 0.00 48.51 chemical nonempirical studies of carbon NTs filled by 2 0.32 0.00 49.53 transition metal atoms [28–30]. Using modifications of (8, 0) 1 0.84 0.00 47.00 the linear augmented plane wave method (the linear augmented cylindrical wave method), a detailed analy- 2 0.84 0.00 47.23 sis of the variation in the energy band structure of NTs ∆ ∆ Note: Q is the charge on a K atom, Eg is the band gap, and Ev is after filling with various intercalants has been per- the valence band width. formed. On the whole, the conclusion was drawn that the density of states at the Fermi level increases for both metal and semiconductor NTs intercalated by various rectly describe a special type of filled (endohedral) NTs transition metals. called quantum nanowires. Analysis of the electronic structure of the above systems shows that intercalation of an NT by K atoms pro- ACKNOWLEDGMENTS ∆ duces a sharp decrease in the band gap Eg, to the point The authors thank A.O. Litinskiœ (Volgograd State of its disappearance, which corresponds to the onset of Technical University) for useful discussions of the metallic conduction in the system (Table 3). In this results and for valuable advice. case, the semiconductorÐmetal transition is observed. This study was supported in part by the Russian This transition occurs because of the appearance of Foundation for Basic Research (project no. 02-03- energy levels in the band gap characteristic of empty 81008) and INTAS (grant no. 00-237). NTs; the energy levels are mainly due to the potassium AOs. The electronic density of states in an intercalated (6, 0) tubelene with one potassium atom per EUC is REFERENCES similar to that for a modified (6, 0) tube (Fig. 3) and 1. S. Iijima, Nature 354, 56 (1991). obviously exhibits peaks at the Fermi level, i.e., in the 2. T. W. Ebbesen and P. M. Ajayan, Nature 358, 220 (1992). region of Ð4.9eV; these peaks generally result in the 3. Z. Ya. Kosakovskaya, L. A. Chernozatonskiœ, and disappearance of the band gap characteristic of carbon E. A. Fedorov, Pis’ma Zh. Éksp. Teor. Fiz. 56, 26 (1992) NTs of this type. A similar situation is observed for [JETP Lett. 56, 26 (1992)]. semiconductor (8, 0) NTs as well. 4. N. Hamada, S. Samada, and A. Oshiyama, Phys. Rev. The charge distribution over the potassium atoms Lett. 68, 1579 (1992). also indicates the electron transfer to the NT surface; 5. J. W. Mintmire and C. T. White, Carbon 33, 893 (1995). just as in the case of surface modification, the charge 6. M. S. Dresselhaus, G. Dresselhaus, and P. C. Eklund, transfer increases the number of majority carriers in Science of Fullerenes and Carbon Nanotubes (Aca- NTs and, therefore, leads to semimetal properties in demic, New York, 1996). semiconductor NTs [e.g., of the (8, 0) type] and to an 7. A. L. Ivanovskiœ, Quantum Chemistry in Materials Sci- increase in the conductivity of metal NTs [e.g., of the ence: Nanotube Forms of Substance (Ural. Otd. Ross. (6, 0) type]. Akad. Nauk, Yekaterinburg, 1999). 8. Yu. E. Lozovik and A. M. Popov, Usp. Fiz. Nauk 167, 752 (1997) [Phys. Usp. 40, 717 (1997)]. 4. CONCLUSIONS 9. A. V. Eletskiœ, Usp. Fiz. Nauk 170, 113 (2000) [Phys. Thus, our calculations have shown that internal Usp. 43, 111 (2000)]. (intercalation) and surface modifications result in inter- 10. A. V. Eletskiœ, Usp. Fiz. Nauk 172, 401 (2002) [Phys. nal or external metallization of carbon NTs and to the Usp. 45, 369 (2002)]. occurrence of the semiconductorÐmetal transition in 11. M. M. J. Trecy, T. W. Ebbesen, and J. M. Gibson, Nature semiconductor NTs and the metalÐmetal transition in 381, 678 (1996). semimetal NTs. To date, a number of researchers have 12. A. M. Rao, P. C. Eklund, S. Bandow, A. Thess, and succeeded in experimentally producing carbon NTs R. E. Smalley, Nature 386, 377 (1997). intercalated by atoms of various metals: potassium, 13. E. G. Gal’pern, I. V. Stankevich, A. L. Chistyakov, and platinum, zirconium, chromium, silver, gold, palladium L. A. Chernozatonskii, Chem. Phys. Lett. 214 (1Ð2), 345 [23Ð27], etc. In particular, the suggested experimental (1993). methods of opening multilayer carbon NTs, filling 14. I. V. Zaporotskova, Candidate’s Dissertation (Volgogr. them with metal atoms, and then closing them are Gos. Univ., Volgograd, 1997). described in detail in [27]. In this way, new carbon NTÐ 15. I. V. Zaporotskova, A. O. Litinskiœ, and L. A. Chernoza- based composite materials (quantum nanowires) are tonskiœ, Pis’ma Zh. Éksp. Teor. Fiz. 66, 799 (1997) obtained. It should be noted that the nanometer system [JETP Lett. 66, 841 (1997)].

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1178 ZAPOROTSKOVA et al.

16. N. G. Lebedev, I. V. Zaporotskova, A. O. Litinsky, and 23. M. Baxendale, V. Z. Mordkovich, and S. Yoshimura, L. A. Chernozatonsky, Aerosols 4c, 143 (1998). Phys. Rev. B 56, 2161 (1997). 17. E. G. Gal’pern, I. V. Stankevich, A. L. Chistyakov, and 24. T. Kyotani, L.-F. Tsai, and A. Tomita, Chem. Commun., L. A. Chernozatonskiœ, Izv. Ross. Akad. Nauk, Ser. Fiz. 701 (1997). 11, 2061 (1999). 25. C. N. R. Rao, B. C. Satiskumar, and A. Govindaraj, 18. L. A. Chernozatonsky, N. G. Lebedev, I. V. Zaporotsk- Chem. Commun., 1581 (1997). ova, A. O. Litinskii, E. G. Gal’pern, I. V. Stankevich, and A. L. Chistyakov, in Proceedings of the Second Paciﬁc 26. F. Okuyama and I. Ogasawara, Appl. Phys. Lett. 71 (5), Basin Conference on Adsorption Science and Technol- 623 (1997). ogy, Ed. by D. Duong (Brisbane, 2000), p. 125. 27. B. C. Satiskumar, A. Govindaraj, J. Mofokeng, 19. I. V. Zaporotskova, N. G. Lebedev, and L. A. Chernoza- G. N. Subbanna, and C. N. R. Rao, J. Phys. B 29, 4925 tonski, in Abstracts of Invited Lectures and Contributed (1996). Papers on Fullerenes and Atomic Clusters (St. Peters- burg, Russia, 2001), p. 325. 28. P. N. D’yachkov and D. V. Kirin, Dokl. Akad. Nauk 369 (5), 639 (1999). 20. A. O. Litinskiœ, N. G. Lebedev, and I. V. Zaporotskova, Zh. Fiz. Khim. 69 (1), 215 (1995). 29. D. V. Kirin and P. N. D’yachkov, Dokl. Akad. Nauk 374 21. M. J. S. Dewar and W. Thiel, J. Am. Chem. Soc. 99, 4899 (1), 68 (2000). (1977). 30. P. N. D’yachkov, Zh. Neorg. Khim. 46 (1), 101 (2001). 22. T. Penkalya, Zarys Krystalochemii (Pa’nstwowe Wydawn. Naukowe, Warszawa, 1972; Khimiya, Lenin- grad, 1974). Translated by I. Zvyagin

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

Physics of the Solid State, Vol. 46, No. 6, 2004, pp. 1179Ð1182. Translated from Fizika Tverdogo Tela, Vol. 46, No. 6, 2004, pp. 1143Ð1146. Original Russian Text Copyright © 2004 by Tomilin, Avramov, Kuzubov, Ovchinnikov, Pashkov.

FULLERENES AND ATOMIC CLUSTERS

Correlation of the Chemical Properties of Carbon Nanotubes with Their Atomic and Electronic Structures F. N. Tomilin1, 2, P. V. Avramov2, 4, A. A. Kuzubov2, 3, S. G. Ovchinnikov2, 4, and G. L. Pashkov1 1 Institute of Chemistry and Chemical Technology, Siberian Division, Russian Academy of Sciences, Krasnoyarsk, 660041 Russia 2 Kirensky Institute of Physics, Siberian Division, Russian Academy of Sciences, Akademgorodok, Krasnoyarsk, 660036 Russia 3 Siberian State Technological University, Krasnoyarsk, 660041 Russia 4 Krasnoyarsk State University, Krasnoyarsk, 660079 Russia e-mail: [email protected] Received September 4, 2003

Abstract—The nature of chemical bonding in carbon nanoclusters is investigated by the PM3 semiempirical quantum-chemical method. The inﬂuence of the atomic structure on the electronic characteristics and chemical properties of nanoclusters is analyzed. A σÐπ model is proposed for the chemical bonding in nanotubes. It is shown that, in the framework of the proposed model, nanotubes are objects characterized by a small contribution of π states to the valence band top. © 2004 MAIK “Nauka/Interperiodica”.

1. CHOICE OF THE OBJECTS ber of works (see, for example, [7, 8]). Moreover, the FOR INVESTIGATION PM3 method makes it possible to calculate the elec- The specific feature of elemental carbon is its ability tronic and atomic structures of sufficiently large-sized to form a great variety of complex spatial structures clusters. consisting of polygons. Nanotubes were theoretically predicted by Kornilov [1, 2] in 1977 and Chernozaton- 2. CALCULATION OF THE ELECTRON skii [3] in 1991 and were found experimentally by Ijima DENSITIES OF STATES [4] in 1991. To date, nanotubes have been used as materials for designing macromolecular structures up to The electronic structure was analyzed in terms of hundreds of nanometers in size. The discovery of nan- densities of states. We constructed an energy spectrum otubes has attracted considerable attention due to the of a molecule in which each molecular orbital was rep- possibility of preparing materials with unusual physic- resented by one line. The intensities of all lines were ochemical properties. taken equal to unity. Then, each line was replaced by a In this work, we investigated the nature of chemical Gaussian distribution whose half-width at half-maxi- bonding in carbon nanoclusters and analyzed how their mum was equal to 0.1 eV. The intensities of all the dis- atomic structure affects the electronic characteristics. tributions at each energy were summarized. For model calculations, we chose two structures with When constructing the partial densities of states of the same set of atoms located in a circle of a carbon cyl- atomic orbitals x, the intensity of each line correspond- inder. These were zigzag (10, 0) and armchair (5, 5) ing to the molecular orbital y was taken equal to the nanotubes, which differ in terms of the mutual arrange- sum of the squares of the coefficients of the atomic ment of hexagons (with respect to the longitudinal axis orbitals x in the LCAO MO expansion of the orbital y. of the tube) and radii. The choice of the tube sizes was Then, the partial density of states was constructed by limited by the computational power. In particular, the the algorithm used to construct the total density of number of atoms in nanotubes was varied from 150 to states. 250 and the tube length was varied from 15 to 25 nm. In standard quantum-chemical programs, the wave Dangling chemical bonds at the ends of carbon nano- functions of the P orbitals for each atom are oriented in clusters were terminated by hydrogen. a global coordinate system. In order to interpret the par- Calculations of the nanotubes were performed by tial densities of states for molecules with spherical and the PM3 semiempirical method [5] with the GAMESS cylindrical symmetry, the basis set calculated for each program package [6]. Semiempirical methods offer an carbon atom is sometimes transformed from the global adequate description of both the atomic structure (bond coordinate system into a local coordinate system lengths, bond angles) and electronic structure (photo- through the Euler transformation. In the case of nano- electron spectra), which has been confirmed in a num- tubes, this is a cylindrical transformation with respect

1180 TOMILIN et al.

Global Global When investigating nanotubes, it is of interest to coordinates coordinates compare the electronic structures of fullerenes (with a XXspherical structure) and benzene (with a planar struc- Local ture) (Fig. 2). It can be seen from Fig. 2 that the upper Y Y Local Z coordinates Z levels in the valence band of benzene are characterized Z' coordinates only by the π (P⊥) bonding, whereas the P|| and P⊥ X X Y' X Z' Z' Z' states are mixed (with approximately equal contribu- Z Z Y Y X' ' Y' Y' tions) for C60 fullerene [9]. The benzene molecule has Y X σ π Z X X' the shape of a regular hexagon, the (P||) and (Pz or Z Y ' Z' P⊥) bonds differ in energy, and the upper orbitals con- X X Y ' sist only of the Pz (P⊥) orbitals. The C60 fullerene mol- Y' Y Z Transformations X' ecule is icosahedral in shape, and undistorted hexagons Z' form faces of the structure. The curvature of the surface leads to mixing of the Pz (P⊥), Px (P||), and Py (P||) orbit- Fig. 1. Transformation of the P wave functions of carbon als, and the upper valence levels are composed of the P⊥ from a global coordinate system into a local coordinate system for nanotubes. and P|| states. Figure 2 shows the partial densities of P|| and P⊥ states for the calculated carbon nanotubes. As can be seen from Fig. 2, the electron wave functions that are aligned normally and tangentially to the surface of the P|| structure and, correspondingly, determine the σ (P||) and π (P⊥) bonding appear to be mixed as a result of the HOMO surface curvature of the carbon nanotube. The nature of C60 chemical bonding in nanotubes is governed by the ratio P⊥ of the contributions from these wave functions to the molecular orbitals. The upper ﬁlled levels in the HOMO valence band of nanotubes involve, for the most part, P|| P⊥ C6H6 the P|| states and a small number of the P⊥ states. The nanotubes have a cylindrical structure. Hence, from P|| general considerations, the ratio of the contributions

Density of states, arb. units P⊥ from the P⊥ and P|| wave functions to the valence band (5, 5) HOMO top should be intermediate between the corresponding ratios for benzene and fullerenes. Let us discuss why this is not the case. P|| (10, 0) P⊥ First, we analyze the nature of chemical bonding in HOMO the (5, 5) and (10, 0) nanotubes. A comparison of the partial densities of states for the (5, 5) and (10, 0) tubes Ð15 Ð10 Ð5 shows that the contribution from the P⊥ bonding is Energy, eV somewhat larger in the electronic structure of the (10, 0) tube. This can be explained by the larger diameter Fig. 2. Partial densities of P|| and P⊥ states for carbon nano- and, hence, by the smaller surface curvature of the tube, structures. which leads to a larger overlap between the P⊥ orbitals and an increase in the contribution of the P⊥ orbitals to to the longitudinal axis of the tube, when one of the P the upper occupied molecular orbital (the valence band orbitals of a carbon atom is oriented normally to the top). However, the aforesaid does not explain the small tube surface (Fig. 1). The transformation of the basis set contribution of the P⊥ bonding to the valence band top, results in a set of orbitals oriented normally (P⊥) and even though these nanotubes are comparable in diame- tangentially (P||) to the tube surface. ter to C60 fullerene [d(n, n) < d(2n, 0), d(5, 5) = 6.78 Å, d(10, 0) = 7.83 Å, d(C60) = 7.09 Å]. 3. RESULTS OF CALCULATIONS OF NANOCLUSTERS 4. DISCUSSION The nature of chemical bonding was investigated by As was noted above, compared to fullerenes, nano- comparing the electron densities of states. The densities tubes should possess a sufﬁciently high density of P⊥ of states were compared for the P orbitals directed both states at the valence band top. Carbon nanotubes are tangentially to the molecular plane (P||, σ bonding) and intermediate in structure between planar benzene (and π normally to the tube surface (P⊥, bonding) (Fig. 2). graphite) and ideally spherical fullerene C60 (symmetry

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 CORRELATION OF THE CHEMICAL PROPERTIES 1181

Ih). The large scatter in the diameters of nanotubes leads 180° – α to mixing of the Pz (P⊥), Px (P||), and Py (P||) orbitals in α α 5 6 the valence band, as is the case with fullerenes. How- 1 120° 6 2 ever, the contribution from the Pz (P⊥) orbitals to the top 180°– 4 1 of the valence band is smaller in nanotubes. 5 120° 3 In order to answer the question as to why the elec- 3 2 α 4 tronic structure of nanotubes possesses such specific α α properties, it is necessary to change over from considering their geometric structure (tube radius, tube length) to a detailed analysis of the atomic structure of hexagons forming a carbon skeleton of the tube. The bond angles in hexagons of all the structures remain constant throughout the tube and are approximately equal to 120° (typical angle for the sp2 hybridization), as is the case in benzene, graphite, and fullerene hexagons. The scatter of internuclear distances in nanotubes also slightly differs from the scatter (10, 0) Z axis (5, 5) characteristic of fullerene molecules and, hence, like the bond angle, does not affect the redistribution of the Fig. 3. Torsion angles of (a) (10, 0) and (b) (5, 5) carbon P|| and P⊥ bonding. nanotubes. An analysis of the torsion (dihedral) angles (Fig. 3) describing polygon distortions (deviations from the planarity) revealed an interesting regularity. In all nan- In the case of planar molecules (benzene, graphite), otubes, the structure of all polygons is strongly dis- the overlap between the Pz orbitals leads to the forma- torted. tion of typical π bonds. Since the fullerene molecule is In the structure of the (10, 0) tube, the torsion angle spherical in shape, the overlap of the Pz orbitals above in hexagons (angle between the planes formed by 1Ð2Ð the molecular plane is not equivalent to that below the molecular plane. This gives rise to mixed states formed 3Ð4 and 4Ð5Ð6Ð1 carbon atoms; see Fig. 3a) is approx- σ π imately equal to 18°. The hexagon plane is folded so by and bonds. The distortion of nanotube hexagons that the fold line is parallel to the tube axis. In the struc- results in a shift of the overlap between the Pz orbitals ture of the (5, 5) tube, the torsion angle in the hexagon toward the low-energy range, and, hence, the contribu- is approximately equal to 24°. The hexagon plane is tion of the Pz orbitals to the valence band top for nano- folded, and the fold line in the hexagon is perpendicular tubes is small. The above analysis of the electronic to the tube axis (Fig. 3b). structure permits us to elucidate particular chemical The torsion angle in the (5, 5) structure is, on the properties of carbon nanoclusters. average, 6° larger than that in the (10, 0) structure. The Fullerene molecules possess an electron affinity and hexagon structure in the former nanotubes is formed by act as weak oxidants in chemical reactions. This can be two well-defined planes. In the (10, 0) nanotubes, two observed, for example, in the hydrogenation of C60 less pronounced planes can also be distinguished in the fullerene with the formation of C H as the reaction distorted structure of hexagons. It can be seen from 60 36 product [10]. At room temperature, C60 fullerene oxi- Fig. 2 that the density of P⊥ states at the valence band dizes only under irradiation with photons, which is top is higher for the (10, 0) nanotube characterized by a Ð smaller distortion of the hexagon structure, and vice explained by the formation of negative O2 ions with a versa. high reactivity [10]. The composition of the products of Therefore, the smaller the distortion of hexagons in fullerene fluorination substantially depends on the reac- the structure of the carbon nanotube, the larger the tion conditions. The reaction of C60 with NaF at T = 500Ð550 K predominantly leads to the formation of overlap of the P⊥ orbitals, which, in turn, results in an C F (with addition of 10Ð15% C F ). Chlorination increase in the density of P⊥ states at the valence band 60 46 60 48 top. As a consequence, strong distortions of carbon of C60, as a rule, results in the formation of compounds hexagons lead to a change in the character and energy containing either 12 or 24 chlorine atoms. Moreover, fullerene participates in addition reactions with the for- of overlap between the P⊥ orbitals and to an increase in the contribution of the σ bonding. mation of products involving hydrogen radicals, phos- phorus, halogens, metals and their oxides, benzene rings Owing to specific features in the geometry, the hexa- and their derivatives, NO , alkyl radicals, etc. [10]. gons forming the surface of carbon nanostructures are 2 distorted in the vast majority of carbon nanoobjects. As a rule, approximately 5Ð8% of carbon atoms in These spatial distortions are responsible for the sub- nanotubes undergo chemical transformation under soft stantial differences in the electronic structure. conditions depending on the type of carbon structures

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 1182 TOMILIN et al. and reactants. Usually, these atoms belong to defects in ences in the electronic structure. The distortion affects the nanostructure. In this case, the terminal atoms have the character and energy of overlap between the Pz a higher reactivity as compared to atoms located on the orbitals, which leads to an increase in the contribution nanotube surface. Under sufficiently severe conditions of σ bonding and, correspondingly, to a decrease in the (strong acids, high temperatures, plasma activation, contribution of π bonding to the upper occupied orbit- etc.), more than 50% of carbon atoms undergo chemi- als. Therefore, the smaller the distortion of hexagons in cal transformation. This leads to a change in the prop- the structure of carbon nanotubes, the larger the overlap erties of nanotubes, in particular, to a decrease in the of the P⊥ orbitals. stability [11Ð16]. Owing to formally identical sp2 hybridization, the same type of carbonÐcarbon bonds, and the predomi- ACKNOWLEDGMENTS nance of hexagons in all structures, fullerenes and nano- This work was performed at the “Quantum-Chemi- tubes should resemble graphite in terms of their chemical cal Calculations of Nanoclusters” Collective Use Cen- properties. Actually, nanotubes (compared to fullerenes) ter of the Krasnoyarsk Center of Science and Education are similar in terms of their chemical properties to graph- in High Technology, which is supported by the Russian ite, because they are rather inactive in reactions. State Federal Program “Integration of Higher Educa- The chemical properties of fullerenes can be tion and Fundamental Science” (projects nos. 31 and explained by the difference in the overlap of the P⊥ orbit- 69) and the 6th Competition of Research Projects of als above and below the molecular plane due to the sphe- Young Scientists of the Russian Academy of Sciences ricity of the molecule. The overlap density above the (project no. 155). fullerene surface is lower. As a result, the corresponding orbitals are more susceptible to attacks by electrophilic REFERENCES agents. For this reason, fullerene more easily enters into chemical reactions as compared to graphite, in which the 1. M. Yu. Kornilov, Dokl. Akad. Nauk Ukr. SSR, Ser. B: overlap densities of the P⊥ orbitals above and below the Geol. Khim. Biol. Nauki 12, 1097 (1977). plane are identical to each other. For benzene, the config- 2. M. Yu. Kornilov, Khim. Zhizn’, No. 8, 22 (1985). uration of the P⊥ orbitals is the same as for graphite. As 3. L. A. Chernozatonskii, Phys. Lett. A 160 (4), 392 (1991). a consequence, reactions of substitution for hydrogen 4. S. Ijima, Nature 354, 56 (1991). rather than reactions of electrophilic addition to the π cloud are characteristic of benzene. 5. J. J. Stewart, J. Comput. Chem. 10 (2), 209 (1989). 6. M. W. Shmidt, K. K. Baldridge, J. A. Boatz, et al., All the aforementioned features of fullerene mole- J. Comput. Chem. 14, 1347 (1993). cules allow us to draw the conclusion that nanotubes should also readily enter into addition reactions. Since 7. S. A. Varganov, P. V. Avramov, and S. G. Ovchinnikov, Fiz. Tverd. Tela (St. Petersburg) 42 (11), 2103 (2000) the geometry of nanotubes is similar to that of [Phys. Solid State 42, 2168 (2000)]. fullerenes, the overlap of the P⊥ orbitals above and below the tube surface should also be similar to the 8. L. G. Bulusheva, A. V. Okotrub, D. A. Romanov, and D. Tomanek, Phys. Low-Dimens. Semicond. Struct., overlap in fullerene. However, this is not the case, No. 3Ð4, 107 (1998). because all hexagons in nanotubes are strongly distorted. As a result, the contribution of the π states to the 9. A. A. Kuzubov, P. V. Avramov, S. G. Ovchinnikov, et al., Fiz. Tverd. Tela (St. Petersburg) 43 (9), 1721 (2001) valence band top is small, since these states are shifted [Phys. Solid State 43, 1794 (2001)]. to the low-energy range. Therefore, the reactivity of nanotubes is considerably lower than that of fullerenes. 10. A. V. Eletskiœ and B. M. Smirnov, Usp. Fiz. Nauk 165, 977 (1995) [Phys. Usp. 38, 935 (1995)]. 11. A. Hirsch, Angew. Chem. Int. Ed. Engl. 41 (11), 1853 5. CONCLUSIONS (2002). Thus, the electronic structure of carbon nanotubes 12. A. Kuznetsova, I. Popova, J. T. Yates, Jr., et al., J. Am. was investigated in the framework of the σÐπ model. It Chem. Soc. 43, 10699 (2001). was shown that carbon nanotubes are objects character- 13. T. W. Odom, J.-L. Huang, P. Kim, and C. M. Lieber, ized by a small contribution of the π states to the J. Phys. Chem. B 104, 2794 (2000). valence band top. The surface curvature of carbon 14. J. Liu, A. G. Rinzler, H. Dai, et al., Science 280, 1253 nanostructures leads to mixing of the atomic orbitals (1998). aligned tangentially and normally to the surface. The 15. J. Chen, M. A. Hamon, H. Hu, et al., Science 282, 95 ratio of the contributions from these wave functions (1998). determines the nature of chemical bonding in the nano- 16. M. Monthioux, B. W. Smith, B. Burteaux, et al., Carbon structures. Owing to specific features in the geometry, 39, 1251 (2001). all hexagons forming the tube wall are strongly distorted in all carbon nanotubes. These spatial distortions of hexagons are responsible for the substantial differ- Translated by N. Korovin

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

METALS AND SUPERCONDUCTORS

Kinetics of Interatomic Correlations in a NiÐ11.8 at. % Mo Alloy S. M. Bokoch and N. P. Kulish Shevchenko National University, Vladimirskaya ul. 64, Kiev, 01033 Ukraine e-mail: [email protected] Received August 18, 2003

Abstract—The kinetics of interatomic correlations in low-concentration NiÐMo (8Ð18 at. % Mo) alloys is studied on the basis of the data on residual-resistivity measurements upon isothermal and isochronous annealings, as well as using x-ray diffuse scattering. The atomic ordering kinetics studied at low temperatures (~373 K) suggests the coexistence of clusters of various ordered structures, which transform in a complex fashion depending on the ﬁxed short-range order structure in various temperature ranges. For example, spinodal ordering of N2M2-, N3M-, and N4M-type structures is observed in low-concentration NiÐMo alloys. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION 1 uous intensity transfer from the special points 1---0 As follows from the phase state diagram, several 2 β γ  equilibrium ordered phases, namely, -Ni4Mo (D1a), - δ 1 Ni3Mo (D022), and -NiMo (P212121) containing 20, to the --- {420} sites observed upon low-temperature 25, and 50 at. % Mo, respectively, are formed in NiÐMo 5 alloys. In this system, ordering processes were studied annealing. This transformation corresponds to a contin- in most detail for stoichiometric and close-to-stoichio- uous transition, which, at the first stage, can promote metric compositions [1Ð6]. A specific feature of these the development of the M2N2 structure accompanied by alloys in the concentration range 8Ð33 at. %, as well as 1 of some other alloys (Au V, Au Fe, Au Mn, Au Cr) is an increase in the intensity of the 1---0 diffuse max- 4 4 4 4 2 1 the presence of 1---0 -type x-ray diffuse scattering ima, followed by their decay and the generation of a 2 1 --- {420} concentration wave. Annealing at tempera- maxima in samples quenched from the disordered fcc 5 phase region. Figure 1 shows that the positions of these tures close to the orderÐdisorder transition temperature maxima differ from the arrangement of superstructural T causes a first-order phase transition, at which the reflections corresponding to long-range order equilib- C rium phases (the superstructure peaks of the D0 phase 22 002 022 1 1 are at the --- {420} and --- {420} sites). In NiÐMo alloys, 5 3 ordering processes are accompanied by nucleation and 1 competition between a number of long-range order 2 structures. All these superstructures [Ni Mo (D1 ), 4 1 3 Ni3Mo (D022), Ni2Mo (Pt2Mo), Ni2Mo2 (I41/amd)] can be described using static concentration waves [7], 4 whose amplitude is related to the long-range order 5 parameters. In the atomic representation, these struc- k1 tures are characterized by a certain filling of the {420}- 6 family planes with Ni and Mo atoms (Fig. 2). k 7 Depending on the alloy stoichiometry, the transition 2 from the quenched state, to which the diffuse maxima 000 020 1 at 1---0 -type sites correspond, to the long-range Fig. 1. Arrangement of the sites, superlattice reflections, 2 and diffuse maxima in the (001)* plane of the reciprocal lat- order state is accompanied by the formation and devel- 1 1 tice: (1) fcc lattice sites, (2, 3) 1---0 sites, (4, 5) --- {420} opment of various intermediate structures. For exam- 2 5 ple, it is assumed that the M N -to-D1 transformation 2 2 a 1 D1 sites, and (6, 7) --- {420} Pt Mo sites. takes place in a Ni4Mo alloy, reasoning from the contin- a 3 2

990 BOKOCH, KULISH

M2341M M231M

M1 2 3 4M M1 2 3M

(420) (420)

MM12 2 1MM

M12M M12M M

(420) (420)

Fig. 2. Filling of the {420} planes, with static concentration waves of subcell clusters for various superstructures observed in Ni−Mo alloys ([001] projection).

nucleation and growth of D1a-structure particles are culated concentration-dependent values of these ener- observed, accompanied by the initiation of superstruc- gies for the first four coordination shells [4]. In addition 1 to thermodynamic factors, the generation of an ordered ture reflections --- {420}. 5 Ni2Mo2 structure is also facilitated by the largest amplification rate of the concentration wave inherent to this The transition from the quenched state to the equi- structure. This rate is determined by the second deriva- librium ordered state in a Ni3Mo alloy is more complex tive of the free energy F '' for the long-range order and is associated with the short-range order structure to Ni2Mo2 η η | 1 parameter = 0, i.e., by the curvature of the F( ) η = 0 which the 1---0 diffuse maxima correspond. Accord- curve. 2 ing to the model developed in [8, 9], the short-range However, in contrast to Ni4Mo alloys, which are order state that arises during quenching is a result of the characterized by the transformation of only one initiation and amplification of concentration waves 1 characterized by the wave vectors k at special high- 1---0 -type concentration wave, Ni3Mo alloys feature 2 1 symmetry points 1---0 of the fcc structure. secondary ordering as a result of instability of the 2 formed nonstoichiometric, partially ordered Ni2Mo2 These waves are caused by instabilities of the fcc structure at T 0 K. The secondary ordering in this phase with respect to such waves and occur below the Ni2Mo2 structure can be accompanied by the develop- instability temperature Ti. Concentration waves of the ment of long-wavelength concentration waves with the wave vector k〈000〉, which results in decay of the struc- 1 1---0 type correspond to a coherent Ni2Mo2 structure into a phase mixture containing a disordered solid 2 solution and a stoichiometric ordered Ni2MO2 structure, whose development is caused by spinodal order- ture. Another type of secondary ordering is the forma- ing and represents an orderÐdisorder transition of section of concentration wave 〈100〉, which gives rise to ond order. The preferred instability of the fcc phase the D022 structure. In the third case of secondary order- with respect to the ordered Ni2Mo2 structure in comparing, there appears a wave analogous to the concentra- ison with other possible ordered structures is confirmed 1 by calculations of the free energy F(η) of various struc- tion wave 1---0 generating the nonstoichiometric tures. This energy was determined by appropriately 2 selecting the pair interaction energies for the three near- Ni2Mo2 structure but with a wave vector k2 perpendic- est coordination shells [2], as well as using ab initio cal- ular to the primary-wave direction with vector k1. The

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 KINETICS OF INTERATOMIC CORRELATIONS 991 interaction between these waves gives rise to the N3M ρ, µΩ cm structure. Thus, at the initial stage, ordering in a 37.44 quenched sample arises as a result of instability with 37.42 respect to the formation of the Ni2Mo2 structure and 1 subsequent annealing at various temperatures brings 37.40 about the formation of various structures caused by sec- 37.38 ondary ordering [3]. 37.36 The above concepts regarding the initial stage of 37.34 2 transformations from short-range order to equilibrium 37.32 long-range order in NiÐMo alloys were developed 37.30 using the results of electron-microscopy studies of sto- 37.28 ichiometric samples. Such studies do not allow quanti- 3 tative description of the short-range order structure in 37.26 the generated structures. This circumstance is espe- 200 300 400 500 600 700 800 900 1000 cially important in the case of low-concentration alloys T, K far from stoichiometry. Taking into account the spinodal ordering dynamics of these structures, which Fig. 3. Isochronous dependences of the residual resistivity should be observed in a wide concentration range, one in a NiÐ11.8 at. % Mo alloy for (1) the annealed, would expect, on the one hand, the formation of con- (2) quenched, and (3) deformed initial states. centration waves similar to those described above and, on the other hand, the absence of certain competing puriﬁed argon. The casts were homogenized at a tem- structures associated with equilibrium intermetallic perature of 1123 K for 75 h. Then, the casts were used ordered phases. A comparison of the ordering processes to obtain rods, which were stretched into wire and in Ni4Mo and Ni3Mo alloys occurring at the initial stage rolled into strips ~0.1 mm thick. The room-temperature shows that they differ substantially. Hence, it is impos- plastic deformation exceeded 1000%. Such an initial sible to predict the short-range order structure at alloy state of the sample corresponds to a deformed state. concentrations below the region of existence of the Then, samples deformed in such a way were annealed ordered Ni4Mo phase. at a temperature of 1273 K for 2 h in vacuum and This study is devoted to the transformation of the slowly cooled (at a rate of 2 K/min) to room tempera- short-range order structure in low-concentration NiÐ ture. Such processing yields another (in comparison Mo alloys. We took samples containing 8, 11.8, 14, and with deformation) initial state of the short-range order, 18 at. % molybdenum. The short-range order structure which corresponds to the annealed state. A third initial was approximated by studying the isochronous and iso- state of the short-range order was obtained by 1-h thermal dependences of the residual resistivity. The annealing of a deformed sample at 1223 K in vacuum, quantitative characteristics were determined using followed by room-temperature quenching in a 10 wt % x-ray diffuse scattering. NaOH solution in water in a special setup. It should be noted that, in the absence of structural All samples in each of the initial states were isoch- changes in the solid-solution region, the residual resis- ronously annealed. Such annealing was carried out in tivity is very sensitive to evolution of the short-range the temperature range 323Ð1073 K in steps of 30 K. order structure [10]. However, since this effect is inte- After 10-min annealings at each temperature in vac- gral (as in the case of electron diffraction studies), it is uum, the samples were quenched. The residual resistiv- impossible to obtain a detailed description of changes ity was measured in liquid nitrogen using the four- in the behavior of interatomic correlations by measur- probe potentiometric method. The accuracy of deter- ing the residual resistivity. The short-range order kinet- mining the relative residual resistivity, ∆ρ/ρ, was ics can be uniquely described only by studying x-ray 0.001%. The kinetics of the residual resistivity for diffuse scattering (or elastic incoherent scattering of annealed, and quenched samples was studied upon isothermal neutrons). thermal annealing at the temperature Ta = 323 and 373 K. 2. RESIDUAL RESISTIVITY Figure 3 shows the isochronous curves of the resid- OF NiÐMo ALLOYS ual resistivity for a NiÐ11.8 at. % Mo alloy in the three NiÐMo alloy samples were prepared differently for initial states indicated above. For the other concentra- various studies. To measure the residual resistivity, tions, the same features in the behavior of the residual polycrystalline samples of all compositions indicated resistivity are observed, which suggests that the order- above were used. The samples were obtained by smelt- ing processes in the NiÐMo composition range under ing a charge that contained vacuum-remelted Ni of study are similar. 99.95% purity and Mo of 99.99% purity in a high-fre- We can see that the residual resistivities in the initial quency induction furnace in an alundum crucible in state differ signiﬁcantly. In the annealed state, the resid-

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 992 BOKOCH, KULISH

ρ, µΩ cm 3. SHORT-RANGE ORDER STRUCTURE 37.360 IN A NiÐ11.8 AT. % Mo ALLOY: RESULTS AND DISCUSSION 37.355 The short-range order structure was studied using x- 37.350 ray diffuse scattering in a NiÐ11.8 at. % Mo single crys- × × 37.345 tal 2.5 6 6 mm in size grown in argon in an alundum crucible with a conic seed region. An outer facet of the 37.340 sample under study was cut parallel to the (100) plane. The sample was studied in the annealed and quenched 37.335 initial states. Both initial states were obtained similarly 37.330 to the states of the samples used for measuring the residual resistivity. To study the intensity kinetics asso- 37.325 ciated with the short-range order, the quenched sample 37.320 was isothermally annealed at Ta = 373 K; this was car- 0 2 4 6 8 10 12 14 16 18 20 ried out in the same way as in the study of the residual t, min resistivity. Fig. 4. Kinetics of the residual resistivity of a NiÐ11.8 at. % X-ray diffuse scattering was measured using hard Mo alloy quenched at Tq = 1223 K. MoKα radiation, which made it possible to separate high harmonics and the fluorescent component from the spectrum of x-rays monochromatized by a one- ual resistivity is larger than that for the other treatments. dimensionally bent LiF crystal through selecting the Obviously, the defect structure is different in all the ini- operating conditions of the setup [11]. The use of tial states; however, the differences observed in the obliquely incident hard x-rays in diffractometry of the residual resistivity are so significant that they cannot be samples and the use of low-absorbing fused quartz as a explained by electron scattering at defects. Therefore, it reference for measuring the intensities in absolute elec- can be assumed that these differences are caused by the tronic units require corrections for the angular depen- difference in the short-range order structure of the solid dence of absorption [12]. Taking into account these cor- solution that takes place in each initial state. As the iso- rections, the diffuse scattering intensity I (θ) from a chronous annealing temperature increases, i.e., for val- sp sample in electronic units expressed in terms of the ues of the residual resistivity measured in the states that θ arise in alloys at various temperatures, the behavior of intensity from the reference at an angle 0 is given by the resistivity depends on the initial state (up to T = ()θ ()µ ()θ ()θ Isp 0 n/2 L sp 773 K for a NiÐ11.8 at. % Mo alloy; see Fig. 3). Isp = Ist 0 ------()θ ()µ - Ist 0 n/2 L st For example, the residual resistivity of annealed (1) γ 2 θ 2 θ samples (curve 1) decreases at low temperatures, in 1 + '2cos M cos 2 0 contrast to that of quenched (curve 2) and deformed × ------, γ 2 θ 2 θ (curve 3) samples. 1 + '2cos M cos 2 θ For the deformed state, the resistivity increases where Ist( 0) is the experimentally measured scattering upon isochronous annealing, whereas it passes through intensity from fused quartz and Lst and Lsp are the cor- a maximum after quenching. rections to the absorptance µ for the reference and the At higher temperatures of isochronous annealing, sample, respectively. The monochromator imperfection beginning from T ≈ 573Ð623 K, the resistivity increases is taken into account by the parameter γ', which is close for all initial states, although its values differ. For to unity for the LiF crystal used. annealing temperatures above T ≈ 773 K, the residual The intensity components associated with the short- resistivities sharply decrease and are identical for all range order and static and dynamic distortions, which initial treatments. modulate diffuse scattering in alloys [13Ð15], were sep- Figure 4 shows the evolution of the residual resistiv- arated using the CohenÐGeorgopoulos method [16Ð ity measured upon isothermal annealing of the 18]. This separation is based on the difference in the quenched sample. The annealing temperature T = symmetry of diffuse background modulation in the 373 K corresponds to the first maximum of the isochro- reciprocal space between these scattering components. nous curve measured after the sample quenching The intensities of two-phonon and multiphonon ther- (curve 2 in Fig. 3). We can see that a sharp increase in mal diffuse scattering were calculated using the tech- the resistivity is initially observed, which is then slowed nique described in [11]. The distribution of the x-ray down and tends to flatten out. diffuse scattering intensity was studied at the positions 1 1 The isochronous and isothermal curves of the resid- equivalent to the points --- (420) (Ni Mo ), --- (420) ual resistivity suggest the complexity of the ordering 4 2 2 5 processes occurring in low-concentration NiÐMo alloys 1 (Ni Mo), and --- (420) (Ni Mo) and in their vicinity cov- at various temperatures and durations of annealing. 4 3 2

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 KINETICS OF INTERATOMIC CORRELATIONS 993

(a) 002 022

(b) 1.00 ~ 110 0.05 I ) r (

α 0 0 2 4 6 8 10 –0.05 Number of coordination shell, r –0.10 –0.15 000 020 002 022 1.00

~ 0.05 ) r (

α 0 110 0 2 4 6 8 10 II –0.05 –0.10 Number of coordination shell, r –0.15

000 020

Fig. 5. (a) Distribution of the short-range order scattering intensity ISRO(h1, h2, h3) in the (001)* plane and (b) the short-range order parameters α(r) for (I) annealed and (II) quenched states of a NiÐ11.8 at. % Mo alloy.

1 1 ering the part of the irreducible Brillouin zone in the at the 1---0 and --- {420} sites and in their vicinities, (001)* plane of the reciprocal space. 2 5 in particular, along the [210] direction. At the point The WarrenÐCowley short-range order parameters 1 α for atomic positions with coordinates lmn were --- {420}, the short-range order intensity is low (Fig. 5a, lmn 3 calculated by taking the inverse Fourier transform of panel I). The short-range order parameters α(lmn) are the determined intensities of the short-range order shown in Fig. 5b, panel I. Despite the smallness of the ISRO(h1, h2, h3) at a given point in reciprocal space with reciprocal space volume in which the intensities coordinates h1h2h3 according to the expression α ISRO(h1, h2, h3) used for calculating (lmn) were sepa-

1 rated, these parameters are in good agreement with α ()π,, () those previously determined with a large number of lmn = kI∑ SRO h2 h2 h3 cos h1 I points h1h2h3 for the same initial state [19Ð21]. ,, (2) h1 h2 h3 = 0 ×π()π() Comparison of these parameters α(lmn) with the cos h2m cos h3n , short-range order parameters for the D1a, D022, N3M, where k is the normalizing factor, which depends on the and N2M2 structures suggests that the short-range order position of the atom. related to the D1a type is dominant in this initial state. 1 A study of the short-range order structure of the NiÐ The rather intense diffuse maxima at the 1---0 sites 11.8 at. % Mo alloy in the initial annealed state shows 2 that the intensity ISRO(h1, h2, h3) is mostly concentrated show that the short-range order structure is more com-

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 994 BOKOCH, KULISH

ISRO, el. units/atom (Fig. 6). At the same time, the intensity increases in 80 1 1 strands directed from the point 1---0 to the positions 70 2 1 1 60 --- {420} and --- {420}. These intensities, especially near 3 5 50 1 the point --- {420}, even exceed I (h , h , h ) at the 40 3 3 SRO 1 2 3 30 positions of the D1a and N2M2 superstructures (Fig. 7a, 2 panel I). 20 Thus, the destruction of the ordered N2M2 structure 10 1 0 described by the unit concentration wave 1---0 is 0 123456789 2 t, min accompanied (as noted for the initial stages of Ni3Mo alloy ordering [2, 3]) by secondary ordering caused by Fig. 6. Kinetics of the short-range order scattering intensity 1 ISRO at the positions (1) N2M2, (2) N4M, and (3) N2M. interference from mutually perpendicular 1---0 con- 2 centration waves, i.e., by the nucleation of the N3M plex and represents a mixed state that also includes the structure. Despite the fact that the short-range order N2M2-type short-range order. parameters α(lmn) (Fig. 7b, panel I) still correspond to the N2M2 structure, their behavior differs from that of When a sample is quenched from 1223 K, the distri- α bution of diffuse scattering associated with the short- (lmn) for the initial state. It is worth noting the small- range order changes abruptly. The intensity I (h , h , ness of the parameter for the fourth coordination shell SRO 1 2 and the relative values of the parameters for other  1 shells, whose behavior is similar to that of the WarrenÐ h3) at the 1---0 sites and neighboring points increases 2 Cowley parameters for an ordered D022 structure. It is by several times. For example, the intensity at this posi- reasonable to expect that the destruction of the ordered tion is 31.4 and 70.9 el. units/atom for the annealed and N2M2structure will be accompanied by an increase in quenched states, respectively. The intensity at the the residual resistivity, which does actually take place 1 at this stage of the ordering kinetics of the quenched --- {420} sites and their vicinities also increases (Fig. 5a, 3 sample (Figs. 3, 4). panel II). The predominance of the N2M2structure is Further isothermal annealing for 6 min is accompa- also confirmed by the calculated parameters α(lmn) nied by a change in the short-range order, which affects (Fig. 5b, panel II). It is worth noting that the parameter the ISRO(h1, h2, h3) distribution (Fig. 7a, panel II). A spe- α(lmn) for the sixth shell appreciably increases, which cific feature of this distribution is the appreciable is inherent to the N M structure. However, in this case, 2 2 1 we cannot argue that only the N2M2 short-range order decrease in intensity at points 1---0 and a certain exclusively exists. 2 1 The distribution of the short-range scattering inten- increase in intensity at the position --- {420} (Fig. 6). sity suggests that the short-range order structure in the 5 1 1 quenched state corresponds to the coexistence of spin- The intensity at points --- {420} and --- {420} exceeds 3 5 odal-ordered N2M2, N2M, and N4M structures. A com- 1 parison of the data on spinodal ordering with the data the intensity at 1---0 , although their difference is not on residual resistivity in the annealed and quenched ini- 2 tial states shows that the predominant D1a short-range as distinct as that for the initial quenched state. order in the annealed alloy samples causes an increase in their resistivity. The preferential formation of the The intensity of the strands in the directions from N M short-range order in the quenched state causes a 1 1 1 2 2 1---0 to --- {420} and --- {420} exceeds the intensity at decrease in resistivity. 2 3 5 Isothermal annealing of the quenched sample for 1 the point 1---0 itself. The most significant increase is 4 min at 373 K results in a sharp decrease in the inten- 2   1 observed near the latter two positions. The change in sity ISRO(h1, h2, h3) at the point 1---0 , while the inten- 2 the short-range order parameters (Fig. 7b, panel II) sug- sities at the two other points remain almost unchanged gests preferential formation of the N3M structure.

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 KINETICS OF INTERATOMIC CORRELATIONS 995

(a) 002 022

(b) 1.00

~ 0.05

I )

r 4 6 (

α 0 0 2 8 10 –0.05 Number of coordination shell, r –0.10 –0.15 000 020 002 022 1.00

~ 0.05 ) r (

α 0 II 110 0 2 4 6 8 10 –0.05 Number of coordination shell, r –0.10 –0.15

000 020

Fig. 7. (a) Distribution of the short-range order scattering intensity ISRO(h1, h2, h3) in the (001)* plane and (b) the short-range order α parameters (r) for quenched NiÐ11.8 at. % Mo alloy samples isothermally annealed at Ta = 373 K for (I) 4 and (II) 6 min.

1 Thus, as time passes, the N M structure is formed in respect to concentration waves --- {420}. At higher tem- 3 5 the quenched sample at 373 K due to secondary order- peratures, the resistivity decreases (Fig. 3), which indi- ing. This structure is a mixture of clusters [2, 3] corre- cates disordering, i.e., the formation of an ordinary dis- sponding to the N4M and N2M structures. The formation ordered fcc structure that is stable to the generation of of this structure is accompanied by a decrease in the various concentration waves. residual resistivity, which results in a change in slope of This scheme of spinodal ordering is conﬁrmed by the kinetic curve (Fig. 4). Upon isochronous annealing the behavior of the isothermal residual resistivity in the (Fig. 3), the resistivity decreases in the temperature case of quenching of isochronously annealed samples ranges ~323Ð573 and ~373Ð623 K for annealed and (for both initial states) from 573 and 673 K. As before, quenched samples, respectively. It is reasonable to isothermal annealing was carried out at Ta = 373 K assume that such a decrease in the residual resistivity is (Fig. 8). We can see that the kinetic curves measured at also caused by interference from concentration waves these quenching temperatures signiﬁcantly differ,  which indicates that the short-range order structures are 1 different at these temperatures. 1---0 , i.e., by the nucleation of the N3M structure. 2 Indeed, as noted above, isochronous annealing tem- From the fact that ordering with the formation of clus- peratures of 573 and 673 K (curves 1, 2 in Fig. 3) cor- ters of the Ni Mo structure results in an increase in the 4 respond to the existence of the N3M and N4M structures, resistivity, it follows that the increase in ρ (observed respectively. In the former case (curve 1 in Fig. 8), the from the isochronous curves) at ~573Ð773 and ~623Ð resistivity slowly increases, while in the latter case 773 K for annealed and quenched samples, respec- (curve 2 in Fig. 8) the ϕ(t) dependence is complex. It is tively, arises from the solid-solution instability with clear that annealing at Ta = 373 K of the sample

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 996 BOKOCH, KULISH

ρ, µΩ cm thermal dependences of the residual resistivity and the distribution of x-ray diffuse scattering intensity in low- 37.406 concentration NiÐMo alloys, the temperature ranges where clusters of various ordered structures exist can be 37.402 determined. 1 37.398 REFERENCES 37.394 1. S. Banerjee, K. Urban, and M. Wilknes, Acta Metall. 32 (3), 299 (1984). 2 2. U. D. Kulkarni and S. Banerjee, Acta Metall. 36 (2), 413 37.390 (1988). 3. S. Banerjee, U. D. Kulkarni, and K. Urban, Acta Metall. 37.386 37 (1), 35 (1989). 0 10 20 30 40 50 4. A. Arya, S. Banerjee, G. P. Das, I. Dasgupta, T. Saha- t, min Dasgupta, and A. Mookerjee, Acta Mater. 49, 3575 (2001). Fig. 8. Kinetics of the residual resistivity at Ta = 373 K for 5. S. Hata, S. Matsumura, N. Kuwano, and K. Oki, Acta isochronously annealed samples quenched at (1) Tq = 573 Mater. 46 (3), 881 (1998). and (2) 673 K. 6. S. Hata, T. Mitute, N. Kuwano, S. Matsumura, D. Shindo, and K. Oki, Mater. Sci. Eng. A 312, 160 (2001). quenched from Tq = 573 K causes destruction of the 7. A. G. Khachaturyan, Theory of Structural Transforma- N3M structure and that the N2M2 structure prevails, although its content seems to be lower than in the initial tions in Solids (Wiley, New York, 1983). 8. D. de Fontaine, Acta Metall. 23 (4), 553 (1975). quenched state. In the case of quenching from Tq = 9. D. de Fontaine, Solid State Phys. 34, 73 (1979). 673 K followed by annealing at Ta = 373 K, the short- range order with the N M structure is initially 10. P. L. Rossiter and P. Wolls, J. Phys. C: Solid State Phys. 4 4, 354 (1971). destroyed, which decreases the resistivity. Then, N3M is destroyed, which increases ρ(t). This N M destruction 11. N. P. Kulish, N. A. Mel’nikova, P. V. Petrenko, 3 V. G. Poroshin, and N. L. Zyuganov, Izv. Vyssh. can be considered a result of the disappearance of the Uchebn. Zaved., Fiz. 32 (2), 82 (1989). 1 interaction between concentration waves 1---0 at low 12. V. G. Poroshin and N. P. Kulish, Metalloﬁz. Noveœshie 2 Tekhnol. 21, 75 (1999). temperatures, i.e., the formation of solitary waves. In 13. M. A. Krivoglaz, Theory of X-ray and Thermal Neutron Scattering by Real Crystals (Nauka, Moscow, 1967; Ple- this case, the nucleation of the N2M2structure is accompanied by a decrease in the residual resistivity. The con- num, New York, 1969). 14. M. A. Krivoglaz, Diffuse Scattering of X-ray and Neu- tent of the N2M2structure formed at this temperature is not optimal; therefore, this structure is partially trons by Fluctuations in Solids (Naukova Dumka, Kiev, destroyed and ρ(t) increases. As the duration of anneal- 1984; Springer, Berlin, 1996). ing increases, the residual resistivity tends to an equi- 15. C. J. Sparks and B. Borie, Local Atomic Arrangements librium value, which seems to correspond to a mixed Studied by X-ray Diffraction, Ed. by J. B. Cohen and J. E. Hilliard (Gordon and Breach, New York, 1966). state of short-range order involving the N2M2 and N3M structures. 16. P. Georgopoulos and J. B. Cohen, J. Phys. (Paris) 38 (12), 7191 (1977). 17. P. Georgopoulos and J. B. Cohen, Acta Metall. 29, 1535 4. CONCLUSIONS (1981). 18. J. B. Cohen, Solid State Phys. 39, 131 (1986). Thus, spinodal ordering of the N2M2, N3M, and N4M structures takes place in low-concentration NiÐMo (8Ð 19. V. G. Poroshin, N. P. Kulish, P. V. Petrenko, 18 at. %) alloys, which is also observed at the initial N. A. Mel’nikova, and S. P. Repetskiœ, Fiz. Met. Metall- oved. 87 (2), 145 (1999). ordering stages in quenched samples of NiÐMo alloys with stoichiometric compositions. Moreover, the order- 20. V. G. Poroshin, N. P. Kulish, P. V. Petrenko, and N. A. Mel’nikova, Fiz. Tverd. Tela (St. Petersburg) 41 ing process scheme is simpler in this case, as it excludes (12), 2121 (1999) [Phys. Solid State 41, 1945 (1999)]. the nucleation of N M- and D0 -structure clusters [2, 2 22 21. N. P. Kulish, N. A. Melnikova, P. V. Petrenko, and 3]. Study of the low-temperature ordering kinetics sug- V. G. Poroshin, Met. Phys. Adv. Technol. 19, 1147 gests the coexistence of clusters of various structures, (2001). which are transformed in a complex fashion depending on the short-range order structure observed in various temperature ranges. Based on the isochronous and iso- Translated by A. Kazantsev

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004

Physics of the Solid State, Vol. 46, No. 6, 2004, pp. 997Ð1000. Translated from Fizika Tverdogo Tela, Vol. 46, No. 6, 2004, pp. 969Ð971. Original Russian Text Copyright © 2004 by Nemov, P. Seregin, N. Seregin, Davydov.

METALS AND SUPERCONDUCTORS

Hyperfine Nuclear Interaction at Copper Lattice Sites of High-Temperature Superconductors Studied by 61Cu(61Ni) Mössbauer Emission Spectroscopy S. A. Nemov*, P. P. Seregin*, N. P. Seregin**, and A. V. Davydov*** *St. Petersburg State Polytechnical University, Politekhnicheskaya ul. 25, St. Petersburg, 195251 Russia **Institute of Analytical Instrumentation, Russian Academy of Sciences, St. Petersburg, 198103 Russia ***Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia Received October 30, 2003

Abstract—Mössbauer emission spectroscopy on the 61Cu(61Ni) isotope has been used to determine the qua- 61 2+ drupole coupling constant C(Ni) and magnetic induction B(Ni) for the Ni probe at copper sites in Cu2O, CuO, La2 Ð xBaxCuO4, Nd2 Ð xCexCuO4, RBa2Cu3O6, and RBa2Cu3O7 (R = Y, Nd, Gd, Yb). The compounds containing divalent copper were found to exhibit linear C(Ni) vs. C(Cu) and B(Ni) vs. B(Cu) relations [C(Cu) and B(Cu) are the quadrupole coupling constant and magnetic induction for the 63Cu probe, respectively, found by NMR], which is interpreted as an argument for the copper being in divalent state. The deviation of the data points corresponding to the Cu(1) sites in RBa2Cu3O6 and RBa2Cu3O7 from the C(Ni) vs. C(Cu) straight line may be due either to the copper valence being other than 2+ (in the RBa2Cu3O6 compounds) or to the principal axes of the total and valence electric ﬁeld gradient being differently oriented (in the RBa2Cu3O7 compounds). © 2004 MAIK “Nauka/Interperiodica”.

Mössbauer spectroscopy is widely used in studies of netic field completely lifts the degeneracy of the 61Ni nuclear quadrupole interactions in high-temperature nuclear levels, to make the 61Ni Mössbauer spectrum a superconductors; by comparing experimentally deter- superposition of 12 lines with a theoretical intensity mined parameters with calculated parameters of the ratio of 10 : 4 : 1 : 6 : 6 :3 : 3 : 6 : 6 : 1 : 4 : 10. electric field gradient (EFG) tensor, one can, in principle, determine the spatial distribution of electronic Samples were prepared by sintering the correspond- defects and the effective charges of atoms [1]. ing oxides, and x-ray diffraction showed them to be single phase. All the starting samples were single phase. These studies acquire particular significance in The La2 Ð x(Sr,Ba)xCuO4 compounds studied for x = 0.1, cases where the Mössbauer probe occupies a copper 0.15, 0.20, and 0.30 yielded Tc = 25, 37, 27, and <4.2 K, site, and this is why the version of Mössbauer emission respectively. The Nd Ce CuO composition with x = 67 67 2 Ð x x 4 spectroscopy (MES) involving the Cu( Zn) isotope 0 did not become superconducting down to 4.2 K, and was proposed and realized [2]. However, determination the composition with x = 0.15 yielded Tc = 22 K. For of the parameters of nuclear quadrupole coupling in YBa Cu O , we obtained T = 90 K, and for the 67 67 2 3 6.9 c magnetically ordered lattices using Cu( Zn) Möss- RBa Cu O compounds T varied from 83 to 90 K. The bauer spectroscopy did not meet with success for a 2 3 7 c samples of RBa2Cu3O6 did not cross over to the super- number of technical reasons [2]. This communication conducting state down to 4.2 K. Mössbauer sources reports on an investigation into the nuclear hyperfine were prepared through diffusion of the short-lived iso- interaction at copper sites in Cu O, CuO, 2 tope 61Cu into the prefabricated ceramic at tempera- La2 − xBaxCuO4, Nd2 Ð xCexCuO4 (x = 0, 0.15), tures ranging from 773 to 923 K for two hours in an RBa2Cu3O6, and RBa2Cu3O7 (R = Y, Nd, Gd, Yb) using oxygen environment. Annealing of check samples in 61 61 61 2+ MES on the Cu( Ni) isotope; the Ni Mössbauer similar conditions did not affect the values of Tc. Four probe forming in the decay of 61Cu is localized in the to six samples were employed to measure an experi- copper sites, and the nuclear and atomic parameters of mental spectrum. 61Cu(61Ni) Mössbauer emission spec- this probe are appropriate for determining the parame- tra were measured on a commercial spectrometer at ters of the combined hyperfine interaction at these sites 80 K with a Ni0.86V0.14 alloy used as an absorber. [3]. Quadrupole splitting of the 61Ni Mössbauer spectrum gives rise to the appearance of five components To compare the possibilities offered by MES on the with an intensity ratio of 10 : 8 : 1 : 6 : 9, and the mag- 67Cu(67Zn) and 61Cu(61Ni) isotopes, Figs. 1 and 2

998 NEMOV et al.

Cu(2) Cu(1) Cu(1)

Cu(2) a

Cu(2)

Cu(1) Cu(1) Relative count rate

Relative count rate b

–90 –45 0 45 90 –4 –2 0 2 4 V, µm/s V, mm/s

67 67 61 61 Fig. 1. Cu( Zn) Mössbauer spectra of (a) YBa2Cu3O7 Fig. 2. Cu( Ni) Mössbauer spectra of (a) YBa2Cu3O7 and (b) YBa2Cu3O6 measured at 4.2 K. The positions of the and (b) YBa2Cu3O6 measured at 80 K. The positions of the quadrupole triplet components for the Cu(1) and Cu(2) sites quadrupole and Zeeman multiplet components for the are speciﬁed. Cu(1) and Cu(2) sites are speciﬁed.

67 67 61 61 present Cu( Zn) and Cu( Ni) spectra of the the Cu(1) sites and the spectrum with the larger eQUzz, 61 2+ YBa2Cu3O7 and YBa2Cu3O6 compounds [the to Ni centers in the Cu(2) sites. The Cu(2) sublattice 67 67 Cu( Zn) Mössbauer sources were prepared through in YBa2Cu3O6 is magnetically ordered; therefore, the diffusion doping of the above compounds with the 67Cu spectrum of this sample may be considered a superpo- radioactive isotope, 67ZnS served as an absorber when sition of a magnetic multiplet [61Ni2+ centers in the measuring the 67Cu(67Zn) spectra, and the spectra were Cu(2) sites] and a broadened singlet [61Ni2+ centers in taken at 4.2 K]. In crystal fields with a symmetry lower the Cu(1) sites]. than cubic, the Mössbauer spectrum splits into three The nuclear hyperfine interaction parameters for the components of equal intensity. As is evident from 61 67 67 Ni probe [the quadrupole coupling constant C(Ni) and Fig. 1, the Cu( Zn) spectrum of YBa2Cu3O7 is a the magnetic induction B(Ni)] are displayed in Figs. 3 superposition of two quadrupole triplets with an inten- and 4. sity ratio of 1 : 2. Based on the x-ray diffraction data, the weaker triplet should be assigned to 67Zn2+ centers The quadrupole coupling constant C(Cu) was mea- 63 in the Cu(1) sites and the stronger one, to the 67Zn2+ sured for the Cu probe in cuprous oxides by NMR [4Ð 67 67 11], which permits one to check the correlation centers occupying the Cu(2) sites. The Cu( Zn) spec- 61 61 63 trum of YBa Cu O is a quadrupole triplet, although between the Cu( Ni) MES data and Cu NMR mea- 2 3 6 | | | | copper in this compound also occupies two structurally surements. Figure 3 presents a C(Ni) vs. C(Cu) graph inequivalent positions. The experimental spectrum for the divalent-copper compounds La2CuO4, relates to the 67Zn2+ centers in the Cu(1) sites, whereas La1.85Sr0.15CuO4, Nd2CuO4, Nd1.85Ce0.15CuO4, and the spectrum due to the 67Zn2+ centers in the Cu(2) sites CuO. We readily see that the experimental points fit a straight line. This linear dependence can be understood is split by magnetic interaction and is not seen in the 2+ experimental spectrum (its observation would require if we recall that the EFG at nucleus sites for the Ni higher Doppler velocities that could not be reached and Cu2+ centers is generated both by the lattice ions with the spectrometer at our disposal). (the crystal field EFG) and by the nonspherical valence As seen from Fig. 2, however, both copper states in shell of the ion itself (the valence EFG), so the quadrupole coupling constant for the probe can be cast as the YBa2Cu3O7 and YBa2Cu3O6 lattices manifest themselves in the 61Cu(61Ni) spectra. The spectrum of ()γ () CeQ= 1 Ð V zz + eQ 1 Ð R0 Wzz, YBa2Cu3O7 is actually a superposition of two quadrupole multiplets; based on the relative intensities of the where eQ is the quadrupole moment of the probe constituents of the spectrum, the spectrum with the nucleus; Vzz and Wzz are the principal components of the 61 2+ smaller eQUzz should be assigned to Ni centers in crystal and valence EFG tensors, respectively; and

PHYSICS OF THE SOLID STATE Vol. 46 No. 6 2004 HYPERFINE NUCLEAR INTERACTION AT COPPER LATTICE SITES 999

C(Ni), MHz B(Ni), T –60 12

8 7 5 6 9 13 3 –50 10 1 12 2 12 11 1 8 11 4 3 5 –40 13 10 17 16 4 –30 15 22 14

–20 18, 19, 20, 21 0 2, 4, 6–9, 14–22 010305070 04812 |C(Cu)|, MHz B(Cu), T

| | | | Fig. 3. C(Ni) vs. C(Cu) diagram for (1) La2CuO4 [4], Fig. 4. B(Ni) vs. B(Cu) diagram. The data points are (2) La1.85Sr0.15CuO4 [5], (3) Nd2CuO4 [6], denoted as in Fig. 3. The straight line is a least squares fit through all points. (4) Nd1.85Ce0.15CuO4 [6], (5) CuO [4], (6) Cu(2) in NdBa2Cu3O7 [7], (7) Cu(2) in GdBa2Cu3O7 [7], (8) Cu(2) in YBa2Cu3O7 [8], (9) Cu(2) in YBa2Cu3O7 [7], (10) Cu(2) in NdBa Cu O [7], (11) Cu(2) in GdBa Cu O [9], 2 3 6 2 3 6 RBa2Cu3O6 compounds is univalent], and the second (12) Cu(2) in YBa Cu O [10], (13) Cu(2) in YbBa Cu O 2 3 6 2 3 6 applies to the Cu(1) sites in the RBa2Cu3O7 compounds [9], (14) Cu(1) in NdBa2Cu3O7 [9], (15) Cu(1) in [it is known that, at least for the Cu(1) sites in GdBa2Cu3O7 [9], (16) Cu(1) in YBa2Cu3O7 [8], (17) Cu(1) YBa2Cu3O7, the z axis of the total EFG tensor is in YbBa2Cu3O7 [9], (18) Cu(1) in NdBa2Cu3O6 [9], (19) Cu(1) in GdBa Cu O [9], (20) Cu(1) in YBa Cu O directed along the b crystallographic axis, whereas the 2 3 6 2 3 6 z axis of the crystal-field EFG is aligned with the a crys- [10], (21) Cu(1) in YbBa2Cu3O6 [9], and (22) Cu2O [11]. The straight line is a least squares fit through points 1Ð13 tallographic axis [13]]. corresponding to divalent copper. The references concern Magnetic induction at the copper sites, B(Cu), was 63Cu NMR data. measured for the 63Cu probe in cuprous oxides by NMR [4Ð11]. As is evident from Fig. 4, the 61Cu(61Ni) MES data are linearly related to the 63Cu NMR data for all the γ and R0 are the Sternheimer coefficients for the probe compounds studied. This is obviously the result of the used. probe occupying the copper sites in the lattice. The linear relation |C(Ni)| vs. |C(Cu)| is actually a consequence of the linear relations C(Ni) vs. Vzz and ACKNOWLEDGMENTS C(Cu) vs. Vzz, which were observed [12] for com- This study was supported by the Russian Founda- pounds of divalent copper. Because the data obtained tion for Basic Research, project no. 02-02-17306. for the Cu(2) sites in the RBa2Cu3O7 and RBa2Cu3O6 compounds fall onto the above straight line, one may conclude that copper in these sites is in the divalent REFERENCES state. 1. V. F. Masterov, F. S. Nasredinov, and P. P. Seregin, Fiz. Tverd. Tela (St. Petersburg) 37 (5), 1265 (1995) [Phys. As seen from Fig. 3, the points obtained for the cop- Solid State 37, 687 (1995)]. per sites in Cu O and for the Cu(1) sites in RBa Cu O 2 2 3 6 2. V. F. Masterov, F. S. Nasredinov, N. P. Seregin, and and RBa2Cu3O7 deviate from a linear relationship. P. P. Seregin, Yad. Fiz. 58 (9), 1554 (1995) [Phys. At. There are two reasons for this deviation, namely, the Nucl. 58, 1467 (1995)]. copper valence state being other than 2+ and different 3. F. S. Nasredinov, P. P. Seregin, V. F. Masterov, N. P. Sere- orientations of the total and valence EFG axes for the gin, O. A. Prikhodko, and M. A. Sagatov, J. Phys.: Con- 61Ni2+ and 63Cu2+ probes. The first reason is valid for the dens. Matter 7 (11), 2339 (1995). Cu(1) sites in RBa2Cu3O6 and for the copper sites in 4. T. Tsuda, T. Shimizu, H. Yasuoka, K. Kishio, and Cu2O [the copper in Cu2O and in the Cu(1) sites of the K. Kitazawa, J. Phys. Soc. Jpn. 57 (9), 2908 (1988).

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