Removal of Asphalt–Paraffin Deposits in Oil Pipelines by a Moving Source

Home , Bergman (crater)

Technical Physics, Vol. 46, No. 9, 2001, pp. 1069Ð1075. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 71, No. 9, 2001, pp. 1Ð8. Original Russian Text Copyright © 2001 by Balakirev, Sotnikov, Tkach, Yatsenko.

THEORETICAL AND MATHEMATICAL PHYSICS

Removal of AsphaltÐParaffin Deposits in Oil Pipelines by a Moving Source of High-Frequency Electromagnetic Radiation V. A. Balakirev, G. V. Sotnikov, Yu. V. Tkach, and T. Yu. Yatsenko Institute of Electromagnetic Investigations, Kharkov, 310022 Ukraine e-mail: [email protected] Received October 11, 2000; in ﬁnal form, March 6, 2001

Abstract—A new method for removing asphaltÐparaffin and gasohydrate plugs in oil pipelines with a movable source of electromagnetic radiation, electromagnetic pig, is suggested. The pig melts the plug when the latter absorbs intense electromagnetic radiation and heats up. Effective melting of the dielectric plug is achieved with the source moving along the pipeline as the solidÐliquid interface propagates. The time of paraffin plug removal and the dependence of this time on the radiation frequency are found with the model suggested. The efficiency of the method is estimated. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION the absorption of EM radiation (α is the absorption Today, the transportation of natural hydrocarbons coefficient, z is the distance to the radiation source). At through oil and gas pipelines is becoming more and high frequencies, the absorption coefficient varies as more popular. However, the conditions of transporta- the radiation frequency (if the dielectric loss tangent is tion often favor the formation of thick asphaltÐparaffin weakly dependent on frequency). Therefore, if the fre- or gas hydrate plugs. The reliable and cost-efficient quency and, accordingly, the attenuation of the HF exploitation of such pipelines under these conditions power in the plug are too low, the heat release is insig- requires advanced plug removal methods to be devised. nificant and the plug is not heated to the desired level. Techniques to prevent plug formation are of no less AsphaltÐparaffin plugs are transparent to low-fre- importance. quency radiation. If, however, the frequency is too high, the radiation is absorbed in the region nearest the The use of one or another currently available plug source and the oil severely overheats. Therefore, the removal method depends on the composition of the frequency is selected such that the absorption coeffi- plug, its structure, properties, etc. AsphaltÐparaffin cient is on the order of the reciprocal paraffin plug plugs in oil pipelines are usually destroyed by heating length. This condition is readily met upon heating oil (with steam or hot water), chemically, or mechanically. pools or upon removing paraffin plugs in a well, which All these approaches are expensive and technically dif- can be viewed as coaxial transmission lines in terms of ficult; moreover, they (e.g., chemical methods) some- electrodynamics. In an actual oil pipeline, however, this times entail adverse side effects. condition is impossible to satisfy. An oil pipeline is a Because of this, the use of high-power electromag- cylindrical waveguide that transmits only waves with netic radiation to thermally destroy asphaltÐparaffin higher-than-cutoff frequencies. For example, a pipeline deposits in oil wells, compressor and pumping plants, of radius 72 cm filled with paraffin with the permittivity and oil pipelines is of indubitable interest. reported in [7Ð9] has the cutoff frequency f = 1.048 × 8 Early (and basic) research on the use of electromag- 10 Hz for the E01 wave. At an operating frequency f = netic radiation in the oil industry has been concentrated 1.4 × 108 Hz, the power attenuation coefficient α = on increasing the oil production capacity by applying 0.08 mÐ1. If the paraffin length is 100 m, the EM power high-frequency electromagnetic (HF EM) radiation to is attenuated e8 times. Obviously, the paraffin plug will oil pools [1Ð5]. When absorbed, the electromagnetic never be melted in this case. field heats oil, making it less viscous. The idea of using HF EM radiation to remove paraffin plugs from oil In this work, we suggest removing paraffin plugs in wells has been put forward in [6, 7]. In [8], the numer- real oil pipelines using a movable EM radiation source. ical simulation of heating a paraffin plug that fills part Its velocity depends on the propagation rate of the liq- of an oil well was performed. In all these works, the fre- uidÐsolid interface during the melting of asphaltÐparaf- quency of EM radiation was relatively small (less than fin deposits by HF EM radiation. The device suggested 100 MHz). This is dictated by the value of the volume will be called an electromagnetic pig analogous to a heat release density, Q ~ αexp(Ðαz), in the plug due to mechanical pig (scraper) used in the oil and gas indus-

1070 BALAKIREV et al.

2 2ε 2 ω ω try for the mechanical cleaning of paraffin and gasohy- In (4), k⊥ = k 0 Ð kz , k = /c, is the circular fre- drate plugs from the pipelines. quency, c is the speed of light, J0 and J1 are the Bessel ε functions of the zeroth and the first order, 0 is the com- BASIC EQUATIONS plex permittivity of paraffin, R is the radius of the waveguide, and σ is the conductivity of the pipeline The heating and melting of a paraffin plug will be ε described in terms of the thermal conduction equation metal. Assuming that the imaginary part 0'' of the per- with a given external heat source. As was noted, a par- ε mittivity is much smaller than the real part 0' , we affin plug in the pipeline can be considered as a dielec- obtain, from (3), the approximate solution tric-filled cylindrical metallic waveguide. It is assumed that the problem is axisymmetric and the waveguide is 2 ε ωε ω 0'' 0' ω filled completely. The thermal conduction equation will α ==------, α ------, V 2 k' S cRk' 2πσ be solved numerically with the through-calculation c z0 z0 method [10]. Because of this, the equation for thermal (5) ω2 µ2 conduction will be written in the general form without ε n kz' = ------0' Ð,----- separating the phases in the explicit form: 0 c2 R2 ∂T 1 ∂ ∂T ∂ ∂T µ ρ λ λ (),, where n is the nth zero of the Bessel function J0. cT ------∂ = ---∂----- r------∂ ++∂----- ------∂ Qrzt, (1) t r r r z z For a stationary source, the volume heat release den- ρ sity can be determined from the HF losses when the EM where is the density of high-paraffin oil, cT is its specific heat, and λ is the thermal conductivity. wave passes through an absorbing medium: The density and the thermal conductivity are ω Q = ------E 2ε''. (6) assumed to be temperature-independent, while the spe- 0 8π 0 cific heat at the phase transition temperature TS has δ-like singularity: For the case of E0n, the other-than-zero components of the electric field are E and E , the expressions for δ() z r cT = c0 + L TTÐ S , (2) which are well known in the case of a dielectric-filled where L is the latent heat of phase transition and δ(T Ð cylindrical waveguide [11]. After necessary transformations, we obtain TS) is the delta function. The volume heat release density is expressed as ε'' µ2 Q = ------0 ------n QQ= Θ()αzzÐ ()t exp()Ð ()zzÐ ()t . (3) 0 π ()ε 4 0 0 0 Re kz 0 R (7) This formula implies that the EM radiation source 2 2 2r k R 2r moves following the law z = z0(t). The explicit form of × PJ µ --- +,------z J µ --- Q is given below [see formula (7)]. In (3), 0n R µ2 1n R 0 n ≥ 1, zz0 where P is the HF power radiated from the source. Θ()zzÐ =  0 < 0, zz0, Thermal conductivity equation (1) must be comple- mented by boundary conditions. At the near end face of and α is the coefficient of HF power attenuation. the plug, z = 0, we specify a boundary condition in the The source is situated in the plane z = 0 and starts form of convective heat exchange obeying Newton’s generating at the time instant t = 0. The EM wave law: decays because of volume losses in the dielectric plug ∂ λ T = κ []Tr(),,0 t Ð T , (8) and surface losses in the metallic walls of the cylindri------∂ 1 0 cal waveguide (the walls of the waveguide have a finite z z = 0 impedance). In view of the axial symmetry of the prob- where T0 is the ambient temperature and the initial tem- lem, it is easy to derive an equation from which the κ perature of the paraffin plug and 1 is the heat exchange α ≡ α α α α attenuation coefficient V + S = 2kz'' ( V and S are coefficient. the attenuation coefficients due to volume and surface At the far end of the plug, z = H, heat exchange is absent: losses, and kz'' is the imaginary part of the longitudinal ∂T wave number kz = kz' + ikz'' ) λ = 0. (9) ------∂ z zH= for the wave E0n can be determined: At the side surface of the cylinder, r = R, we also J ()k⊥ R ω 0 1 + i ε specify convective heat exchange but with a different κ. k⊥------()- = ------k 0 ------πσ . (4) J1 k⊥ R 2 2 In addition, we take into account heat release due to the

TECHNICAL PHYSICS Vol. 46 No. 9 2001 REMOVAL OF ASPHALTÐPARAFFIN DEPOSITS 1071

–3 Imkz Q(r, z = 0)/P, m 1.00 100

0.75 80 4

0.50 60 3

40 0.25 2 20 1 0 1 2 3 4 5 f, GHz 00.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 r, m Fig. 1. Frequency dependence of the imaginary part of the longitudinal wave number for a paraffin-filled metallic Fig. 2. Transverse distribution of the heat loss power density cylindrical waveguide. The relative permittivity ε = 2.3 + normalized to the source power in the paraffin-filled cylin- i0.0276 [12, 13]. drical waveguide. f = (1) 1.4, (2) 2, (3) 3, and (4) 4 × 109 Hz. absorption of the EM wave at the surface of the metallic solving Eq. (1) with boundary conditions (8)Ð(10) and waveguide: a complex spatial dependence of Q(r, z), we will ∂ assume that, during motion, the spacing between the Ðλ T = κ[]TRzt(),, Ð T Ð qzt(), , (10) ------∂ 0 source and the interface remains constant. r rR= where κ = Nuλ/R is the coefficient of heat exchange with the environment and Nu is the Nusselt number. NUMERICAL RESULTS The power absorbed in the metallic walls of the For the numerical analysis of asphaltÐparaffin plug pipeline can be found by solving the rigorously posed removal, we took advantage of the physical parameters electromagnetic problem in the metal and in the dielec- ρ 3 of high-paraffin oil [8]: = 950 kg/m , c0 = 3 kJ/(kg K), tric. Having determined the electric field components ° λ and applying a formula like (6) to the metallic area with TS = 50 C, L = 300 kJ/kg, and = 0.125 W/(m K). The ° a appropriate permittivity, we find the power absorbed ambient temperature was set equal to T0 = 20 C. The ε by the metal volume. Actually, the skin depth is much real and the imaginary parts of the dielectric constant 0 less than the pipeline wall thickness; therefore, we can were taken for dehydrated oil, since they are weakly assume that the HF power is absorbed at the inner sur- dependent on frequency over a wide range [12, 13]. In face of the pipeline. With the skin depth tending to zero, ε' ≈ δ ε'' ε' ≈ the expression for the surface heat release density is the calculations, we used 0 2.3 and tan = 0 /0 × Ð2 κ given by 1.2 10 . The heat exchange coefficients were 1 = 2 κ 2 κ 2 0.2 W/(m K) and = 1.613 W/(m K). The value of ωε ω P q = ------0 ------exp()Ðα()zzÐ ()t corresponds to the Nusselt number Nu = 1 (the pipeline ()ε πσ 2 0 cRe kz 0 2 2πR (11) in dry soil). To numerically solve Eq. (1) with boundary ×Θ()() conditions (8)Ð(10), we applied an explicit difference zzÐ 0 t . scheme on a uniform rectangular mesh. The singularity Here, as in (3), we take into account the fact that the at the point r = 0 in the Laplacian was bypassed in the source radiates only in a forward direction and moves standard way [14]: the δ function in Eq. (2) for the spe- in the pipeline longitudinally, following the law z = cific heat was approximated by a step with a half-width ° z0(t). of 0.4 C. To close the set of equations that characterizes the To check the accuracy with which the difference removal of a paraffin plug by a moving EM radiation scheme approximates the set of equations, we used the source, it is necessary to specify the law of its motion equation for energy balance that is directly derived z0(t). We noted in the introduction that z0(t) is defined from Eq. (1) and relationships (7)Ð(10): by the motion of the solidÐliquid interface. However, the motion of the interface is governed also by differen- T tial equation (1). In the explicit form, an equation for ∂ ρ----- dVcT()' dT' = P[]1 Ð exp()Ðα()HzÐ ()t interface velocity can be written only for the one- ∂t∫ ∫ 0 dimensional case [10]. Therefore, when numerically T0

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1072 BALAKIREV et al.

Q, kW/m3 R πκ [](), 250 Ð2 1∫drr T r z = 0 Ð T 0 (12) 0

200 H πκ [](), 150 Ð2 RzTrR∫d = z Ð T 0 . 0 100 Energy balance (12) coincides with that in [15] if the EM radiation source is stationary [z (t) = 0]. 50 0 The results given below were obtained for a model 0 pipeline of radius R = 0.0775 m. It was assumed to be 0 made of steel with a conductivity σ = 0.37 × 1017 sÐ1. The 1 0.02 2 0.04 parafﬁn plug length H was set equal to 5 m. r, m z, m 3 0.06 4 Figure 1 plots the imaginary part kz'' of the longitu- 5 dinal wave number against frequency for the above parameters of the plugged cylindrical pipeline. Hereaf- Fig. 3. Distribution of the heat loss power density over the ter, we will consider only the wave E01. The cutoff fre- Ӎ × 9 volume of the parafﬁn plug. quency for E01 for this pipeline is f0 0.97 10 Hz.

(a) (b) T, °C T, °C 200 200

160 160

120 120

80 80 40 40 0 0 0.02 1 0.02 1 2 2 0.04 3 0.04 3 0.06 z, m 4 r, m 4 0.06 r, m z, m 5 5 (c) (d) T, °C T, °C 200 200

160 160

120 120

80 80

40 40 0 0 1 1 2 0.02 2 0.02 3 0.04 3 0.04 z, m 4 z, m 4 0.06 r, m 0.06 r, m 5 5

Fig. 4. Temperature proﬁles along the pipeline length H = 5 m at different time instants during the removal of the asphaltÐparafﬁn plug by the electromagnetic pig. R = 0.0775 m, f = 1.4 × 109 Hz. t = (a) 15, (b) 75, (c) 135, and (d) 180 min.

TECHNICAL PHYSICS Vol. 46 No. 9 2001 REMOVAL OF ASPHALTÐPARAFFIN DEPOSITS 1073

(a) (b) T, °C T, °C 200 200

160 160

120 120

80 80

40 40

0 0.02 0 1 1 0.02 2 0.04 2 0.04 3 0.06 3 0.06 z, m 4 r, m z, m 4 r, m 5 5 (c) (d) T, °C T, °C 200 200

160 160

120 120

80 80

40 40 0 0 1 1 2 0.02 2 0.02 3 0.04 3 0.04 z, m z, m 4 0.06 r, m 4 0.06 r, m 5 5

Fig. 5. The same as in Fig. 4 for f = 2 × 109 Hz (in panel “d”, t = 150 min).

≈ Ð1 Ӎ × The value of kz'' 0.37 m has a minimum at f 1.38 verse component of the electric field E01, whose maxi- 109 Hz and then grows with frequency. Although the mum is closer to the waveguide walls (Fig. 2; curves 3, depth of HF power penetration into the plug is of minor 4). The wave with a frequency f = 2 × 109 Hz provides importance in our method of plug removal, it should be the most uniform distribution of the heat density over taken as long as possible. A penetration depth that is too the cross section. In this case, the thermal energy den- short would make the on-line control of source move- sity in the peak Q(r, z = 0) is smaller than in the other ment difficult. cases in Fig. 2; however, it is the E01 wave with a frequency of 2 × 109 Hz that is the most preferable for The phase transition substantially depends on the removing a paraffin plug with radius R = 7.75 cm. In distribution of the volume heat release density Q(r, z). this case, heating is uniform; hence, the paraffin uni- In Fig. 2, the distributions Q(r, z = 0) in the waveguide formly melts over the cross section of the plug. The cross section that are normalized to the power of the overheating of separate oil layers, which reduces the stationary source are shown for various frequencies. At efficiency of the method and causes some engineering the minimum of the attenuation coefficient (and smaller troubles, is prevented. In the longitudinal direction, the frequencies), the longitudinal component of the electric heat release density drops exponentially. As the fre- field E01 with a maximum at the axis of the cylindrical quency rises, the density Q(r, z) in the longitudinal waveguide (Fig. 2, curve 1) plays a dominant part in the direction drops faster, as demonstrated in Fig. 1. Figure 3 distribution of the heat release density Q. As the fre- exemplifies Q(r, z) for a stationary source z0(t) = 0 and quency increases, so does the transverse component of P = 5 kW. the electric field. At f = 2 × 109 Hz, the transverse and the longitudinal components are comparable in magni- The results of the numerical simulation of paraffin tude (Fig. 2, curve 2). At higher frequencies, the heat plug removal with the electromagnetic pig are shown in release density distribution is governed by the trans- Figs. 4Ð6 for f = 1.4 × 109 Hz, 2 × 109 Hz, and 3 × 109,

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1074 BALAKIREV et al.

(a) (b) T, °C T, °C 200 200

160 160

120 120

80 80

40 40 0 0 1 0.02 1 0.02 2 0.04 2 0.04 3 r, m 3 z, m 4 0.06 z, m 4 0.06 r, m 5 5 (c) (d) T, °C T, °C 200 200

160 160 120 120 80 40 80 0 1 0 0.02 1 2 40 2 0.04 3 3 r, m z, m z, m 4 0.06 4 0.02 5 5 0.04 0.06 r, m

Fig. 6. The same as in Fig. 4 for f = 3 × 109 Hz (in panel “d”, t = 195 min). respectively, at different time instants. In all cases, the teau near the source. This plateau is the widest at f = source power is P = 5 kW. As was expected, the melting 2 × 109 Hz, because the distribution of the thermal process strongly depends on the Q(r, z) distribution energy density over the plug cross section is the most (Figs. 2, 3). Initially, the temperature surface has a pla- uniform in this case. The region where melting starts strictly correlates with the peak of the Q(r, z) curve. At z, m a low frequency f = 1.4 × 109 Hz, melting begins at the 5 center of the dielectric plug. At high frequencies, f ≥ 3 × 109 Hz, the process starts closer to the periphery of the 4 cylinder. At f = 2 × 109 Hz, the process starts propagat- 3 ing from the middle of the cylinder radius. In the last case, the distribution of the oil and parafﬁn tempera- 2 tures over the cross section turns out to be the most uniform. For this frequency, the maximal temperature of ° 1 the melted parafﬁn and oil does not exceed 75 C, while at f = 1.4 × 109 Hz, it approaches 175°C and at f = 3 × 9 0 10 Hz, the maximal temperature is ≈120°C. The time of plug removal for H = 5 m strongly depends on the 00.5 1.0 1.5 2.0 2.5 t, h frequency in the range 1Ð5 GHz under study. At f = 1.4 × 109 Hz, this time is 3 h; at f = 2 × 109 Hz, it is the Fig. 7. Position of the electromagnetic pig relative to its initial location in the pipeline at different time instants. least, 2.5 h; and at f = 3 × 109 Hz, it equals 3.25 h.

TECHNICAL PHYSICS Vol. 46 No. 9 2001 REMOVAL OF ASPHALTÐPARAFFIN DEPOSITS 1075

With further increase in the radiation frequency, the 2. F. L. Sayakhov, G. A. Babalyan, and A. N. Al’mesh’ev, time of plug removal grows, because the penetration Neft. Khoz., No. 12, 33 (1975). depth of the HF power in the direction of pig motion 3. F. L. Sayakhov, M. Kh. Fatykhov, and O. L. Kuznetsov, decreases and the radial distribution of the heat release Izv. Vyssh. Uchebn. Zaved., Neft’ Gaz, No. 3, 36 (1981). density becomes less uniform. A progressively larger 4. Ngok Haœ Zyong, A. G. Katushev, and R. N. Nigmatulin, part of the useful power is spent on local overheating Prikl. Mat. Mekh. 51, 29 (1987). rather than on melting the whole plug, and is dissipated through the side surface. 5. Ngok Haœ Zyong, N. D. Musaev, and R. I. Nigmatulin, Prikl. Mat. Mekh. 51, 973 (1987). Figure 7 depicts the source position along the pipeline relative to its initial location. The source power is 6. F. L. Sayakhov, M. A. Fatykhov, and N. Sh. Imashev, × USSR Inventor’s Certificate No. 1314756, Byull. Izo- P = 5 kW, and the EM radiation frequency is f = 1.4 bret., No. 20 (1986). 109 Hz. Within t < 0.5 h, the electromagnetic pig remains stationary until the early layer is melted in its 7. A. T. Akhmetov, A. I. D’yachuk, A. A. Kislitsyn, et al., cross section. Then, the pig velocity increases and RF Inventor’s Certificate No. 1707190, Byull. Izobret., No. 3 (1992). becomes constant in 0.5 h. 8. A. A. Kislitsyn, Prikl. Mekh. Tekh. Fiz. 37 (3), 75 (1996). CONCLUSION 9. A. A. Kislitsyn, Prikl. Mekh. Tekh. Fiz. 34 (3), 97 Let us estimate the efficiency of our method. We (1993). define the efficiency of the pig as the useful-to-total 10. A. A. Samarskiœ and B. D. Moiseenko, Zh. Vychisl. Mat. work ratio. By useful work, we mean the work spent on Mat. Fiz. 5, 816 (1965). the complete melting of the paraffin. The total work is 11. L. A. Vaœshteœn, Electromagnetic Waves (Radio i Svyaz’, the energy emitted by the source into the oil pipeline. Moscow, 1988). For f = 2 × 109 Hz and the plug parameters listed above, the efficiency is ≈70%. 12. S. I. Chistyakov, F. L. Sayakhov, and L. N. Bondarenko, Neft. Khoz., No. 11, 51 (1969). Thus, our investigations have shown that the removal of extended plugs in pipelines is possible if the 13. S. I. Chistyakov, N. F. Denisova, and F. L. Sayakhov, Izv. velocity of the EM source is self-consistently derived Vyssh. Uchebn. Zaved., Neft’ Gaz, No. 5, 53 (1972). from the law of motion of the liquidÐsolid interface. 14. A. A. Samarskiœ and A. V. Gulin, Numerical Methods (Nauka, Moscow, 1989). 15. V. A. Balakirev, G. V. Sotnikov, Yu. V. Tkach, and REFERENCES T. Yu. Yatsenko, Elektromagn. Yavleniya 1, 552 (1998). 1. F. L. Sayakhov, S. I. Chistyakov, G. A. Babalyan, and B. I. Fedorov, Izv. Vyssh. Uchebn. Zaved., Neft’ Gaz, No. 2, 47 (1972). Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 46 No. 9 2001

Technical Physics, Vol. 46, No. 9, 2001, pp. 1076Ð1081. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 71, No. 9, 2001, pp. 9Ð14. Original Russian Text Copyright © 2001 by Bulyarskiœ, L’vov, Svetukhin. THEORETICAL AND MATHEMATICAL PHYSICS

The Effect of Clustering on the Melting of IIIÐV Semiconductors S. V. Bulyarskiœ, P. E. L’vov, and V. V. Svetukhin Ul’yanovsk State University, Ul’yanovsk, 432700 Russia e-mail: [email protected] Received December 14, 2000

Abstract—A thermodynamic model of cluster formation in the melts of binary semiconductor compounds is proposed. Cluster formation in the gallium arsenide melt is studied, and expressions for equilibrium concentrations of clusters of different size, as well as a liquidus equation, are obtained. In the limiting case of small cluster concentration, the latter equation is consistent with the results of the theory of quasi-chemical interaction. The system of equations derived enables the satisfactory description of experimental data on aftermelting and liquidus curves in gallium arsenide. The calculated values of the melting enthalpy and entropy agree well with data for the energy of dissociation. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION aggregated into clusters: Melting and crystallization of solids remain some of GG= L + GS. (1) the least-studied problems in solid-state physics. As the temperature increases, the atomic arrangement in a (1) The free energy of melt GL can be written in the solid becomes disordered and the atoms pass into liquid form [3] (melt). The long-range order breaks down in a certain L temperature range and some intermediate state exists L L i L i N ! between the solid and the melt. This state is character- G = N A∑gA pi + N B∑gB pi Ð kTln ------, (2) N L!N L! ized by the presence of small-sized ordered regions, or i i A B atomic clusters. The existence of the clusters at temper- L L atures above melting points (the aftermelt phenome- where N A and N B are the numbers of particles A and non) has been proved experimentally [1, 2]. In many i i materials, the percentage of particles forming clusters B, respectively, in the melt; gA and gB are the energies at melting points may reach 10%. of particles A and B that have i bonds with particles of L L L A liquidus curve, or the dependence of the phase the same sort; and N = N A + N B is the total number of transition temperature on the melt composition, is of sites in the melt. great importance for semiconductor technology. Liqui- Formula (2) involves the probabilities pi, which are dus curves are most often derived in terms of the theory given by of quasi-chemical interaction. This theory relies on the equality of the chemical potentials upon the formation L ziÐ L i z! ()N ()N of interatomic bonds in solution components and p = ------A B -, (3) i i!()ziÐ ! ()L L z ignores clustering, thus deeling with the total concen- N A + N B trations of atoms. However, such an approach is valid only at temperatures far exceeding the melting point. where z is the number of nearest neighbors. L L In this paper, a model of interparticle interaction in Let gAA and gBB be the energies of pair interaction the liquid phase is proposed. It includes cluster forma- L tion and enables the description of both liquidus and between like particles and gAB be the energy of pair aftermelt curves. interaction between particles A and B. In this notation, i i the energies gA and gB are expressed in the form MELT FREE ENERGY WITH RESPECT i 1 L 1 L 1 L L z L TO CLUSTERING g ==---g i + ---g ()ziÐ ---()g Ð g i + ---g , A 2 AA 2 AB 2 AA AB 2 AB Let us consider clustering in the melt of a IIIÐV (4) i 1 L 1 L ()1()L L z L semiconductor. Results obtained here can readily be gB ==---gBBi + ---gAB ziÐ --- gBB Ð gAB i + ---gAB. generalized to any other binary compound. 2 2 2 2 We represent the Gibbs energy of melt as the sum of The substitution of (3) and (4) into (2) makes it pos- the energy of free particles and the energy of particles sible to sum over i and considerably simplify further

THE EFFECT OF CLUSTERING ON THE MELTING 1077 calculations: and necessarily depends on the energy of clustered particles. L L L z L z L L N G = N ---g + ---()g Ð g ------B (2) The free energy of clusters in a system is given A 2 AB 2 AA AB L L by [4] N A + N B S L G = ∑gm Nm (10) L z L z L L N + N ---g + ---()g Ð g ------A m B 2 AB 2 BB AB L L N A + N B

N! Nm ()L L () N A + N B ! z L L L Ð kTln ------∏ NPm . Ð kTln ------= ---g ()N + N (5) L A B B A Nm L L 2 A A B N !∏ Nm!()mA!mB!mA!mB! m N A!N B! m L L z N N Here, gm is the energy of a cluster of m particles, Pm is + ---()g + g Ð 2g ------A B 2 AA BB AB L L the formation probability of a cluster of m particles, the N A + N B factor N takes into account that a cluster may be arbi- β ()L L trarily located within the system, and mα is the number N A + N B ! Ð kTln ------L L - . of particles of sort α = {A, B} at the site β = {A, B} in N A!N B! a cluster of size m [5]. Now the melt energy per particle of either sort can Suppose that the clusters are stoichiometric and do be obtained [3]: not contain antisite defects. Then, for each cluster of m particles, the following relationships are true: L L ∂ z L L 2 L G Ω() [] A B gA ==------gAB ++xB kTln xA , ∂ L 2 mA ==mA m/2, mB ==mB m/2, N A (11) (6) A B ∂ L mB ==mA 0. L G z L Ω()L 2 []L gB ==------gAB ++xA kTln xB , ∂N L 2 This approximation can be used, because the homo- B geneity region for IIIÐV semiconductors is very nar- where xL and xL are the relative concentrations of the row; i.e., the variation of the component ratio in the A B melt practically does not change the composition of the particles in the melt and solid phase. z L L L Ω = ---()g + g Ð 2g The formation probability of a cluster having m par- 2 AA BB AB ticles of one sort is given by is the parameter of quasi-chemical interaction. 1 m1 N' Ð m L L L Pm = ---- 1 Ð ---- . (12) The energies gAA , gBB , and gAB are expressed in N N terms of the entropy and enthalpy of pair interaction; ӷ ≈ Ω Since N' m and N N' (this follows from the therefore, the interaction parameter should linearly experimentally observed incompressibility of liquids), depend on temperature the second factor in (12) can be omitted as having the Ω = abTÐ , (7) order of unity. where a and b are constants. In the case of clusters consisting of two unlike particles occupying sites of a certain type (in IIIÐV semi- Equation (6) is independent of the absolute number conductors, these are sites in the two sublattices), the of particles in the melt; thus, Eq. (5) can be written in probability will have the form the form m L L L L L 1 G = g N + g N . (8) P = ------. (13) A A B B m 2N L L It is important to note that gA and gB are not the Using (11) and (13), we obtain the thermodynamic chemical potentials of particles A and B, because the probability W appearing in (10): chemical potential is deﬁned as the derivative of total energy (1) with respect to the total number of particles Nm N! 1 of a given sort, W = ------∏N------m . (14) ()()2Nm ()2N ∏ Nm! m/2 ! m ∂G ∂GL µ = ------≠ ------, (9) m A ∂N ∂ L A N A With the Stirling formula, formula (14) can be rear-

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1078 BULYARSKIŒ et al. ranged to the form To calculate the sum in (21), several assumptions on the free energy of m particle clusters should be made. Nm N! 1 We will assume that the clusters are strictly stoichio- W = ------∏N------m- . (15) Nm metric and their energy is proportional to the number of N !()m! ()N ∏ m m particles m z S The conservation equations for the number of parti- g = ---g m. (22) cles of both sorts are m 2 AB

L 1 Also, the clusters are assumed to be large enough ϕ ==N Ð N Ð0,---∑mN A A A 2 m that the surface energy in Eq. (22) is much smaller than m (16) the binding energy inside the clusters and is neglected. Substituting (20) and (22) into (21) yields ϕ L 1 B ==N B Ð N B Ð0.---∑mNm 2 z S m λ λ ---gAB ++A/2 B/2 S 2 Using the Lagrange method of multipliers, we con- N = expÐ------struct a functional that will take into account Eqs. (16): kT 

N (23) N! 1 m z S Φ ---g ++λ /2 λ /2 = ∑gm Nm Ð kTln ------∏------m Ð -1 AB A B ()Nm N 2 m ∏ Nm! m! m expÐ------expkT Ð 1 m  × .  exp ------Ð 1 L 1 N + λ N Ð N Ð ---∑mN (17)  AA A 2 m m To construct the liquidus curve, it is necessary to specify the melting point, since the meltÐcrystal phase L 1 + λ N Ð N Ð ---∑mN . transition actually occurs in a certain temperature range BB B 2 m that may be as wide as several tens of degrees. m Minimizing functional (17) yields the following We will assume that an [AB] crystal melts if the expressions for the number of clusters and free particles energy of interparticle interaction upon clustering of both sorts: changes by a certain value. It follows from (23) that the melt crystallization starts at a certain point where the z L Ω()L 2 λ S λ λ ---gAB ++xB A expression gAB + A/2 + B/2 Ð kTlnN takes a particular L 2 N = exp Ð,------(18) value. Thus, the equation A kT z S ---g +λ /2 +λ /2 Ð kTln N = const (24) z L Ω()L 2 λ 2 AB A B ---gAB ++xA B L 2 N = exp Ð,------(19) B kT is the liquidus equation for IIIÐV compound semiconductors. This equation can be rewritten in a more famil- λ iar form. To do this, we express the parameters A and m m λ g ++----λ ----λ B from (18) and (19) as 1 m 2 A 2 B Nm = ------expÐ------, (20) m Ð 1 z L L 2 L m!N kT λ = Ð---g Ð Ω()x + kTln N ,  A 2 AB B A (25) where parameters λ and λ are determined from the z L L 2 L A B λ = Ð---g Ð Ω()x + kTln N . solution of Eqs. (16). B 2 AB A B The substitution of expressions (25) into (24) results LIQUIDUS CURVES in the most general liquidus equation It is evident from Eqs. (16) that the total number of z S L 1 L 2 L 2 particles in the clusters NS is ---()g Ð g Ð ---Ω()()x + ()x 2 AB AB 2 A B ∞ (26) S N = ∑ mNm. (21) 1 []L L Ð---kTln XA XB = const, m = 2 2

TECHNICAL PHYSICS Vol. 46 No. 9 2001 THE EFFECT OF CLUSTERING ON THE MELTING 1079

L L L L L L where X A = N A /N and XB = N B /N are the relative con- composition of IIIÐV semiconductors is xA = xB = 0.5. centrations of free particles in the melts. The substitution of the melting temperature T = TF and Let us demonstrate that, from (26), the liquidus L the corresponding composition xA = 0.5 into Eq. (27) equation, well known from the theory of quasi-chemi- makes it possible to find the constant on the right-hand cal interaction [6, 7], can be derived. In the general side of (26). L L L case, due to the presence of the clusters, xA = X A /( X A Now initial liquidus equation (24) can be rewritten L ≠ L L L L L ≠ L L in the form + XB ) X A , xB = XB /(X A + XB ) XB , and X A + L []L ()L ()∆ F ()F X ≠ 1. The first approximation stems from the fact kTln 4xA 1 Ð xA + S + k ln4 T Ð T B (28) that the fraction of clusters in the melt is small, which Ω[]Ω()L 2 L ()F is true for many semiconductor melts [2]. If the fraction + 2 xA Ð12xA + Ð0,T /2 = L ≈ L L ≈ L which exactly coincides with the well-known liquidus of clusters is small, xA X A and xB XB ; therefore, the liquidus equation takes the form equation for IIIÐV semiconductors [7]. Thus, a family (27) of liquidus curves transforms into the equation (27) F 1 L 2 L 2 1 L L emerging from the theory of quasi-chemical interaction ∆g +---Ω()()x + ()x +---kTln[] x x = const, if the cluster concentration is small. AB 2 A B 2 A B ∆ F S L where gAB = Ðz(gAB Ð gAB )/2 is the change in the SIMULATION OF LIQUIDUS energy of interaction between unlike particles upon the AND AFTERMELTING CURVES IN GALLIUM solidÐliquid phase transition. ARSENIDE This change takes into account only the AÐB inter- Using the above-proposed model, we simulated action; however, in the melt, the AÐA and BÐB interac- cluster formation in the gallium arsenide melt. Both tions, which depend on the parameter Ω, also take experimental liquidus curves [8] and data on the after- ∆ F melting phenomenon [2] were taken into account. place. Hence, gAB is not equal to the melting energy It is evident from Eqs. (7), (23), and (27) that the ∆ F ≠ ∆ melt of the crystal [AB]: gAB GAB . specific position of the aftermelting and liquidus curves Equation (26) defines a family of curves according depends on the values of a, b, ∆HF, and ∆SF. By varying to the constant on its right. However, such an uncer- these parameters, both experimental dependencies can tainty can be avoided by using experimental data on be described satisfactorily. melting semiconductor crystals, i.e., on melting points Substituting Eqs. (25) into (23) yields the equation and stoichiometric compositions. The stoichiometric for the aftermelt curves

∆ F ∆ F abTÐ ()L 2 ()L 2 kT []L L S H Ð T S ++------()xA + xB ------ln X A XB S N 2 2 X ==------expÐ ------N kT  (29) F F abTÐ L 2 L 2 kT L L ∆H Ð T∆S ++------()()x + ()x ------ln[]X X 2 A B 2 A B × expexpÐ ------Ð 1 . kT 

Equation (29) should be solved jointly with The simulation of the aftermelting curves can be Eqs. (16), which are conveniently represented in terms simpliﬁed if it is remembered that gallium arsenide of the relative values melts congruently. This means that the relative concen- L L total L 1 S trations of the particles in the melt are equal, x = x = X = X + ---X , A B A A 2 (30) Experimental and theoretical values of the enthalpy and total L 1 S X = X + ---X , entropy of melting for gallium arsenide B B 2 F ∆ melt total total h, eV s, k T , K Tmelt, K [7] Edis, eV [9] HAB , eV [1] where X A and XB are the full particle concentra- total total 3.36 25.7 1504 1511 3.26 1.16 tions in the melt: X A + XB = 1.

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1080 BULYARSKIŒ et al.

XS, % F a F b 18 ∆H + --- Ð T ∆S + ---  --- 4 4 × 1 S expÐ------exp---1()Ð X --- kT 2 14  (31) F a F b  10 ∆H + --- Ð T ∆S + --- 4 4  × expÐ------ Ð.1 kT  6 

2 It follows from (31) that cluster formation during 0 10 20 30 40 50 60 70 melting of the stoichiometric crystal can be character- ∆ T, K ized by some values of enthalpy h and entropy s: S Fig. 1. Percentage of clusters in gallium arsenide X F F vs. ∆T = T Ð TF (continuous curve). Data points are taken h ==∆H + a/4, s ∆S + b/4. (32) from [2]. Thus, to describe the aftermelting curves, it is necessary to vary only two parameters: h and s. For h = 0.5, and the total numbers of the particles are also 3.36 eV and s = 25.7 k, experimental aftermelting total total equal, X A = XB = 0.5. Then, Eq. (29) can be trans- curves are adequately described by Eq. (31) (Fig. 1).1 formed to The simulation of the liquidus curves with regard S 1 S X = ---()1 Ð X for clustering was carried out using the expression 2 resulting from relationships (27) and (30):

()total TXA 2 2 2 a/2(() 1 Ð Xtotal Ð XS()T + ()Xtotal Ð XS()T )Ð () const Ð ∆HF ()1 Ð XS()T = ------A A ------, ∆ F []()total S total S() ()total S 2 ()total S 2 S Ð k ln 1 Ð X A Ð X ()T ()X A Ð X T + b/2() 1 Ð X A Ð X ()T + X A Ð X ()T

HEAT OF MELTING S const X ()T = expÐ------kT By definition, the heat of melting is the energy needed  for a material to pass from the solid phase to the melt. (33) Hence, the heat of melting per particle is written in the const form × expexpÐ------Ð 1 . kT melt 1 L L S ∆C = ---()g + g Ð 2g . (34) AB 2 A B AB In our case, by the solid phase, we mean a set of The constant in Eq. (33) was determined from for- clusters. Using (6) and (22), (34) can be recast as mula (27) and the stoichiometric crystal melting condi- melt F F L L ∆C = ()∆H Ð T∆S tions x = x = 0.5 and T = TF with XS obtained from AB A B (35) abTÐ L 2 L 2 kT L L Eq. (31). + ------()()x + ()x + ------ln[]X X . 2 A B 2 A B The eventual simulation of the liquidus curve was carried out by fitting parameters a and b. In (32), the From this expression, it follows that the heat of melting of a compound semiconductor depends not ∆ F ∆ F values of H and S were varied in such a way as to only on the binding energy between its particles but on satisfy the values of h and s obtained. The parameters a, the temperature and composition of the melt as well. ∆ F ∆ F b, H , and S for gallium arsenide are a = Ð13.76 eV, Since gallium arsenide melts congruently, expres- b = Ð8.32 × 10Ð3 eV/K, ∆HF = 6.8 eV, and ∆SF = sion (35) can be transformed into the form 99.4271 k. With such fitting, the aftermelt curve is ∆ melt []S described automatically at any values of the parameters CAB = hTsÐ + kTln 1 Ð X , (36) a and b. The final liquidus curve for gallium arsenide is shown in Fig. 2. 1 Both parameters are given per particle.

TECHNICAL PHYSICS Vol. 46 No. 9 2001 THE EFFECT OF CLUSTERING ON THE MELTING 1081

T, K ing point TF calculated from the relationship between 1600 the enthalpy and entropy h = TFs, are also presented. The enthalpy h calculated is almost three times higher than the experimental enthalpy value and approaches the energy of dissociation. The melting point obtained agrees well with the experimental data. 1200 ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research (grants nos. 00-01-00209 and 00- 800 01-00283).

REFERENCES 1. A. A. Aœvazov, V. M. Glazov, and A. R. Regel’, Entropy 400 of Melting of Metals and Semiconductors (TsNII Élek- 0 0.2 0.4 0.6 0.8 tronika, Moscow, 1978). XGa 2. A. R. Regel’ and V. M. Glazov, Élektron. Tekh., No. 9 (194), 7 (1984). Fig. 2. Simulated liquidus curve for gallium arsenide (con- 3. S. V. Bulyarskiœ and V. V. Prikhod’ko, Pis’ma Zh. Tekh. tinuous curve). Data points are taken from [8]. Fiz. 25 (7), 33 (1999) [Tech. Phys. Lett. 25, 263 (1999)]. 4. P. E. L’vov and V. V. Svetukhin, Khim. Fiz. 18 (2), 93 (1999). where the values of h and s are determined by relation- 5. S. V. Bulyarskiœ, V. V. Svetukhin, and P. E. L’vov, Fiz. ships (32). Tekh. Poluprovodn. (St. Petersburg) 34, 385 (2000) Experiments on aftermelting in gallium arsenide [2] [Semiconductors 34, 371 (2000)]. show that the relative concentration of the particles 6. I. Prigogine, The Molecular Theory of Solutions (North- forming clusters at temperatures exceeding the melting Holland, Amsterdam, 1957; Metallurgiya, Moscow, point is less than 16%; therefore, when evaluating the 1990). entropy and the enthalpy of melting, one can neglect the 7. N. C. Casey, Jr. and M. B. Panish, Heterostructure logarithmic term. Thus, the parameters h and s obtained Lasers (Academic, New York, 1978; Mir, Moscow, from the simulation of the aftermelting curves are noth- 1981), Vol. 2. ing else than the enthalpy and the entropy of melting, 8. C. Hilsum and A. C. Rose-Innes, Semiconducting IIIÐV respectively, of gallium arsenide. Compounds (Pergamon, Oxford, 1961; Inostrannaya Literatura, Moscow, 1963). The enthalpy and the entropy of melting of gallium 9. W. A. Harrison, Electronic Structure and the Properties arsenide obtained from simulating the liquidus and of Solids: The Physics of the Chemical Bond (Freeman, aftermelting curves are presented in the table. For com- San Francisco, 1980; Mir, Moscow, 1983), Vol. 1. parison, the energy of dissociation Edis and the heat of ∆ melt melting H AB of gallium arsenide, as well as the melt- Translated by M. Lebedev

TECHNICAL PHYSICS Vol. 46 No. 9 2001

GASES AND LIQUIDS

An Interferometric Technique for Measuring Gas Flow Velocity at Large Peclet Numbers A. A. Pikulev All-Russia Research Institute of Experimental Physics (VNIIEF), Russian Federal Nuclear Center, ul. Mira 37, Sarov, Nizhni Novgorod oblast, 607190 Russia Received November 29, 2000

Abstract—A technique for determining the gas flow velocity at large Peclet numbers using an interferometric approach has been developed. It is shown that if a heat source is introduced in a uniform gas flow, areas under curves describing shifts of interference maxima are approximately equal to each other and tend to a certain limiting value. The limiting area is inversely proportional to the gas flow velocity and does not depend on the heat conduction and the type of flow. A theoretical expression has been derived that gives the relative error of velocity determinations by the method developed. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION In this work, it has been shown that at large Peclet numbers, the gas velocity can be determined on the At present, interferometric methods are widely used basis of interferometric experiments with a heated wire for investigations of thermophysical and gasdynamic placed across the stream. The proposed method has parameters of gas flows. For example, in experiments several advantages, namely (1) the measurement results [1, 2], the authors investigated a laser model using do not depend on the heat conduction and the type of Michelson and MachÐZehnder interferometers [3]. The flow; (2) additional experiments are not necessary (the laser model was a channel consisting of two sections same interference pattern can be used that is taken in with heat exchangers in between. The gas flowing the study of the heat exchanger wake); and (3) calibra- through the channel was infused with uranium fission tion is unnecessary. fragments. The interferometers were employed to measure the gas temperature and to investigate the thermal boundary layer contiguous to the heated channel wall BASIC THERMODYNAMIC EQUATIONS and characteristics of the heat exchanger wake. In this Let us consider the thermodynamic aspects of the last experiment, a thin Nichrom wire heated by an elec- gas flow. In the region considered, the flow is assumed tric current was placed across the flow downstream of to be isobaric and the stream lines are assumed to be the heat exchanger. A schematic of the section with the parallel to the x axis. The equation of stationary transfer installed wire is shown in Fig. 1. Study of the thermal of the energy under constant pressure, ignoring the dis- wake of the heated wire makes it possible to determine sipative terms, has the form [8] the heat conduction in the gas as a function of the dis- ρ(), ∇ ()µ∇ tance from the heat exchanger and, consequently, to c p u T = div T + q, (1) identify the type of the flow and the size of convective where ρ, u, and T are the density, the velocity, and tem- cells forming in the heat exchanger wake [4, 5]. perature of the gas, respectively; µ is the thermal con- To process the results of such experiments, it is nec- z essary to know the average gas flow velocity. In exper- 2 y iments [1, 2], the velocity was determined from the pressure differential. Such an approach demands cali- U 1 bration of the measuring instrument, and the measure- x ment accuracy is not high. Therefore, it is desirable to d carry out independent measurements of the velocity by other methods. Measurements of the velocity with Ven- L turi and Pitot tubes [6] are difficult in gas-tight chan- 4 b nels. Use of a pulsed heat-loss anemometer [7] specially designed for this purpose has two disadvantages: 3 (1) it must be calibrated; and (2) the measurement Fig. 1. Schematic of the gas section (the second heat results depend on the heat conduction of the gas, that is, exchanger is not shown): (1) gas flow direction, (2) wire, on the type of flow. (3) wire holders, and (4) heat exchanger.

AN INTERFEROMETRIC TECHNIQUE 1083 ductivity; cp is the specific heat of the gas at constant in a uniform gas flow. The power of the heat source is pressure; and q is the power of the volume heat source. equal to W. The flow disturbance caused by the heat At the channel inlet, the gas has the velocity U , source is assumed to be negligible. Let us represent the 0 χ χ χ density ρ , and temperature T . Let U denote the only thermal diffusivity of the gas in the form = l + t, 0 0 χ χ nonzero component of the gas velocity u . From the where l and t are the laminar and turbulent thermal x χ χ ρ ρ diffusivities, respectively [9], and t = t(x). This makes continuity equation [8] it follows that U = 0U0. For convenience we will use dimensionless variables. To it possible to take into account the effect of small-scale convective mixing, for instance, of the heat exchanger this end, let us introduce a characteristic linear dimen- χ sion, the distance h along which the temperature distri- wake. In a laminar flow, t = 0. Taking into account the bution will be studied. As the characteristic temperature definition of the Peclet number (2), we obtain and velocity, let us use T0 and U0, respectively, and rep- 1 1 1 ∆ ------= ------+ ------, (6) resent the temperature in the form T = T0 + T. Then we () Pe Pel Pet x have x = hx , y = hy , z = hz , and ∆T = T ∆T . We 0 where Pe and Pe are the laminar and turbulent Peclet assume the temperature variation to be small, that is, l t |∆T | Ӷ T . The energy equation in dimensionless numbers. 0 ∆ form is Perturbation of the temperature field T satisfies two boundary conditions: (1) the condition before the ∆ ∪ ∂∆T ∇2∆T hq heat source and (2) the condition at infinity T(x < 0 ------==------+ q, q ------, y, z ±∞) = 0. Note that in addition to these condi- ∂x Pe c ρ U T p 0 0 0 (2) tions, ∆T also satisfies the integral condition (5). The ρ c p 0U0h solution of Eq. (3) has the form of the function of a Pe = ------µ -, point source [10]: 2 p W Ðp2()y2 + z2 where Pe is the Peclet number. ∆T = ------σ()x e , (7) Below, unless otherwise indicated, we shall use the π dimensionless variables and omit the upper bars. x 1 dx Let us assume that the convective heat transfer along -----= 4 ∫------, (8) the x axis far exceeds the diffusive heat transfer, that is, p2 Pe Pe ӷ 1. In this case, the energy equation can be simpli- 0 fied: where σ(x) is the Heaviside function (11). ∂∆T 1 ∂2 ∂2 Let us also give a solution to Eq. (3) for a linear heat ------= ------+ ------∆Tq+ . (3) source with a uniform energy release along its length, ∂ 2 2 x Pe ∂y ∂z when the source coincides with the y axis. In this case, Let a localized heat source of power W be located in the temperature does not depend on the y coordinate the vicinity of the origin of the coordinates. Let us inte- and the solution is ∈ ∈ 2 2 grate Eq. (3) over a volume of V = {x [Ða, a]; y, z pw Ðp z ∞ ∞ ∆T = ------σ()x e , (9) (Ð , + )} of the rectangular parallelepiped including π the source region. We obtain +∞ +∞ where w is the power of the linear heat source per unit length. ∫ dy ∫ ∆Tzd = W. (4) Ð∞ Ð∞ MEASUREMENT OF THE GAS Consequently, the flux of energy transferred by the FLOW VELOCITY gas through any plane x = const > 0 is equal to the power of the heat source. It is obvious that the energy Changes in the temperature of a medium, as a rule, flux through some surface S that deviates negligibly are accompanied by changes in its refractive index, from the plane x = const > 0 is also approximately equal which can be detected using the interferometric tech- to W, nique. For an ideal gas under constant pressure, the refractive index is related to the temperature by a |∆ | Ӷ ∫∫∆T()is, d ≅ W, (5) known formula [12] (at T 1) ∆ ()∆ S nn= Ð 0 Ð 1 T. (10) where i is the unit vector of the x axis and ds is the unit Phase incursion of a ray of light in the test channel is area of the surface S, whose orientation coincides with the positive direction of the normal. L L ϕ ∆ ()∆ Let us consider the following problem: a point heat ==k∫ nyd Ðkn0 Ð 1 ∫ Tyd , (11) source located at the origin of the coordinates is placed 0 0

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1084 PIKULEV ∆ ξ ∆ tions m; then, m(z) = xm(z) Ð m. An equation for the 2 shifts of the maxima of fringes of ﬁnite width has the 1 8 form 3 1 ϕ()(), π ξ () xm z z = Ð2 N m z . (14) The main quantity characterizing the shifts of the 4 interference fringes from their undisturbed positions is ξ the area under the curve m. This area is +∞ ξ 56 Sm = ∫ mdz. (15) Ð∞ Integrating expression (14) with respect to the z coordinate between inﬁnite limits and using formu- 7 las (11) and (15), we get ∞ ∞ + + π 3 1 ∆ ()(),, 2 NSm 1 ∫ dy ∫ Txm z yzdz = ------(). (16) kn0 Ð 1 2 Ð∞ Ð∞

Fig. 2. Diagram of the nonequal-branch MachÐZehnder The integration in (16) is performed over a surface interferometer: (1) focusing lenses, (2) semitransparent of cylindrical shape. Comparing with (15), we find mirrors, (3) mirrors-reflectors, (4) optical wedge, (5) test π ≅ π () channel, (6) reference channel, (7) probing laser, and 2 NSm 2 NS = kn0 Ð 1 W. (17) (8) recording camera. Now we have the following result: all areas Sm are approximately equal to each other and tend to S. The where k is the wave number and L is the length of the farther downstream of the heat source the interference test channel. maximum is located, the more precise expression (17) The scheme of the experiment with the use of a is. This is associated with an increase in the local Peclet MachÐZehnder interferometer is shown in Fig. 2. The number downstream and weakening of the role of heat plane-parallel light beam of the probing laser is split in conduction in heat transfer. Hence, the law of equality two; one beam passes through the test channel and the of the areas Sm is an interferometric expression for the other, through the reference channel. For convenience, law of conservation of the heat energy at large Peclet an optical wedge is inserted in the reference beam in the numbers. This expression does not depend on the shape direction of gas flow (x axis). It is desirable to equalize of the heat source and the type of flow. The gas velocity the intensities of the light beams to achieve maximum is determined by the formula (in dimensional variables) contrast of the interference pattern. Positions of the () ≅ kn0 Ð 1 W interference maxima are defined by the following equa- U0 ------π ρ . (18) tion [3]: 2 c p 0 NST0 ϕ()x ()z , z ++22πNx ()z ϕ = πm, (12) Let us determine the phase incursion caused by a m m 0 point heat source. We assume that the laser beam ϕ where N is the number of fringes per unit length, 0 is probes the entire region of the thermal wake of the heat the initial phase difference, and xm(z) is the position of source along the y and z axes. From formulas (8) and the mth maximum. (11), it follows that When the gas is not heated, the interference maxima ()n Ð 1 kW ϕ = Ð------0 px()exp()Ðp2z2 . (19) are evenly spaced straight lines parallel to the z axis. π Their positions are given by the formula 2πN∆ +2ϕ ==πm, ϕ 2πN∆ , (13) The shifts of interference maxima of finite-width m 0 0 0 fringes from their undisturbed positions are calculated ∆ where m is the undisturbed position of the mth maxi- using formula (14), from which, using (19), we get mum. px()S ξ = ------m exp()Ðpx()2z2 . (20) At N = 0, we have a so-called infinite-width fringe. m π m The problem associated with the use of an infinite- width fringe is the evaluation of the initial phase incur- Formulas (19) and (20) are valid not only for a point ϕ sion 0, because it is hard to suggest an effective source but for any linear source located along the y axis ξ approach for determining it. Let m(z) denote the shifts with an arbitrary distribution of the heat release along of the interference maxima from their undisturbed posi- the source length. In this case, as before, the entire

TECHNICAL PHYSICS Vol. 46 No. 9 2001 AN INTERFEROMETRIC TECHNIQUE 1085 wake of the source must be probed by the laser beam in In the case when the Peclet number is constant along the planes normal to the x axis, and W is the total power the gas stream, (24) takes the form of the heat source. α α2 Equation (20) is transcendental; that is, its solution ()2 m m Sm = S1 Ð ------+ ------, cannot be represented in the form of elementary func- 42 63 tions. For simplicity, let us assume that the relation (25) ξ Ӷ ∆ m m is satisfied and seek a solution to (20) by the S Pe α = ------Ӷ 1. successive approximations method. As an initial m 2∆ π∆ approximation, we use the following expression: m m Thus, the accuracy of the law of areas is governed by ()0 p()∆ S 2 2 the second terms in expansions (24) or (25). The accu- ξ = ------m exp()Ðp()∆ z . (21) m π m racy increases rapidly with distance downstream from the heat source; therefore, practically under any exper- Integrating (21) with respect to the z coordinate gives imental conditions it is possible to find fringes for which the law of conservation of areas is valid to a high +∞ degree of accuracy. ()0 ξ ()0 Sm ==∫ m dz S. (22) The results obtained above are illustrated in Figs. 3Ð6. Ð∞ The interferograms were obtained under the following conditions: the working gas was air at atmospheric con- ()0 Consequently, all areas S are equal to each other ditions, the flow velocity was U0 = 5 m/s, the initial m ϕ phase was 0 = 0, the source power was W = 6 W, and and do not depend on the thermal conductivity, which λ coincides with the above result expressed by (17). the laser wavelength was = 632 nm. For a characteristic length h = 1 cm under these conditions, the Peclet Analysis of Eq. 20 shows that the areas Sm satisfy the following conditions: number is Pe = 2704; that is, the restriction imposed on the magnitude of Peclet number is satisfied. As shown <<<… < <… < in [4, 5], the turbulent Peclet number in the heat 0 S1 S2 Sm S, lim Sm = S. (23) m → ∞ exchanger wake may be several tens of times less than the laminar Peclet number. This fact should be taken From inequality (23), it is seen that the use of for- into account when employing the above model of the mula (18) gives a somewhat overestimated gas velocity. thermal wake of a linear source. Figure 3 displays an To clarify the issue of the accuracy of formula (18), infinite-fringe interferogram of the flow with a point let us expand the right-hand side of expression (20) into (linear) heat source at χ = 0. The effect of the convec- ξ t a Taylor series with respect to m, retaining the terms up tive wake of the heat exchanger under the same tuning to the second order of smallness inclusively. After inte- of the interferometer is illustrated in Fig. 4. Intensifica- grating and collecting like terms, we obtain tion of the heat exchange in the wake was simulated by inserting immediately behind the heat exchanger a 2 region of length x , where χ > 0. At x > x the turbulent ()2 px ppxx Ð px 2 t t t Sm = S1 ++------S ------S , thermal conductivity was assumed to be zero. The 22π 3π 3 ∆ χ x = m (24) lengths xt and magnitudes of t in the wake of the plate p()∆ S heat exchanger under different conditions can be found ------m Ӷ 1. in [4, 5]. In this study, we accepted x = 1 cm and χ = 5χ . ∆ π t t l m Comparison between Figs. 3 and 4 shows that the pres-

0 , mm z

–3 0 1 2 3 4 5 6 x, cm

χ Fig. 3. An inﬁnite-fringe interferogram of the gas ﬂow with a linear heat source at t = 0.

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1086 PIKULEV

0 , mm z

–5 0 1 2 3 4 5 6 x, cm

χ χ Fig. 4. Same as in Fig. 3 at t = 5 l and xt = 1 cm.

0 , mm z 5.57 6.33 6.66 6.83 6.93 7.00 7.04 7.07 7.10 7.12 7.13 7.14 –3 0 1 2 3 4 5 6 x, cm

χ Fig. 5. A ﬁnite-fringe interferogram of the gas ﬂow with a linear heat source at t = 0.

0 , mm z 6.31 7.14 7.15 7.16 7.17 7.18 7.19 7.20 7.21 7.21 7.22 7.22 –5 0 1 2 3 4 5 6 x, cm

χ χ Fig. 6. Same as in Fig. 5 at t = 5 l and xt = 1 cm. ence of the heat exchanger causes faster expansion of shows that the presence of the zone of intensive mixing the thermal wake of the source in the zone of convective behind the heat exchanger (Fig. 6) makes the approach mixing. of the areas Sm to the limiting value S faster than in the Finite-fringe interferograms in Figs. 5 and 6 were case of its absence (Fig. 5). At m > 4 the relative error ε = (S Ð S )/S does not exceed 4% and is quite ade- obtained for N = 2 cmÐ1, other experimental conditions m m being the same as those in Figs. 3 and 4. Beside each quately described by the function interference maximum the areas Sm are given in units of ()1 α 2 <<ε ε()1 SSÐ m m mm . The areas were obtained by numerical integration 0 m m ==------. (26) of Eq. 20 using the successive approximation proce- S 42 dure. The limiting values of the areas in the both case 2 ε()1 are S = 7.3 mm . Comparison between Figs. 5 and 6 The magnitude of error m found under the

TECHNICAL PHYSICS Vol. 46 No. 9 2001 AN INTERFEROMETRIC TECHNIQUE 1087 assumption of laminarity of the flow at Pe = 0 turned REFERENCES out to be maximal; therefore, it can be used as an esti- mation of the accuracy of velocity determinations for 1. V. V. Borovkov, B. V. Lazhintsev, S. P. Mel’nikov, et al., any type of flow. Izv. Akad. Nauk SSSR, Ser. Fiz. 54, 2009 (1990). It was assumed above that nonuniformity of the 2. V. V. Borovkov, B. V. Lazhintsev, V. A. Nor-Arevyan, refractive index in the test channel only causes phase et al., Kvantovaya Élektron. (Moscow) 22, 1187 (1995). changes in the probing laser beam and does not influ- 3. A. Sommerfeld, Vorlesungen über theoretische Physik, ence its rectilinear propagation. However, if gas density Bd. 4: Optik (Dieterich, Wiesbaden, 1950; Inostrannaya variations are large enough, the light rays are deflected Literatura, Moscow, 1953). from their initial direction toward a denser medium. 4. F. S. Milos and A. Acrivos, Phys. Fluids 29, 1353 (1986). This leads to an increase of S areas for fringes located m 5. F. S. Milos, A. Acrivos, and J. Kim, Phys. Fluids 30, 7 immediately downstream of the wire and violation of (1987). the law expressed by (23). This fact should be taken into account when carrying out experiments. 6. G. K. Batchelor, An Introduction to Fluid Dynamics (Cambridge Univ. Press, Cambridge, 1967; Mir, Mos- To summarize, let us describe the procedure of cow, 1973). velocity measurement. (1) In the course of an interferometric experiment, 7. V. M. Kul’gavchuk and V. Yu. Mat’ev, in Proceedings of the 2nd International Conference on Physics of Nucle- the power of a heat source W and the gas temperature arly Excited Plasma and Problems of Nuclear-Pumped T0 in front of the wire are measured. Lasers, Arzamas-16, 1995, Vol. 2, p. 93. (2) In the interferogram, the quantity of undisturbed 8. L. D. Landau and E. M. Lifshitz, Course of Theoretical fringes per unit length N is determined and the areas Sm Physics, Vol. 6: Fluid Mechanics (Nauka, Moscow, under the curves describing shifts of the interference 1988; Pergamon, New York, 1987). maxima are calculated. 9. L. G. Loœtsyanskiœ, Fluid and Gas Mechanics (Nauka, (3) Values of the velocities Um corresponding to Moscow, 1973). maxima m are calculated using formula (18). 10. S. L. Sobolev, Equations of Mathematical Physics (4) From the set of Um, the magnitude of the velocity (Nauka, Moscow, 1992). is selected which corresponds to the minimal sum of the 11. A. N. Kolmogorov and S. V. Fomin, Elements of the The- ε()1 relative error of measurement of the areas Sm and m ory of Functions and Functional Analysis (Nauka, Mos- (see (26)). cow, 1989). 12. Tables of Physical Quantities: A Handbook, Ed. by I. K. Kikoin (Atomizdat, Moscow, 1976). ACKNOWLEDGMENTS The author expresses his gratitude to L.V. L’vov for helpful discussions. Translated by N. Mende

TECHNICAL PHYSICS Vol. 46 No. 9 2001

GAS DISCHARGES, PLASMA

A Spherical Probe with a Time-Varying Potential in a Stationary Collisional Plasma A. V. Kashevarov Zhukovsky Central Institute of Aerohydrodynamics, Zhukovskiœ, Moscow oblast, 140180 Russia e-mail: [email protected] Received March 31, 2000; in ﬁnal form, October 10, 2000

Abstract—The problem of a spherical probe in a stationary, weakly ionized, collisional plasma is studied analytically under the assumption of a slight harmonic temporal modulation of the constant probe potential, which is equal to the potential of the environment. The solution is obtained in two limiting cases of a thick and a thin space-charge sheath. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION measured impedance of a spherical probe in a station- The diagnostic data from probe measurements are ary plasma. In that paper, an analysis was carried out fairly difficult to interpret when the plasma is collision- for a thin collisionless space-charge sheath around the probe and the Debye screening radius was assumed to dominated, i.e., when the characteristic mean free path λ Ӷ λ Ӷ of the plasma particles is small compared to the charac- satisfy the condition D R. teristic probe size, λ Ӷ R, so that the plasma can be The objective of the present paper is to analytically regarded as a continuous medium. For a long time, investigate the impedance of a spherical probe in two of research effort has been focused on the interpretation of the other possible characteristic operating modes in a data from probes operating in steady modes in both sta- continuous medium, specifically, in the limiting cases λ Ӷ λ Ӷ λ Ӷ Ӷ λ tionary and flowing plasmas (see, e.g., [1Ð4]), whereas of a thin ( D R) and thick ( R D) colli- a more complicated theory of unsteady operating sional space-charge sheath. modes of the probes is still far from being completely elaborated. FORMULATION OF THE PROBLEM The probe operation in unsteady modes was investigated only in a few papers, which can be divided into We treat the problem with the following simple for- two groups. The first group involves papers aimed at mulation. We consider a spherical probe in a stationary, studying the behavior of the probe current during the weakly ionized, high-density plasma consisting of neu- evolution of the probe from one steady state to another tral particles, electrons, and singly charged positive because of rapid change in the applied potential. For ions and neglect chemical reactions between the plasma example, in my recent paper [5], an analytic solution to components. Such a plasma is thermally equilibrium the problem of the evolution of the current to a spheri- and remains unchanged in transport properties. Under cal probe in a stationary, weakly ionized, high-density these conditions, the basic set of equations for the probe plasma was obtained in the limiting case of a thick [1] can be generalized to describe unsteady operating space-charge sheath when a small potential difference modes as follows: between the probe and the plasma drops to zero in a βÐ1∂ ∂ ∇∂()∇ψ jumplike manner. n+/ t Ð0,n+ + n+ = (1) ∂ ∂ ∇∇()∇ψ The papers of the second group [6Ð8] were intended nÐ/ t Ð0,nÐ Ð nÐ = (2) to determine the impedance of the probes with modu- α2∇ψ lated potentials. A slight sinusoidal temporal modula- = nÐ Ð n+. (3) tion of the constant potential of a probe substantially extends the possibilities of the probe diagnostics. In Equations (1)Ð(3) are written in dimensionless form principle, the data from probes with modulated poten- and describe non-steady-state diffusion of charged par- tials make it possible not only to measure the electron ticles in an electric field. Here, the ratio of the diffusion β temperature and density and the plasma space potential coefficient of the ions to that of the electrons, = D+/DÐ, β Ӷ (which are all traditionally obtained from the steady- is assumed to be much smaller than unity, 1; n+ and state currentÐvoltage characteristic of the probe), but nÐ are the ion and electron number densities normalized also to estimate other important plasma parameters. to the charged-particle density at infinity, N∞; and the Thus, Baksht [6] arrived at the conclusion that the dimensionless electric potential ψ is related to the ϕ ψ ϕ ion diffusion coefficient can be determined from the dimensional potential by = e /kTÐ, where e is the

A SPHERICAL PROBE WITH A TIME-VARYING POTENTIAL 1089 α ∞ charge of an electron, k is Boltzmann constant, TÐ is the SOLUTION FOR ϕ electron temperature, and is the difference between For α ∞, Eq. (3) degenerates into the Laplace the probe and plasma potentials. The radial coordinate equation ∆ψ = 0. The solution to the Laplace equation r is in units of the probe radius R, and the time t is in that satisfies the boundary conditions (4) and (6) is ψ = 2 units of the ratio R /DÐ. The dimensionless parameter ar Ð1sinωt, or, in complex form, α λ = D/R characterizes the thickness of the space- charge sheath around the probe. ψ = arÐ1 exp()iωt . (9) Far from the probe (at r ∞), Eqs. (1)Ð(3) are In this limit, Eqs. (1) and (2) are independent of one supplemented with the boundary conditions another, so we can consider only one of them, e.g., Eq. (2). We represent the electron number density n ()∞ ==n ()∞ 1, ψ∞() =0. (4) + Ð n−(r, t) as At the probe surface (r = 1), the charged particles are n ()rt, ==n ()r + n˜ ()rt, n ()r assumed to be completely absorbed, Ð 0 Ð 0 ∞ (10) () () () ()ω n+ 1 ==nÐ 1 0. (5) + ∑ nm r expim t . and the probe potential is specified as m = 1 Substituting expressions (8)Ð(10) into Eq. (2) and ψ() ψ ω 1 = p + asin t, (6) singling out the coefficients of the fundamental har- ω ψ Ӷ monic of , we arrive at the following ordinary differ- where p and a are constants (a 1). ential equation for n1(r): Let the modulation frequency ω be low enough to Ð1 ω Ð4 guarantee the satisfaction of the underlying assump- n1'' + 2r n1' Ð i n1 = Ðar . (11) tions of our problem, in particular, the validity of the The boundary conditions on Eq. (11) follow from Einstein relationship between the diffusion coefficients ∞ and the mobilities of charged particles and the assump- conditions (4) and (5): n1(1) = n1( ) = 0. The solution tion that the plasma is in thermal equilibrium (the latter to Eq. (11) can be found in accordance with [9]: allows us to eliminate the electron energy equation () Ð1 Ð2 Ð1 Ð1 []ω() from the mathematical formulation of the problem). n1 r = Ð2 ar + 2 ar exp Ð i r Ð 1 × { Ð1 ω[ ()ω ()ω The task now is to calculate the electron (IÐ) and ion 12+ i exp i E1 i (I ) currents to the probe as functions of the modulation (12) + ()ω (ω )]} Ð1 Ð1 ω Ð exp Ð i E1 Ð i Ð 4 ar i frequency ω. The dimensionless current I+− is normal- π × []()ω ()ω ()ω ()ω ized to 4 eN∞RD+− and is defined as exp ri E1 ri Ð exp Ðri E1 Ðri ,

− − where I+ = dn+ /dr r = 1. (7) z Ðu With allowance for the fact that the currents IÐ and () e E1 z = ∫------du I+ are oppositely directed, the total dimensionless cur- u ∞ rent IΣ to the probe is equal to β is the integral exponent [10]. Differentiating solution IΣ = IÐ Ð I+. (12) with respect to r at r = 1 yields the complex ampli- The time-dependent problem (1)Ð(7) can be solved tude of the electron current associated with the funda- ψ mental harmonic of the electron density oscillations: analytically for p = 0, i.e., for the case in which the probe potential oscillates with a small amplitude about I = 2Ð1a Ð 2Ð1aiωexp()iω E ()iω . (13) the plasma potential. Regardless of the value of α, the 1 1 ω ψ steady solution ( = 0) has the form 0(r) = 0, which Clearly, the amplitude I2 of the electron current corresponds to associated with the second harmonic of the electron density oscillations is on the order of ~a2, so that for () Ð1 n0+− r = 1 Ð r . (8) a Ӷ 1, this harmonic, as well as all higher harmonics, can be neglected. Under these conditions, no charge separation occurs. Consequently, the cases of thick and thin space- It is convenient to represent the complex quantity I1 charge sheaths are merely formal limits, because, in as reality, no sheath forms around a probe operating in a I = aM()ω exp[]iΦω(). (14) steady mode. However, for a probe with a modulated 1 potential, we may again speak of an unsteady space- The absolute value and the phase shift of the time- ω charge sheath (either thick or thin). varying current I1 are related to the impedance Z( ) of

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1090 KASHEVAROV

M Φ 0.6 30 2 1 3 1 4 0 20 40 60 0.4 2 ω; Ω

–30 3 0.2 4 –60 2 2 4 4 0 204060 –90 ω; Ω

Fig. 1. Frequency dependence of the amplitude of the time- varying electron current for (1) α ∞, (2) α = 0.1, Fig. 2. Frequency dependence of the phase shift of the time- (3) α = 0.01, and (4) α = 0.001. Curve 1 represents the M(ω) varying electron current. Curves 1Ð4 indicate the same dependence, and curves 2Ð4 are for the M(Ω) dependence. dependences as in Fig. 1. the probe by SOLUTION FOR α 0 In this limit, we must solve the full set of Eqs. (1)Ð Ð1 ()Φ ZM= exp Ði . (3). However, for β 0, Eq. (1) degenerates into the ∂ ∂ equation n+/ t = 0, which indicates that the ion distri- We thus arrive at the ﬁnal expression for the sought- bution in space is steady-state and thus can be described for electron current to the probe: by the time-independent expression (8). For this reason, []()ω we can only consider Eqs. (2) and (3). IÐ = I0 + Im I1 exp i t (15) We again represent the electron number density ()ω [] ω Φω() = 1+ aM sin t + . n−(r, t) by formula (10) and substitute it into Eqs. (2) ψ ω and (3). In this way, the potential should also be rep- Figures 1 and 2 illustrate the time evolutions M( ) resented in a form similar to expression (10): and Φ(ω), which were determined from formula (13) with the help of the tables presented in [10]. ∞ ψ(), ψ() ()ω Ӷ ≅ γ γ rt = ∑ m r exp im t . For z 1, we have E1(z) Ð Ð lnz, where = 0.5772… is Euler’s constant. For z ∞, the corre- m = 1 Ð1 sponding asymptotic expression is E1(z) ~ z exp(Ðz) As a result, we arrive at the equations [10]. Taking into account the identity i = (1 + i)/ 2 ξ4 '' ξ4 ()ξξ 4 ' ω and using formula (13), we can obtain explicit expres- n1 Ð1E1 + Ð E1 = i n1, (16) sions for M(ω) and Φ(ω). For ω Ӷ 1, we have α2ξ4 E1' = Ðn1, (17) Ð1 M()ω ==()1 Ð ωπ/4 /2, Φω() ωγ()+ 2 lnω . ξ ψ where = 1/r and E1 = Ðd 1/dr. Equation (17) implies α ≅ For ω ∞, we have that for 0, we may set n1 0 over almost the entire space except for a thin sheath near the probe sur- ξ ≅ ()ω Ð1[]ωω()ω()π face ( 1). According to Eq. (16), the quasineutral M = 2 1 Ð exp Ð /2 sin /2 + /4 , plasma region is described by the equation Φω() ω()ω ω ()π ()ξ = exp Ð /2 sin /2 + /4 . 1 Ð E1' Ð0,E1 =

Clearly, the ion current I+ is also described by whose solution is ω expression (15), but with Ða in place of a and M( ) and E = C/1()Ð ξ , (18) Φ(ω) in place of M(ω/β) and Φ(ω/β). Since the dimen- 1 sionless currents I+ and IÐ are of the same order of mag- where C is an integration constant. Taking into account nitude, the total current is approximately equal to IΣ ≈ the boundary conditions (4), we can obtain from solu- β Ӷ ψ IÐ, provided that 1. tion (18) the amplitude 1 of the fundamental harmonic

TECHNICAL PHYSICS Vol. 46 No. 9 2001 A SPHERICAL PROBE WITH A TIME-VARYING POTENTIAL 1091 of the probe potential: tion (20) to the integral in the latter expression gives

0 ψ = Cln()1 Ð ξ . (19) 1 α2/3 ζ ()ζζ Ω aC= ln + Cln * + A ∫ Ai + i d ψ ∞ ξ ζ We can see that 1 as 1, which contra- * (26) dicts boundary condition (6). This indicates that, near 0 α2 the probe surface, the product E1' remains finite as + πC Gi()ζζ + iΩ d , α ∫ 0; in other words, in a thin space-charge sheath ζ ≠ * around the probe, we have n1 0. An unsteady space- charge sheath can be analyzed in essentially the same where ζ∗ is the coordinate of the conditional boundary way as for a steady sheath in the classical paper by of the space-charge sheath. The first two terms on the Cohen [11]. right-hand side of formula (26) were obtained by apply- We substitute Eq. (17) into Eq. (16) and transform to ing transformation (20) to expression (19); they deter- new variables in the resulting equation: mine the potential at the boundary of the quasineutral plasma region. The last two terms describe the voltage drop across the space-charge sheath near the probe sur- ζαÐ2/3()ξ ()ξ αÐ2/3 ()ζ ==1 Ð , E1 F1 . (20) face. For ζ∗ ∞, we can use the familiar asymptotic For α 0, we neglect terms on the order of formula [13] O(α2/3) to obtain ζ * 2γ + ln3 1 F''' = ()ζ + iΩ F' + F . (21) Gi()ζζ d ∼ ------+ --- lnζ (27) 1 1 1 ∫ 3π π * 0 Here, we keep the term with Ω = α4/3ω, because, for α 0, we have Ω = O(1) as ω ∞. For ζ ∞, and the relationship [12] ζ the solution to Eq. (21) should approach solution (18) * for the quasineutral plasma region. In other words, in Ai()ζζ d 1/3. (28) terms of variables (20), we have ∫ 0 ()ζ ζ ζ∞ F1 C/ , . (22) We can show that formulas (27) and (28) are also valid for the integrals in expression (26) (see [14]). We Integration of Eq. (21) gives [9] insert formulas (25), (27), and (28) into expression (26) to find the constant C: ()ζ Ω F1'' Ð + i F1 = ÐC. (23) C = 3a/2[]lnαπ+ Gi'()iΩ /Ai'iΩ Ð32γ Ð ln .

The solution to Eq. (23) that satisﬁes condition (22) As a result, the complex amplitude I1 = ζ ζ| | has the form dn1( )/d ζ = 0 = F1'' ζ = 0 of the electron current associ- ()πζ Ω ()ζ Ω ated with the fundamental harmonic of the electron F1 = AAi + i + CGi + i . (24) density oscillations is equal to ()Ω Ω ()Ω Ωπ ()Ω Here, A is an integration constant and Ai and Gi are I1 = ÐCi++AAi i i CGi i . (29) Airy functions [12]; moreover, the function Gi(z) satis- Ω πÐ1 For = 0, we have I1 = ÐC; moreover, we can see ﬁes the equation y'' Ð zy = Ð . The constant A can be that the amplitude I is real: expressed in terms of C with the help of condition (5), 1 | which takes the form n1 ξ = 1 = 0, or with the help of () ()αÐ2 Ð1/2π γ I1 0 = 3a/ ln+++ 3 2 ln3 . (30) Eq. (17) with transformation (20) such that F1' (0) = 0. As a result, from solution (24), we obtain Note that the ratio I1(0)/a is the slope of the steady- state currentÐvoltage (IÐV) characteristic of the probe ψ A = ÐπCGi'()iΩ /Ai'()iΩ . (25) for p = 0. The question of the slope of the IÐV characteristic of a probe whose potential is equal to the poten- For Ω = 0 (i.e., for low modulation frequencies ω ~ tial of the environment was studied by Benilov and Tir- ski [15] in connection with the possibility of determin- 1), we have Gi'(0) = Ð3Ð1/2Ai'(0) and A = 3Ð1/2πC. œ ing the electron temperature. In particular, for an ψ | isothermal plasma, they obtained the relationship At the probe surface, we have 1 ξ = 1 = a; on the other hand, ψ (1) = 1 E (ξ)dξ. Applying transforma- ()ψ ψ ()Ð1 αÐ2 Ð1 1 ∫0 1 dIÐ /d ψ = 0 = 1.587+ 3 ln . (31)

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1092 KASHEVAROV According to the analytic solution (30), the ﬁrst zero, Φ(ω) 0, as α ∞ (curve 1); this indicates term in parentheses in expression (31) is approximately that the reactive part of the impedance vanishes. For equal to (2γ + ln3 + 3Ð1/2π)/3 ≈ 1.356. α 0, we have

For Ω ≠ 0, it is difficult to calculate the amplitude Ð2 Ω Φω()arctan[] π/()lnα ++ 2γ ln3 I1( ) from formula (29) because the tables of the Airy functions in the range of complex arguments are incom- as ω ∞. In this case, the reactive part of the imped- plete or because the numerical algorithms for calculat- ance is capacitive; moreover, the phase difference ing these functions are lacking. The calculations were between the current and voltage increases with α. Con- carried out with the help of the four-digit tables of the sequently, for high modulation frequencies, the depen- Airy functions Ai(z) and Bi(z) and their derivatives dence of the phase shift on the ratio α is nonmonotonic. Ai'(z) and Bi'(z) [16]. The Airy function Gi(z) and its derivative Gi'(z) were calculated from the formulas [12] Since the analytic treatment presented here has been performed only for two particular limiting cases, the z z results obtained are rather difficult to apply in practice. 1 Gi()z = ---Bi()z + Ai()z ∫Bi()u du Ð Bi()z ∫Ai()u du, However, it would hardly be possible to carry out a 3 complete analytic study of the general case of arbitrary 0 0 values of the constant component of the probe poten- z z tial. To do this, it might be necessary to apply numerical 1 Gi'()z = ---Bi'()z + Ai'()z ∫Bi()u du Ð Bi'()z ∫Ai()u du. methods. The above analytic solutions may be useful in 3 verifying the computational results. 0 0 Unfortunately, the tables in [16] do not contain the values of the functions Ai(iΩ) and Bi(iΩ)and their REFERENCES derivatives in the range Ω > 2.4. For this reason, the 1. P. Chung, L. Talbot, and K. Touryan, Electric Probes in Ω Ω ӷ dependence I1( ) for 1 was calculated from the Stationary and Flowing Plasmas (Springer-Verlag, Ber- large argument asymptotic expansions of the Airy func- lin, 1975; Mir, Moscow, 1978). tions. Note that Gi(iΩ) ~ Bi(iΩ) for Ω ӷ 1 [14]. The 2. M. S. Benilov, Teplofiz. Vys. Temp. 26, 993 (1988). asymptotic formulas for the functions Ai(iΩ) and Ω 3. M. S. Benilov and B. V. Rogov, J. Appl. Phys. 70, 6726 Bi(i ) and their derivatives are presented in [12]. As a (1991). result, for Ω ӷ 1, we have 4. A. V. Kashevarov, Teplofiz. Vys. Temp. 36, 700 (1998). ()Ω { πΩ3/4 I1 = ÐC 12Ð 5. A. V. Kashevarov, Teplofiz. Vys. Temp. 38, 147 (2000). × exp[]Ð32Ω3/2()1 Ð i /3 + πi/8 }, 6. F. G. Baksht, Zh. Tekh. Fiz. 48, 2019 (1978). 7. E. F. Prozorov and K. N. Ul’yanov, Teplofiz. Vys. Temp. C = 3a/2{ lnα Ð32γ Ð ln + iπ Ð 2π 21, 538 (1983). × []}3/2Ω3/2() 8. E. F. Prozorov and K. N. Ul’yanov, Teplofiz. Vys. Temp. exp Ð2 1 + i /3 . 21, 1179 (1983). Recall that it is convenient to represent the complex 9. E. Kamke, Differentialgleichungen, Bd. I: Gewöhnliche Ω Ω quantity I1( ) by formula (14). The functions M( ) Differentialgleichungen (Geest and Portig, Leipzig, and Φ(Ω) are shown Figs. 1 and 2 for several values of 1964; Nauka, Moscow, 1976). α Ӷ 1. 10. Tables of Integral Exponential Functions in Complex Domain. Library of Mathematical Tables, Ed. by K. A. Karpov (Vychisl. Tsentr Akad. Nauk SSSR, Mos- DISCUSSION cow, 1965), Vol. 31. From Fig. 1, we can see that curve 1 (α ∞) is 11. I. M. Cohen, Phys. Fluids 6, 1492 (1963). α Ӷ α 0 very similar to curves 2Ð4 ( 1). For , we 12. Handbook of Mathematical Functions, Ed. by have M. Abramowitz and I. A. Stegun (Dover, New York, 1971; Nauka, Moscow, 1979). ()ω ()α γ 2 π2 M 3/ 2lnÐ3 2 Ð ln + 13. M. Rothman, Q. J. Mech. Appl. Math. 7, 379 (1954). as ω ∞. For α ∞, we have M(ω) 1/2 as 14. F. W. J. Olver, Asymptotics and Special Functions (Aca- ω ∞. In other words, for high modulation frequen- demic, New York, 1974; Nauka, Moscow, 1990). ω α cies , the current amplitude increases with . On the 15. M. S. Benilov and G. A. Tirskiœ, Dokl. Akad. Nauk SSSR other hand, at small and large values of α, the time evo- 240, 1324 (1978) [Sov. Phys. Dokl. 23, 395 (1978)]. Φ Ω lutions ( ) are significantly different (Fig. 2). For low 16. P. M. Woodward and A. M. Woodward, Philos. Mag., modulation frequencies, curves 1Ð4 are similar to each Ser. 7 37 (267), 236 (1946). other: the current lags behind the voltage in phase and the reactive part of the impedance of the probe is induc- tive. For high frequencies ω, the phase shift approaches Translated by O. Khadin

TECHNICAL PHYSICS Vol. 46 No. 9 2001

GAS DISCHARGES, PLASMA

Porous Piezoelectric Composites with Extremely High Reception Parameters V. Yu. Topolov and A. V. Turik Rostov State University, pr. Stachki 194, Rostov-on-Don, 344004 Russia e-mail: [email protected] Received November 13, 2000

FC FC, E Abstract—The reception parameters Q33* and Qh* of modified fiber composites based on high-e33 /c33 fer- FC FC, E roelectric piezoceramics (e33 is the piezoelectric constant, c33 is the elastic modulus) are considered as functions of the electromechanical properties and the porosity of the components. Pore configurations at which 2 ≈ 2 ≈ × Ð9 Ð1 the values (Q33* ) ()Qh* 7 10 Pa for the polymer matrices of 1Ð0Ð3 composites are much higher than for conventional ternary composites are analyzed. © 2001 MAIK “Nauka/Interperiodica”.

Composites based on ferroelectric piezoelectric tration dependences of (Q* )2(m) and (Q* )2(1) for 1Ð3 ceramic (FPC) materials offer many physical properties 33 h composites “Pb(Zr, Ti)O (volume concentration m)Ð of practical value. Of binary piezoactive composites, 3 fiber materials (extended FPC filaments in a polymer polymer matrix (volume concentration 1 Ð m)” are sev- 2 matrix that have 1Ð3 connectivity) are most commonly eral tens of times higher than those for (Q33* ) (1) and α α α α used. In 1Ð 2 connectivity, the integers 1, 2 = 0Ð3 2 )2 = ()Qh* (1); for various 1Ð3 composites, max(Qh* 1Ð3 (m) define the number of axes in an (X1X2X3) Cartesian sys- α α )2 = tem along which the first, , and the second, 2, com- 2000Ð9500 and max(Q33* 1Ð3 (m) 20000Ð50000 (in ponents are distributed continuously [1, 2]. Numerous units of 10Ð15 PaÐ1) [9, 11, 13, 14].1 The calculated val- experiments and theoretical studies suggest that FPC- ||ζ* ||ζ* * based 1Ð3 composites exhibit a nonmonotonic concen- ues of e , d [6], and gh [9, 11, 13] of the 1Ð3 com- tration dependence of all four piezoelectric constants posites are frequently much greater than experimen- FC ||FC FC FC FC FC dij* [3], eij* [3Ð6], gij* [7], and hij* [4]; planar, kp* , and tally found e33 / e31 , d33 /|d31 |, and g33 + 2g31 of FPC composites. Moreover, the complication of the thickness, kt* , electromechanical coupling coefficients [3Ð6, 8]; anisotropy coefficients of the piezoelectric composite geometry may raise gh* and g33* by more ζ* * * * * ζ* * * FC FC constants e = e33 /e31 = h33 /h31 and d = d33 /d31 = than one order of magnitude above g33 + 2g31 and FC g33* /g31* [3, 6]; reception parameters Q33* and Qh* g33 , respectively [15] (hereafter, the superscript FC is [9−13]; and other effective constants. For the compos- ∞ assigned to the electromechanical constants of ferro- ites polarized along the OX3 axis ( mm symmetry), the electric ceramics used as the components of the com- reception parameters squared are given by posites). The above effective properties of binary fiber ()2 ()2 ε σ composites and like materials are finding wide applica- Q33* ==d33* g33* d33* / 33* , (1) tion in piezoelectric transducers, as well as in hydroa- ()2 ()2 ε σ coustic, medical, and other devices. Qh* ==dh*gh* dh* / 33* , The possibility of producing ternary composites where dh* = d33* + 2d31* and gh* = g33* + 2g31* are the involving FPC filaments has been discussed in [16–18]. ε σ )2 hydrostatic piezoelectric constants and 33* is the per- In particular, it has been shown [18] that (Qh* 1Ð0Ð3 may mittivity of the mechanically unloaded material. exceed 10 000 (PPa)Ð1 for 1Ð0Ð3 connectivity and * * )2 Ð1 The parameters Q33 and Qh in (1) are related to the (Qh* 1Ð2Ð2 may attain 50 500 (PPa) for 1Ð2Ð2 connec- power densities and the signal-to-noise ratios of piezo- tivity (FPC filaments in a 2–2 polymer matrix) [2]. The electric elements, and dh* and gh* characterize the 1 * 2 * 2 hydrostatic stability of the elements and piezoelectric According to [2, 10, 11, 14], (Q33 ) and (Qh ) will hereafter be devices. It has been found that the peaks of the concen- expressed in (PPa)Ð1.

1 – ν ν

1 3

1 – t 1 – f

f t X2 4 B1 C1 A D O X1 m 1 – m 1 1 r n C B AD

Fig. 1. Schematic diagram of the composites with various connectivities (insets 1Ð3). The volume concentration of spherical pores in the ﬁlaments is t and in the surrounding matrix it is f (spherical pores) or nr (parallelepiped-like pores arranged within the Banno cubic cell [20, 21] ABCDA B C D as shown in inset 4), where n and r are concentration parameters: n equals the ratio of the par- 1 1 1 |1 | | | | | ν allelepiped height to the cube edge AA1 ; and r is the ratio of the parallelepiped base area to the cube face area AB á BC . In inset 2, and 1 Ð ν are the volume concentrations of the layers of the ﬁrst and second types, respectively.

)2 Ð1 σM )2 × 6 Ð1 latter value exceeds the value (Qh* 2Ð2Ð0 = 50000 (PPa) 13 > 0 and max(Qh* GT = 5.655 10 (PPa) for (for 2Ð2Ð0 connectivity), which until recently has been σM < 0 [12], which were obtained for the 1Ð0Ð3 com- the highest for ternary composites [2]. Of much interest 13 posite “PZT-5A filaments–polyurethane matrix with is also the effect of pore configuration on the effective conical pores,” are two or three orders of magnitude properties of FPC [19Ð22] and polymer [12] materials. * 2 Although the averaging methods applied to ferroelec- higher than the values of (Qh ) for Pb(Zr,Ti)O3-based tric piezoceramics (FPCs) and porous polymers are composites used in practice [2, 14]. The above-men- )2 3 4 Ð1 approximate and the elastic and dielectric properties of tioned jump from (Qh* 1Ð3 = (10 Ð10 ) (PPa) to the solid components (FPCs and polymers) and air sub- * )2 ≈ × 4 Ð1 stantially differ (at least by 103 times and by 10Ð (Qh 1Ð2Ð2 5 10 (PPa) , which is associated with 103 times, respectively), the results sometimes agree modification of the fiber structure without allowance 2 well with experimental data. Agreement is observed for for porosity [18], as well as unique data [12] for (Qh* ) the porous media that can be considered as composite for 1Ð0Ð3 porous composites with sgnσM = ±1, are with connectivity 0Ð3 (i.e., the pores are closed) [19, 13 strong impetuses to extend works on the prediction of 20] or 3Ð3 (the pores are open) [21]. It has been shown the effective properties of the ternary composites. In [12] that the reception parameters of the fiber compos- this article, we elaborate upon and generalize theoreti- FC FC ites can be much improved compared with d33 g33 and cal concepts [18] and analyze factors that can improve FC FC the reception parameters and the hydrostatic sensitivity dh gh by producing a complex configuration of pores of the porous composites with 1−3 connectivity. in the polymer matrix. The pores provide a negative In the model we put forward, the composite consists Poisson’s ratio σM for the matrix and are responsible for of extended cylindrical piezoactive filaments running 13 parallel to the polarization axis OX and surrounded by the nonuniform fields of internal mechanical stresses. 3 a nonpiezoelectric matrix (Fig. 1). Two cases are con- )2 × 6 Ð1 The values of max(Qh* GT = 2.200 10 (PPa) for sidered: porous FPC filaments of connectivity 0Ð3 sur-

TECHNICAL PHYSICS Vol. 46 No. 9 2001 POROUS PIEZOELECTRIC COMPOSITES 1095

Table 1. Electromechanical constants of ferroelectric and polymer components measured at room temperature

FC, E FC, E FC, E FC, E FC, E FC FC FC FC FC FC FC , ξ , ξ FPC c11 , c12 , c13 , c33 , c44 , e31 , e33 , e15 , εFC ε εFC ε d33 g33 , dh gh , 11 / 0 33 / 0 material 1010 Pa 1010 Pa 1010 Pa 1010 Pa 1010 Pa C/m2 C/m2 C/m2 (PPa)Ð1 (PPa)Ð1 PKR-7M [ 25] 13.3 9.2 9.1 12.5 2.28 Ð9.5 31.1 20.0 1980 1810 13000 84 PZT-5A [12, 27] 12.1 7.54 7.52 11.1 2.11 Ð5.4 15.8 12.3 916 830 9300 67

cp , cp , εp ε Polymer 11 12 kk/ 0 , 1010 Pa 1010 Pa k = 1; 2; 3 Elastomer [9] 0.114 0.0931 5.0 Araldit [26] 0.78 0.44 4.0 Polyurethane 0.442 0.260 3.5

||ζ ӷ rounded by a homogeneous polymer matrix (Fig. 1, of m and the resulting high anisotropy, e* 1 [4, 6, inset 1) and FPC filaments surrounded by a heteroge- 18], expressions (1) can be recast as neous polymer matrix. The heterogeneous matrix may ()2 ≈ η ()E 2 ε σ contain layers of two dissimilar polymers (connectivity Qh* elas* , h e33* /c33* / 33* 2Ð2, Fig. 1, inset 2), consist of a polymer with air-filled ≈η []FC ()FC, E M 2 ()εFC, σ (2) pores in the form of a sphere or a parallelepiped (cellu- elas* , h me33 / mc33 + c33 / m 33 , lar matrix of connectivity 0Ð3; Fig. 1, insets 3 and 4), or ()2 ≈ ()η η ()2 be laminated and porous. In the last case, it is assumed Q33* elas* , 33/ elas* , h Qh* , that the layers of porous and homogeneous polymers of where the same chemical composition alternate, with the interfaces parallel to the X OX plane. The effective Ð1 Ð1 Ð1 1 2 η* , = {[]1 + ()β* Ð 2()β* /1[ + ()β* constants of the porous composites are determined it elas h 12 13 12 two steps. First, the effective constants of the spherical- ()β Ð1(β β )]}2 Ð2 13* 33* / 13* , pore medium (Fig. 1; insets 1, 3) are found by averag- (3) Ð1 Ð1 FC, E η {[]()β [ ()β FC elas* , 33 = 1 + 12* /1+ 12* ing the elastic, cab ; piezoelectric, eij ; and dielec- εFC, ξ ()β* Ð1(β* β* )]}2 tric, kk , constants of the FPC material or by averag- Ð2 13 33/ 13 , p p ε E E M ing the elastic, cab , and dielectric, ff , constants of the β * * ab* = c11 /cab , and c33 is the elastic modulus of the polymer using formulas for connectivity 0Ð3 (the self- matrix. consistency method for piezopassive [23] and piezoactive [24] media). The effective constants of the 2Ð2 From (2), we obtain that the reception parameters of matrix (Fig. 1, inset 1) are calculated from formulas for the composite depend on the ratio , laminated media [17, 25]; and those for the 0Ð3 matrix γ = meFC/()mcFC E + cM , (4) with parallelepiped-like pores (Fig. 1, inset 4), using 33 33 33 the matrix method [3, 26]. Then, the effective elastic εFC, σ the permittivity 33 of the FPC material, and the dif- *E moduli cab , piezoelectric constants eij* and dij* , and ferences between the elastic moduli of the FPC fila- ε σ FC, E M dielectric permittivities *ff of the modified fiber com- ments, cab , and the matrix, cqr . Among perovskite- posite are found by averaging the associated constants like FPCs [25, 27], PKR-7M (one of the so-called Ros- for the filaments and the matrix using the self-consis- FC FC, E tov piezoceramics) has the highest ratio e33 /c33 at tency method [9] for connectivity 1Ð3. Subsequently, room temperature; hence, high values of γ in (4) at m Ӷ the effective constants thus obtained are employed to 1. An elastomerÐaraldit laminated structure (Fig. 1, determine the reception parameters [from (1)] and M M other material properties. inset 2) features a large difference between c11 /c33 and M M c /c [18], so that η* , and η* , in (3) are appre- FC, E FC εFC, ξ p 12 13 elas h elas 33 From experimental data for cab , eij , kk , cab , ciable. Below, we will consider how the shape of pores and ε p [25, 27], and also from the results of computa- in the filaments and the matrix affects parameters (1) of kk the composite based on the PKR-7M ceramic. The elec- 2 2 tion [18], it follows that (Q33* ) and (Qh* ) peak at a fil- tromechanical constants of the ceramics and polymers ament concentration m Ӷ 1. Because of the smallness are listed in Table 1.

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1096 TOPOLOV, TURIK

η Table 2. Correlation between the elastic parameters of the laminated porous matrix, coefﬁcient elas,* h in (3), and the peaks 2 2 ()* ν ()* ν of the functions Qh 1Ð0Ð3 (m, , f ) and maxQ33 1Ð0Ð3 (m, , f ) all calculated for the 1–0–3 “PKR-7M ﬁlaments–laminated porous elastomer matrix with spherical pores” composite f 0.90 0.80 0.70 ν 0.10 0.20 0.10 0.20 0.30 0.10 0.20 0.30

M M c11/c12 1.71 1.88 1.52 1.69 1.80 1.43 1.57 1.67

M M c11/c13 3.15 4.86 2.07 2.81 3.19 1.71 2.14 2.50

M M c11/c33 2.45 3.59 1.61 2.09 2.43 1.34 1.60 1.78 η elas,* h 0.552 0.657 0.339 0.465 0.528 0.250 0.331 0.401

2 ()* Ð1 4.94 × 105 1.11 × 106 1.67 × 105 4.00 × 105 6.10 × 105 7.97 × 104 1.89 × 105 3.10 × 105 maxQh 1Ð0Ð3 , (PPa) 2 ()* Ð1 1.38 × 106 2.09 × 106 9.76 × 105 1.32 × 106 1.72 × 106 8.31 × 105 1.04 × 106 1.25 × 106 maxQ33 1Ð0Ð3 , (PPa) f 0.60 0.50 ν 0.10 0.20 0.30 0.40 0.10 0.20 0.30 0.40 0.50

M M c11/c12 1.37 1.48 1.57 1.64 1.33 1.42 1.49 1.56 1.62

M M c11/c13 1.53 1.80 2.04 2.23 1.42 1.60 1.76 1.89 1.99 M M c11/c33 1.20 1.35 1.46 1.53 1.12 1.21 1.28 1.32 1.33 η elas,* h 0.175 0.254 0.291 0.335 0.137 0.181 0.225 0.242 0.268 2 ()* Ð1 4.47 × 104 1.01 × 105 1.64 × 105 2.29 × 105 2.77 × 104 5.73 × 104 9.10 × 104 1.25 × 105 1.56 × 105 maxQh 1Ð0Ð3 , (PPa)

2 ()* Ð1 7.62 × 105 9.00 × 105 1.03 × 106 1.18 × 106 7.18 × 105 8.10 × 105 9.07 × 105 1.00 × 106 1.09 × 106 maxQ33 1Ð0Ð3 , (PPa)

2 η ν ()* Note: The coefﬁcient elas,* h in (3) was calculated for volume concentrations mj, , and f that correspond to maxQh 1Ð0Ð3 . The mean porosity of the matrix is speciﬁed in the interval of 0 < νf ≤ 0.25.

The introduction of FPC filaments with spherical aments of the same composition are embedded in the pores into the homogeneous polymer matrix (Fig. 1, heterogeneous (laminated [18] or porous) polymer FC FC, E matrix. This statement is confirmed by the calculation inset 1) causes an increase in e33 /c33 . As a result, of (Q* )2 and (Q* )2 for 1Ð0Ð3 composites with * *E * h 0Ð1Ð3 33 0Ð1Ð3 e33 /c33 and the hydrostatic piezoelectric constant gh spherical pores distributed throughout the polymer of the 0Ð1Ð3 composite also grow. These circum- matrix (Fig. 1, inset 3; Figs. 2c, 2d) or over similar lay- ||ζ ≥ 2 stances, as well as the high anisotropy e* 10 at ers (Table 2). In the latter case, the laminated porous m Ӷ 1, raise, according to formulas (2) and (3), elastomer matrix has the structure shown in inset 2 of Fig. 1 and the layers of concentration ν have pores with )2 )2 max(Qh* 0Ð1Ð3 (m, t) and max(Q33* 0Ð1Ð3 (m, t) (Figs. 2a, the volume concentration f (Fig. 1, inset 3). 2b) in comparison with max(Q* )2 (m) = 6240 (PPa)Ð1 h 1Ð3 )2 The calculated values of max(Qh* 1Ð0Ð3 = and max(Q* )2 (m) = 6.24 × 105 (PPa)Ð1 for the 1Ð3 33 1Ð3 ((Q* )2 m , f) and max(Q* )2 = (Q* )2 (m , f) “PKR-7M–elastomer” composite. However, the h 1Ð0Ð3 0 33 1Ð0Ð3 33 1Ð0Ð3 0 improvement of parameters (1) of the 0–1–3 porous fil- of the 1–0–3 “PKR-7M filaments–spherical-pore elas- aments embedded in the homogeneous matrix is less tomer matrix” composites correlate, at f = const, with M M M M M M appreciable than in the case when the homogeneous fil- the ratios c11 /c12 = c11 /c13 = c33 /c13 of the elastic

TECHNICAL PHYSICS Vol. 46 No. 9 2001 POROUS PIEZOELECTRIC COMPOSITES 1097

2 –1 (Q*h ) , (PPa) (a) (b) 6000 600000

5 4000 4 300000 5 4 3 3 2 2 2000 1 200000 1

50000 5 5 (c) 4 (d) 40000 800000 3 4 2 30000 3 600000 20000 2 1 1 10000 400000 200000 0 12345 012345 m, % m, %

2 2 Fig. 2. Concentration dependences of the reception parameters squared (a) (Q* ) (m, t), (b) (Q* ) (m, t), h 0Ð1Ð3 33 0Ð1Ð3 2 2 (c) (Q* ) (m, f), and (d) (Q* ) (m, f) calculated for the (a, b) 0–1–3 “PKR-7M spherical-pore ﬁlaments–elastomer matrix” h 0Ð1Ð3 33 0Ð1Ð3 composite and (c, d) 1–0–3 “PKR-7M ﬁlaments–porous elastomer matrix” composite. (a, b) t = (1) 0.05, (2) 0.10, (3) 0.15, (4) 0.20, and (5) 0.25. (c, d) f = (1) 0.05, (2) 0.10, (3) 0.15, (4) 0.20, and (5) 0.25.

η ν moduli of the isotropic porous matrix and with elas* , h computed for various concentrations mj, , and f corre- in (3). For example, as f increases from 0.05 to 0.25, )2 )2 sponding to max(Qh* 1Ð0Ð3 and max(Q33* 1Ð0Ð3 , we con- cM /cM monotonically grows from 1.30 to 1.59 and 33 13 clude that cM /cM has the most signiﬁcant effect on η 11 33 elas* , h (m0, f), grows from 0.0721 to 0.109. The values η* , in (3) and parameters (1), while the effect of of max(Q* )2 and max(Q* )2 also rise monoton- elas h h 1Ð0Ð3 33 1Ð0Ð3 M M ically (Figs. 2c, 2d). One reason for the correlation c11 /c13 is somewhat weaker. This directly follows from 2 relationship (5) and is obvious, for example, for between these concentration dependences is that (Qh* ) η * )2 * )2 strongly depends on elas* , h according to (2). In addi- max(Qh 1Ð0Ð3 = (Qh 1Ð0Ð3 (m1; 0.70; 0.10) and η* )2 )2 tion, as follows from (3), elas, h considerably depends max(Qh* 1Ð0Ð3 = (Qh* 1Ð0Ð3 (m2; 0.50; 0.20) or for β β on the ratio 33* /13* , which is reduced to )2 )2 max(Qh* 1Ð0Ð3 = (Qh* 1Ð0Ð3 (m3; 0.80; 0.10) and β β ≈ M )2 )2 33* / 13* c13* /c33 (5) max(Qh* 1Ð0Ð3 = (Qh* 1Ð0Ð3 (m4; 0.70; 0.20) (in every Ð3 -2 ≈ Ð2 case 10 < mj < 10 ). As for the differences between at m0 10 and f = const. * )2 ν Similar considerations hold for the coefﬁcient the values of max(Qh 1Ð0Ð3 (m, , f) that are obtained at η* * 2 M M η elas, 33 in (3), which governs (Q33 ) [see formulas (2)] nearly equal c11 /c33 and elas* , h (cf., e.g., the estimates )2 for f = 0.60, ν = 0.20 and f = 0.70, ν = 0.10 in Table 2), and the values of max(Q33* 1Ð0Ð3 calculated. σ they can be explained by the effect of ε* (m, ν, f) on )2 33 The computed values of max(Qh* 1Ð0Ð3 and 2 Ӷ ()Qh* at m 1 [see formulas (1) and (2)]. Note also max(Q* )2 in Table 2 are much larger than those in 33 1Ð0Ð3 )2 ν η ν ≥ Figs. 2c and 2d, because the elastic modulus ratios that the values of max(Qh* 1Ð0Ð3 (m, , f) and elas* , h at M M M M M M 0.70 given in Table 2 are considerably larger than c11 /c12 , c11 /c13 , and c11 /c33 differ in the laminated )2 ν η ν porous matrix. Comparing the data in Table 2 that were max(Qh* 1Ð2Ð2 (m, ) and elas* , h (m, ) for the 1Ð2Ð2

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1098 TOPOLOV, TURIK

η Table 3. Correlation between the elastic parameters of the porous matrix, coefﬁcient elas,* h in (3), and the peaks of the func- 2 2 ()* ()* tions Qh 1Ð0Ð3 (m, n, r) and maxQ33 1Ð0Ð3 (m, n, r) all calculated for the 1–0–3 PKR-7M ﬁlaments–elastomer matrix with plane-parallel pores composite n 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 r = 0.99 M M 2.16 2.19 2.20 2.21 2.21 2.21 2.22 2.22 2.22 2.22 c11/c12

M M c11/c13 33.8 66.5 99.1 132 164 197 230 262 295 327

M M c11/c33 27.4 53.2 78.6 103 128 151 175 197 219 241 η elas,* h 0.933 0.961 0.972 0.978 0.982 0.985 0.986 0.988 0.989 0.990

2 ()* maxQh 1Ð0Ð3 , 3.96 5.05 5.57 5.90 6.15 6.35 6.52 6.66 6.80 6.92 106 (PPa)Ð1 2 ()* maxQ33 1Ð0Ð3 , 4.31 5.28 5.74 6.04 6.27 6.45 6.61 6.75 6.88 7.00 106 (PPa)Ð1 r = 0.90 M M 2.16 2.19 2.20 2.21 2.21 2.21 2.21 2.22 2.22 2.22 c11/c12

M M c11/c13 30.9 60.5 90.2 120 150 179 209 239 268 298

M M c11/c33 25.0 48.6 71.7 94.4 117 138 160 181 201 221 η elas,* h 0.928 0.958 0.970 0.977 0.981 0.983 0.985 0.987 0.988 0.989

2 ()* maxQh 1Ð0Ð3 , 3.79 4.91 5.45 5.80 6.05 6.24 6.41 6.55 6.69 6.81 106 (PPa)Ð1 2 ()* maxQ33 1Ð0Ð3 , 4.16 5.16 5.64 5.95 6.17 6.36 6.51 6.64 6.77 6.89 106 (PPa)Ð1

2 η ()* Note: The coefﬁcient elas,* h in (3) was calculated for volume concentrations mj, n, and r that correspond to maxQh 1Ð0Ð3 . The mean porosity of the matrix is speciﬁed in the interval of 0 < nr < 0.10.

)2 )2 “PKR-7M filaments–laminated elastomer–araldit max(Qh* 1Ð0Ð3 and max(Q33* 1Ð0Ð3 are attained and (2) matrix” composite [18]. the possibility of electrical breakdown because of the electric field concentration at pore edges [28] normal to The extremely high (Q* ) and (Q* ) are 33 1Ð0Ð3 h 1Ð0Ð3 the OX3 axis and the external field vector E. Known cal- observed in the case of plane-parallel air-filled pores in culated data for max(Q* )2 and max(Q* )2 indicate that the homogeneous polymer matrix (Fig. 1, inset 4). h 33 When the concentration of these pores varies in the the corresponding volume concentrations of the FPC range 0.01 ≤ n ≤ 0.10 and 0.90 ≤ r ≤ 0.99, the values of filaments in 1–3 [9–13], 1–2–2 [18], and 1–0–3 [12, 18] composites are small (m ≈ 1%). This seems likely to )2 )2 max(Qh* 1Ð0Ð3 and max(Q33* 1Ð0Ð3 change only slightly retard the use of piezoelectric composites of this sort. In (Table 3): they remain roughly one order of magnitude the 1–0–3 “PKR-7M filaments–elastomer matrix with higher than those computed for the laminated porous plane-parallel pores” composite, (Q* )2 (m, n, r) and matrix with a mean porosity νf Շ 0.10 (Table 2). The h 1Ð0Ð3 )2 Ӷ disadvantages of the 1Ð0Ð3 composites with plane-par- ((Q33* 1Ð0Ð3 m, n, r) vary (at m > mj, n 1, and r 1) allel pores in the matrix are (1) small (mj < 0.5%) vol- in such a way that parameters (1) remain roughly one ume concentrations of the PKR-7M filaments at which order of magnitude higher than those for the 1Ð3 com-

TECHNICAL PHYSICS Vol. 46 No. 9 2001 POROUS PIEZOELECTRIC COMPOSITES 1099

)2 FC FC × 4 posites [9Ð11, 13, 14] even at m ≈ 10%. A comparison max(Qh* 1Ð0Ð3 /(dh gh ) estimated by us at 8.24 10 with the calculated data for the 1–2–2 “PKR-7M–lam- for n = 0.10 and r = 0.99 (“PKR-7M filaments–elas- inated elastomer–araldit matrix” [18] shows that tomer matrix with plane-parallel pores” composite) * )2 ≈ * )2 * )2 ≈ )2 FC FC × (Qh 1Ð0Ð3 10max(Qh 1Ð2Ð2 at m = 2% and (Qh 1Ð0Ð3 also exceeds the value max(Qh* GT /(dh gh ) = 3.33 )2 104 obtained in [12]. Finally, for the “PKR-7M fila- 4 max(Qh* 1Ð2Ð2 at m = 5%. ments–polyurethane matrix with plane-parallel pores” M M 2 The interrelation between c /c (a = 1, 3), η* , ) 11 a3 elas h composite, max(Qh* 1Ð0Ð3 at n = 0.10 and r = 0.99 is no 2 2 6 Ð1 in (3), and the parameters (Q33* ) and (Qh* ) in (1) and higher than 7.20 × 10 (PPa) ; this slight rise (cf. data (2), which is observed in the composites with different in Table 3) in the piezoelectric sensitivity is explained pore configurations (Tables 2, 3), suggests that the dif- primarily by the fact the permittivity ε p of polyure- ference between the elastic properties of the FPCs and kk the matrix are essential for the composites to have a thane is smaller than that of the elastomer (Table 1). high piezoelectric sensitivity. The presence of porous layers in the homogeneous matrix or the formation of CONCLUSIONS plane-parallel pores in it (Fig. 1, insets 2, 4) causes a system of alternating compacted and rarefied nonpiezo- (1) Factors that govern the reception parameters electric regions to occur along the OX3 axis. Simulta- Q33* and Qh* of “porous FPC filaments–polymer M M matrix, FPC filaments–porous polymer matrix, and neously, the elastic moduli c33 and c13 of the heterogeneous matrix noticeably decrease in comparison with FPC filaments–porous laminated polymer matrix” were studied. It was shown that max(Q* )2 and max(Q* )2 c p and c p of the homogeneous polymer. As a result, 33 h 11 12 FC FC, E M M M M the configuration of the mechanical stresses within the strongly depend on e33 /c33 , c11 /c33 , and c11 /c13 η* η* εFC, σ ε p composite changes so that elas, h and elas, 33 in (3) and to a lesser extent on kk , kk , and the difference become larger than in the matrices shown in insets 2 FC, E p and 3 in Fig. 1. From Tables 2 and 3, it follows that between cab and cab . 2 the highest values of η* , ≈ 1, max(Q* ) , and (2) For the “PKR-7M filaments–elastomer matrix elas h h 1Ð0Ð3 with plane-parallel pores” composite, the values of )2 max(Q33* 1Ð0Ð3 are observed in the porous matrix )2 )2 max(Q33* 1Ð0Ð3 and max(Qh* 1Ð0Ð3 calculated indicate M M ӷ (Fig. 1, inset 4) with c11 /ca3 10 much higher than that our model involving the more effective redistribu- for the other matrices (Fig. 1, insets 2, 3). tion of internal mechanical stresses in the composite than that considered in [12] is valid. It was demon- The values of max(Q* )2 (Table 3) for the con- η h 1Ð0Ð3 strated that the coefficient elas* , h can be used as a quan- centration ranges 0.01 ≤ n ≤ 0.10 and 0.90 ≤ r ≤ 0.99 titative measure of this redistribution. The correlation exceed max(Q* )2 = 2.200 × 106 (PPa)Ð1, which has M M η )2 h GT between c11 /ca3 , elas* , h and max(Q33* 1Ð0Ð3 , been obtained [12] for “PZT-5AÐporous polyurethane max(Q* )2 that was established for the different matrix” composites at σM > 0. According to our esti- h 1Ð0Ð3 13 pore configurations and concentrations in the matrix mates, for a similar PZT-5A-based composite whose will help to optimize the search for FPC and polymer ≤ ≤ matrix contains plane-parallel pores with 0.01 n materials that provide high piezoelectric sensitivity of )2 0.10 and r = 0.99 (Fig. 1, inset 4), max(Qh* 1Ð0Ð3 = related composites. × 6 Ð1 )2 3.26 10 (PPa) > max(Qh* GT at n = 0.10, REFERENCES max(Q* )2 = 2.90 × 106 (PPa)Ð1 > max(Q* )2 at h 1Ð0Ð3 h GT 1. R. E. Newnham, D. P. Skinner, and L. E. Cross, Mater. )2 × 6 Ð1 n = 0.05, and max(Qh* 1Ð0Ð3 = 1.94 10 (PPa) < Res. Bull. 13, 525 (1978). )2 2. R. E. Newnham, Mater. Res. Bull. 22 (5), 20 (1997). max(Qh* GT at n = 0.01. Thus, the assumption that 3. F. Levassort, V. Yu. Topolov, and M. Lethiecq, J. Phys. D )2 max(Qh* GT is the upper theoretical limit of the hydro- 33, 2064 (2000). static sensitivity of the porous composite with σM > 0 4. H. Jensen, IEEE Trans. Ultrason., Ferroelectr. Freq. 13 Control 38, 591 (1991). needs correction. In our opinion, the authors of [12] have underestimated the effect of pores, which may 5. H. Taunaumang, I. L. Guy, and H. L. W. Chan, J. Appl. Phys. 76, 484 (1994). M M η* markedly decrease c13 and c33 and increase elas, h in 6. V. Yu. Topolov and A. V. Turik, J. Appl. Phys. 85, 372 (3) (see, e.g., Tables 2, 3). In addition, the ratio (1999).

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1100 TOPOLOV, TURIK

7. M. J. Haun and R. E. Newnham, Ferroelectrics 68, 123 19. W. Wersing, in Proceedings of the 6th International (1986). Symposium on Applications of Ferroelectrics, ISAF’86, 8. J. Mendiola and B. Jimenez, Ferroelectrics 53, 159 New York, 1986, p. 212. (1984). 20. H. Banno, Ceram. Bull. 66, 1332 (1987). 9. A. A. Grekov, S. O. Kramarov, and A. A. Kuprienko, 21. H. Banno, Jpn. J. Appl. Phys., Part 1 32 (9B), 4214 Mekh. Kompoz. Mater., No. 1, 62 (1989). (1993). 10. G. Hayward, IEEE Trans. Ultrason., Ferroelectr. Freq. 22. C.-W. Nan, J. Appl. Phys. 76, 1155 (1994). Control 43, 98 (1996). 23. M. Dunn, J. Appl. Phys. 78, 1533 (1995). 11. J. Bennett and G. Hayward, IEEE Trans. Ultrason., Fer- 24. V. I. Aleshin, Candidate’s Dissertation (Rostov. State roelectr. Freq. Control 44, 565 (1997). Univ., Rostov-on-Don, 1990). 12. L. V. Gibiansky and S. Torquato, J. Mech. Phys. Solids 25. V. Yu. Topolov and A. V. Turik, J. Phys. D 33, 725 45, 689 (1997). (2000). 13. L. Li and N. R. Sottos, J. Appl. Phys. 77, 4595 (1995). 26. F. Levassort, M. Lethiecq, D. Certon, and F. Patat, IEEE 14. S. Schwarzer and A. Roosen, J. Eur. Ceram. Soc. 19, Trans. Ultrason., Ferroelectr. Freq. Control 44, 445 1007 (1999). (1997). 15. V. M. Petrov, Zh. Tekh. Fiz. 57, 2273 (1987) [Sov. Phys. 27. Landolt-Börnstein: Zahlenwerte und Funktionen aus Tech. Phys. 32, 1377 (1987)]. Naturwissenschaften und Technik. Neue Serie (Springer- 16. V. Yu. Topolov and A. V. Turik, Pis’ma Zh. Tekh. Fiz. 24 Verlag, Berlin, 1984; 1990), Gr. III, Bd. 18; Bd. 28. (11), 65 (1998) [Tech. Phys. Lett. 24, 441 (1998)]. 28. E. I. Bondarenko, V. Yu. Topolov, and A. V. Turik, Kri- 17. V. Yu. Topolov and A. V. Turik, J. Electroceram. 3, 347 stallograﬁya 37, 1572 (1992) [Sov. Phys. Crystallogr. 37, (1999). 852 (1992)]. 18. V. Yu. Topolov and A. V. Turik, Pis’ma Zh. Tekh. Fiz. 27, 81 (2001) [Tech. Phys. Lett. 27 (2001)]. Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 46 No. 9 2001

GAS DISCHARGES, PLASMA

High-Pressure-Induced Baroelastic Effects and Martensitic Transformations in Two-Layer Micro- and Nanocomposites G. A. Malygin Ioffe Physicotechnical Institute, Russian Academy of Sciences, ul. Politekhnicheskaya 26, St. Petersburg, 194021 Russia e-mail: [email protected] Received December 28, 2000

Abstract—The theory of smeared diffusionless martensitic transitions is applied to analyzing martensitic transformation and relaxation of baroelastic stress in a thin shape-memory alloy layer included in a two-layer microcomposite. When omnidirectional pressure is applied to the composite, baroelastic stress arises in the alloy because of the different bulk compression moduli of the alloy and the substrate material. Baroelastic strain in the microcomposite undergoing martensitic transformation is found to acquire nonlinear and hysteretic properties, which can be used in pressure microtransducers and special-purpose miniature actuators. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION layer microcomposite under high pressure. Nonlinear In [1], martensitic transformation in a constrained characteristics exhibited by microcomposites with a shape-memory alloy (SMA) [2] is treated within the thin SMA layer can be used to develop pressure theory of smeared martensitic transitions. The con- microtransducers and special-purpose microactuators. straint arises, for example, in a two-layer microcom- Section 1 of this paper contains the basic formulas posite that has the form of a narrow strip and consists of 3 of the theory of smeared martensitic transitions, which a thin (50Ð10 nm thick) SMA layer applied on a are necessary to calculate the martensitic relaxation of ≈100-µm-thick substrate that does not undergo struc- tural transformation under the given conditions [3]. The the baroelastic stress that arises in SMAs from the dif- difference in the thermal expansion coefficients of the ference in the bulk compression moduli of the alloy and composite components causes thermoelastic stress in a the substrate material. In Section 2, the results of calcu- composite when its temperature varies. The theory of lation are outlined, and Section 3 is devoted to relax- smeared martensitic transitions [4, 5] makes it possible ation-induced size effects. to self-consistently calculate the stress relaxation through martensitic transformation taking place in the thin layer under temperature (and stress) variations and 1. MATHEMATICAL BACKGROUND thus to determine the strain in the strip [1]. We will consider a microcomposite in the form of a Recently, much attention has been given to thin-film Ӷ two- and multilayer microcomposites, since they offer narrow strip of length l and width w l composed of a promise for microtransducers and microactuators in thin SMA layer of thickness h and a substrate of thick- microelectromechanical systems (MEMS) [6]. SMAs ness H ӷ h. The substrate does not undergo the trans- are usually employed in these systems as active temper- formations in a given temperature range. Because of the ature [6–9], stress [5, 8], pressure [10, 11], and field- difference in the bulk compression moduli Ki (i = 1.2) (magnetic and electric) sensitive elements [12, 13]. of the composite components, placing the strip in a σ A variety of external and internal factors that can con- high-pressure vessel generates a baroelastic stress P = trol SMA properties provides the basis for designing ∆ε ∆ε Ð1 smart multifunctional microdevices with intriguing (1/3)Y1 in the thin SMA layer. Here, = (K1 Ð characteristics resulting from reversible martensitic Ð1 K )P is the difference in the volume compressive transformations in their active elements. 2 strains in the thin SMA layer and the substrate under It would be appropriate to calculate the characteris- ν ν tics of these devices using a theory of diffusionless pressure P; Y1 = E1/(1 Ð 1); E1, K1, and 1 are the martensitic transformations. The theory of smeared Young modulus, the bulk compression modulus, and martensitic transitions [4, 5] seems to be the most suit- Poisson’s ratio of the SMA layer; K2 is the bulk com- able and efficient tool for the calculation. In this study, pression modulus of the substrate. With the assumption it will be used to investigate the deformation of the two- that both materials are isotropic, the baroelastic stress is

1102 MALYGIN given by dominates in the crystal if ∆u > 0 (∆u < 0). From (3), the pressure range (smearing) of the transformation K Ð K Y σ δ δ 12 1 1 ∆ ϕ Ð1 ∆ P ==K P, K ---------. (1) PM0 = d /dP PP= , its hysteresis Pf 0, and the crit- 3 K K c0 2 1 σ ical pressure Pc0 can be expressed at = 0 as Similarly to the case of thermoelastic stress [12, 13], the strip with baroelastic stress bends to a radius of cur- ξ ∆ q 4 ∆ ± τ vature of R and its edges deflect from the initial plane PM0 ==δ------, P f 0 δ----- f , 0 B 0 by the distance z: (4) 2 2 2 σ q TTÐ c0 H Y2 l 3hl p P = ------. ------c0 δ R ===σ , z ------, (2) 0 T c0 6h P 8R 4 H Y2 ν ν ω where Y2 = E2/(1 Ð 2), E2, and 2 are the respective Here, B = q /kTc0 is the structure-sensitive parameter parameters of the substrate. defining the smearing of the transition. The curves for the direct and reverse martensitic transformations pro- As follows from (2), the deflection of the strip edges σ increases directly with baroelastic stress and, hence, ceeding in free conditions ( = 0) under the action of with pressure P. the dimensionless pressure P/K1 are shown in Fig. 1a. A rise in pressure (and stress) may initiate martensi- The curves were calculated according to formulas (3) tic transformation and thereby the relaxation of the with the dimensionless parameters B = 50, T/Tc0 = 1.15, δ ∆ τ τ baroelastic stress in the thin alloy. The reduction of the c = ( 0/q)K1 = 1.5, Pc0/K1 = 0.1, Pf 0/K1 = f /c M = τ τ τ ξ stress, in turn, changes the amount of martensite in the 0.013, f / M = 0.02, and M = q/ . alloy. To perform the self-consistent calculation of the pressure and temperature dependences of the baroelastic stress, as well as the magnitude of its relaxation, we 2. MARTENSITIC RELAXATION will make use of the theory of smeared martensitic tran- OF BAROELASTIC STRESS sitions [1, 4, 5]. According to the theory, for given temperature T, The reversible shear strains attendant to martensite ε ε ϕ ε ξ stress σ, and pressure P, the relative volume fraction of formation, M = m M, where m = m , cause the relax- ϕ martensite M in the material is defined as ation of the baroelastic stress in the thin SMA layer by σ σ ε ε σ M(T, , P) = Y1 m M(T, , P). Therefore, in view of (1), ∆U Ð1 ϕ ()T,,σ P = 1 + exp ------. (3a) we can write for the current stress in the layer: M kT σσ= ()P Ð σ()T,,σ P . (5) Here, ∆U = ω∆u; ω is the elementary volume of trans- P M formation, which depends on the bulk density of obsta- Since both parts of Eq. (5) contain the stress σ, a cles that hinder the motion of the phase boundaries; self-consistent solution should be found for every value of pressure P. It is convenient to reduce (5) to the equa- ∆ TTÐ c0 ξ()δστ± uq= ------Ð m f Ð 0P (3b) tion for the part of the stress that has relaxed: T c0 δ is the change in the internal energy of the SMA unit σ () ε   TTÐ c0 0 volume as the alloy passes from the austenitic to the M P = Y1 m1 + expB------Ð -----P T c0 q martensitic state; q and Tc0 are, respectively, the heat of (6a) ε τ Ð1 transition and the critical (characteristic) temperature mδ σ ()± f    of transition at σ = τ = P = 0; ξ is the spontaneous Ð ----- KP Ð M P τ------   . f q M shear deformation of the lattice undergoing martensitic transformation; m is the crystallographic orientational Introducing the dimensionless stress SM = factor of the martensite type most favorable for relax- |σ | |ε | τ M /Y1 m , pressure p = P/K1, and temperature t = T/Tc0, ation of the baroelastic stress; f is the dry friction δ |δ | stress, which arises upon moving the phase boundaries and putting K = Ð K in (6a), one can rearrange and specifies the force hysteresis of the transformation; Eq. (6a) to the form δ and 0 is the lattice dilatation upon transformation. ()    () Expressions (3) suggest that the amount of marten- SM p = 1 + expBt Ð 1 Ð caÐ p site in the alloy is a function of both the absolute value ∆ (6b) and the sign of the energy u. Therefore, it also τ Ð1 depends on the crystal temperature, as well as on the ()() ± f    Ð bSM p Ð SM0 τ------   , stress and pressure applied. Austenite (martensite) M

TECHNICAL PHYSICS Vol. 46 No. 9 2001 HIGH-PRESSURE-INDUCED BAROELASTIC EFFECTS 1103

ϕ where M(P) 1.5 ε ε2 (a) a ==-----m δ K , b -----mY , q K 1 q 1 (6c) δ a K K1 () 1.0 --- ==------ε ------, SM0 SM 0 . b m Y1 2 The graphical solution of Eq. (6b) is illustrated by 0.5 Fig. 2, where straight line 1 shows the dependence 1 L(SM) on the left-hand side of the equation and curves 2Ð4 depict the functions R(SM) on its right-hand side for different dimensionless pressures with B = 50, c = 1.5, 0 a = 0.7, b = 0.07, and τ = 0. The intersection of line 1 f ( ) with curves 2Ð4 yields the amount of the martensitic SM P 1.5 relaxation of the baroelastic stress under a given pres- (b) sure. The solutions of Eq. (6b) for direct and reverse martensitic transformations are presented in Fig. 1b by 1.0 curves 1 and 2, respectively. Bearing in mind that ≡ ϕ SM(p) M(p), one can treat the curves as the pressure dependences of the amount of martensite emerging in 2 1 the constrained alloy during transformation. When 0.5 Fig. 1b is compared with Fig. 1a, it is apparent that ϕ under constraint (1), the curves M(p) shift to higher pressures; (2) the martensitic transformation occupies a narrower temperature range, i.e. the smearing 0 0.1 0.2 0.3 decreases; and (3) the hysteresis loop widens. P/K1 In fact, the differentiation of Eq. (6b) with respect to ∆ Fig. 1. Pressure dependences of the relative volume fraction p yields the dimensionless values of smearing pM, ∆ of martensite at (1) direct and (2) reverse martensitic transi- hysteresis 2 pf, and critical (characteristic) pressure pc tions under (a) free conditions and (b) under constraint. The of the transition in the following form: dashed curves depict the transformation in the absence of hysteresis. 4 Ð bB 1 τ ∆ p ==------, ∆ p ±------f- , M ()caÐ B f caÐ τ M (7) Eqs. (5) and (6a) with respect to P yields the sensitivity t Ð1 Ð 0.5b (at P = Pc) pc = ------. caÐ σ δ Ð δ ε ()Y /4q B d = ------K 0 m 1 . (9) ------ε2 () Putting a = b = 0 in (7), we arrive at the expressions dP PP= c 1 Ð m Y1/4q B for martensitic transformation under free conditions (4). ε In the absence of martensitic transformation ( m = 0), δ According to (5), the dimensionless total stress the sensitivity depends only on K, given by (1), developing in the thin SMA layer at a given pressure is whereas during the martensitic relaxation of the deﬁned as baroelastic stress, it is governed by the combination of the parameters δ , ε , and B, which are related to the σ()p a 0 m () () () thermodynamics and the kinetics of the transition. ------ε - ==Sp, Sp --- pSÐ M p . (8) Y1 m b Figure 3 illustrates the associated dependence for 3. SIZE EFFECTS direct and reverse martensitic transformations (curves 1 In [3], the size effect observed in a thin NiTi alloy and 2, respectively). It is evident that microcomposites during the martensitic relaxation of thermoelastic stress with the active SMA layer acquires nonlinear and hys- was associated with the effect of the layer thickness h teretic properties, which are absent in those without the on the relaxation parameters. As the thickness SMA layer. decreased from 1 µm to 50 nm, the transformation One more point to mention is that in the pressure became more and more smeared and the degree of range of martensitic transformation, the sensitivity of stress relaxation progressively lowered. It was argued stress to pressure changes the sign. Differentiating [1] that the size effects may arise because the elemen-

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1104 MALYGIN

L(SM), R(SM) S(P) 1.5 2 1 1 1.0

4 3 2 1

0.5 2

012 0 0.1 0.2 0.3 / SM P K1 Fig. 3. Pressure dependences of the baroelastic stress for Fig. 2. Dependences of (1) left-hand and (2Ð4) right-hand (1) direct and (2) reverse martensitic transformations. The σ ε sides of Eq. (6b) on the stress SM = M/Y1 m for the pressure dashed curve corresponds to the transformation in the P/K1 = (2) 0.1, (3) 0.15, and (4) 0.2. absence of hysteresis.

S(P) ∆p(h), ∆S(h), Q(h) 2 0.50 1

2 1 1 0.25

0 0.1 0.2 0.3 0510 15 20 P/K λ 1 h/ m Fig. 5. Dependences of (1) the pressure range of stress Fig. 4. Pressure dependences of the baroelastic stress for relaxation, (2) the amount of stress relaxation, and (3) the λ ӷ SMA layers with the thickness h/ m (1) 1 and (2) = 1.5. coefﬁcient Q on the layer thickness. tary volume of transformation ω becomes dependent on ation is smaller. Taking into account that dσ/dP = ε the alloy thickness when the latter is comparable to the (Y1 m/K1)(dS/dp), the sensitivity can be expressed as average distance λ between obstacles limiting the m 4aÐ bcB() h mobility of the phase boundaries. Then, for the param- dS ==Qh() ------. (11) ω ------b[]4 Ð bB() h eter B ~ , we have [1] dp pp= c

λ The dependence of Q(h)/Qm, where Qm is the value () h/ m Bh = Bm------λ , (10) of Q(h) in the thicker layer, is shown in Fig. 5 by 1 + h/ m curve 3. ӷ λ As follows from (11), the sensitivity Q < 0 when 4 > where Bm is the value of B in a thick (h m) layer. bB > 4a/c, goes to zero at B = Bc = 4a/bc, and may The pressure dependences of the baroelastic stress become positive in a very thin layer. In the opposite λ for thick (B = Bm = 50) and thin (B = 30, h/ m = 1.5) lay- case of a relatively thick layer, Q also becomes positive ers are presented in Fig. 4 (curves 1 and 2, respec- when the conditions bB > 4 and bB > 4a/c are met tively). It is seen that in the thinner layer, the sensitivity simultaneously. In this situation, in addition to the main of stress to pressure in the region of martensitic relax- hysteresis observed in the S(p) curves (Figs. 3, 4), a

TECHNICAL PHYSICS Vol. 46 No. 9 2001 HIGH-PRESSURE-INDUCED BAROELASTIC EFFECTS 1105 local loop should appear in the region of martensitic smeared martensitic transitions allows the calculation transformation, still further underlying the bistability of of the deformation characteristics of microdevices the curves. based on a microcomposite with a thin SMA layer sub- Let us ﬁnd the amount of martensitic relaxation of jected to high pressure. With the parameters of marten- the baroelastic stress, i.e., the stress difference ∆σ = sitic transformation shown in Figs. 3 and 4, the operat- σ σ σ σ σ σ 1 Ð 2, where 1 = (P1), 2 = (P2), and P1 and P2 ing pressure range where the microcomposite can be correspond, respectively, to the maximum and mini- used as a pressure microtransducer or microactuator is ≈ mum of the curve σ(P). Differentiating Eq. (8) with (0.1Ð0.2)K1 1Ð20 GPa. respect to dimensionless pressure p in view of (6b) and equating the derivative of the total stress to zero, REFERENCES dS/dp = 0, we ﬁnally obtain the following system of 1. G. A. Malygin, Fiz. Tverd. Tela (St. Petersburg) 43, 822 two equations for dimensionless stresses S1 and S2 and (2001) [Phys. Solid State 43, 854 (2001)]. pressures p1 and p2: 2. K. Shimizu and K. Otsuka, Shape Memory Effects in τ a± f c a z1 1 Alloys (Nauka, Moscow, 1979) (translated from Japa- S12, = ---t Ð 1 τ------Ð ------Ð ------ln------, (12a) nese). b M 1 + z , bB z caÐ 12 2 3. A. L. Roytburd, T. S. Kim, Q. Su, et al., Acta Mater. 46, τ 5095 (1998). − f b 1 z1 1 p12, = t Ð 1 + ------Ð ------Ð --- ln------, (12b) 4. G. A. Malygin, Fiz. Tverd. Tela (St. Petersburg) 36, 1489 τ 1 + z , B z caÐ M 12 2 (1994) [Phys. Solid State 36, 815 (1994)]. where 5. G. A. Malygin, Zh. Tekh. Fiz. 66 (11), 112 (1996) [Tech. Phys. 41, 1145 (1996)]. 2 1/2 1α ± 1α α bc 6. S. M. Spearing, Acta Mater. 48, 179 (2000). z12, ==--- Ð 1 --- Ð 1 Ð 1 , ------B. (12c) 2 2 a 7. Materials for Smart Systems II, Ed. by E. P. George, R. Gotthardt, K. Otsuka, et al. (Materials Research Soci- According to (12), the amount of stress relaxation ety, Pittsburg, 1997), Vol. 459. ∆S and the pressure range ∆p where it occurs are 8. J. E. Bidaux, W. J. Yu, R. Gotthardt, and J. A. Manson, 1 1/2 z c J. Phys. IV 5 (C2), 453 (1995). ∆SS==Ð S --- ()αα()Ð 4 Ð ln----1 ------, (13a) 1 2 α z caÐ 9. S. P. Belyaev, S. A. Egorov, V. A. Likhachev, and 2 O. E. Ol’khovik, Zh. Tekh. Fiz. 66 (11), 36 (1996) [Tech. Phys. 41, 1102 (1996)]. ∆ 1 c z1 ()αα()1/2 b pp==2 Ð p1 α------ln---- Ð Ð 4 ------.(13b) 10. T. Kikashita, T. Saburi, K. Kindo, and S. Endo, Jpn. J. a z2 caÐ Appl. Phys. 36, 7083 (1997). Since B = B(h), Eq. (12c) implies that α = α(h); 11. R. D. James and K. F. Hane, Acta Mater. 48, 197 (2000). therefore, both the amount of relaxation and the associ- 12. G. G. Stoney, Proc. R. Soc. London, Ser. A 82, 172 ated pressure range depend on the SMA layer thick- (1909). ness. The functions ∆S and ∆p [see (13)] are plotted in 13. L. E. Andreeva, Elastic Elements in Devices (Mashinos- Fig. 5 (curves 1 and 2, respectively). troenie, Moscow, 1981). In conclusion, we have demonstrated that, as in the case with thermoelastic stress [1], the theory of Translated by A. Sidorova-Biryukova

TECHNICAL PHYSICS Vol. 46 No. 9 2001

GAS DISCHARGES, PLASMA

Interaction of a Long Pulsed High-Density Electron Beam with a Jet of Solid-Target Destruction Products A. V. Arkhipov and G. G. Sominski St. Petersburg State Technical University, St. Petersburg, 195251 Russia e-mail: [email protected], [email protected] Received December 6, 2000

Abstract—Results are presented from experimental studies of the destruction of a solid target by a high-density nonrelativistic electron beam at a deposited power density of 20 MW/cm2 in millisecond pulses. Results of studying beam transportation under these conditions are presented as well. © 2001 MAIK “Nauka/Interperi- odica”.

INTRODUCTION vapor and other destruction products of the surface material is formed in front of the surface. This layer can High-density electron beams are traditionally used substantially attenuate the electron beam. The vapor, to for the transportation of energy to solid objects and for a greater or lesser extent, can be ionized thermally or controlled energy release in the surface region. Besides via individual interactions of atoms with the beam elec- classical applications of electron beams, such as electrons. Due to the long duration of irradiation pulses, the tron-beam welding, cutting, and nonthermal hardening role of slow processes (such as heat conductivity, diffu- of metals and alloys, we can mention attempts using sion, and expansion of the vapor into a vacuum) electron beams in nuclear fusion, the excitation of increases substantially. Since the theoretical descrip- shock waves, the generation of intense X radiation, etc. tion of nonequilibrium systems including a solid body, Practical applications stimulated intensive studies in an electron beam, and a multicomponent plasma is different ranges of the power density transported by the rather complicated, an experimental study of the inter- electron beam. Note, however, that the beams with action between a solid target and an electron beam in extreme parameters have been studied most exten- the given parameter range is of great importance. sively. One of these cases is typical of technological applications: the relationship between the deposited energy density and the area of the irradiated region is EXPERIMENTAL TECHNIQUES such that, during the melting and evaporation of the material to be processed, the vapor density in front of Experiments were carried out on the EPVP device the target is insufficiently high to significantly hamper of St. Petersburg State Technical University [12Ð15]. electron transport toward the target surface [1Ð3]. On A schematic diagram of the experimental device is the other hand, in inertial fusion research [4, 5], the typ- shown in Fig. 1. An electron beam with an electron ical power densities are so high that the target material energy up to 45 keV and a current up to 5 A was pro- is completely and rapidly (in several nanoseconds or duced with a Pierce gun, transported through the drift even faster) converted into an overdense, highly ionized channel in a pulsed magnetic field of a solenoid, and plasma that absorbs the electron beam energy. then deposited on a solid target in a mode of long (from tens to thousands of microseconds) single pulses. The processes occurring at intermediate beam power densities have been studied less intensively. The One of the most serious engineering problems that practical necessity of studying these processes arose in we faced in our study was to ensure the electric strength the early 1990s in connection with some new applica- of the high-voltage gap of the electron gun in the pres- tions, such as the electronic ablation acceleration of ence of vapor and plasma produced due to the intense macroparticles up to extremely high velocities [6, 7] (in destruction of the target. The solution to this problem particular, for the power supply and diagnostics in large was found empirically. tokamaks [8, 9]) and the modeling of the processes In the first version of the device, the distribution of occurring on the first walls and divertors of such toka- the magnetic field B(z) near the electron gun was con- maks during discharge disruptions [10, 11]. In these trolled by an individual solenoid placed near the anode applications, the beam power density is typically on the plane. The aim of the optimization was to match the order of several units to several tens of MW/cm2 at a electric and magnetic field lines in order to minimize nonrelativistic electron energy in millisecond current electron losses at the entrance to the transport channel. pulses. Under these conditions, a layer consisting of However, it turned out that with this matching, the

INTERACTION OF A LONG PULSED HIGH-DENSITY ELECTRON BEAM 1107

12 34 567 8

10 9

Fig. 1. Schematic diagram of the EPVP device: (1) cathode and (2) anode of the electron gun, (3) magnetic screen, (4) solenoid of the transport channel, (5) sections of the transport channel, (6) target, (7) solenoid of the target, (8) X-ray detectors, (9) optical detectors (fibers), and (10) electron beam. problem of the electric strength of the high-voltage gun levels of the beam current and accelerating voltage, we gap was aggravated. In contrast, if the magnetic field of could obtain high values of the specific energy charac- the gun solenoid compensated for the field penetrating teristics: the power density deposited on the target into the gun from the transport-channel solenoid, the attained 20 MW/cm2, and the total energy density per electric breakdown occurred at higher values of the pulse attained tens of kJ/cm2. energy load on the target. This observation showed that To obtain information on the parameters of the elec- the magnetically shielded gun was better protected tron beam produced and its interaction with a medium from the penetration of the plasma from the target side: formed in the device volume, we measured the elec- it was protected by the transverse (radial) components trode currents, signals from optical detectors positioned of B in the transient region, where the magnetic field at different points of the drift channel, and signals from lines arrived at the walls of the transport channel, thus detectors of soft X radiation emitted from the vicinity guiding the expanding plasma to it. For this reason, of the target. The dynamics of target destruction was when modifying the experimental device, we placed a studied by measuring the shape and dimensions of cra- 40-mm-thick ferromagnetic shield with a large-diameters (Fig. 2) produced by single current pulses of differ- ter aperture (20 mm) between the gun and the transport ent duration τ . For this purpose, the irradiated region channel. This ensured a relatively slow increase in the 0 magnetic field along the z axis and, consequently, a was displaced over the sample surface before every shot fairly long transient region. This innovation enhanced by varying the inclination of the axis of solenoid 7 the electric strength of the system, although it posed (Fig. 1). some additional problems: an appreciable fraction of Most of the results presented below were obtained the beam could be lost at the entrance to the channel. for graphite targets. This choice is motivated by the fact We also protected the gun from the neutral component that carbon materials are commonly used in the produc- of the jet of the destruction products of the target. The tion of electrodes undergoing high energetic loads and target was placed a large distance away (about 1 m), also by the convenience of operation with this material: which allowed us, with refractory targets, to substan- graphite is nontoxic, does not produce irreversible poi- tially decrease the density of the vapor neutral compo- soning of cathodes, has a low vapor pressure, and does nent near the gun. not form a liquid phase. Due to the described modifications of the experimental device, the proper choice of the optimum oper- EXPERIMENTAL RESULTS ating modes, and the training of the components of the AND DISCUSSION electronic-optical system that underwent the intense bombardment (see [14, 15] for details), we prevented 1. Dynamics of the Target Destruction breakdowns in the electron gun. The maximum dura- in the Low-Current Mode tion of the beam-current pulse (~5 ms) was only limited In the course of experiments, we measured the by the parameters of the solenoid power supply system. dependence of the mass M of the material lost by the The electrostatic focusing in the gun in combination target during one electron-irradiation pulse on the total with an additional compression by the magnetic field energy transported by the electron beam: made it possible to compress the area of the electron τ beam by a factor of ~1500. Thus, at rather moderate wU= 0 I 0, (1)

TECHNICAL PHYSICS Vol. 46 No. 9 2001

1108 ARKHIPOV, SOMINSKI

d, mm numerical values corresponding to the experiment, the (a) 0.2 empirical formulas [3] give ~5 and ~15 µm for the mean and maximum penetration depths, respectively. As is seen from the relief of the graphite surface, the (b) 0.4 material leaves the target in the form of grains (monoc- 0.2 rystals). We believe that, at low deposited energy densities, the loss of the target material is caused by the cracking of the surface layer due to nonuniform thermal expansion [11]: only at w > wt does the material begin (c) to evaporate, which is reflected as a bend in the M(w) 0.2 dependence. Intense sublimation occurs only after the surface layer of thickness dt becomes heated to the critical temperature: for graphite, this temperature is equal (d) 0.2 to ~4000 K [16]; for metals, this is the temperature of the triple point. This occurs at –1.2 –0.8 –0.4 0 0.4 0.8 1.2 ≈ ρ ∆ r, mm ww= t dtS c p T, (2) where S is the irradiated area, ρ is the mass density of Fig. 2. Typical shapes of the craters produced by a pulsed ∆ electron beam in different sublimation regimes for U0 = the target material, cp is its specific heat, and T is the 36 kV and B = 0.8 T: (a) the irradiation dose is insufficient difference between critical and room temperature. τ for the beginning of sublimation (I = 0.8 A, 0 = 0.24 ms, w ω = 6.9 J, and M = 31 µg); (b) the beam current is below the Determining the value of the threshold dose t from τ discharge threshold (I = 0.38 A, 0 = 1.6 ms, w = 21.6 J, and the plots, we can estimate dt. The values obtained in this M = 323 µg); (c) the target is screened by the discharge (I = ≥ µ τ η ≈ µ manner (dt 100 m) substantially exceed the calcu- 0.96 A, 0 = 1.6 ms, w = 56.2 J, i 1.8, and M = 908 g); (d) the degree of target screening is substantial (I = 2.6 A, lated electron penetration depth and depend on the τ η ≈ µ 0 = 1.45 ms, w = 135 J, i 10, and M = 240 g). deposited power density. They are close to the spatial scale length characteristic of thermal conductivity (σ τ)1/2, where σ is the thermal diffusivity and τ is the M, µg t t time needed to reach the dose wt for given parameters 400 of the electron beam. wt The slope of the dependence M(w) at w > w corre- 300 t sponds to the specific energy spent on graphite destruction, ~40 kJ/g (or 5 eV/atom), which agrees well with 200 the results of the thermodynamic analysis performed in [17, 18]. In particular, it was shown there that, soon 100 after the beginning of evaporation, a time-independent distribution of all quantitative characteristics of the vaporÐsolid system (including the temperature) is 0510 15 20 , J w established in the reference frame related to the phase Fig. 3. Mass of material evaporated from the graphite target boundary. This occurs because the propagation velocity vs. the total energy of the pulsed electron beam in the of the heat conduction wave becomes equal to the absence of a discharge. velocity with which the phase boundary is displaced. As a result, the heat conduction ceases to affect the where U is the voltage accelerating the electrons and I value of the specific energy spent on sublimation and 0 the material (graphite) mass evaporated per unit time is is the current of the beam fraction reaching the target. related to the power P of the energy source (for an elec- At given values of U0 and I, the energy of the beam was tron beam, this is U I) by a simple equation: varied by varying the pulse duration. 0 dM/dt= P/()∆H + c ∆T (3) The destruction of the graphite target begins at very subl p low values of the energy deposited on its surface or 2 (~10 J/cm ). In a typical plot of the dependence M(w) Mww= ()Ð /()∆H + c ∆T , (4) (Fig. 3), we can notice two almost linear segments sep- t subl p arated by a characteristic bend at w = wt. At w < wt, the where Hsubl is the specific sublimation energy of graph- thickness of the evaporated layer of the target material ite. dp depends slightly on the irradiation dose and is equal Since the term determining the energy spent on sub- to 5Ð30 µm for the entire surface irradiated by electrons limation is dominant, it is expected that (Fig. 2a). This value is comparable with the electron ≈ ∆ penetration depth into the target. After substituting the dM/dw 1/ Hsubl. (5)

TECHNICAL PHYSICS Vol. 46 No. 9 2001 INTERACTION OF A LONG PULSED HIGH-DENSITY ELECTRON BEAM 1109 ∆ However, for graphite, the value of Hsubl is not tions would lead to a stronger interaction of the beam uniquely determined [11]: it is equal to 59 kJ/g for with vapor and to a noticeable absorption of its energy, atomic carbon sublimation, and it is only 30 kJ/g for which was not observed. sublimation in the form of three-atom aggregations. The value of dw/dM obtained in the experiment is nearly the average of these results, probably because of 2. Influence of the Discharge Phenomena on Beam the complicated composition of the formed vapor. Transportation and on the Dynamics of Target Destruction The effect of surface-layer cracking, which causes the destruction of the target at small deposited energies, Under our experimental conditions, the neutral ceases to be important after sublimation begins. This component of the jet of the target destruction products can be explained, e.g., by the short sublimation time of is insufficiently dense for the parameters of the passing the destroyed layer of thickness dp (we assume that electron beam to change appreciably. However, effi- dp > dt) cient screening of the target can be detected due to the interaction of the beam with the ionized component of τ ≈ ρ∆ p d pS Hsubl/P, (6) the target material vapor if an RF discharge develops in the system. The value of the threshold beam current I* which is equal to several microseconds under our causing the RF discharge ignition depends on the elec- experimental conditions. The produced powder has no tron energy, the magnitude and distribution of the mag- time to leave the region irradiated by electrons and, netic field (which determines the irradiated area), and until it is completely evaporated, hampers the energy the target material. For example, for graphite at eU0 = transfer to the subsequent target layers. For this reason, 36 keV and B = 0.8 T (which corresponds to r ≈ 1 mm), the specific destruction energy does not decrease. 0 the current I* is equal to ~0.8 A, whereas at B = 0.4 T Good agreement between the experimental value of (r0 ~ 1.5Ð2 mm), this current is equal to 1.5 A. Typical dM/dw and the predictions of thermodynamic theory waveforms obtained for I > I* are shown in Fig. 4. Dur- allows us to conclude that, in this case, we do not ing a rather long period of time (τ* = 30Ð1000 µs, observe a substantial screening of the target by the depending on the experimental conditions), the signals products of its destruction, in spite of the density of the remain constant. Then, the current is redistributed and evaporated material in front of the target being very a new quasi-steady state is established. This state is high. This density can be estimated as characterized by two effects: substantial noise components with frequencies from 1 MHz and higher appear n = PN /µSv ∆H , (7) 0 A j subl in all the current signals, and a great number of slow µ particles with energies no higher than several electron- where NA is Avogadro’s number, is the material molar mass, and v is the vapor jet velocity. volts arrive at the electrodes. The current of positive j ions onto the wall of the transport channel can be sev- According to [17, 18], the jet velocity depends only eral times higher than the injected beam current. The slightly on the parameters of the energy source and, slow plasma electrons escape from the device volume over a wide range of parameters, is almost equal to the along the magnetic field lines onto the magnetic shield. thermal velocity at the critical temperature. For graph- During the transient process, the detectors of the inte- ite in the case of atomic sublimation, the mean value of gral optical radiation indicate the expansion of the ≈ the normal velocity component is vj 3.5 km/s. Near high-density plasma along the transport channel at a the target surface, at a distance z less than the size of the irradiated region r0, the vapor density appears to be τ* ≈ × 19 Ð3 n0 5 10 cm for the maximum deposited power 0 density. Based on a simple model of the expansion of Is 0.5A the products of target destruction (atomic sublimation, I isotropic expansion within a solid angle of 2π, a high i adhesion rate of vapor to the surfaces of surrounding It electrodes, and the low degree of their ionization that I eliminates the influence of the magnetic field), the d decrease in the material density with distance from the 0.5A target at z > r0 can be assumed to be approximately quadratic. In this case, the total amount of the evaporated 0 material in front of the target under steady-state condi- µ tions will be equivalent to a solid layer ~1 m in thick- 0 400 800 1200 1600 t, µs ness, which is significantly smaller than the electron penetration depth. Therefore, it is not surprising that the Fig. 4. Typical time dependences of the target current I and target screening was not observed in the experiment. t transport-channel current Id in the presence of a discharge Note that the violation of any one of the above condi- that screens the target.

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1110 ARKHIPOV, SOMINSKI

Is, A discharge to be ignited, it is necessary that, in front of the target, a sufficiently extended plasma be formed 1.0 I* with the density corresponding to the efficient interaction with the electron beam (on the order of the beam electron density) and with a density gradient that is not too high. For such a plasma to be formed, the vapor 0.5 should expand with the thermal velocity at least over a distance of several centimeters from the target. The loss of the beam energy during the beam propagation through the discharge region is substantial and 0 0.5 1.0 1.5 2.0 Ii, A increases with increasing current. Figure 5 shows the steady-state value of the fraction of the beam current Fig. 5. Steady-state electron-beam current reaching the tar- reaching the target, Is, as a function of the electron gun get, Is, vs. the injected current Ii. In the absence of a discharge (at I < I*), these currents coincide. current Ii, which varies only slightly during the pulse. When the injected current exceeds the threshold value M, µg I*, the current that reaches the target under quasi-steady conditions decreases abruptly severalfold and contin- 1500 wt ues to fall as Ii further increases. This means that the current loss in the developing discharge increases with Ii more rapidly than linearly. In the pulses characterized 1000 η by a high current loss factor I = Ii/Is, the decrease in the η X-ray signal from the vicinity of the target, r , is even 500 η η more pronounced ( r > I), which can be explained by a decrease in the mean energy of electrons reaching the target. 050100 150 200 w, J Target screening by the discharge affects the dynam- Fig. 6. Dependence M(w) for a lower magnetic field (B = ics of target disruption. Figure 6 shows the dependence 0.4 T) and the beam current exceeding the threshold dis- M(w) obtained at a reduced magnetic field B = 0.4 T charge current by 5Ð10%. and the beam current Ii exceeding the threshold discharge current I* (which is equal to ~2 A in this case) M, µg by nearly 5Ð10%, which corresponds to the current loss η wt factor I = 1.5Ð2. To determine the irradiation dose w, we used the value of the injected current I . The form of 300 i the dependence is similar to that shown in Fig. 3, but its ≈ 200 slope at w > wt corresponds to the value dw/dM 80 kJ/g; i.e., the sublimation efficiency reduces by one- 100 half. Figure 7 shows similar data obtained at B = 0.8 T for 1.5I* < I < 2.6I*. As compared to the case illustrated in Fig. 3, the specific sublimation energy dw/dM at 03060 90 120 w, J w > wt increases substantially (by a factor of 10Ð20). Another specific feature of beam transportation in Fig. 7. Dependence M(w) for B = 0.8 T and the beam currents substantially (by a factor of 1.8Ð2.6) exceeding the the presence of a discharge was the increase in both the threshold discharge current. crater size and the degree of its asymmetry (Figs. 2c, 2d); sometimes, the prints even had an annular shape [13]. These effects may be attributed to a lower effi- velocity of ~106 cm/s. The time at which the plasma ciency of the magnetic confinement of the beam in the front arrives at the electron gun nearly coincides with presence of the RF field of the discharge. In addition, the time at which the system comes to a quasi-steady the electrons at the periphery of the beam have a better state. chance of reaching the target and transfer a higher energy onto the target surface than the axial electrons, All these observations allow us to interpret the phe- for which the trajectory of motion toward the target nomenon observed as a beamÐplasma discharge in the passes through the discharge region. We can assume ionized plume of the target-destruction products. The that, using an electron beam with a larger cross section fact that the time at which intense target sublimation (for the same density and particle energy and an begins, wt/U0I, does not coincide with the time of dis- increased total current), we would observe even more charge ignition τ* can be explained as follows. For the efficient screening of the target by the discharge

TECHNICAL PHYSICS Vol. 46 No. 9 2001 INTERACTION OF A LONG PULSED HIGH-DENSITY ELECTRON BEAM 1111 because of a lower relative contribution from the into the target) attains 0.1Ð1 m/s and the vapor density peripheral regions. in front of the target attains 3 × 1019 cmÐ3. In the interaction of an electron beam with a metal 3. Experiments with Metal Targets target, the dynamics of the melt filling the crater plays In addition to the experiments with graphite an important role. described above, we attempted to carry out similar experiments with metal (copper) targets. As was ACKNOWLEDGMENTS expected, the methods used in the previous experiments This work was supported in part by the Russian for studying the formation of craters were difficult to Foundation for Basic Research (project no. 98-02- employ because copper had a liquid phase. The craters 18323) and by the Ministry of Education of the Russian formed were surrounded by a parapet whose volume Federation (a grant from the 1997 Competition in the could be several times greater than the volume of the Field of Nuclear Engineering and Physics of Ionizing material that left the target, which reduced the accuracy Radiation). of measuring this volume. In addition, a substantial portion of the material left the target as drops of up to REFERENCES several tenths of a millimeter in size. Such drops were observed through the entire transport channel, which 1. V. V. Bashenko, Electron-Beam Facilities (Mashinostro- indicates a rather high velocity. Accordingly, the quan- enie, Leningrad, 1972). 2. S. Schiller, U. Heisig, and S. Panzer, Elektronenstrahl- tity of the material that left the target was several times technologie (Technik, Berlin, 1976; Énergiya, Moscow, greater than the estimates from formula (4). 1980). The features of electron-beam transport in experi- 3. I. A. Abroyan, A. N. Andronov, and A. I. Titov, Physical ments with a metal target do not differ qualitatively Principles of Electronic and Ionic Technology from those observed for graphite. Here again, when the (Vysshaya Shkola, Moscow, 1984). current exceeded the threshold level I*, we observed 4. Generation and Focusing of High-Current Relativistic the beamÐplasma discharge absorbing a substantial Electron Beams, Ed. by L. I. Rudakov (Énergoatomiz- fraction of the initial electron beam energy; however, dat, Moscow, 1990). other conditions being the same, the value of I* for cop- 5. V. P. Smirnov, Prib. Tekh. Éksp., No. 2, 7 (1977). 6. J. Linhart, in The Physics of High Energy Density, Ed. by per turned out to be lower. P. Caldirola and H. Knoepfel (Academic, New York, 1971; Mir, Moscow, 1974). CONCLUSIONS 7. F. V. Bunkin and A. M. Prokhorov, Usp. Fiz. Nauk 119, In summary, one of the main results of this study is 425 (1976) [Sov. Phys. Usp. 19, 561 (1976)]. the production of an electron beam with high values of 8. S. L. Milora, J. Vac. Sci. Technol. A 7, 925 (1989). 9. B. V. Kuteev, Zh. Tekh. Fiz. 69 (9), 63 (1999) [Tech. differential energy parameters (a power density up to Phys. 44, 1058 (1999)]. 2 20 MW/cm and a density of the energy transferred in a 10. B. Bazylev, L. Landman, and H. Wurz, in Proceedings of single millisecond pulse up to 100 kJ/cm2), which were the Conference “Physics and Technique of Plasma,” sustained under conditions when the beam-forming Minsk, 1994, p. 451. system was affected by the target-destruction products. 11. V. Engelko, A. Andreev, O. Komarov, et al., in Proceed- The study of the physical processes occurring when ings of the 11th International Conference on High- such a dense, long, pulsed beam interacts with a target Power Particle Beams, Prague, 1996, Vol. II, p. 793. have allowed us to draw the following conclusions: 12. A. V. Arkhipov and G. G. Sominski, Pis’ma Zh. Tekh. Fiz. 20 (11), 6 (1994) [Tech. Phys. Lett. 20, 431 (1994)]. The steady state of this interaction is reached in tens or 13. A. V. Arkhipov and G. G. Sominski, in Proceedings of even hundreds of microseconds. Such high values of the the 11th International Conference on High-Power Parti- delay times can be determined by thermal processes in the cle Beams, Prague, 1996, Vol. 2, p. 789. surface layers of the target or by the motion of particles 14. A. V. Arkhipov and G. G. Sominski, in Proceedings of evaporated from the surface over distances comparable to the 18th International Symposium on Discharges and the characteristic scale length of the problem (e.g., the Electrical Insulation in Vacuum, Eindhoven, 1998, beam diameter or the length of the device). Vol. 2, p. 762. 15. A. V. Arkhipov and G. G. Sominski, IEEE Trans. The only effect hampering beam propagation Dielectr. Electr. Insul. 6, 491 (1999). toward the target under our experimental conditions is 16. Handbook of Physical Quantities, Ed. by I. S. Grigor’ev the beamÐplasma discharge in the flow of the evapo- and E. Z. Meœlikhov (Énergoatomizdat, Moscow, 1991). rated material. This effect results in a substantial (up to 17. I. V. Nemchinov, Prikl. Mat. Mekh. 31 (2), 300 (1967). 90Ð95%) absorption of the energy of the electron beam. 18. Yu. V. Afanas’ev and O. N. Krokhin, in The Physics of For the beam current below the threshold discharge High Energy Density, Ed. by P. Caldirola and current, the beam propagates toward the target without H. Knoepfel (Academic, New York, 1971; Mir, Moscow, appreciable losses even under conditions such that the 1974). evaporation velocity (i.e., the velocity at which the phase boundary of the solid and gas phases propagates Translated by N. Larionova

TECHNICAL PHYSICS Vol. 46 No. 9 2001

GAS DISCHARGES, PLASMA

The Effect of Ultraviolet Irradiation on the Properties of Zr/Si Contacts I. S. Nuprienok and A. N. Shibko Institute of Electronics, Belarussian Academy of Sciences, Minsk, 220841 Belarus e-mail: [email protected] Received December 13, 2000

Abstract—The effect of ultraviolet (UV) irradiation on the properties of Zr/Si contacts subjected to heat treatment is investigated. It is established that, due to variations in the phase composition of the contact, its electrophysical parameters are modiﬁed. The application of the combined treatment allows one to form Zr/Si contacts with speciﬁed properties. © 2001 MAIK “Nauka/Interperiodica”.

Thin-film materials are widely used in semiconduc- chosen from the results of tentative experiments. At tor devices. In modern microelectronics, semiconduc- annealing temperatures T > 500°C, the Zr/Si system tor elements based on refractory metals (including zir- was oxidized, whereas at T < 500°C, the phase modifi- conium) are considered as promising high-temperature cations in it were virtually absent. Therefore, tempera- building blocks. They offer the electrophysical proper- tures above 500°C are of interest in investigating phys- ties of the metals and the high melting points of their ical processes occurring in the system under the joint oxides and silicides. One of the methods modifying the action of the treatments. electrophysical parameters of a metalÐsemiconductor The samples were examined by electron diffraction system is heat treatment, which changes electron states using a JEM-120 electron microscope and by electron at the interface. In polycrystalline films, heat treatment spectroscopy for chemical analysis (ESCA). The stimulates recrystallization processes, which bring the Schottky barrier height was determined from the IÐV system to a thermodynamically more equilibrium curves by the method described in [5]. The Zr/Si con- state. Recently, however, nonthermal processes that 2 occur in metalÐsemiconductor systems subjected to tact area was equal to 0.7 mm . heat treatment and simultaneously irradiated by a pho- To obtain deeper insight into the phase transforma- ton beam with a certain energy have attracted much tions in the Zr/Si system, we performed a similar treat- attention [1Ð3]. ment of thin zirconium films. For the purity of experi- In this study, we investigated the phase transforma- ment, the Zr films were deposited on the as-cleaved sur- tions and modifications of the electrophysical parame- face of single-crystal NaCl in a single process cycle ters in the Zr/Si system subjected to thermal treatment with the Zr/Si system. Next, they were removed from and simultaneously exposed to UV radiation with λ = the NaCl crystal surface and placed on a molybdenum 0.312, 0.365, and 0.552 µm. mesh. Zr films with a thickness of ≈80 nm were deposited It was found that the heat treatment (T = 500°C) of onto Si(111) n-type substrates by electron-beam evap- the Zr films causes the oxide phase to appear. It is worth oration at a pressure of 3 × 10Ð5 Pa. During the process, the substrate temperature was kept at 373 K. Prior to 3 deposition, the silicon wafer was chemically cleaned using the method described in [4]. The films grown had fine-grained polycrystalline structure with average grain sizes of 15Ð20 nm. 5 7 6 The samples were placed into a vacuum chamber 1 2 4 where they were heat-treated at a pressure of 5 × 10−5 Pa at a temperature of 500°C for 1, 5, 10, 15, and 30 min. As a UV source, we used a DRSh-250 mer- curyÐquartz lamp. The energy density of incident radiation was E = 0.01 J/cm2. Its value was monitored by Setup for sample treatment: (1) DRSh-250 mercuryÐquartz an IMO-2 power meter. The desired wavelengths were lamp, (2) semitransparent mirror, (3) IMO-2 power meter, obtained with the help of special light filters. Our setup (4) light filter, (5) focusing lens, (6) vacuum chamber, and is shown in the figure. The annealing temperature was (7) annealing furnace.

THE EFFECT OF ULTRAVIOLET IRRADIATION 1113 noting that when the treatment time was 1Ð5 min, Table 1. Modification of the phase composition in the Zr/Si ° oxides with a small amount of oxygen, ZrO0.35 or ZrO, system under thermal annealing at T = 500 C were formed. However, when the annealing time was τ, min 10Ð30 min, the electron diffraction patterns contained d, Å the reflections of zirconia ZrO2 alone. 1 5 10 15 30 Under the joint action of thermal annealing and UV irradiation in a vacuum, the phase composition of the 3.61 ÐÐÐÐZrSi2 films is modified. The zirconia phase appears even at 3.28 ÐÐÐÐZrSi2 annealing times of 5 and 1 min for the UV irradiation 3.21 Ð Ð Zr Si ÐÐ wavelengths λ = 0.312 and 0.552 µm, respectively. For 5 3 λ = 0.365 µm, zirconia appears only after annealing for 2.92 Ð ZrO2 ZrO2 ZrO2 ZrO2 30 min. Note that the kinetics of the phase transforma- 2.81 ZrO0.35 ZrO0.35 ÐÐÐ tions was virtually independent of the energy density of 2.79 Zr Ð Ð Ð the incident radiation. The change in the phase composition of the Zr films under the treatments can be repre- 2.67 ZrO ZrO Ð Ð Ð sented in the following way: 2.60 ZrO0.35 ZrO0.35 ÐÐÐ (1) heat treatment (T = 500°C) without UV irradia- 2.58 Zr ÐÐÐÐ tion: 2.56 Ð Ð Ð ZrSi ZrSi 1 min Zr ZrO0.35; 2.55 Ð Ð Zr3Si2 Zr3Si2 Ð 5 min 10Ð30 min 2.53 Ð ZrO ZrO ZrO ZrO ZrO ZrO0.35; ZrO2 ZrO2; 2 2 2 2 2.48 ZrO ZrO ÐÐÐ (2) heat treatment with irradiation (λ = 0.312 µm): 0.35 0.35 2.46 Zr ÐÐÐÐ 1 min 5Ð30 min Zr ZrO0.35; ZrO ZrO2; 2.40 Ð Ð Zr3Si2 Zr3Si2 Ð (3) heat treatment with irradiation (λ = 0.365 µm): 2.34 Ð Ð Zr5Si3 ÐÐ 2.30 ÐÐÐÐZrSi2 Zr1 min Zr; ZrO 5 min ZrO ; 0.35 0.35 2.29 Ð Ð Ð ZrSi ZrSi ZrO10Ð15 min ZrO30 min ZrO ; 2 2.27 Ð Ð Zr5Si3 ÐÐ λ µ (4) heat treatment with irradiation ( = 0.552 m): 1.86 ÐÐÐÐZrSi2

1 min 5Ð30 min 1.80 Ð ZrO2 ZrO2 ZrO2 ZrO2 Zr ZrO; ZrO2 ZrO2. 1.63 ZrO ZrO Ð Ð Ð The growth of the zirconium oxides is stimulated by the reaction of the film with oxygen that adsorbs during 1.53 Ð Ð Ð ZrSi ZrSi the deposition and diffuses from the environment. 1.50 Ð ZrO2 ZrO2 ZrO2 ZrO2 Since zirconium is a getter, the reaction between the 1.43 Ð Ð Zr Si Zr Si Ð film heated to 500°C and the oxygen proceeds with a 3 2 3 2 high rate. From our results, it appears that during the 1.39 ZrO ZrO Ð Ð Ð combined treatment with λ = 0.365 µm, the chemical Note: d is the interplanar spacing; τ is the annealing time. activity of oxygen decreases and the oxidation reaction slows down. This is explained by the fact that the zirconium film absorbs photons having an energy of 3.39 eV. by the metal (Zr5Si3 and Zr3Si2). An increase in the Our data also indicate that when the Zr/Si system is annealing time to 15 min results in the formation of annealed without UV irradiation, the oxide phases with ZrSi; and to 30 min, of ZrSi2. different contents of oxygen (ZrO , ZrO, and ZrO ) 0.35 2 The combined vacuum treatment insignificantly and metal-enrinched silicides (ZrSi3, Zr3Si2, ZrSi, and ZrSi ) form on the surface. Their appearance depends modifies the phase composition of the surface layer 2 (Table 2). However, note the appearance of the oxide on the annealing time (Table 1). As was noted, the λ µ τ oxides arise when zirconium combines with the oxygen Zr3O at = 0.365 m and = 5 min. The higher oxide that adsorbs on the film during deposition and diffuses ZrO2 appears only at a heat treatment time of 30 min. from the environment. As is seen from Table 1, after Also, at τ = 30 min, the reflections of ZrSi appear but annealing for 1 min, the electron diffraction patterns those of ZrSi2 disappear. The combined treatment with also contain reflections of the initial Zr. The formation λ = 0.312 and 0.552 µm (the photon energies are 3.96 of the silicides depends on the diffusion of silicon to the and 2.25 eV, respectively) does not change the phase sample surface and its interaction with the Zr film. At composition in the Zr/Si system as compared with ther- an annealing time of 10 min, the silicides are enriched mal annealing alone (Table 2).

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1114 NUPRIENOK, SHIBKO

Table 2. Modiﬁcation of the phase composition in the Zr/Si system under combined treatment at T = 500°C and E = 0.01 J/cm2 τ, min λ, µm 1 5 10 15 30

Thermal annealing Zr, ZrO0.35, ZrO ZrO, ZrO0.35, ZrO2 ZrO2, Zr5Si3, Zr3Si2 ZrO2, ZrSi, Zr3Si2 ZrO2, ZrSi, Zr3Si2 without irradiation

0.312 ZrO, ZrO0.35 ZrO, ZrO2 ZrO2, ZrSi, Zr3Si2 ZrO2, ZrSi ZrO2, ZrSi2 0.365 ZrO, ZrO0.35 Zr3O, ZrO0.35, ZrO ZrO, Zr5Si3 ZrO, Zr5Si3, Zr3Si2 ZrO2, ZrSi, Zr3Si2 0.552 ZrO, ZrO0.35 ZrO2, Zr5Si3 ZrO2, Zr3Si2 ZrO2, ZrSi, Zr3Si2 ZrO2, ZrSi2

Table 3. Modiﬁcation of the phase composition in the Zr/Si system under combined treatment at T = 500°C and E = 0.05 J/cm2 τ, min λ, µm 1 5 10 15 30

0.312 ZrO, ZrO0.35 ZrO2 ZrO2, ZrSi, Zr3Si2 ZrO2, ZrSi2 ZrO2, ZrSi2 0.365 Zr, ZrO0.35 ZrO, ZrO0.35 ZrO, Zr5Si3 ZrO, Zr3Si2 ZrO2, ZrSi 0.552 ZrO, ZrO2 ZrO2, Zr3Si2 ZrO2, Zr3Si2, ZrSi ZrO2, ZrSi, Zr3Si2 ZrO2, ZrSi2

Table 4. Modiﬁcation of the electrophysical parameters of the Zr/Si contact under combined treatment τ ϕ ϕ ϕ , min 1, eV 2, eV 3, eV U1, eV U2, eV U3, eV n1 n2 n3 As-prepared 0.57 6 1.16 Thermal annealing 1 0.55 6 1.18 5 0.55 7 1.17 10 0.57 8 1.14 15 0.58 9 1.13 30 0.59 11 1.10 E = 0.01 J/cm2 1 0.56 0.55 0.56 6 6 8 1.17 1.18 1.18 5 0.56 0.55 0.57 7 7 8 1.15 1.17 1.16 10 0.58 0.56 0.58 10 7 9 1.12 1.16 1.13 15 0.59 0.57 0.59 11 8 10 1.10 1.14 1.12 30 0.60 0.57 0.60 12 8 11 1.09 1.14 1.10 E = 0.05 J/cm2 1 0.57 0.55 0.57 10 6 10 1.12 1.18 1.12 5 0.57 0.55 0.58 10 7 12 1.12 1.17 1.09 10 0.58 0.56 0.59 12 7 13 1.09 1.16 1.08 15 0.60 0.56 0.60 14 8 14 1.07 1.16 1.07 30 0.60 0.56 0.61 14 7 15 1.07 1.15 1.06 Note: τ is the treatment time; ϕ, Schottky barrier height; U, breakdown voltage; and n, ideality factor. Subscripts: 1, λ = 0.312; 2, 0.365; and 3, 0.552 µm.

By increasing the energy density of the incident down the oxidation process and the formation of the radiation to 0.05 J/cm2, the surface layer oxidizes more silicides. As follows from Tables 2 and 3, the formation rapidly (Table 3). At λ = 0.312 and 0.552 µm, the higher of zirconia takes place when the annealing time τ = τ λ µ zirconium oxide ZrO2 forms when = 5 and 1 min, 5 min or more ( = 0.312 and 0.552 m). Photons with respectively. The UV irradiation of the Zr/Si system at hν = 3.96 and 2.25 eV affect the chemical activity of λ = 0.365 µm (the photon energy hν = 3.39 eV) slows oxygen, increasing the oxidation rate. Because of the

TECHNICAL PHYSICS Vol. 46 No. 9 2001 THE EFFECT OF ULTRAVIOLET IRRADIATION 1115 gettering properties of zirconium, its deposition is and the density of electron traps is reduced. However, accompanied by the dissolution of the residual gases oxygen diffusion makes the dielectric layer thicker. including oxygen. The process of deposition goes in This, in turn, raises the breakdown voltage. As the parallel with the ionization of the soluble elements by annealing time increases, so does the Schottky barrier the metallic solvent. In the lattice of the metals, the cat- height. During the thermal annealing of the Zr/Si sys- ions of light elements are formed and their valence tem, the density of surface states on the silicon at the electrons pass into the collective state. The residual metalÐsemiconductor contact changes, which affects oxygen in the surface layer seems to be responsible for the barrier height. The ideality factor n of the IÐV char- the lower oxides ZrO0.35 and ZrO. The ordered solid acteristic is given in Table 4. The value of n was deter- solution of oxygen, Zr3O, is associated with the redis- mined from the experimental IÐV curves [5]. It is shown tribution of the oxygen cations over tetrapores. that, when the annealing time increases, the ideality During the irradiation of the heated Zr/Si system factor decreases. This is associated with the change in with a photon beam of a particular flow density, the the dielectric parameters of the oxide layer at the inter- radiation is absorbed by the oxygen present in the surface and with that in the concentration of the electron face layer. This absorption is due to the transition traps under the thermal treatment. Thus, as the annealing time increases, the IÐV curves of the Zr/Si contact 1+Σ 3ÐΣ g g , which is described by the diagram of tend to their ideal shape. The modification of the elec- energy levels for certain states of O2 [6]. It should be trophysical parameters can be related to the modifica- noted that the flow density of photons with the given tion of the phase composition of the interface: from energy is insufficient to neutralize the chemical activity Zr/SiO2/Si for the as-prepared sample to ZrxOy/ZrnSim of the oxygen. This density stimulates the formation of or ZrSi/SiO2/Si for the sample treated. the oxides in the surface layer of the Zr/Si system under Thus, the combined treatment of the Zr/Si system combined treatment. To suppress the oxidation, a makes it possible to form rectifying contacts with the higher energy density of the incident radiation is desired phase composition and, hence, with the desired required, because thermal annealing causes the oxygen electrophysical properties. to interact with the zirconium surface. Therefore, to neutralize the M–O bonds, a higher flow density of the photons is necessary. Under UV irradiation, the photo- REFERENCES chemical processes in the surface layer are associated 1. D. T. Alimov, V. K. Tyugaœ, P. K. Khabibulaev, et al., Zh. with the electron transitions in the dissolved and Fiz. Khim. 61, 3065 (1987). adsorbed oxygen. The electron diffraction data corre- 2. A. M. Chaplanov and A. N. Shibko, Kvantovaya Élek- late with those obtained by the ESCA method. tron. (Moscow) 20, 191 (1993). The change in the phase composition of the Zr/Si 3. A. M. Chaplanov and A. N. Shibko, Izv. Akad. Nauk system results in the modification of the electrophysical SSSR, Ser. Fiz. 53, 1111 (1989). parameters of the contact. The IÐV characteristic of the 4. Handbook of Thin-Film Technology, Ed. by L. I. Maissel initial contact is asymmetric, and the breakdown volt- and R. Glang (McGraw-Hill, New York, 1970; Sov. age is 6 V. This indicates that there is natural silicon Radio, Moscow, 1977), Vol. 1. dioxide SiO2 between the Zr film deposited and the sil- 5. E. H. Rhoderick, MetalÐSemiconductor Contacts icon, as demonstrated by the ESCA data. The electro- (Claredon, Oxford, 1978; Radio i Svyaz’, Moscow, physical parameters after the thermal and combined 1982). treatments are listed in Table 4. The annealing 6. J. G. Calvert and J. N. Pitts, Jr., Photochemistry (Wiley, decreases the concentration of different defects in the New York, 1966; Mir, Moscow, 1968). oxide film at the metal–semiconductor interface. The structure of the SiO2 tends to a more equilibrium state, Translated by Yu. Vishnyakov

TECHNICAL PHYSICS Vol. 46 No. 9 2001

Technical Physics, Vol. 46, No. 9, 2001, pp. 1116Ð1120. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 71, No. 9, 2001, pp. 49Ð53. Original Russian Text Copyright © 2001 by Gorelov, Mel’nikov, Ogenko, Shalyapina.

GAS DISCHARGES, PLASMA

Suppressed Diffusion of Metal Atoms upon Excitation of Slightly Damped Plasmons B. M. Gorelov, V. S. Mel’nikov, V. M. Ogenko, and G. M. Shalyapina Institute of Surface Chemistry, National Academy of Sciences of Ukraine, Kiev, 252022 Ukraine e-mail: [email protected] Received December 5, 2000

Abstract—In the YBa2Cu3O7Ðδ superconductor, the excitation of slightly damped plasmons upon the nonuniform heating of electron gas in a microwave electromagnetic ﬁeld and upon perturbation of the charge density by the thermal desorption of oxygen atoms from the Cu1ÐO plane suppresses surface and volume diffusion of nickel and gold interstitials. The interstitial atoms aggregate in the surface layer of the crystallites. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION the l and h carriers located in nonoverlapping quasi- two-dimensional bands acquire different velocities In high-temperature superconductors (HTSCs), whose structures are characterized by a combination of eE ˜ () 0 ω wide and narrow bands [1, 2], low-frequency collective vhl, t = Ð------cos 0t , m*, ω υ , excitations, acoustic plasmons, may exist in the heavy- hl 0 hl carrier (h) subsystem. Under normal conditions, due to because m* ӷ m* (m* is the effective mass of the l light- (l) and h-carrier Landau damping, the number of h l l plasmons is apparently small, their propagation in the carriers) [4]. The l-carrier Landau damping decreases crystal is hardly probable, and their experimental with increasing field amplitude. At a certain E0, the col- observation is difficult. However, slightly damped plas- lective excitation modes of the h carriers with phase mons propagating in an HTSC with the velocities velocities () Ӷ Ӷ () Ӷ Ӷ v˜ h t u v˜ l t (3) vFh u vFl (1) can propagate in the crystal nondissipatively. or In order to excite plasmons with velocities satisfy- ӷ u vFh, vFl (2) ing (2), oxygen atoms in YBa2Cu3O7 are removed from the Cu1ÐO chains by heating. When the 01 atoms are may manifest themselves in the subthreshold formation δ thermally desorbed, the perturbations Zie of the elec- of defects, the aggregation of a large number of defects tron concentration appear near the resulting vacancy in in the surface crystallite layer, and suppressed diffusion δ the interlayers ( Zi is the effective perturbation charge of metal atoms. Here, vFh and vFl are the Fermi veloci- Ω near the ith 01 vacancy). The perturbations produce the ties of h and l carriers, u = q/q are the excitation rates, shielding effect, which increases the spectral density of Ω2 π 2 = 4 e nh/,mh* nh and m* are the concentration and the plasmon states [5, 6]: the effective mass of h carriers, and q is the wave vec- 1 Ð1 tor. Of special interest is the suppression of volume dif- S()ω, t = Ð--- V ()q Imε ()q, ω . fusion of nickel atoms [3]. If this is the case, the sup- π∑ c pression of metal diffusion may indicate the presence iq, πδ 2 2 ω of variously excited slightly damped plasmons. Here, Vc(q) = 4 Zie /q is the permittivity, N( ) = In order to initiate slightly damped excitations, S (ω, T)f(ω, T)dωdr is the number of slightly external fields can be used. These fields must not cause ∫∫ defect formation, as well as the transport and aggrega- damped plasmons, and f(ω, T) is the BoseÐEinstein ω ӷ tion of atoms in the surface layer. Also, they must not distribution function with vFhq, VF, lq. Ӎ suppress diffusion. Under normal conditions, vFh vFl. It is worth noting that the numbers of slightly Therefore, a microwave electromagnetic field can be damped plasmons N(ω) probably differ for the two employed to stimulate plasmons with velocities satisfy- ways of excitation. In the microwave field, the plas- ω ω Ӷ υ ing (1). In an electric field E = E0sin 0t at 0 h, l mons form only in the skin layer, where Landau damp- υ ( h, l are the collision frequencies of h and l carriers), ing is weak and the plasmons are excited by local field

1063-7842/01/4609-1116 $21.00 © 2001 MAIK “Nauka/Interperiodica” SUPPRESSED DIFFUSION OF METAL ATOMS 1117 effects. In the case of thermal desorption of 01 atoms, respectively, 104 W, 2.5 µs, and 400 Hz. Then, the sam- the plasmons are excited in the whole volume due to ples were placed in the antinode of the microwave elec- enhanced electronÐplasmon interaction. However, irre- tric field of a single-pass cavity. spective of the way of excitation, the propagation of the slightly damped plasmons in an YBa Cu O crystal Desorption of the oxygen atoms from YBa2Cu3O7 Ð δ 2 3 7 was accomplished by heating the samples with δ = 0 at may make itself evident in the formation of defects, the ° transport and aggregation of atoms in the surface layer, a rate of 5 C/min. We removed the 01 atoms in amounts ≈ × 21 × 21 Ð3 and suppressed diffusion. Note that both microwave of 1.7 10 and 2.6 10 cm , which corresponds δ Ӎ irradiation and the oxygen thermal desorption do not to 0.3 and 0.45, respectively. The amount of the directly generate defects (except for 01 vacancies), oxygen removed was determined by X-ray diffraction since the energy of a microwave field photon and the analysis and from data on oxygen absorption due to kinetic energy of the 01 atoms are small compared with heating with a Q-1500 derivatograph. the energy required for Y, Ba, Cu, or O(2, 3) atoms to Water molecules were adsorbed on YBa2Cu3O6.55 leave the lattice sites (5.5Ð20 eV [7Ð9]) and with the samples (preheated in a vacuum at ≈140°C for 2.5 h) same energy for the 01 and 04 atoms (4.5Ð10 eV [9]). from saturated steam at room temperature for 360 min. The energy of the microwave photon is also smaller The annealing of water molecules was performed by than their migration energy, which excludes the aggre- heating the samples to 400°C. gation of the atoms near the surface and the suppression effect. However, in YBa2Cu3O7 Ð δ, oxygen absorption The diffusion coefficient of nickel and gold atoms and desorption may induce the diffusion of Ba and Cu was measured by the radioactive tracer (63Ni and 195Au) atoms to and from the surface, respectively [10]. This method with successive layer removal. The diffusion to may enhance or weaken the suppression effect induced a depth of 150Ð250 µm proceeded in air at temperatures by collective excitations. In addition, the desorption of of 200Ð500°C for 5Ð45 h. The concentrations were 01 atoms destroys the band formed by the pd orbitals of determined with a step of 3Ð5 µm. the 01, 04, Cu1, and Cu2 atoms where the h carrier excitations may propagate. The desorption may even eliminate high-temperature superconductivity because RESULTS AND DISCUSSION of a decrease in the number of holes in the cuprate layers. Therefore, the excitation of slightly damped plas- The temperature dependences of the diffusion coef- mons is possible only in the range of oxygen concentra- ficient for the gold atoms are shown in Fig. 1. In the as- δ prepared samples, the slow and rapid components of tion 0 < < 0.3, where the effect of high-temperature ° superconductivity is likely to take place [11]. surface diffusion between 200 and 410 C are The aim of this work is to observe the effect of sup- s × Ð11 () Ds = 2.8 10 exp Ð0.07/kT pressed diffusion of metal atoms in YBa2Cu3O7 Ð δ when slightly damped acoustic plasmons are excited by microwave irradiation or the thermal desorption of –logD[cm2 s–1] 01 atoms. Ni and Au atoms were used as diffusants. The 7 diffusion of these atoms has both surface and the volume components [12, 13]. This made it possible to observe the suppression of the volume diffusion and the 3' arrest of surface diffusion. 2' EXPERIMENTAL We studied polycrystalline single-phase (according 1' to X-ray diffraction analysis) YBa2Cu3O7 samples with a density of 5.5 g/cm3 and a grain size of 5Ð15 µm. They were produced by solid-state synthesis from a mixture 2 of Y2O3, Ba2CO3, and CuO powders. The lattice 11 3 parameters were a = 3.821 Å, b = 3.889 Å, and c = 11.667 Å. The parameter η = (C01 Ð C05)/(C01 + C05) 1 was equal to 0.09 (C01 and C05 are the oxygen concentrations in the 01 and 05 states, respectively). This 1.5 2.0 parameter characterizes the filling of these states with 103/T, K–1 oxygen [14]. Fig. 1. Temperature dependences of the diffusion coeffi- The samples were exposed to a pulsed microwave cient for Au atoms (1, 1') before and (2, 2') after 10-min (9.4 GHz) field for 10 min at room temperature. The microwave irradiation and (3, 3') of thermal desorption of pulse amplitude, duration, and repetition rate were, oxygen atoms.

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1118 GORELOV et al.

R(T)/R(300) defects and the aggregation of a large number of atoms 1.2 in the surface layer of the grains. Note that after microwave irradiation, the dc resistance R near the normal-superconducting transition 2 increases, whereas the transition temperature Tc does not vary (Fig. 2). The irradiation does not influence the 1 lattice parameters a and b, whereas the parameter c either remains constant (after 1 min of exposure) or 0.4 decreases by 0.022 Å (after 5 and 30 min of exposure). The parameter η varies from 0.05 to 0.36. The increase in R means that the microwave irradiation generates extra defects, since [6] 100 200 300 T, K 4π Ð1 Ð1 RT()= ------[]τ ++τ ()T γ()T . (8) Ω2 d ph q Fig. 2. Temperature dependences of the YBa2Cu3O7-resis- pl tance (1) before and (2) after microwave irradiation for Ω τ 10 min. Here, pl is the plasma frequency of the l carriers, d τ and ph are the times of scattering by defects and phonons, and γ is the coefficient of damping of the l and q carriers by acoustic plasmons. Note also that τÐ1 ~ T, r × Ð9 () ph Ds = 1.9 10 exp Ð0.13/kT . (4) γ γ 2 q ~ T for nondegenerate and q ~ T for generate h car- For volume diffusion [13], which is observed at tem- riers. The increase in R at Tc < T < 150 K is caused by ≥ peratures of the thermal desorption of the 01 atoms, T the rise in both τÐ1 and the number of defects. On the 410°C, d other hand, the migration of oxygen between the 01 and s Dv = 6.6exp()Ð 1.24/kT 05 states with an energy of activation of 2.03 eV [15] indicates that microwave irradiation triggers the low- and energy mechanism of defect migration. The formation r Ð2 and redistribution of the defects may have no effect on Dv = 1.9× 10 exp()Ð1.08/kT . (5) Tc, which depends on the density p of holes [16]: The microwave irradiation arrests the volume diffu- , () []()2 sr T c p = T cm 1Ð 82.6 p Ð 0.16 , (9) sion of Au and suppresses both Dv components. In addition, the preexponentials and the energy of activa- where Tcm is the maximal value of Tc at a constant num- tion of the surface diffusion coefficient grow. The com- ber of holes in the cuprate layers, which are responsible ponents of the surface diffusion coefficient are given by for high-temperature superconductivity. s1 × Ð10 () Ds = 8.0 10 exp Ð0.17/kT The effect of vacancies of the 01 atoms on the diffu- and sion of Au atoms is similar to that of microwave irradiation. The desorption of the 01 atoms in an amount of r1 × Ð7 () 1.7 × 1021 cmÐ3 suppresses the volume diffusion of the Ds = 1.8 10 exp Ð0.18/kT (6) Au atoms and increases both the preexponential and the in the temperature range 200Ð305°C, and by energy of activation of the surface diffusion: s2 × Ð7 () Ds = 2.3 10 exp Ð0.45/kT s3 × Ð10 () Ds = 4.0 10 exp Ð0.16/kT and and r2 × Ð6 () Ds = 2.0 10 exp Ð0.30/kT (7) r3 × Ð6 () at T ≥ 305°C (Fig. 1, curves 2, 2'). Note that the irradi- Ds = 1.45 10 exp Ð0.24/kT (10) ation affects neither the temperature nor the rate of thermal desorption of the 01 atoms from the bulk of (Fig. 1, curves 3, 3'). The introduction of the vacancies in an amount of 2.6 × 1021 cmÐ3 into the Cu1ÐO chains YBa2Cu3O7. This follows from the fact that the curves describing weight losses upon heating the irradiated has a similar effect on the diffusion of the Ni atoms and the as-prepared samples are identical, which is (Fig. 3). In the as-prepared YBa2Cu3O7 samples, the indicative of the constant oxygen concentration in the diffusion has a surface component samples. The arrest of volume diffusion upon the × Ð10 () microwave irradiation implies the formation of volume Ds = 3.16 10 exp Ð0.17/kT

TECHNICAL PHYSICS Vol. 46 No. 9 2001 SUPPRESSED DIFFUSION OF METAL ATOMS 1119 ° in the temperature range 200Ð410 C and a volume –logD[cm2 s–1] ∆ component – m, % 0 Ð2 1 Dv = 1.0× 10 exp()Ð0.13/kT 2 at T > 410°C [12]. Once the oxygen has been desorbed, the volume diffusion of Ni is suppressed and the surface 3 diffusion is characterized by the slow and rapid compo- 1.5 nents, 3' 7 0 800 r4 × Ð6 () 2' T, °C Ds = 2.0 10 exp Ð0.21/kT and 4' s4 × Ð9 () Ds = 1.4 10 exp Ð0.27/kT (11) in the whole temperature range (Fig. 3, curves 2, 2'). These surface components are apparently due to diffu- 3 sion in various crystallographic directions [17]. Note that the effect of the slightly damped plasmons excited 4 by the perturbations δZe on the grain surface on Ni dif- 2 fusion is similar [3]. 11 The arresting effect shows up after the desorption of 1 01 atoms in amounts of more than 0.9 × 1021 cmÐ3. This may be achieved by heating to T > 500°C. The desorption of a lesser number of 01 atoms or the generation of 1.5 2.0 103/T, K–1 a lesser number of perturbations ∑ δ Z e upon heating i i Fig. 3. Temperature dependences of the diffusion coeffi- to T Ӎ 500°C does not affect the diffusion of the Ni and cient for Ni atoms (1) before and (2, 2') after thermal des- Au atoms. In addition, in the YBa2Cu3O7 Ð δ samples with orption of oxygen atoms, (3, 3') after adsorption of water δ molecules for 360 min, and (4, 4') after subsequent anneal- = 0.3 and 0.45, the diffusion is accompanied by oxy- ing of water at 400°C. The insert shows the temperature gen absorption in the range 200 < T ≤ 350°C and oxy- δ dependences of the YBa2Cu3O7 Ð δ weight change. = (1) gen desorption at T > 400°C (Fig. 3, insert). However, 0, (2) 0.3, and (3) 0.45. the formation of oxygen vacancies in the planes Cu1ÐO in an amount of ≈2.6 × 1021 cmÐ3 and their filling are rs, the activation energy of Dv and are independent of the not reflected in the temperature dependences of Ds concentration of the oxygen vacancies in the Cu1ÐO and the volume diffusion Dv does not recover in this layers inside the grains. case. At the same time, atom diffusion due to oxygen desorption and adsorption during the formation of the The suppression of the volume diffusion, which can diffusion profile [10] seems to insignificantly affect the be described by the expression [18] Dv = D0(1 Ð number of atoms in the surface layer. This indicates that q)exp(ÐE/kT) (here, q = m/M, m is the number of filled the arresting effect after both microwave irradiation and interstices, and M is the total number of interstices), oxygen desorption is caused by atoms that are trans- may be explained by an increase in the factor q when ferred to the surface with a rate far exceeding the diffu- the interstices are filled by defects in the surface layers sion rate at the temperatures of diffusion profile forma- of the grains, as well as by the fact that Dv 0 for tion. This follows from the fact that this effect takes q 1. The rise in the preexponentials in the expres- , place after the 10-min irradiation or heating for no more sions for Drs can also be explained by filling the inter- than 50 min at a rate of 5°C/min. The transfer may be s associated with the migration of atoms in the electric stices with the defects. For instance, the introduction of ≈ × 20 Ð3 field of the propagating weakly damped plasmons [3]. water molecules in an amount of 1.4 10 cm , which occupy the interstices of the YBa Cu O lattice Thus, both the microwave irradiation and the intro- 2 3 6.55 duction of oxygen vacancies into the Cu1ÐO chains [19], results in an increase in the preexponentials for s r affect the diffusion of Au and Ni atoms in a similar way. Ds and Ds and weakly increases the activation energy: Namely, they arrest volume diffusion, suppress the sr, r5 × Ð6 () components Dv and Dv , and increase the preexponen- Ds = 6.3 10 exp Ð0.30/kT s tial and activation energy of surface diffusion. The Ds and and Dr components of the surface diffusion have close s s5 × Ð9 () activation energies that are substantially smaller than Ds = 7.6 10 exp Ð0.22/kT (12)

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1120 GORELOV et al. (Fig. 2, curves 2, 2'). Surface diffusion is nearly com- REFERENCES pletely recovered annealing of the water molecules: 1. G. P. Shveœkin, V. A. Gubanov, A. A. Fotiev, et al., Elec- r6 × Ð6 () tron Structure and Physicochemical Properties of High- Ds = 1.2 10 exp Ð0.26/kT Temperature Superconductors (Nauka, Moscow, 1990). 2. High-Temperature Superconductivity. Fundamental and Research and Applications, Ed. by A. A. Kiselev (Mash- inostroenie, Leningrad, 1990). Ds6 = 1.6× 10Ð9 exp()Ð0.20/kT ; (13) s 3. B. M. Gorelov, Zh. Éksp. Teor. Fiz. 116, 586 (1999) [JETP 89, 311 (1999)]. however, the volume component Dv does not recover (Fig. 2, curves 3, 3'). Consequently, the localization of 4. A. F. Alexandrov, L. S. Bogdankevich, and A. A. Ru- defects in the interstices increases both the preexponen- khadze, Principles of Plasma Electrodynamics (Vysshaya Shkola, Moscow, 1978; Springer-Verlag, sr, tial and the activation energy of Ds . Berlin, 1984). 5. É. A. Pashitskiœ, Fiz. Nizk. Temp. 21, 995 (1995) [Low In the case of diffusion by two types of interstices Temp. Phys. 21, 763 (1995)]. [18], 6. É. A. Pashitskiœ, Fiz. Nizk. Temp. 21, 1092 (1995) [Low sr, 2 λ + qµ Ð K Temp. Phys. 21, 837 (1995)]. D = αl ω ------exp()ÐE/kT , (14) s µ2 7. R. C. Baetzold, Phys. Rev. B 38, 11304 (1988). q 8. N. N. Degtyarenko, V. F. Elesin, and V. L. Mel’nikov, where α is the geometric factor, l is the transfer length, Sverkhprovodimost: Fiz., Khim., Tekh. 3, 2516 (1990). ω is the oscillation frequency of the atom in an inter- 9. V. V. Kirsanov, N. N. Musin, and E. I. Shamarina, Sverkhprovodimost: Fiz., Khim., Tekh. 7, 427 (1994). 2 stice, µ = 1 Ð ε, K = ()λ + 3qµ Ð 12qµ , λ = 1 + 2ε, 10. S. I. Sidorenko, K. I. Barabash, S. M. Volos’ko, and ε = exp[(u0 Ð uT)/kT], and u0 and uT are the potential B. V. Egorov, Metallofizika 14 (10), 81 (1992). energies of the atom in different interstices. The 11. É. A. Pashitskiœ, Fiz. Nizk. Temp. 21, 405 (1995) [Low increase in the preexponential factor for q 1 results Temp. Phys. 21, 315 (1995)]. from the changes in the potential barriers u0 and uT. 12. P. P. Gorbik, V. V. Dyakin, F. A. Zaitov, et al., Sverkhpro- vodimost: Fiz., Khim., Tekh. 3, 1654 (1990). If the atoms diffuse from a site to an interstice [20], 13. V. N. Alfeev, P. P. Gorbik, V. V. Dyakin, et al., Dokl. sr, 1 2 Ð1/2 Akad. Nauk Ukr. SSR, No. 1, 41 (1991). D = ---l ()gτ τ exp()ÐE/kT , (15) s 6 g z 14. V. N. Lysenko, V. T. Adonkin, V. V. Dyakin, et al., Super- cond.: Phys., Chem., Technol. 5, 341 (1992). τ τ where g is the number of vacant sites and g and z are 15. R. C. Baetzold, Phys. Rev. B 42, 56 (1990). potential energy and interstice, respectively. The behav- 16. G. V. M. Williams, J. L. Tallon, R. Michalak, and s r R. Dupree, Phys. Rev. B 54, R6909 (1996). ior of D and D is defined by a decrease in τ and τ s s g z 17. V. T. Adonkin, A. P. Galushka, P. P. Gorbik, et al., when the defects fill the interstices, as well as by an Sverkhprovodimost: Fiz., Khim., Tekh. 5, 1901 (1992). increase in g due to the formation of the defects upon 18. A. A. Smirnov, Theory of Diffusion in Interstitial Alloys irradiation and thermal desorption. (Naukova Dumka, Kiev, 1982). Thus, nonuniform heating of the l and h carriers in 19. B. M. Gorelov, D. V. Morozovskaya, V. M. Pashkov, and the microwave field and the perturbations of the charge V. A. Sidorchuk, Zh. Tekh. Fiz. 70 (9), 50 (2000) [Tech. density in the YBa2Cu3O7 interlayers suppress the vol- Phys. 45, 1147 (2000)]. ume and surface diffusion of metal atoms. The suppres- 20. W. Seith, Diffusion in Metallen: Platzwechselreaktionen sion is apparently caused by the aggregation of a large (Springer-Verlag, Berlin, 1939; Inostrannaya Literatura, number of interstitials in the surface layers of the grains Moscow, 1958). and may indicate that both effects excite slightly damped collective perturbations of h carriers. Translated by M. Fofanov

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Technical Physics, Vol. 46, No. 9, 2001, pp. 1121Ð1124. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 71, No. 9, 2001, pp. 54Ð57. Original Russian Text Copyright © 2001 by Gritsyuk, Lyakhov, Mel’nichuk, Strebezhev.

SOLID-STATE ELECTRONICS

Semiconductor Compound Thin Films Obtained with Capillary Evaporators B. N. Gritsyuk, A. A. Lyakhov, S. V. Mel’nichuk, and V. N. Strebezhev Chernovtsy State University, Chernovtsy, 58012 Ukraine e-mail: [email protected] Received September 22, 2000

Abstract—The thickness of decomposable semiconductor thin films that are obtained with capillary evaporators is measured. In the viscous flow approximation, an expression that relates the film thickness, the distance between the capillary and the deposition area, and the angle between the capillary axis and the flow direction is derived. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION capillaries complicates the preparatory stage. The Obtaining semiconductor compound thin films by charge weight is small; therefore, it rapidly evaporates, thermal evaporation is a challenge, since the compo- so that the service time of the evaporator is very short. nents vaporize with various rates that depend on their Capillary evaporators that do not suffer from these vapor pressures and composition in the melt. The disadvantages and allow the growth of high-grade sto- resulting composition and the properties of the con- ichiometric films have been reported in [5, 6]. The cap- densed material differ from those of the initial compo- illaries act as a single evaporating system, since the nents. melt is delivered to the capillaries from the noncapillary The use of specially designed evaporators that pro- container common to all of them. The container, much duce a steady vapor flow condensing to stoichiometric like communicating vessels, is made by connecting the films is a promising approach in this field. bottoms of the capillaries and has a large volume. This makes it possible to extend the service time of the system and to reduce the number of the capillaries, since MATERIAL EVAPORATION USING they do not contain a charge. The evaporator design can CAPILLARY EVAPORATORS be both compound- and process-specific. Evaporators for obtaining films of multicomponent We experimented with capillary evaporators with alloys and compounds have been described in [1Ð3]. An crucibles made of quartz glass (Fig. 1a) and graphite original design [4] involves a crucible in the form of a (Fig. 1b). Capillaries 1 (inner diameter 0.8Ð1.0 mm) set of capillaries closed at one end, with their diameter communicated with the charge container 2 (diameter such that it prevents convective mixing in the melt. 10Ð12 mm). The quartz container is sealed on the top Each of the capillaries is loaded by the initial multicom- and has one capillary. In the graphite container, which ponent semiconductor charge and the crucible is heated is plugged by spacer 3 and threaded stopper 4 from the by coaxial cylindrical furnaces. Since melt mixing in bottom, the number of capillaries varies depending on the fine capillaries is prevented, the melt becomes the vapor flow rate and trajectory. The crucibles were depleted in the high-volatility component over time and heated by cylindrical tungsten heaters 5 surrounded by the steady state that provides its congruent evaporation tantalum foil thermal shields 6. is established. Once melt 7 has entered the capillary, the evaporat- However, the evaporator suggested in [4] is not free ing layer becomes depleted by the high-volatility com- of disadvantages. To load the crucible, the initial single ponent after a time and the steady state is established; crystal should be finely ground. As a result, the total that is, the difference in the vapor pressures of the com- contaminated and oxidized surface area of the charge ponents is balanced by their amounts in the melt. The greatly increases; accordingly, this adversely affects the outgoing vapor has an appropriate ratio of the high- and purity and the properties of the films. In addition, the low-volatility components and condenses to the sto- melting time varies from capillary to capillary because ichiometric films. The evaporator with the quartz cruci- of different charge weights and the nonuniform distri- ble is conveniently used to watch the melt level for bution of the thermal field over the capillaries. Because selecting evaporation conditions and also when the of this, the steady state in the capillaries is established contact of the melt with materials other than quartz is at different time instants, which makes obtaining near- undesirable. However, the quartz crucible can be used stoichiometric films difficult. The need load each of the only once and is difficult to make if the number of cap-

1122 GRITSYUK et al.

(a) (b) 1

6 1 2 2 5 7 7 6

5 3

Fig. 1. (a) Single-capillary quartz and (b) four-capillary graphite evaporators. illaries is more than one. Hence, the deposition rate of In our case, Knudsen’s law fails, since the material the films remains low. evaporates from the capillary interior rather than from The graphite crucible can be loaded many times. It the open surface. Before entering the vacuum chamber, is significant that the capillaries are filled by the melt the vapor should pass through the capillary. simultaneously and evaporation begins from the steady Consider the capillary flow of the vapor of a binary state. The design of the evaporator is known to mark- compound (e.g., CdSb). Let the capillary radius and edly influence the condensate thickness over the substrate. By the vapor direction and the vapor flow rate, length be a and L, respectively. For a vapor pressure of the evaporators are classified into surface-type evapora- about 1 torr, the ratio of the free path of the molecule to tors, open-type crucible evaporators, closed-type the capillary diameter is 0.017 in our case; therefore, pulsed evaporators, and quasi-closed-type crucible we must consider viscous flow [10]. In addition, the evaporators [7]. Capillary evaporators fall into the last flow velocity is much less than the sound velocity, so category. The distribution of the condensate thickness that the Mach number, the ratio of the flow and sound over the substrate for the evaporation from a point velocities, is very small, indicating that the flow is source, a small-area evaporator, ring- and disk-shaped incompressible. At such pressures, the Reynolds num- evaporators, as well as from actual evaporators (effu- ber is also very small (Re < 10); hence, the flow is lam- sion cells and cone crucibles), has been reported in [8]. inar. The distance Le over which the flow becomes com- At the same time, data on the condensate thickness for pletely equilibrium is given by Le = 0.227aRe [9]. In capillary evaporation are lacking in the literature. In our case, it is much shorter than the capillary part along this article, we try to bridge this gap, since thickness which the flow moves. However, even for such low uniformity is very important for optical and other appli- pressures, the flow velocity near the walls is other than cations. zero [10] and comprises about 6% of the velocity calculated. We define the flux Q as

CALCULATION OF THICKNESS QPS= v , (2) DISTRIBUTION The film thickness as a function of the source type where v is the flow velocity, S is the cross section of the and the position of the source relative to the deposition capillary, and P is the mean pressure that specifies this area has been considered in detail in [8] for point, one- velocity. dimensional (wire), and planar sources. For the planar source, it is assumed that evaporation obeys Knudsen’s Under the above conditions, Poiseuille’s law is cosine law. The mass dM deposited per unit time on the valid: surface area dS whose position is defined by the angle π 4 ()()η Θ is given by Q = a Pp1 Ð p2 /8 L , (3) mcos2 Θ dM = ------dS, (1) where p1 and p2 are the vapor pressures near the evapo- πr2 rating area and at the end of the capillary, respectively, and η is the dynamic viscosity of the vapor. where m is the mass of the material evaporated per unit time and r is the distance from the source to the depo- The pressure p2 at the exit from the capillary is far sition site. smaller than the pressure p1 near the evaporating area,

TECHNICAL PHYSICS Vol. 46 No. 9 2001

SEMICONDUCTOR COMPOUND THIN FILMS OBTAINED 1123 ≈ so that we can put P p1. Another expression for the center of the capillary end) to a point on the deposition flux is area where the film thickness H(Θ, L) is measured. If the flow velocity v inside the capillary is zero, N A dm QkT= ------µ------, (4) expression (9) passes to Knudsen’s law for the evapo- dt rating area: where k is the Boltzmann constant, N is the Avogadro A Ccos4 Θ Ccos2 Θ number, µ is the atomic mass of the evaporating mate- H ==------. rial, and dm/dt is the evaporation rate. The values of P L2 r2 and v are found from expressions (2)Ð(4). Let our coordinate system move with the flow velocity v. Separate an elementary flow layer bounded by EXPERIMENTAL DATA two concentric spheres with radii r and r + dh, where dh Using the original capillary evaporators, we depos- is a small increment. The number of atoms in this layer ited II–V semiconductor (CdSb and ZnSb) films. When is these compounds evaporate from evaporators of usual π 2 design (boats, helical and strip evaporators, and cruci- dN = 4 r ndh, (5) bles), their dissociation takes place [7]. As a result, where n is their surface concentration. early in the process, the vapor is enriched by the high- In the layer, the number of atoms remains constant volatility component (Zn or Cd). Then, multiphase (dN = const); therefore, mixtures of metastable cadmium and arsenic compounds form. Near-stoichiometric CdSb films have dN 1 n = ------. been obtained by pulsed evaporation [11], double- 4πdhr2 source evaporation [12], and laser-induced evaporation [13]. These techniques are, however, very difficult to The atoms are deposited on the substrate; conse- apply in practice. CdSb and ZnSb films with properties quently, for the thickness of the deposit, we have close to those of the starting single crystals have been C obtained in our previous works by using quartz [5, 14] H ==----2, rVt. (6) and graphite [6] capillary evaporators. The properties r and the composition of the films successively applied in Here, a single process cycle were shown to vary from metal to semiconducting during the transient in the capillary dm 1 C = ------evaporator. Once the steady-state evaporation condi- dt πρ tions have been established in the evaporator, the properties of the films become identical to those of CdSb is an evaporation-rate-dependent constant, t is the time and ZnSb single crystals. This effect has been most viv- it takes for an atom to reach the substrate, V is the ther- ρ idly demonstrated with temperature dependences of the mal velocity of the atoms, and is their density inside conductivity that were taken from the films in the order the capillary. From Fig. 2, of their application [14]. vtL= Ð () Vt2 Ð.x2 (7) In this work, we studied the thickness uniformity of CdSb films deposited on pyroceramic substrates from a Solving (7) for t, we find four-capillary graphite evaporator. The process was × 2 2 2 2 2 2 2 2 carried out in a VUP-5 installation at a pressure of 1 tL= Ð v + L v Ð.()V Ð v ()x + L (8) 10Ð6 torr, and the substrates were 12 cm away from the Here, we leave only the positive root. As V, we can take the mean velocity of atoms; that is, V = (3RT/µ)1/2, x where R is the gas constant. The thermal velocities of cadmium and arsenic atoms differ only by 4.37%. vt Therefore, in calculations, they were set equal to each other: V = (VSb + VCd)/2. Taking into consideration that x2 + L2 = L2/cos2Θ, we obtain from (6) and (8) L H()Θ, L Θ Vt 2 CV()2 Ð v 2 cos4 Θ (9) = ------, V 2 L2()cos2Θv 2 Ð 2cosΘv V 2 Ð v 2 sin2 Θ + V 2 where L is the capillary endÐsubstrate distance and Θ is the angle between the capillary axis and the radius vector from the center of the evaporating area (from the Fig. 2. Particle deposition for capillary evaporation.

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1124 GRITSYUK et al.

µm REFERENCES

0.8 1. Yu. M. Bukker, A. S. Valeev, A. A. Vanin, et al., USSR Invetror’s Certificate No. 659642, Byull. Izobret., No. 16 0.6 (1979). 2. V. M. Kuznetsov, Vacuum Technology (VINITI, Mos- 0.4 cow, 1977), No. 8. 3. B. A. Chenov, S. P. Oles’kiv, and B. S. Oles’kiv, Fiz. Éle- 0.2 ktron. (Lvov) 38, 66 (1989). 0 4. G. F. Dolidze, D. V. Yakashvili, V. N. Vigdorovich, and –30 –20 –10 0 10 20 30 G. A. Ukhlinov, USSR Invetror’s Certificate No. 544711, mm Byull. Izobret., No. 4 (1977). 5. B. N. Gritsyuk and V. N. Strebezhev, Prib. Tekh. Éksp., Fig. 3. Film thickness distribution over the substrate sur- No. 5, 157 (1997). face. 6. B. N. Gritsyuk and V. N. Strebezhev, Problems of Atomic Science and Technology (VANT, Kharkov, 1998). 7. Handbook of Thin-Film Technology, Ed. by L. I. Maissel open end of the capillary. The film thickness distribu- and R. Glang (McGraw-Hill, New York, 1970; Sov. tion was determined by an MII-11multiple-beam inter- Radio, Moscow, 1977), Vol. 1. ferometer and also by examining the substrate–film 8. L. Holland and W. Steckelmacher, Vacuum 11 (4), 346 interface (on the cross section of the structure) with an (1952). RÉM-100U scanning electron microscope. 9. H. L. Langhaar, J. Appl. Mech. 9, A-55 (1942). 10. S. Dushman, Scientific Foundations of Vacuum Tech- Figure 3 shows the film thickness distribution over nigue, Ed. by J. M. Lafferty (Mir, Moscow, 1960; Wiley, the substrate (dashed line). The continuous line repre- New York, 1962). sents the calculated distribution of the condensate 11. S. I. Baranov, I. V. Glavatskiœ, K. R. Zbigli, and A. G. Cheban, Thermoelectric Devices and Films: Pro- thickness for the case when four capillaries are ceedings of the All-Union Conference, Leningrad, 1976. arranged at the vertices of a square. It is seen that four- 12. M. Komatsu, N. Matsuda, Y. Kashiwaba, and H. Saito, capillary evaporation improves the film thickness uni- Mater. Bes. Bull. 13, 835 (1978). formity over a substantially large area. The quality of 13. B. N. Gritsyuk and S. V. Nicheœ, Prib. Tekh. Éksp., No. 2, the films obtained, as well as agreement between the 114 (1997). experimental and calculated thickness distribution 14. B. N. Gritsyuk, V. V. Zolotukhina, I. M. Rarenko, and curves, suggests that the use of capillary evaporators is V. N. Strebezhev, Fiz. Élektron. (Lvov) 39, 19 (1989). a promising approach and that our technique for determining film thickness is appropriate. Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 46 No. 9 2001

Technical Physics, Vol. 46, No. 9, 2001, pp. 1125Ð1127. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 71, No. 9, 2001, pp. 58Ð60. Original Russian Text Copyright © 2001 by Grebenshchikova, Zotova, Kizhaev, Molchanov, Yakovlev.

SOLID-STATE ELECTRONICS

InAs/InAsSbP Light-Emitting Structures Grown by Gas-Phase Epitaxy E. A. Grebenshchikova, N. V. Zotova, S. S. Kizhaev1, S. S. Molchanov, and Yu. P. Yakovlev Ioffe Physicotechnical Institute, Russian Academy of Sciences, St. Petersburg, 194021 Russia 1e-mail: [email protected] Received October 16, 2000

Abstract—Using the metal-organic chemical vapor decomposition technique, light-emitting diodes based on InAs/InAsSbP double heterostructures emitting in a wavelength range around 3.3 µm have been fabricated. The external quantum yield of the diodes is 0.7%. In laser diodes, stimulated emission at a wavelength of 3.04 µm has been obtained at T = 77 K. © 2001 MAIK “Nauka/Interperiodica”.

At present, a demand exists for devices that monitor Consider a symmetric double heterostructure grown the maximum admissible concentrations of a number of for use in fabricating LEDs and lasers (Fig. 1). The hydrocarbons (methane, propane, ethylene, and oth- structure consisted of an InAs(111)B substrate (n ~ 3 × ers). Molecules of methane, as well as other hydrocar- 1017 cmÐ3) on which were grown a 1.2 µm thick unin- bons, can absorb infrared radiation. Methane has strong tentionally doped InAsSbP layer (n ~ 1017 cmÐ3), an absorption bands in a wavelength range around 3.3 µm unintentionally doped active layer of n-InAs (1 µm [1]; therefore, semiconductor light-emitting diodes thick), and a p-InAsSbP layer doped with zinc to p ~ (LEDs) emitting in this spectral range can be used as 8 × 1017 cmÐ3 of a thickness 1.2 µm. Subsequent mea- radiation sources for portable gas analyzers. Com- surements have shown that zinc diffusion from the last pounds based on lead salts (IVÐVI) [2], as well as nar- barrier layer of InAsSbP took place, making the con- row band HgCdTe (IIÐVI) semiconductors [3], have ductivity of the InAs active layer the p-type. The pÐn low thermal conductivity and considerable metallurgi- junction was formed in the first barrier layer of cal instability, which makes them less suitable for the InAsSbP 0.5 µm from the n-InAs substrate. The phos- fabrication of infrared emitters than IIIÐV solid solu- phorus content in the barrier layers was 25%. The band- tions. Light-emitting structures for the 3.3 µm spectral gap (Eg) of the InAsSbP solid solution calculated with range based on IIIÐV semiconductors are usually the use of data in [10] was found to be 580 meV. grown by the liquid-phase epitaxy (LPE) method [4Ð6] InAsSbP/InAs/InAsSbP structures were grown by or sometimes by the method of metal-organic chemical MOCVD in a standard horizontal-type reactor under vapor deposition (MOCVD) [7] and molecular beam atmospheric pressure. The reactor was similar in design epitaxy (MBE) [8]. to a system considered earlier [11]. The rate of hydro- The main drawback of the LEDs for the 3Ð5 µm gen flow through the reactor was 18 l/min. The sources spectral range is that their power output is insufficient of indium, arsenic, antimony, and phosphorus were, for practical applications, being ≤0.1 mW in the continuous mode and ~ 1 mW in the pulsed mode. This work Eg, eV is a continuation of our research on LEDs for the 3Ð np 5 µm spectral range grown by MOCVD [9], with the aim of increasing the LED efficiency by about an order InAsSbP0.25 InAsSbP0.25 of magnitude compared with LEDs grown by LPE. Our 0.58 expectations are based on the possibility of more effi- InAs InAs cient use of the potential of the MOCVD technique in growing InAsSb/InAsSbP heterostructures, first of all 0.36 by growing InAsSbP layers having a wider bandgap (for example, in the immiscibility region); also, heterostructures with better electron and optical confinement and more perfect morphology can, hopefully, be grown. This work deals with the use of MOCVD for the fabri- Fig. 1. Energy diagram of the grown symmetrical cation of LEDs emitting at 3.3 µm with high emitted InAsSbP/InAs/InAsSbP double heterostructure at room optical power. temperature.

1126 GREBENSHCHIKOVA et al.

J, arb. units J, arb. units 1.0 1.0 2 1 0.8 0.8 (× 25) 0.6 0.6 0.4 0.4 0.2

0.2 0 400 420 E, meV 0 350 400 450 500 550 E, meV

Fig. 2. Electroluminescence spectra of the diodes at different temperatures. T = (1) 77, (2) 300 K. Inset: stimulated emission spectrum of the laser diode at T = 77 K.

respectively, trimethylindium (TMIn), arsine (AsH3) 500 Hz). In the spectra taken at 77 K, two peaks are diluted to 20% with hydrogen, trimethylstibin (TMSb), clearly distinguished. The short-wavelength peak cor- and phosphine (PH3) diluted to 20% with hydrogen. responds to the n-InAs substrate (hvmax = 410 meV, The InAsSbP barrier layers were grown at a sub- Dhv1/2 = 50 meV) with an electron concentration of strate temperature of 580°C. The flow rates of hydrogen ~3 × 1017 cmÐ3 [12], and the other peak is due to the p- through bubblers with TMIn and TMSb were 435 and InAs active layer (hvmax = 380 meV, Dhv1/2 = 26 meV) 50 cm3/min, respectively. TMIn and TMSb were main- with p ~ 8 × 1017 cmÐ3 [13]. These results suggest that tained at temperatures of 27 and 6°C, respectively. The zinc diffusion took place from the top barrier layer into flow rates of AsH3 (20%) and PH3 (20%) were 6 and the semiconductor bulk and the contributions into radi- 50 cm3/min, respectively. InAsSbP was made p-type by ative recombination come from the n-InAs substrate doping with zinc. The zinc source was diethylzinc and the InAs active layer having p-type conductivity (DeZn). The bubbler with DeZn was held at a tempera- due to zinc diffusion. To confirm these observations, the ture of 4.7°C. The hydrogen flow through the DeZn location of the pÐn-junction was determined using vaporizer was 20 cm3/min. scanning electron microscopy. The pÐn-junction was found to be located in the first wide-bandgap InAsSbP The InAs active layer was grown at a substrate tem- µ perature of 620°C and the V/III ratio in the gas phase layer at a distance of 0.5 m from the substrate (Fig. 1). equal to 40. At T = 295 K, the peak in the EL spectrum corresponds to hv = 364 meV and Dhv = 56 meV. At room tem- The LEDs were fabricated in the form of mesas max 1/2 using standard photolithography. The mesa diameter perature, InAs has Eg = 360 meV [10]; the shift of the was 300 µm. A solid ohmic contact was applied on the peak to higher energies confirms that the heavily doped side of p-InAsSbP. The diameter of the point contact to n-InAs substrate contributes to the radiative recombina- the substrate was 100 µm. Ohmic contacts were pre- tion. pared by evaporaton of gold with tellurium (onto the WÐI characteristics of the diodes were measured in layer of n-type conductivity) or gold with zinc (onto the pulsed and continuous modes. Shown in Fig. 3 is a WÐ layer of p-type conductivity). I characteristic of the diode in pulsed mode (t = 5 ms, The laser diodes had a mesa-stripe width of 30 µm. f = 500 Hz). Nonlinearity of the characteristic is not The contact stripe was made on the p-InAsSbP layer. caused by heating of the diode, as evidenced by the The width of the contact stripe was 5 µm. Resonators coincidence of the characteristics measured in continu- 300 µm in length were made by cleaving. Properties of ous and pulsed current modes. The external quantum the structures grown were studied using electrolumi- yield of the diodes was ~0.7%. Fast saturation of the nescence (EL). EL was registered by a cooled InSb power versus the pump current curve is evidently due to photodiode in the phase-lock detection mode. the small thickness of the active region (~1 µm), caus- First, consider the characteristics of the LEDs. Fig- ing a rapid rise in the charge carrier concentration and, ure 2 shows EL spectra at 77 and 300 K. The diodes as a consequence, enhancement of the Auger recombi- were powered with a pulsed current of 1 A (t = 5 ms, f = nation.

TECHNICAL PHYSICS Vol. 46 No. 9 2001 InAs/InAsSbP LIGHT-EMITTING STRUCTURES 1127

P, µW enhance the optical and electronic confinement in 180 InAs/InAsSbP DHSs. Although the diode structure 160 investigated is far from optimal, the emission power of the first LEDs grown by MOCVD is found to be com- 140 parable to LEDs prepared by other methods [4Ð8]. 120 100 ACKNOWLEDGMENTS 80 The authors are grateful to M. A. Remenny for his 60 interest in this work and measurements of the emission 40 power of the LEDs; T. B. Popov for determining the chemical composition of InAsSbP; and V. A. Soloviev 20 for determining of the position of the pÐn-junction. 0 100 200 300 400 S. S. Kizhaev thanks the Robert Haveman Foundation I, mA for financial support during this study.

Fig. 3. WÐI characteristic of the laser diode measured in pulsed regime (τ = 5 µs, f = 500 Hz). P is the optical power. REFERENCES 1. L. S. Rothman, R. R. Gamache, R. H. Tipping, et al., J. Quant. Spectrosc. Radiat. Transf. 48, 469 (1992). The efficiency of the diodes can be improved by 2. Z. Feit, D. Kostyk, R. J. Woods, and P. Mak, Appl. Phys. optimizing the DHS parameters. So, in [14], where the Lett. 58, 343 (1991). recombination mechanism of nonequilibrium carriers 3. E. Hadji, J. Bleuse, N. Magnea, and J. L. Pautrat, Appl. in InAs/InAs0.16Sb0.84 structures (with the active region Phys. Lett. 67, 2591 (1995). of p-InAs) was considered, it was found that at small 4. M. Aœdaraliev, N. V. Zotova, S. A. Karandashev, et al., injection currents when the radiative recombination Fiz. Tekh. Poluprovodn. (St. Petersburg) 34, 102 (2000) efficiency is independent of current, the efficiency in p- [Semiconductors 34, 104 (2000)]. InAs reached 24% at 300 K and the external quantum 5. A. A. Popov, M. V. Stepanov, V. V. Sherstnev, and yield was ~9%. These values were measured for an Yu. P. Yakovlev, Pis’ma Zh. Tekh. Fiz. 23 (21), 24 optimum hole concentration of p ~ 3 × 1017 cmÐ3 and an (1997) [Tech. Phys. Lett. 23, 828 (1997)]. µ active region thickness of ~3 m. With increasing active 6. M. K. Parry and A. Krier, Electron. Lett. 30, 1968 region thickness, reabsorption of the emission (1994). increased (absorption length in p-InAs at the emission µ 7. A. Stein, D. Puttjer, A. Behres, and K. Heime, IEE Proc.: spectrum maximum was ~4.8 m) and the efficiency Optoelectron. 145 (5), 257 (1998). dropped. At less than optimum thicknesses, the recom- 8. B. Grieteus, S. Nemeth, and G. Borghs, in Proceedings bination at the InAs/AlAsSb interface was more proof the International Conference on Midinfrared Opto- nounced. electronics, Materials and Devices, Lancaster, 1996. The inset in Fig. 2 shows the stimulated emission 9. N. V. Zotova, S. S. Kizhaev, S. S. Molchanov, et al., Fiz. spectrum at T = 77 K of a laser diode fabricated using Tekh. Poluprovodn. (St. Petersburg) 34, 1462 (2000) the structure grown. Its peak position at 3.04 µm corre- [Semiconductors 34, 1402 (2000)]. sponds to an Eg value in InAs of 408 meV for this tem- 10. S. Adachi, J. Appl. Phys. 61, 4869 (1987). perature. The observation of stimulated emission pro- 11. W. J. Duncan, A. S. M. Ali, E. M. Marsh, and P. C. Spur- vides evidence of a perfected InAs/InAsSbP heterointer- dens, J. Cryst. Growth 143, 155 (1994). face. The high value of the threshold current Jth = 12. A. A. Allaberenov, N. V. Zotova, D. N. Nasledov, and 330 mA is explained by insufficient thickness of the L. D. Neuœmina, Fiz. Tekh. Poluprovodn. (Leningrad) 4, InAsSbP barrier layers, nonoptimal for the laser struc- 1939 (1970) [Sov. Phys. Semicond. 4, 1662 (1970)]. ture size of the active region and the position of the 13. N. P. Esina and N. V. Zotova, Fiz. Tekh. Poluprovodn. p−n-junction. (Leningrad) 14, 316 (1980) [Sov. Phys. Semicond. 14, 185 (1980)]. In the future, in order to increase the efficiency of 14. M. J. Kane, G. Braithwaite, M. T. Emeny, et al., Appl. the light-emitting diodes based on the InAs/InAsSbP Phys. Lett. 76, 943 (2000). DHS, the active layer thickness must be increased to the optimal value and the hole concentration reduced. Also, 15. M. J. Jou, Y. T. Cherng, H. R. Jen, and G. B. Stringfellow, because of the higher probability of radiative recombi- J. Cryst. Growth 93, 62 (1988). nation in n-InAs than in p-InAs [13], the active region 16. A. Behres, D. Puttjer, and K. Heime, J. Cryst. Growth should be made n-type. In addition, with the MOCVD 195, 373 (1998). technique, InAsSbP solid solutions in the immiscibility region can be grown [11, 15, 16]; these can be used to Translated by B. Kalinin

TECHNICAL PHYSICS Vol. 46 No. 9 2001

SOLID-STATE ELECTRONICS

Transition Processes Occurring under Continuous and Stepwise Heating of GaAs Surface-Barrier Structures Yu. A. Gol’dberg and E. A. Posse Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia Received October 31, 2000.

Abstract—Evolution of the capacitanceÐvoltage (CÐU) and currentÐvoltage (If ÐU and IrÐU) characteristics of solid metalÐsemiconductor structures (Ni/GaAs) in the process of their continuous and stepwise heating are studied. Properties of the initial structures obey the theory of thermionic emission. It has been shown that as a result of continuous heating, the rectifying structures become ohmic at a temperature of TOhm = 720 K, which is substantially lower than the melting points of the metal or the metalÐsemiconductor eutectic. For comparison, properties of the structures annealed at different temperatures Tann are measured after cooling to room temperature (stepwise heating). In this case, IÐU characteristics are closer to the initial ones for annealing temperatures Tann < T0 = 553 K; for Tann > T0, the characteristics display excess currents; and, ﬁnally, for Tann exceeding T0 by 200Ð300 K, the characteristics become purely ohmic. It is suggested that these effects are due to a chemical interaction between Ni and GaAs, which changes the properties of the semiconductor surface. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION To study the evolution of properties of the Shottky diodes, some specimens were heated continuously It is known that a metalÐsemiconductor contact is from room temperature to 870 K at a slow rate either rectifying if there is a potential barrier between (5 deg/min) in a neutral atmosphere (helium); currentÐ the metal and the semiconductor impervious to tunnel- voltage (IÐ U) and capacitanceÐvoltage (CÐU) charac- ing, or ohmic if there is no potential barrier or the bar- teristics were measured in the course of heating (i.e., rier is transparent for tunneling (see, for example, without cooling to room temperature). − [1 3]). In our studies [4, 5] of liquid metalÐsemicon- Other specimens were subjected to a stepwise heat- ductor structures (Ga/GaP, In/GaP, Ga/GaAs), it was ing; i.e., they were annealed at different temperatures found that as a result of heating the structure, the recti- with cooling down to room temperature after anneal- fying contact becomes ohmic because the semiconduc- ing; then IÐU and CÐU characteristics were measured. tor surface layer by the metal dissolves. In [6], we The parameter characterizing the asymmetry of the IÐU showed that such a transition also occurs in the case of characteristic was the rectiﬁcation coefﬁcient Kr = If /Ir a semiconductorÐsolid metal contact. measured at U = ±0.5 eV. In the present work, we studied the evolution of the capacitanceÐvoltage (CÐU) and currentÐvoltage (IfÐU RESULTS AND DISCUSSION and IrÐU) characteristics of solid metalÐsemiconductor structures (Ni/GaAs) caused by continuous and step- 1. Let us consider the evolution of the differential wise heating. capacitanceÐvoltage and currentÐvoltage characteristics in the case of continuous heating of the structures. 1.1. CÐU dependences in CÐ2ÐU coordinates were EXPERIMENTAL TECHNIQUE linear over a temperature range of 290Ð470 K (Fig. 1a), in agreement with Shottky’s theory For preparation of the structures, epitaxial GaAs 2()U Ð U Ð kT/q layers (n = 1015 cmÐ3) grown on GaAs substrates (n = CÐ2 = ------d -, (1) 18 Ð3 ε ε 2 () 10 cm ) were used. Orientation of all crystals was in s 0S qNd Ð Na the (100) plane. First, an ohmic contact was applied by ε alloying In into a GaAs substrate. The epitaxial GaAs where s is the static dielectric constant of the semicon- ε layer was processed in the way usually used in prepar- ductor; 0 is the dielectric constant of a vacuum; q is the ing rectifying surface-barrier structures: mechanical electron charge; Nd Ð Na is the concentration of uncom- lapping, chemical lapping, and washing. Then a layer pensated donors in the semiconductor; Ud is the diffu- of Ni was deposited onto the chemically treated surface sion potential; k is the Boltzmann constant; and T is the [7]. The obtained structures were rectifying. temperature (K).

TRANSITION PROCESSES OCCURRING UNDER CONTINUOUS AND STEPWISE HEATING 1129

From these characteristics, the concentration of (a)ϕ , V (b) 1 BO uncompensated donors, N Ð N = 1Ð2 × 1015 cmÐ3, and d a 2 1.0 6 values of the diffusion potential difference at various 3 temperatures were determined. The potential barrier 4 0.9 –2 height not corrected for its lowering by image forces 0.8 pF (ϕ ) was determined from the formula ϕ = U + µ/q, 0 200 400 4 –6 BO BO d T, K where µ is the Fermi level energy in the semiconductor. 5 , 10 ϕ –2 From the plots of BO = f(T) (Fig. 1b), the temperature C α × Ð4 ϕ coefficient = 2.4 10 V/deg, and BO extrapolated 2 ϕ to 0 K, BO(0 K) = 0.96 V, were determined. The CÐ2ÐU dependence measured at temperatures T ≥ 470 K deviated from a straight line (Fig. 1a) and the measured capacitance was significantly lower. In this –3 –2 –1 01 case the capacitance measured by the bridge method U, V differed substantially from its true value because of the considerable drop in the differential resistance of the Fig. 1. (a) Dependence of the capacitance C on voltage U structure, giving evidence that the barrier contact for structure N1 at temperatures T, K: 1—295; 2—368, became ohmic. 3—413; 4—465; 5—493. (b) The temperature dependence ϕ Ð2 of the potential barrier height BO calculated from C ÐU 1.2. Measured curves of the forward current If vs. characteristics by the cut-off voltage on the ordinate. voltage (Fig. 2a) had an exponential portion in the temperature range T = 290Ð430 K, If, A ()β –2 I f = I0 exp qU/ kT , (2) 10 7 6 where β = 1.02Ð1.03. 10–3 (a) Analysis of these characteristics has shown that the dependence of I on U obeys the theory of thermionic –4 f 10 5 4 3 2 1 emission with correction for the effect of image forces ϕ on the potential barrier height B(U): 10–5 K 106 II= []exp()qU/kT Ð 1 , (3) 0 –6 (b) 10 4 2 ()ϕ 10 I0 = A*ST exp Ðq B/kT , (4) 10–7 2 1 ϕ ϕ ()α 102 B = B 0K Ð T, (5) 10–8 where the Richardson constant is A* = 120m /m , 100 e 0 300 500 700 T, K me being the effective mass of majority charge carriers 10–9 α and m0 the free electron mass in vacuum; and is the 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 barrier height temperature coefficient. Uf, V If it is taken into account that the true barrier height Fig. 2. (a) Dependences of the forward current If on voltage is U for structure N1 at temperatures T, K: 1—296; 2—343, ϕ () ϕ ∆ϕ () 3—373; 4—393; 5—428; 6—473; 7—563. (b) The temper- B U = BO Ð B U , (6) ature dependence of the rectification coefficient Kr for then the preexponential factor I0 can be written as (1) a Ni/GaAs structure and (2) a Ga/GaAs liquid metal- semiconductor structure [4]. ()∆ϕ () I0 = Is expq B U /kT , (7) where the saturation current is value of U by Eq. 9. Values of Is were determined by ∆ϕ 2 ()ϕ extrapolating to [U + B(U)] = 0 the exponential por- Is = A*ST exp Ðq BO/kT , (8) tions of the dependences plotted in the form If = f[U + and the barrier lowering due to image forces is ∆ϕ 2 B(U)]. The dependence of Is/ST on 1/T (Richard- 3 1/4 son’s plot in Fig. 3) turned out to be linear, in accor- q ()N Ð N ∆ϕ d a ()dance with the theory, with the Richardson constant B = ------2 2 - Ud Ð U Ð kT/q . (9) π ε ε 2 2 8 s 0 equal to A* = (8.2 ± 1.0) A/cm deg , as predicted by the theory (the electron effective mass for GaAs is m* = To determine Is at different temperatures, If ÐU char- e acteristics were plotted in coordinates If = f[U + 0.068m0). The potential barrier height not corrected for ∆ϕ ∆ϕ B(U)]. Values of B(U) were determined for each the image forces determined from Richardson’s plot,

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1130 GOL’DBERG, POSSE

2 2 2 leakage currents. At 490 ≤ T ≤ 550 K, the I ÐU charac- Is/ST , A/cm K r 104 teristics had a portion close to saturation (Fig. 4). In order to determine the reverse current mechanism corresponding to these portions of the characteristic, a 100 Richardson plot was built for the reverse current. Is values that would have been in effect in the absence of the 10–4 image forces were determined by dividing the reverse current values at different voltages by ∆ϕ exp(q B(U)/kT) (solid lines in Fig. 4). The plot of –8 2 10 Is/ST as a function of 1/T for the reverse current turned out to be a linear continuation of the Richardson plot for the forward current (Fig. 3). This means that the 10–12 reverse current in the structures studied follows the thermionic emission theory at temperatures up to T = 550 K. At T > 550 K, an excess bulk current appeared, 10–16 012 3 4 caused by the irreversible transition of the barrier con- 103/T, K–1 tact into an ohmic one. Thus, in the Ni/GaAs surface-barrier structures Fig. 3. A Richardson plot drawn using measured If ÐU data studied, the forward current at T ≤ 430 K and the ᭝ ᭺ ᮀ ᭡ ᭹ ( , , and ) and IrÐU data ( , ) for the NiÐGaAs struc- ≤ tures. reverse current T 550 K are due to thermal electron emission. At higher temperatures, excess currents arise that are caused by transition of the contact from barrier ϕ to ohmic. BO(0 K) = 0.96 V, coincided with the value obtained from the C-V characteristics. 1.4. Let us consider next the behavior of the ratio of At T > 430 K, the exponential portion could not be the forward and reverse currents (rectiﬁcation ratio, K = observed because the voltage dropped almost entirely If /Ir) in the process of continuous heating of the struc- across the residual resistance (resistance of the semi- tures. The measurement results for K = f(T ) at U = conductor bulk and the ohmic contact) as a result of a ±0.5 V (Fig. 2b) can be summarized as follows. substantial reduction in the differential resistance of the (a) Similar to semiconductorÐliquid metal contacts structure. [4], when a certain temperature is reached in the pro- 1.3. The dependence of the reverse current (Ir) on cess of continuous heating, the Ni/GaAs rectifying con- voltage (Fig. 4) was measured in the temperature inter- tacts become ohmic (Fig. 2b). This transition takes val 290Ð580 K. At T < 350 K, the reverse current was place before the possible formation of a heavily doped very low. At 350 ≤ T ≤ 450 K, the reverse current was or graded-gap recrystallized semiconductor typical of strongly dependent on voltage, probably because of traditional ohmic contacts.

C–2, pF–2 , V Ur 8 5 4 3 2 1 0 10–8

1 6 1 10–7 2 4 10–6 –4 3 2 10 2 3 10–3

10–2 –4 –3 –2 –1 0 1 4 U, V Ir, A Fig. 5. Dependence of the capacitance C on voltage U for structure N2 measured immediately after fabrication (1) Fig. 4. Dependence of the reverse current Ir on voltage U and after annealing. The annealing temperatures Tann, K: for structure N1 at temperatures T, K: 1—363; 2—493, 2—573, 3—582. Curves for annealing at temperatures 533, 3—533; 4—553. Points, experiment; curves, calculation by 543, 553, and 563 K coincided with curve 1. Measured at the thermionic emission theory. 295 K.

TECHNICAL PHYSICS Vol. 46 No. 9 2001 TRANSITION PROCESSES OCCURRING UNDER CONTINUOUS AND STEPWISE HEATING 1131

(b) The temperature at which the rectifying If, A Ni/GaAs contact became ohmic (TOhm) is 720 K, which 10–2 is about 100 K higher than the temperature of such a 5 transition for contacts between liquid metal and GaAs. 10–3 At this temperature the contact metal does not melt (the ° melting temperature of Ni being Tm = 1453 C), so the 10–4 4 dissolution of the semiconductor in the liquid metal is not a possibility. We suggest that at these temperatures 10–5 a chemical interaction takes place between the semi- 3 conductor and the metal, resulting in the disappearance 10–6 of the thin near-surface layer of the semiconductor, 1 2 where the density of surface states is high and which is 10–7 responsible for the rectifying properties of the contact. –8 Note that the interaction between Ni and GaAs, as 10 determined from measurements of the backscatter of α 10–9 particles [8], starts at ~470 K. As a result of this inter- 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 action, a new phase is formed, which appears at ~570 K U , V at the structure surface. f Fig. 6. Dependence of the forward current If on voltage U In [9], it was found that this new phase is a metasta- for structure N2 measured immediately after fabrication (1) ble compound Ni2GaAs; its formation raises the barrier and after annealing. Tann, K: 2—533, 543, and 553; 3—563, height to 0.84 eV. At temperatures of 620Ð820 K, this 4—573; 5—582. Measured at 295 K. phase decomposes into two compounds, NiGa and NiAs, and the barrier height drops signiﬁcantly. At a Ur, V temperature of 870 K, this process leads to the forma- 8 7 6 5 4 3 2 1 0 tion of the ohmic contact. According to our data, forma- 10–9 tion of the ohmic contact occurred at 720 K. However, this result was obtained not for annealing, as in [9], but 3 for continuous heating. 2 1 2. Consider the evolution of the capacitanceÐvoltage 10–8 and currentÐvoltage characteristics of the Ni/GaAs –6 structures in the case of stepwise heating. The struc- 10 tures were heated to different temperatures and cooled to room temperature; measurements were then carried out. 4 2.1. The results on CÐU characteristics can be sum- 10–5 marized as follows. ≤ (a) After annealing at temperatures of Tann 563 K, the CÐ2ÐU plots coincided with the starting characteristics (Fig. 5). –4 5 10 ≥ (b) After annealing at temperatures of Tann 580 K, Ð2 the C ÐU plots differed markedly from the starting Ir, A characteristics (Fig. 5), which is evidence of the irreversible nature of the processes in the barrier contact. Fig. 7. Dependence of the reverse current Ir on voltage U for structure N2 measured immediately after fabrication (1) 2.2. The results on the evolution of the I ÐU charac- and after annealing. Tann, K: 2—543; 3—553, 4—563; f 5—573. Curve for annealing at the temperature 364 K coin- teristics are as follows. cided with curve 1. Measured at 295 K.

(a) After annealing at temperatures of Tann below a certain temperature T0 = 553 K, the If ÐU characteristics ≥ (b) After annealing at temperatures of Tann T0, the coincided with the starting characteristics. (Fig. 6). I ÐU characteristics showed excess currents although Note that in the structures which had an ideality factor f β the structures remained rectifying (Fig. 6). for the starting If ÐU characteristic somewhat different from the theoretical value (for curve 1 β = 1.05), after 2.3. The results on the evolution of the IrÐU charac- annealing β was close to the theoretical value (curve 2, teristics are as follows. β = 1.01Ð1.02) due to the disappearance of the interme- (a) After annealing at temperatures of Tann below a diate dielectric layer. certain temperature T0 = 553 K, the IrÐU characteristics

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1132 GOL’DBERG, POSSE were similar to the starting characteristics. Note that the electrons does not form and the contact becomes structures in which an intermediate layer remained ohmic. between the metal and the semiconductor during fabrication had lower reverse currents after annealing due to the disappearance of this layer during annealing (Fig. 7, REFERENCES curves 2 and 3). 1. E. H. Rhoderick, MetalÐSemiconductor Contacts (Claredon, Oxford, 1978; Radio i Svyaz’, Moscow, (b) After annealing at temperatures of Tann > T0, the 1982). reverse current increased by more than two orders of 2. P. K. Kupta and W. A. Anderson, J. Appl. Phys. 69, 3623 magnitude due to the beginning of an irreversible tran- (1991). sition of the rectifying contact into an ohmic contact 3. N. Mochida, T. Honda, T. Shirasawa, et al., J. Cryst. (Fig. 7). At temperatures of 200Ð300 K higher than T0, Growth 129/130, 716 (1997). the contact remained ohmic after cooling as well. 4. Yu. A. Gol’dberg, E. A. Posse, and B. V. Tsarenkov, Fiz. Tekh. Poluprovodn. (Leningrad) 20, 1510 (1986) [Sov. Phys. Semicond. 20, 947 (1986)]. CONCLUSION 5. Yu. A. Gol’dberg, M. V. Il’ina, E. A. Posse, and B. V. Tsarenkov, Fiz. Tekh. Poluprovodn. (Leningrad) The metalÐsemiconductor contact, initially rectify- 22, 555 (1988) [Sov. Phys. Semicond. 22, 342 (1988)]. ing, becomes ohmic in the process of heating even 6. Yu. A. Gol’dberg and E. A. Posse, Fiz. Tekh. Polupro- before the recrystallized layer is formed. A chemical vodn. (St. Petersburg) 32, 200 (1998) [Semiconductors interaction takes place between the metal and the near- 32, 181 (1998)]. surface region of the semiconductor (in the case of 7. Yu. A. Gol’dberg, E. A. Posse, B. V. Tsarenkov, and Ni/GaAs structures) or dissolution of this near-surface M. I. Shul’ga, Fiz. Tekh. Poluprovodn. (Leningrad) 25, region in liquid metal (in the case of Ga/GaAs struc- 439 (1991) [Sov. Phys. Semicond. 25, 266 (1991)]. tures [4]). From this fact, it follows that the newly 8. V. G. Bozhkov, V. M. Zavodchikov, K. V. Soldatenko, et formed surface of the semiconductor acquires proper- al., Élektron. Tekh., Ser. 2: Poluprovodn. Prib., No. 7, 41 ties varying from those of the starting surface. It can be (1978). assumed that on this surface, in contact with the metal, 9. F. Lahav, M. Eizenberg, and Y. Komem, J. Appl. Phys. states differing from the initial ones arise that pin the 60, 991 (1986). surface Fermi level either in the conduction band or close to its bottom so that a potential barrier for the Translated by B. Kalinin

TECHNICAL PHYSICS Vol. 46 No. 9 2001

OPTICS, QUANTUM ELECTRONICS

Optical Nonlinearity in Thin Films Exposed to Low-Intensity Light A. V. Khomchenko and E. V. Glazunov Institute of Applied Optics, Belarussian Academy of Sciences, Mogilev, 212793 Belarus e-mail: [email protected] Received August 2, 2000; in ﬁnal form, February 19, 2001

Abstract—Optical nonlinearity in semiconducting and insulating thin-film structures is studied by waveguide methods under self-action conditions at a light intensity less than 0.1 W/cm2 and a wavelength of 630 nm. The optical properties vs. light intensity are similar for semiconducting and insulating films, multilayer thin-film structures, and semiconductor-doped glass films. It is demonstrated that the optical nonlinearity and the nonlinear optical constants depend on the interface condition. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION evaporation time. The refractive indices of the quartz Optical devices for data processing employ materi- glass, lithium niobate, and tin dioxide films were 1.476, als with the refractive index and the absorption coeffi- 2.160, and approximately 2.0, respectively. The tin cient dependent on the light intensity. The highest val- dioxide and lithium niobate layers of the multilayer ues of the nonlinear optical constants have been found structure were obtained by sputtering the ceramic and in semiconductor materials. Therefore, at present, single-crystal (z cut) targets, respectively. The glass emphasis is on developing new semiconductor materi- layers were produced by sputtering OS-12 and KV als and structures with still higher optical nonlinearity glasses. KV and K8 optical glasses were used as substrates. In the course of deposition, the substrate tem- [1]. At a relatively low intensity of incident light, one ° can easily achieve the desired level of the controllable perature did not exceed 250 C. All films were applied power in optical waveguides owing to their small trans- in argonÐoxygen (4 : 1) atmosphere except for the ZnSe verse dimensions [2]. That is why optical nonlinearity films and the semiconductor-doped glass films, which in waveguide semiconductor structures is of great were obtained in argon atmosphere. interest. The refractive indices and the absorption coeffi- In semiconductors, optical nonlinearities due to var- cients of the films were determined by the waveguide ious mechanisms are observed at a light intensity of no method at the wavelength of an HeÐNe laser. The less than ~10 W/cm2. In this article, we observed the waveguide mode in the structure was excited by a cou- nonlinear variation in the refractive index and the pling prism. We measured the spatial distribution of the absorption coefficient of thin films at an incident light intensity in the cross section of the beam reflected from intensity lower than 0.1 W/cm2. The nonlinear refrac- the base of the coupling prism. A photodetector array tive index and the absorption coefficient were measured measured the spatial distribution of the intensity of the Ð3 2 reflected beam in the focal plane of the objective. From to be ~10 cm /W. the distribution obtained, we calculated the real h' and the imaginary h'' parts of the propagation constant h of EXPERIMENTAL the waveguide mode (h = h' + ih'', where h' = Reh and h'' = Imh) [3]. Using the values of the complex h for any Amorphous and polycrystalline thin-film structures two modes, one can determine the refractive index, the were obtained by vacuum deposition. The arsenic sul- absorption coefficient, and the thickness of the film [4]. fide films were deposited by thermal evaporation. The zinc selenide, tin dioxide, and quartz glass films were We determined the nonlinearity parameter of the produced by rf sputtering of ceramic targets. The GaSe waveguide structure by recording the shape of the sig- films were prepared by electron-beam evaporation. The nal when the intensity of the incident light I was gradu- low-dimensional thin-film structures were obtained by ally increased under self-action conditions. Based on sputtering semiconductor-doped glasses that were sub- the refractive index and the thickness of the film, and sequently coated with lithium niobate, tin dioxide, and knowing the field of the waveguide mode, one can find quartz glass. In the multilayer structures, the thick- the nonlinear refractive index n2 and absorption coeffi- nesses of the quartz glass, lithium niobate, and tin diox- cient k2 [5]. The incident power was varied from 0.5 to ide layers were 70, 50, and 10Ð80 nm, respectively. The 500 µW. The radius of the beam at the base of the cou- thickness of each of the layers was controlled by the pling prism was no more than 200 µm.

1134 KHOMCHENKO, GLAZUNOV RESULTS AND DISCUSSION their values [6]. This fact points to the nonthermal character of the nonlinearity observed. We measured the nonlinear optical constants of thin vitreous arsenic sulfide films at incident light intensities Thin polycrystalline zinc selenide films also exhibit nonlinear changes in their optical properties as the opti- ranging from 10 to 100 W/cm2. Experiments were car- cal power was varied [8]. Curve 2 in Fig. 1 shows the ried out under self-action conditions at wavelength of µ propagation constant of the mode vs. incident light 0.63 m [6]. The value of the nonlinear constant n2 = intensity. The significant spread of the h' values far × Ð5 2 1.5 10 cm /W agrees with data in [7]. However, the exceeding the experimental error (δh' = 5 × 10Ð6) necessi- measurements at intensities lower than 0.1 W/cm2 tated a more thorough study of the nonlinear properties revealed a strong nonlinear dependence of the optical of the zinc selenide films. The h' versus I curves were parameters of the thin films on the incident intensity. nonmonotonic, and their run was very complex. It was ∆ Figure 1 plots h'(I) = h'(I) Ð h'(I0) versus the incident found that the nonmonotonicity of the curves and the light intensity for the TE polarization waveguide mode value of the constant n2 correlate with the crystal per- of zero order excited in the As2S3 film (I0 is the highest fection of the film. All films were polycrystalline with intensity at which the nonlinear effects are not yet their grains having cubic structure and preferred (022) observed). Hereafter, we are dealing with the values of orientation parallel to the substrate. Diffraction patterns h' that are related to the measured resonance angle ϕ of did not show any other structures [8]. Figure 1 waveguide mode excitation by the expression h' = (curves 2Ð4) plots ∆h'(I) versus I for the ZnSe films at ϕ ° k0npcos , where np is the refractive index of the cou- substrate temperaturess of 140, 180, and 250 C, respectively. The mean grain sizes in the films varied between pling prism and k0 is the wave number. In the linear case, h' is the real part of the propagation constant of the 19.7 and 12 nm. waveguide mode. The nonlinear constant was found to The thin-film structures obtained by sputtering the × Ð3 2 be n2 = 2.65 10 cm /W. Similar curves were semiconductor-doped OS-12 glasses and by sputtering obtained upon exciting waveguide modes of higher the ceramic targets containing SiO2 and CdSe also orders and other polarizations. The changes in the opti- exhibit the nonmonotonic dependence of the cal parameters of the film due to its heating by light waveguide properties on the incident intensity (Fig. 2). absorption were five orders of magnitude smaller than In these structures, h' also tends to decrease and the absorption tends to rise. The peaks in the curves again exceed the experimental error. The optical nonlinearity ∆ × 5 ∆ × 5 ( h'/k0) 10 ( h'/k0) 10 here depends on the size of the semiconductor grains in 2 the glass matrix [9]. One can change the size of these grains by thermal annealing or by varying the condi- 1 tions for film deposition. The curves ∆h'(I) in Fig. 2 were obtained for the OS-12 glass films prior to and 5 after 6 h of thermal annealing at 400°C. The increase in the substrate temperature also affects the nonmonotonic dependence of the optical properties on the light intensity (curves 1Ð3 in Fig. 2). 1 3 Similar studies were carried out for glassy arsenic *** * * * * * ** sulfide films; polycrystalline zinc selenide, zinc oxide, * * and gallium selenide films; and the doped glass films. * * * * * The radiation wavelength 0.63 µm does not necessarily coincide with the absorption edge of these materials, –3 * * * * * * * 4 yet all the structures exhibit nonlinear optical proper- * ** 4 ties. Recently, Gaponenko [10] proposed that the rea- * * son for the optical nonlinearity is the effect of the sur- * face and the interfaces and made an attempt to simulate ** such a nonlinear medium with a multilayered structure. * ** The structure was produced by sequentially deposit- * ing lithium niobate and fused quartz. Figure 3 plots h' 0 versus the incident radiation intensity for the multilayer structures. Note the nonmonotonic run of these curves 0 40 80 120 2 and also the fact that the number of peaks (five or six) I, mW/cm coincide with the number of layers in the structure produced by sputtering the lithium niobate target. For the Fig. 1. Propagation constant versus intensity of incident intensity ranges I and II indicated in Fig. 3, the nonlin- light for (1) As2S3 film and ZnSe films deposited at the substrate temperature (2) 180, (3) 240, and (4) 250°C. ear refractive index n2 and the nonlinear absorption

TECHNICAL PHYSICS Vol. 46 No. 9 2001 OPTICAL NONLINEARITY IN THIN FILMS 1135

∆ × 5 ∆ × 5 ∆ × 5 ( h'/k0) 10 ( h'/k0) 10 ( h'/k0) 10 1 4 III

* * –1 * 2 2 0 * *

1 –3 * 3 0 * * * * * * * * * * * –4 * * * * 2 * * * * * * * 4 –5 * * * –2 * * * * * * * 1 * * 3 * * * * * * * * –8 0 40 80 120 0 40 80 120 2 I, mW/cm2 I, mW/cm Fig. 3. ∆h'(I) versus intensity of incident light for the zero- Fig. 2. ∆h'(I) versus intensity of incident light for the color order mode in multilayer structures containing (1) ﬁve and glass ﬁlms deposited at the substrate temperature (1) 140 (2) six layers of lithium niobate, and (3) three and (4) one ° and (3) 190 C. (1) Prior to and (2) after thermal annealing. conducting SnO2 layers.

() () I × Ð3 2 I coefficient k2 were n2 = Ð2.1 10 cm /W, k2 = assume that the optical nonlinearity in the structures () () considered is associated with electron processes at the − × Ð3 2 II × Ð3 2 II 6.2 10 cm /W, n2 = 3.1 10 cm /W, and k2 = semiconductorÐinsulator interfaces. Ð5.1 × 10Ð3 cm2/W (curve 1). The large values of n and 2 ||()I ||()II k2 at low intensities make it possible to use these struc- If the condition k2 > k2 is met, the processes tures as nonlinear optical media. taking place at the interfaces increase in the total absorption of the multilayer structure as the incident Another type of multilayer structure was produced light intensity grows. The additive nature of this effect by the sequential deposition of the linear (in the usual explains the tendency for the absorption to grow. In the sense) materials: conducting tin dioxide and insulating () () opposite case (||k I < ||k II ), the film will become SiO2. Such a structure with a thickness of the layers of 2 2 about 10 nm simulates a low-dimensional nonlinear more and more transparent with an increase in the medium. In Fig. 3, ∆h' is plotted against the light inten- intensity. We suggest that these processes are responsi- sity for a structure containing three tin dioxide layers ble for the nonmonotonic behavior of the optical prop- separated by silicon dioxide layers. It is seen that the erties of the structures at low incident intensities. behavior of the curve correlates with the number of lay- For the multilayer structures, the nonlinearity of the ers in the structure. The thicknesses of the conducting optical properties substantially depends on the optical (tin dioxide) layers were 12, 24, and 36 nm. Curve 3 in quality of the insulating layer. Figure 4 shows the ∆h'(I) Fig. 3 has three peaks with different widths. It can be curves for SnO films deposited onto three different concluded that the thicker the layer, the wider the cor- 2 responding peak. The third peak in curve 3 (d = substrates in a single process cycle. The substrates were layer quartz glass and quartz glass covered by an SiO film. 36 nm) is similar in shape to that in curve 4 for the x thicker film (d = 120 nm). The film was applied by sputtering fused quartz under layer various conditions. The difference in the composition The decrease in h' with increasing intensity corre- of the SiOx films accounts for the difference in their sponds to the decrease in the refractive index of the absorption coefficients: 1.5 × 10Ð5 (curve 2) and 5 × 10Ð6 film. In this case, the absorption coefficient grows. The (curve 3). The thickness of these films was 1 µm. Even increase in the absorption coefficient and the decrease an inert amorphous substrate is known to affect the in the refractive index are usually related to the rise in properties of thin-film waveguides [12]. In our case, the the charge carrier concentration in the semiconductor nonlinearity of the optical properties is also more pro- material [11]. Based on our experimental data, we nounced in the waveguide structures where the sub-

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1136 KHOMCHENKO, GLAZUNOV

∆ × 5 ∆ × 5 ( h'/k0) 10 ( h'/k0) 10 strate and the waveguide are separated by the imperfect 1 buffer SiOx films with the higher absorption coeffi- 2 cients. * * * * * * Note that the insulating films also exhibit optical * 1 nonlinearity at low incident intensities. Nonlinear opti- 0 * * * * * 9 cal properties were observed in the films obtained by * * * sputtering fused quartz. The value of the nonlinear con- * 3 stant n (about 10Ð7 cm2/W) was significantly smaller * * 2 –1 than those for the semiconductor films or for the semiconductor-doped glass films. However, the nonlinear * dependence of the optical parameters on the light intensity was reliably detected only in the nonstoichiometric –2 * 0 films. We did not observe this effect when the optical losses in the waveguide film were about 2 dB/cm (Fig. 5). 4 The range I of photoinduced absorption (Fig. 3) is –3 most likely to be related to the capture of light-induced 5 charge carriers by localized states in the imperfect SiO2 film. The nature of the localized states is usually speci- –4 –9 fied by defects in the film. Therefore, it is expedient to 0 40 80 study the spatial distribution of defects near the inter- I, mW/cm2 faces in thin-film structures. If the nonmonotonicity of the curves is related to the Fig. 4. ∆h'(I) versus intensity of incident light for multilay- surface-induced modification of the electron states, the ered structures (1) SnO Ðsubstrate, (2) SnO ÐSiO with the run of the curves can be varied by introducing gas 2 2 × x Ð5 absorption coefficient of the SiOx film 1.5 10 , and admixtures into the medium surrounding the film. Gas (3) SnO ÐSiO with the absorption coefficient of the SiO 2 Ð6 x x molecules adsorbed on the surface of a thin-film struc- film 5 × 10 . Curves 4 and 5 refer to prismÐbuffer layerÐ ture can both induce and remove the surface states [13]. SnO structures in (4) air and (5) water vapor, respectively. 2 To study these phenomena, we made a waveguide structure consisting of silicon dioxide and tin dioxide films sequentially deposited on the base of a glass ∆ × 5 ( h'/k0) 10 prism. In this case, the surface of the semiconductor 6 film can be exposed to the gas admixtures. Water vapor is known to intensely passivate the surface states [14]. Therefore, it is logical to assume that the presence of water vapor would suppress the decrease in the absorp- 1 tion (and the increase in h'). Curves 4 and 5 in Fig. 4 show the variation in the parameters of the waveguide 4 film with incident intensity in air and in the water vapor atmosphere, respectively. The low-intensity portion is where the refractive index drops and the absorption increases is retained. However, when exposed to water 2 * * * vapor, the film did not become transparent. Thus, this * * 2 * can be considered as proof that our assumption of the effect of the interfaces on the nonlinear behavior of the * * optical properties of the multilayer structures is valid. * In our opinion, the above considerations are also 3 applicable to nonlinear structures based on doped * glasses (Fig. 2). In this case, by a semiconductorÐinsu- 0 * lator interface, we mean the surface of the semiconduc- * tor grains embedded in the glass matrix. Immediately after deposition, the films have statistically equal grain 0 20 40 60 2 sizes (curve 1). Thermal annealing changes the grain I, mW/cm size distribution (curve 2). Similar charges are observed when the substrate temperature rises (curve 3). By anal- Fig. 5. ∆h'(I) versus intensity of incident light for the quartz ogy with the multilayer structures (curve 3 in Fig. 3), films with the absorption coefficients (1) 2 × 10Ð5, (2) 9 × we can assume that the wider the peaks, the larger the 10Ð6, and (3) 3 × 10Ð6. grain size.

TECHNICAL PHYSICS Vol. 46 No. 9 2001 OPTICAL NONLINEARITY IN THIN FILMS 1137

Our results may indicate the similarity of the pro- to the generation, capture, and recombination of charge cesses that take place in these thin-film structures and carriers at the interfaces of the semiconductor film. are responsible for the nonmonotonic dependence of the optical properties on the radiation intensity. ACKNOWLEDGMENTS Irradiation of semiconductors by light generates electronÐhole pairs, which diffuse into the semiconduc- We are grateful to A. I. Voœtenkov for fruitful discus- tor. As the thickness of the semiconductor film is much sions. This work was supported by the Belarussian smaller than that of the screening layer, we are dealing Foundation for Basic Research. with the surface generation of charge carriers or, more precisely, with the generation of carriers in the near- REFERENCES surface layer of the semiconductor. Then, the equation for the charge carrier balance at the surface is given by 1. H. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985; Mir, Moscow, 1988). 1 g = --- j ()O + Sδn , 2. H. Haus, Waves and Fields in Optoelectronics (Prentice- s e s s Hall, Englewood Cliffs, 1984; Mir, Moscow, 1988). δ 3. V. P. Red’ko, A. A. Romanenko, A. B. Sotskiœ, and where gs is the rate of carrier surface generation, S ns is A. V. Khomchenko, RF Patent No. 2022247, Byull. Izo- the rate of carrier surface recombination, and js(O) is bret., No. 20 (1994). the current of the carriers near the surface [15]. 4. A. B. Sotskiœ, A. A. Romanenko, A. V. Khomchenko, and This equation implies that gs is independent of the I. U. Primak, Radiotekh. Élektron. (Moscow) 44 (5), 1 absorption coefficient of the film. Since the wavelength (1999). of the incident light does not fall into the fundamental 5. A. B. Sotskiœ, A. V. Khomchenko, and L. I. Sotskaya, absorption band and the concentration of surface states Pis’ma Zh. Tekh. Fiz. 20 (16), 49 (1994) [Tech. Phys. can reach the concentration of atoms at the surface of Lett. 20, 667 (1994)]. 6. A. B. Sotskiœ, A. V. Khomchenko, and L. I. Sotskaya, the film, js(O) < 0. In this case, the charge carriers move toward the surface of the semiconductor. Filling the Opt. Spektrosk. 78, 502 (1995) [Opt. Spectrosc. 78, 453 (1995)]. free energy levels of the surface states decreases the absorption in the film. Note also that the surface states 7. A. Yu. Vinogradov, É. A. Smorgonskaya, and E. I. Shi- frin, Pis’ma Zh. Tekh. Fiz. 14, 642 (1988) [Sov. Tech. do not all take part in recombination. Some of them are Phys. Lett. 14, 287 (1988)]. involved in bandÐenergy level transitions, so that the 8. A. V. Khomchenko, Zh. Tekh. Fiz. 67 (9), 60 (1997) energy level is a trap. Then, photoinduced (enhanced) [Tech. Phys. 42, 1038 (1997)]. absorption in the thin-film structures can be treated 9. S. Sh. Gevorkyan and N. V. Nikonorov, Pis’ma Zh. Tekh. within the two-level model of optically recharging deep Fiz. 16 (13), 32 (1990) [Sov. Tech. Phys. Lett. 16, 494 impurity levels in the band gap. (1990)]. The competition of these processes at the interfaces 10. S. V. Gaponenko, Optical Properties of Semiconductor gives rise to the nonmonotonic dependence of the opti- Nanocrystals (Cambridge Univ. Press, Cambridge, cal properties on the intensity of the probing beam. 1998). 11. A. V. Khomchenko and V. P. Red’ko, Proc. SPIE 1932, 14 (1993). CONCLUSION 12. R. G. Hunsperger, Integrated Optics: Theory and Tech- With the waveguide methods, we studied optical nology (Springer-Verlag, Berlin, 1984; Mir, Moscow, nonlinearity in semiconductor and dielectric thin-film 1985). structures at radiation intensities below 0.1 W/cm2 and 13. A. Zangwill, Physics at Surfaces (Cambridge Univ. at a wavelength of 630 nm. The general tendencies in Press, Cambridge, 1988; Mir, Moscow, 1990). the dependences of the optical properties on the radia- 14. The Physics of Hydrogenated Amorphous Silicon, Vol. 2: tion intensity for multilayer structures, polycrystalline Electronic and Vibrational Properties, Ed. by J. D. Joan- films, and semiconductor-doped glass films were con- nopoulos and G. Lucovsky, with contributions by sidered. In low-dimensional semiconductor structures, D. E. Carlson et al. (Springer-Verlag, New York, 1984; optical nonlinearity depends on the quality of the semi- Mir, Moscow, 1988). conductorÐdielectric interface; and in polycrystalline 15. V. L. Bonch-Bruevich and S. G. Kalashnikov, Physics of films, on the degree of crystallinity and the grain size. Semiconductors (Nauka, Moscow, 1977). Our approach suggests that the nonlinear variation of the optical properties in the thin-film structures is due Translated by A. Chikishev

TECHNICAL PHYSICS Vol. 46 No. 9 2001

Technical Physics, Vol. 46, No. 9, 2001, pp. 1138Ð1142. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 71, No. 9, 2001, pp. 73Ð78. Original Russian Text Copyright © 2001 by Parygin, Vershubskiœ, Filatova.

ACOUSTIC, ACOUSTOELECTRONICS

Optimization of the Transfer Function of an Acoustooptic Cell with an Apodized Piezoelectric Transducer V. N. Parygin, A. V. Vershubskiœ, and E. Yu. Filatova Moscow State University, Vorob’evy gory, Moscow, 119899 Russia Received October 27, 2000

Abstract—It is shown that an apodized piezoelectric transducer can signiﬁcantly reduce the side lobe level of the acoustooptic cell transfer function. Series and symmetric connections of the transducer sections and mea- sures to suppress the effect of spurious elements arising in the electric circuit are proposed. In particular, the effect of spurious capacitances and inductances on the frequency response of the transducer is studied. It is shown that they violate the optimal condition for the suppression of the transfer function side lobes, especially at high frequencies. It is shown by calculations that the effect of spurious elements can be eliminated by the insertion of additional capacitors at 80 and 150 MHz. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION tooptic devices. Recently, a number of works [5Ð10] The field of knowledge of acoustooptics lies at the aimed at suppressing the side lobes have appeared. border between physics and technology. It studies the Acoustooptic filters based on acoustooptic cells can interaction between electromagnetic and acoustic be collinear and noncollinear. In the former, the transfer waves and offers a variety of applications based on this function is controlled by applying appropriate acoustic phenomenon. The interaction between light and acous- pulses instead of a continuous signal to the cell. The tic waves is used for controlling coherent light in mod- width of the control pulse specifies the filter passband, ern optics, optoelectronics, and laser technology. while its waveform defines the shape of the transfer Acoustooptic devices are capable of controlling the function. This has been demonstrated experimentally amplitude, frequency, polarization, spectrum, and and theoretically in [8, 10]. direction of a light beam. An important field of applica- Unfortunately, these methods for optimizing the tion of acoustooptic effects is data processing systems, transfer function cannot be applied to the usual orthog- where microwave signals are processed in real time [1]. onal geometry of the acoustooptic interaction. There- Acoustooptic effects are employed in active optical fore, in this paper, we consider the possibility of reduc- devices that can entirely control a light beam and pro- ing the side lobes by apodizing the transducer with cess information carried by optical and acoustic waves. regard for the arising parasitic elements. The core of these devices is an acoustooptic cell that consists of a working medium where lightÐacoustic BASIC EQUATIONS FOR AN APODIZED wave interaction occurs and of an acoustic radiator TRANSDUCER (usually, a piezoelectric transducer). There are a number of acoustooptic devices designed for various pur- Consider the typical orthogonal geometry of the poses: deflectors, modulators, filters, processors, etc. acoustooptic interaction, where the sound beam is gen- [2Ð4]. erated by a rectangular piezoelectric transducer and the The acoustooptic interaction phenomenon relies on light wave is considered to be plane. Then, the system the photoelasticity effect, i.e., on the property of a of equations that describes the acoustooptic interaction medium to change its refractive index under the action by relating the amplitudes of the transmitted, Et, and of elastic stress. Due to this effect, an acoustic wave diffracted, Ed, light waves is written as propagating in an optically transparent medium is, in dEt q () ()η essence, a phase grating that moves with the speed of ------= Ð---Ed x exp Ð j x , v dx 2 sound . When passing through the acoustic field, the (1) light is diffracted by refractive index nonuniformities. dE q ------d- = ---E ()x exp()jηx . The acoustooptic cell is characterized by its transfer dx 2 t function, i.e., by the dependence of the intensity of diffracted light on its wavelength at a given sound fre- Here, q is a coefficient proportional to the acoustic η quency f. Along with the main lobe, the transfer func- wave intensity; = (kt + K Ð kd) á ex is the detuning tion usually exhibits significant side lobes. The side parameter; kt, kd, and K are the wave vectors of the lobes substantially narrow the dynamic range of acous- incident and diffracted light waves and of the acoustic

OPTIMIZATION OF THE TRANSFER FUNCTION 1139 wave, respectively; ex is the unit vector along the x axis, amplitudes Et(Lk) and Ed(Lk) for the previous section by and (kt + K Ð kd) á ex is the scalar product. A solution to the recurrent formulas system (1) under the boundary conditions  () ()η () ξ () () Et Lk + 1 = exp Ð j lk + 1/2 Et Lk cos k - Ed 0 ==0, Et 0 Ei (2)  is given by η η () ()Ð j Lk  j Et Lk Ð qk + 1Ed Lk e ξ + ------sin k , () ()η qEi 2 η2 l 2 η2  Ed l = exp j l/2 ------sin q + ---, qk + 1 + 2 2 2 q + η (4)  () ()η () ξ Ed Lk + 1 = exp j lk + 1/2 Ed Lk cos k -  2 2 l  E ()l = E exp()Ð jηl/2 cos q + η --- (3) t i  2 jηL jηE ()L Ð q E ()L e k  Ð ------d k k + 1 t k sinξ . jη 2 2 l  k + ------sin q + η --- , q2 + η2  2 2 2 k + 1 q + η Here, qk is a parameter proportional to the amplitude of ξ where Ei is the amplitude of incident light. Formulas (3) the acoustic wave in the kth section, k = define the amplitude of the light waves at the exit from 2 2 k q + η (l /2), and L = ∑ l is the total the cell (at x = l). The diffracted light intensity at the exit k + 1 k + 1 k i = 1 i length of k sections of the piezoelectric transducer. from the cell, Id = Ed(l)(Ed* l), depends on the RamanÐ η Relationships (4) are the solution of system (1) at Nath parameter ql and the detuning parameter l. The q = q for arbitrary amplitudes of the incident and latter depends on the sound frequency f = Kv/2π and k + 1 λ π diffracted light at the entrance to the (k + 1)th section of the wavelength of incident light = 2 nt/kt. Here, v is the piezoelectric transducer. For example, at k = 0 and the acoustic velocity and nt is the refractive index of the E (0) = 0, formulas (4) yield solution (3). Relation- η η d medium for incident light. Thus, P( l) = Id( l)/Id(0) is ships (4) are valid at any qk. However, for weak acous- the transfer function of the acoustooptic cell. tooptic interaction, characterized by a small RamanÐ Nath parameter (q l Ӷ 1), the terms containing (q l )2 Relationship (1) shows that E (η) is the Fourier k k k k d can be neglected and relationships (4) can be written in transform of the product qEt, i.e., of the product of the a simpler form transverse amplitude distribution of the acoustic field Ð jζ ζ π by the amplitude of incident light. To change the trans- Et(Lk + 1 ) = Et(Lk )Ð 0.5qk + 1lk + 1e Ed(Lk )sinc(),/ fer function, it is therefore necessary to vary the distri- (5) jζ ζ π bution of the acoustic field amplitude. In the case of Ed(Lk + 1 ) = Ed(Lk )+ 0.5qk + 1lk + 1e Et(Lk )sinc(),/ orthogonal geometry, this problem is very difficult. where ζ = ηl /2 and sinc(x) = sin(πx)/(πx). With However, we can divide the transducer into several sec- k + 1 tions and apply different voltages to them. With such an boundary condition (2) at the entrance to the first section, we successively calculate the amplitudes at the approach, the acoustic beam will consist of several lay- entrance to each section. As a result, Eqs. (5) yield ers, each having its own sound intensity. E ()L = E , Mathematically, the solution to the problem can be t k + 1 i described, as before, by Eqs. (1) if q is assumed to be k (6) () jζm ()ζ π constant within each section. At the exit from the first Ed Lk + 1 = 0.5Ei ∑ qmlme sinc m/ , section, the solution is thus given by formulas (3) by q m = 1 and l replaced with q1 and l1, respectively. Then, ζ η where m = lm/2. Eqs. (1) can be solved for the case q = q2 under the initial conditions E = E (l ) and E = E (l ). As a result, we Thus, successively applying relationship (4) to each t t 1 d d 1 of the layers of the acoustic beam, we can calculate the obtain the amplitudes of the incident and diffracted distribution of the transmitted and diffracted light light at the exit from the second section: Et = Et(l2) and amplitudes at the exit from the acoustooptic cell. This Ed = Ed(l2), where L2 = l1 + l2. These amplitudes are the distribution may be rather complex. By appropriately initial conditions as applied to Eqs. (1) for the third sec- choosing the lengths lk of the acoustic beam layers and tion, and so on. The amplitudes at the exit from the the amplitudes qk of the acoustic field, the side lobes of (k + 1)th section can thus be expressed in terms of the the transfer function may be significantly suppressed.

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1140 PARYGIN et al.

P, dB THE EFFECT OF SPURIOUS ELEMENTS ON THE TRANSFER FUNCTION 0 OF THE ACOUSTOOPTIC CELL 2 Curve 1 in Fig. 1 shows a typical transfer function of –10 1 the acoustooptic cell with a nonapodized piezoelectric transducer. Here, P is the relative intensity of the dif- η π –20 fracted light in decibels, l/ is the detuning parameter, and l is the length of acoustooptic interaction. The transfer function of this form has been widely discussed –30 in the literature. A stepwise distribution of the acoustic field amplitude and a modified transfer function (curve 2 –40 in Fig. 1) can be obtained by dividing the piezoelectric transducer into several electrically insulated series-connected sections of different lengths. –50 Figure 2 schematically illustrates a sectional piezo- –10 –5 0 5 10 ηl/π electric transducer to which a power supply is connected (a) in series and (b) symmetrically. It should be noted that, in the symmetric connection (Fig. 2b), there Fig. 1. Transfer function of the acoustooptic cell for strong interaction: (1) nonapodized cell and (2) piezoelectric trans- is no need for dividing the central part of the transducer ducer divided into nine sections of optimal lengths. into two equal sections; therefore, in this case, the number of cuts and connections is less than in Fig. 2a by (a) one. Each of the sections can be viewed as a capacitor I whose value is proportional to the length of the section. The amplitudes of the acoustic field are proportional to ~ the voltage applied to the sections of the transducer. Because of the series connection, the acoustic amplitude in each section is in inverse proportion to its capac- 12kk + 1 N itance and, consequently, to its length. II R R R Such a transducer produces an acoustic beam con- 1 2 N sisting of several layers with different amplitudes. The LL Llonger the section, the lower the acoustic amplitude in it and conversely. Our calculations have shown that C1 C2 CN C C optimal proportions exist between the section lengths ~ s s that minimize the side lobes of the transfer function. We calculated the parameters of optimal (in terms of side lobe suppression) piezoelectric transducers with various numbers of sections [11]. For example, curve 2 in (b) I ~ Fig. 1 plots the transfer function for a piezoelectric transducer with strong acoustooptic interaction that is divided into nine sections (l1 = l9 = 0.187l, l2 = l8 = 0.112l, l3 = l7 = 0.086l, l4 = l6 = 0.0785l, and l5 = 0.073l, where lk is the length of the kth section). 12k, k + 1 N – 1 N This design reduces the side lobe level from Ð9.3 dB II for an acoustooptic cell with a nonapodized transducer Rk Rk+1 to Ð21.7 dB for an acoustooptic cell with an apodized transducer. Experimentally, the side lobe suppression LL with the optimally selected stepwise voltage distribu- R1 RN tion has been demonstrated in [12]. The larger the num- C1 Ck Ck+1 CN C ~ C ber of sections, the lower the side lobe level of the trans- s s fer function. However, a large number of sections requires a very high cutting accuracy, which may cause Fig. 2. Connection diagram for a multisection piezoelectric problems if the transducer is short. Moreover, a large transducer: I, upper electrode; II, lower electrode. (a) The number of electrical connections induces parasitics, power supply is connected in series and (b) the power sup- which adversely affect the performance of the device. ply is connected to the central section of the transducer. The parasitics resulting in these circuits may sub- C1, …, CN are the capacitances of the transducer sections; R , …, R are the radiation resistances of the sections; L is stantially change the amplitudes and the phases of the 1 N voltages applied to the sections, because wires that con- the spurious inductance of the wires; and Cs is the ground capacitance of a section. nect adjacent sections always exhibit a certain induc-

TECHNICAL PHYSICS Vol. 46 No. 9 2001 OPTIMIZATION OF THE TRANSFER FUNCTION 1141 tance. Also, each section exhibits a spurious ground U0 capacitance. At low acoustic frequencies, the effect of 40 these reactive components may be neglected. However, at high frequencies, they may significantly change the 30 amplitudeÐphase distribution of the acoustic field. As a 12 consequence, the condition for suppressing the side lobes will be violated. Figures 2a and 2b, respectively, 20 show the equivalent circuits of piezoelectric transduc- 3 ers with the series and the symmetric connection of the power supply. Here, C1, …, CN are the capacitances of 10 the sections; R1, …, RN are the radiation resistances of the sections, which describe the electric-to-acoustic energy conversion efficiency; L is the inductance of 0 50 100 150 200 wires between the adjacent sections; and Cs is the f, MHz ground capacitance of the sections. Fig. 3. Voltage U0 across the piezoelectric transducer neces- It has been shown that, in these circuits, electric sary a 100% diffraction efficiency versus acoustic fre- amplitude resonance occurs in the transducer sections quency f. π at a frequency f0 = 1/2 LC, where C = C0/N, N is the U number of the sections, and C = N C is the total k 0 ∑k = 1 k (a) 6 capacitance of the transducer. Figure 3 shows the volt- 3 age U0 across the transducer providing a 100% diffrac- 5 tion efficiency versus ultrasonic frequency f at N = 7, 1a L = 15 nH, Cs = 10 pF, and C0 = (1) 20 300, (2) 2100, 4 and (3) 700 pF. These values are typical of modern 3 acoustooptic piezoelectric transducers. The respective 2 resonance frequencies are f0 = 24, 75, and 130 MHz. It 2 is seen that the voltage required for the 100% incident- 1 to-diffracted light conversion is minimal at a frequency close to the resonance frequency for a particular C0. 0

However, the higher C0, the narrower the operating (b) frequency band of the device, because at higher-than- 6 2 resonance frequencies, the supply voltage necessary to 5 obtain a high diffraction efficiency is large. The voltage amplitude distribution between the sections also 4 1b depends on the sound frequency. Figures 4a and 4b 3 illustrate this distribution for the sections connected as 3 shown in Figs. 2a and 2b, respectively, with N = 8, C0 = 2 700 pF, L = 15 nH, and Cs = 10 pF. In Fig. 4, Uk is the voltage amplitude across the kth section. The ampli- 1 tudes were calculated for 100% diffraction efficiency. 0 123 4 5 6 7 8 It should be noted that the curves in Fig. 4 are arbi- k trarily drawn to some extent, because the distributions are actually stepwise and the voltages indicated are Fig. 4. Voltage amplitudes Uk on the sections of the trans- constant within the respective section. The data points ducer for the (a) series (Fig. 2a) and (b) symmetric (Fig. 2b) are connected only for the purpose of illustration. connections at N = 8, C0 = 700 pF, L = 15 nH, Cs = 10 pF. Curves 1 correspond to the optimal suppression of the f0 = (1a) 142.5 and (1b) 133 MHz; f = (2) 100 and (3) 160 MHz. transfer function side lobes at the resonance frequencies. At off-resonance frequencies, the optimal voltage amplitude distribution will be different. For the circuit The significant change in the voltage amplitude dis- shown in Fig. 2a, this distribution becomes asymmet- tribution increases the side lobe level, especially at high ric: at frequencies below or above the resonance frequency, the maximal amplitude tends to the left (curve frequencies. Our calculations have shown that the 2) or to the right (curve 3). With the sections connected effect of the spurious reactive components may be as shown in Fig. 2b, the voltage amplitude distribution reduced by introducing additional trimming capacitors. ∆ among the sections remains symmetric but the ampli- These capacitors of value Ck should be connected in tudes change. parallel with Ck in Fig. 2. By appropriately selecting the

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1142 PARYGIN et al.

P, dB CONCLUSION –10 Thus, the spurious elements arising in the apodized –12 piezoelectric transducer may significantly affect the amplitude and phase distributions of the acoustic field –14 on its sections, particularly at high frequencies, and, 2 hence, violate the optimal condition for side lobe sup- –16 1 pression. Such a situation should be avoided, because 3 the use of the apodized transducer can significantly –18 reduce the side lobes of the transfer function of the acoustooptic cell. However, the effect of the spurious –20 elements can be eliminated by inserting additional –22 capacitors into the circuit. We calculated their values at 0 50 100 150 200 frequencies both below and above the resonance fre- f, MHz quency. The symmetric connection of the supply voltage was shown to provide the most complete elimina- Fig. 5. Maximum side lobe level of the transfer function tion of the spurious elements. For a seven-section versus acoustic frequency f for the symmetric connection of piezoelectric transducer, the side lobe level as low as the transducer sections at N = 7, C0 = 700 pF, L = 15 nH, and less than Ð20 dB within a certain frequency range can Cs = 10 pF: (1) without parasitic suppression and (2, 3) with parasitic suppression at 80 and 150 MHz, respectively. be reached.

∆ REFERENCES values of Ck, the section-to-section voltage amplitude distribution can be optimized at a particular frequency. 1. V. N. Parygin and V. I. Balakshiœ, Optical Information It should be noted that the trimming capacitors shift Processing (Mosk. Gos. Univ., Moscow, 1987). the characteristics shown in Fig. 3 towards lower fre- 2. A. Korpel, Acousto-optics (Marcel Dekker, New York, quencies. The best compensation has been shown to be 1988; Mir, Moscow, 1993). achieved at the symmetric connection of the piezoelec- 3. L. N. Magdich, Izv. Akad. Nauk SSSR, Ser. Fiz. 44, 1683 tric sections (Fig. 2b). For this case, the maximum side (1980). lobe level of the transfer function versus frequency is 4. V. I. Balakshiœ, V. N. Parygin, and L. E. Chirkov, Physi- shown in Fig. 5 at N = 7, C0 = 700 pF, L = 15 nH, and cal Principles of Acousto-optics (Radio i Svyaz’, Mos- Cs = 10 pF. Here, curve 1 results in the absence of the cow, 1985). trimming capacitors, while curves 2 and 3 show the 5. M. M. Mazur, V. N. Shorin, et al., Opt. Spektrosk. 81, suppression of the parasitics at 80 (C1 = C7 = 2 pF, C2 = 521 (1996) [Opt. Spectrosc. 81, 475 (1996)]. C6 = 4 pF, C3 = C5 = 10 pF, and C4 = 30 pF) and 6. A. K. Zaœtsev and V. V. Kludzin, Izv. Vyssh. Uchebn. 150 MHz (C1 = C7 = 60 pF, C2 = C6 = 25 pF, C3 = C5 = Zaved., Élektron. 41 (10), 75 (1998). 6 pF, and C4 = 0), respectively. 7. V. Pustovoit and N. Gupta, EOS Topical Meetings Digest From these data, we may conclude that, near the fre- Series 24, 35 (1999). quency for which the trimming capacitances are calcu- 8. V. N. Parygin and A. V. Vershubskiœ, Akust. Zh. 44, 615 ∆ lated, the side lobe level is lower than when Ck = 0. (1998) [Acoust. Phys. 44, 529 (1998)]. There exists a finite frequency band within which the 9. V. N. Parygin and A. V. Vershubskiœ, Radiotekh. Élek- side lobe level is sufficiently low, e.g., less than Ð20 dB. tron. (Moscow) 43, 1369 (1998). In particular, for the elimination of the parasitics at 10. V. N. Parygin, A. V. Vershubskiœ, and K. A. Kholostov, 80 MHz, this band is about 100 MHz wide, while at Zh. Tekh. Fiz. 69 (12), 76 (1999) [Tech. Phys. 44, 1467 150 MHz, its width is only about 20 MHz. Note also (1999)]. that at a high C , the elimination at above-resonance 0 11. V. N. Parygin, A. V. Vershoubskiy, and E. Yu. Filatova, frequencies requires a high supply voltage (see Fig. 3). J. Mod. Opt. 47, 1501 (2000). Nevertheless, it could be expected that, if the parameters of the spurious elements are known, for example, 12. V. N. Parygin, V. Ya. Molchanov, and E. Yu. Filatova, EOS Topical Meetings Digest Series 24, 45 (1999). C0 = 700 pF, one can find an operating frequency such that the side lobe level in its vicinity will be sufficiently low, e.g., below 20 dB. Translated by A. Khzmalyan

TECHNICAL PHYSICS Vol. 46 No. 9 2001

ELECTRON AND ION BEAMS, ACCELERATORS

RF Focusing of Ion Beams in the Axisymmetric Periodic Structure of a Linac É. S. Masunov and N. E. Vinogradov Moscow State Engineering Physics Institute (Technical University), Kashirskoe sh. 31, Moscow, 115409 Russia e-mail: [email protected]; [email protected] Received November 3, 2000

Abstract—A new approach to the analysis of charge particle motion in periodic resonant RF accelerators is suggested. A three-dimensional equation of motion in the Hamiltonian form is derived. This equation makes it possible to carefully study the relationship between the transverse and the longitudinal dynamics of the ion beams at low initial energies. General conditions for RF focusing in ion linacs are formulated. Basic results are compared with numerical simulation data for the beam dynamics in the polyharmonic ﬁeld of the accelerating cavity. A version of an RF-focusing proton accelerator in which the current transmission coefﬁcient is close to that in an accelerator with radio frequency quadrupole is described. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION It was shown that, in the smooth approximation, the analysis of three-dimensional beam dynamics is It is known that the stable motion of an ion beam in reduced to the analysis of the Hamiltonian function. a linac can be provided by using external focusing The Hamiltonian of the system takes a simple form if devices or specially configured accelerating fields (rf some properties of solutions to the Maxwell equations focusing). For low-energy ion accelerators, the latter are employed [8]. Later, such an approach was used to approach seems to be more promising. Today, several study beam acceleration and focusing in the polyhar- approaches to RF focusing are known: alternating- monic field of the cavity in the one-particle approxima- phase focusing (APF), radio frequency quadrupole tion [9, 10]. (RFQ), and undulator RF focusing (UrfF). The basic This work extends the studies of ARF in periodic principles of APF in the single-wave approximation resonant structures [9, 10] with respect to the space have been formulated in [1Ð3]. Later, APF has been charge field of the beam. considered in terms of a two-wave model where one wave is in synchronism with the beam [4, 5]. Subse- quently, it has been shown [6] that the second nonsyn- EQUATION OF MOTION chronous harmonic of the RF field should be taken into IN SMOOTH APPROXIMATION account in some cases. In [7], the situation with a large First we consider the equation of motion in the one- number of the spatial harmonics of a standing wave was particle approximation without including the self-field of treated in the short-gap approximation. The methods the beam. The external RF field of a periodic structure is used in these works to describe APF suffer from disad- represented as an expansion in spatial harmonics: vantages. The relationship between the longitudinal and transverse motions of the beam are considered ∞ ()()ω() incorrectly. The same is true for the effect of fast longi- Ez = ∑ En I0 hnr cos ∫hndz cos t , tudinal oscillations on the beam dynamics. The averag- n = 0 ing method used in [5] to analyze RF focusing is also ∞ (1) not well-defined. Thus, the available APR theory E = ∑ E I ()h r sin()ωh dz cos()t , incompletely discovers the RF focusing potentialities. r n 1 n ∫ n n = 0 Currently, the problem of increasing the current and where En are the amplitudes of the harmonics at the the current transmission coefficient in low-energy π µ µ linacs is becoming more and more important. It cannot axis, hn = h0 + 2 n/D, h0 = /D, is an oscillation phase be tackled within conventional APF theory. Because of advance, D is the period of the structure, and I0 and I1 are this, interest in axisymmetric RF focusing (ARF) has the modified Bessel functions of the zeroth and the first waned in recent years. In [8], RF focusing systems were order. considered with the method of averaging over fast In a polyharmonic field, the particle path can be rep- oscillations (the so-called smooth approximation). resented as the sum of slow- and fast-varying compo-

1144 MASUNOV, VINOGRADOV nents. Averaging over fast oscillations, as was done in ics at which the radial and phase stability of the beam is [8, 9], we come to the equation of motion achieved, as well as to find restrictions imposed on the amplitudes. d2R ∂ ------= Ð------Ueff, (2) dτ2 ∂R GENERAL ANALYSIS OF THE EFFECTIVE POTENTIAL FUNCTION where Ueff = U0 + U1 + U2 + U3 is the effective potential function. Here, Let us turn to expression (3) for Ueff . The term U0 1 describes the interaction of the particle with the syn- U = Ð---e β []I ()ψχη/β sin()+ /β Ð ()ψχ/β cos , 0 2 s s 0 s s s chronous harmonic, which accelerates the beam and simultaneously defocuses it radially; that is, the extre- 2 2 1 en 1 en mum of U0 is a saddle point. The term U1 contributes U1 = ------∑ ------g , ()η + ------∑------g , ()η , 16 ()∆Ð 2 sn 16 ()∆+ 2 sn only to the transverse motion, always focusing the ns≠ sn, n sn, beam in the radial direction. Its value depends neither on the amplitude nor on the phase of the synchronous 1 ene p U2 = ------∑ ------wave. It will be shown that taking these two terms into 16 ()∆Ð 2 account allows us to substantiate several types of APF ns≠ sn, np+2= s mentioned above. With the condition n + p = 2s (where s is also an integer) satisfied, the term U appears in the × []()1 ()η ()ψ χ β ()χ β ()ψ 2 f snp,, cos2 + 2 / s + 2 / s sin 2 , (3) expression for Ueff . This term affects both the longitudinal and the transverse motion of the particle and 1 ene p U3 = ---∑ ------arises in the presence of two or more asynchronous 8 ()∆Ð 2 ns≠ sn, field harmonics. Finally, if the condition hn Ð hp = 2hs

()2 (n ≠ s) is met, the term U3 becomes nonzero. It also × []f ,, ()η cos()2ψ + 2χ/β + 2()χ/β sin()2ψ , snp s s affects the radial and the longitudinal motion. Unlike

hn Ð2h p = hs. U2 , U3 cannot equal zero even in the two-wave We use the following designations: approximation (one synchronous and one asynchronous waves). The extrema of U and U , like that of []χη, χ π()slow slow λ 2 3 R ==, 2 z Ð zs / , U0 , are also saddle points. Therefore, the necessary slow ± condition for the beam to be stable in both the longitu- η ===2πr /λ, τωt, ∆ , ()h ± h /h , sn s s s dinal and transverse directions is the existence of the h η h η ()η 2n 2n absolute minimum of Ueff . In this case, Ueff is, in gsn, = I0------β- + I1-----β----- Ð 1, hs s hs s essence, a moving three-dimensional well at the bottom of which the synchronous particle is situated. η η η η ()1 ()η hn h p hn h p For generality, the synchronous velocity β is conve- f snp,, = I0------β- I0-----β----- Ð I1------β- I1-----β----- , s hs s hs s hs s hs s nient to eliminate from all the expressions in (3). To do χ χ β η η β ú this, we pass to the new variables = / s, = / s, () h η h η h η h η 2 ()η n p n p e = e /β , and R = R /β . With the new variables, the f snp,, = I0-----β----- I0-----β----- + I1-----β----- I1-----β----- . n n s s hs s hs s hs s hs s three-dimensional well U = U /β2 retains its shape. ψ β ω eff eff s The variables and s = /hsc are the phase and the velocity of the synchronous particle, and the variables We begin our analysis with the simplest case and χ χ η expand Ueff in the vicinity of its minimum { = 0, ( and ) mean the transition to the coordinate system η related to the particle. = 0}: 2 2 2 2 The effective potential Ueff provides the full three- U ()χη, = U()ω00, ++χχ /2 ωηη /2 eff (5) dimensional description of the particle dynamics in the δχη2 γχ3 … smooth one-particle approximation. It is related to the + /2 ++/3 . Hamiltonian of the system as The expansion coefﬁcients here depend on the 1dR 2 amplitudes en of the harmonics (n = 0, 1, …). Obvi- ---------τ + Ueff = H. (4) ously, radial and longitudinal phase focusing will be 2 d provided when the amplitudes are such that Analysis of the effective potential makes it possible ω2 > ω2 > to set relations between the amplitudes of the harmon- χ 0, η 0. (6)

TECHNICAL PHYSICS Vol. 46 No. 9 2001 RF FOCUSING OF ION BEAMS 1145

Note that parametric coupled resonances, which dis- its effect is weak. This set of harmonics, as well as turb the stability of the beam, may occur for certain {s = 1, n = 2}, provides effective transverse focusing relationships between ωχ and ωη. To study these reso- provided that the amplitudes are properly selected. nance effects, it is necessary to take into consideration The condition for transverse focusing can be the last two terms in (5). expressed in the form In the general case, the analysis of the effective 2 3 1 1 hn 3 α potential as a function of the harmonic composition of αψsin < ---e ------+ ------+ ---e --- . (7) 8 maxÐ 2 + 2 h 8 max 4 the field allows us to study the particle behavior not ()∆ , ()∆ , s only near the minimum but also throughout the ranges sn sn of the coordinates χ and η and velocities. The first The energy increases with an acceleration gradient important restriction on the amplitudes of the spatial dWs 1α ψ field harmonics can be derived from the condition of ------= --- Emax cos , (8) nonoverlapping resonances for various waves when the dz 2 phase portrait of the beamÐwave dynamic system is α β χ which is proportional to ; therefore, this parameter is considered in the ( χ, ) plane. On the one hand, this bounded from below. In the simplest case of APF [2], condition specifies the applicability limits of the aver- the phase velocity of the focusing wave is lower than aging method; on the other hand, it prevents possible that of the accelerating velocity (n > s) and the ampli- disturbances of the longitudinal stability of the beam. tudes of the harmonics decrease with increasing har- Using Hamiltonian (4) and analyzing the shape of the monic number (es > en). Therefore, condition (7) can be four-dimensional phase space, one can easily find a satisfied only if the synchronous phase is small (sinψ Ӷ relationship between a given longitudinal channel 1). This type of RF focusing has a small longitudinal acceptance and the limit value of the transverse emit- acceptance and a high acceleration gradient. tance in order to obtain the maximal current transmission coefficient. To illustrate the aforesaid, we will con- In one of the cases considered, it is also assumed sider specific accelerating and focusing channels with that n > s. Here, however, condition (7) can be met even at a large capture efficiency (sinψ ≈ 1) if the accelerat- different forms of Ueff. ing harmonic amplitude es is smaller than the amplitude α Ӷ of the focusing harmonic en; that is, if 1. This ver- IMPLEMENTATIONS OF RF-FOCUSING sion of focusing can be implemented by selecting the ACCELERATORS structure period in such a way that it covers two or more The amplitudes of the spatial harmonics at the sys- accelerating gaps. From condition (7), it also follows tem axis are represented as e = α e (α ≤ 1). The that all versions with n > s are inefficient for transverse n n max n focusing. value of emax related to the breakdown amplitude of the accelerating field is usually defined by the cavity design Another important restriction on the amplitudes fol- and will be considered fixed. Thus, our goal is to find lows from the condition that the resonances (separa- the weighting coefficients {α } for various focusings. trices) of adjacent waves do not overlap. The swing of n the separatrices over longitudinal velocities grows with the harmonic amplitudes. When the amplitudes reach 1. One Synchronous and One Asynchronous some values, the separatrices overlap. This disturbs the Field Harmonics condition for the phase stability of the beam. Except for the case {µ = 0, s = 0, n = 1}, U can be This system is the simplest. The addition of U1, eff which provides radial focusing, to the saddle-like term represented as U0 yields, under certain conditions, a three-dimensional U = ÐAF[]()η sin() ψ+ χÐ χΨ)cos + F ()η . (9) potential well. In this case, the weighting coefficients eff 1 2 α α α are { s = , n = 1}. The initial values of the Hamiltonian and Ueff First we consider the field with the phase advance strongly depends on the channel aperture and the syn- µ ≥ chronous velocity. Figure 1 shows sections obtained = 0 and s 1. The set {s = 1, n = 0} corresponds to η χ the usual Alvarez structure. The effect of focusing here when Ueff is cut by the planes = 0 and = 0. To provide efficient particle capture by phase and efficient is absent, since U1 = const. The set {s = 2, n = 0} corresponds to the same structure but with a doubled period. radial focusing, it is desirable that the cut depths in Of the systems with {s = 1, n = 2} and {s = 2, n = 1}, Fig. 1 be equal. Mathematically, this requirement is which do not have the zero harmonic, the former can written as provide efficient focusing. []ψ ψψ () ψ () A sinÐ 2 cos = Ð AF1 a sin + F2 a , (10) The implementation of ARF is much easier for the π λβ structures with the phase advance µ = π and s ≥ 0. The where a = 2 R/ s is a dimensionless aperture. sets {s = 0, n = 1} and {s = 1, n = 0} are Wideröe struc- For low-energy (100Ð300 keV) proton accelerators, tures, where tubes of different lengths and inner diam- requirement (10) is difficult to satisfy over the entire eters alternate. In the case {s = 0, n = 1}, U3 appears but channel length. As follows from the results of numeri-

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1146 MASUNOV, VINOGRADOV

0.010 –0.002 –2 –10 1 2 χ, η –2 –10 1 2 χ, η 0.005 –0.004 –0.002

–2 –10 1 2 χ, η –0.006 –0.004

–0.005 –0.008 –0.006

–0.010 –0.010 –0.008 β = 0.025 β = 0.028 β = 0.030

η χ Fig. 1. Sections obtained by cutting Ueff by the planes = 0 (continuous curve) and = 0 (dashed curve) at different velocities of the synchronous particle. ψ = 0.25π. cal simulation of the beam dynamics (see below), this 2. The Effect of the Second Asynchronous is a reason for inefficient particle capture by the phase Harmonic at the beginning of acceleration. When implementing optimized accelerating resona- In the one-particle approximation, the choice of the tors (even with a simple period structure), one must RF structure and the optimization of the field composi- take into account the contribution of higher spatial har- tion are performed as follows. Phase advances µ, the monics whose amplitude usually rapidly drops as the synchronous phase ψ, the maximal field e , and the max harmonic number grows. The analysis of the Ueff parameter α satisfying condition (7) are selected. In this behavior allows the elucidation of the effect of these way, the longitudinal motion of the particles, including harmonics on the particle dynamics. Consider the effect the longitudinal acceptance, are completely defined in of the second nonsynchronous harmonic in the above-con- the smooth approximation. The parameters α and ψ are sidered system with α Ӷ 1. In this case, the set of the selected such that the rate of acceleration and the longi- α α α weighting coefficients { s = , n = 1) is supplemented tudinal acceptance have reasonable values. The param- α ≡ ε µ by p < 1. Two cases are possible depending on the eters emax and depend on the design of the accelerator. ω harmonic number and the type of the structure. The first Next, the transverse oscillation frequencies η are cal- one is U = 0. This means that the second nonsynchro- culated for various s and n. Systems with large values 2, 3 ω nous harmonic adds only to U1, i.e., enhances focusing, of η are the best. It should be noted that, first, the field without affecting the longitudinal motion, the addition amplitude at which the separatrices overlap rapidly to the focusing power being proportional to ε2. In the drops with increasing harmonic number. Second, the other case, the combination of the asynchronous waves implementation of a field with high harmonic numbers generates the cross terms U (see table) and the phase (n > 2) at small β is difficult because of the need for 2, 3 complicating the structure period: a short period D portrait of the dynamic system becomes qualitatively different. Figure 2 shows the section obtained by cut- must have two or more accelerating gaps. For example, η the system with {s = 2, n = 3}, which has the maximum ting the effective potential by the plane = 0 at different ε. At ε 0, U is given by (9). As ε increases, the ωη for any µ, cannot be implemented if the beam energy eff is small. longitudinal capture width decreases and the rate of the acceleration grows in proportion with ε. The structure of the period D depends on the amplitude of the second Table ε harmonic. At some , one more minimum of Ueff Composition appears, hence, the second bunch in the period. The Structure type U of RF field eff general view of the effective potential and the map of its isolevels for this case are depicted in Fig. 3. The phase µ ≥ = 0, s 1 s = 1, n = 2, p = 0 U0 + U1 + U2 + U3 portrait upon forming the second region of stable s = 2, n = 4, p = 0 U0 + U1 + U2 + U3 motion is demonstrated in Fig. 4. Special investigations s = 2, n = 1, p = 3 U0 + U1 + U2 showed that the accelerating structure with the second µ π ≥ ε ≈ = , s 0 s = 0, n = 2, p = 1 U0 + U1 + U3 large-amplitude ( 1) asynchronous wave is not effi- s = 1, n = 3, p = 0 U + U + U cient for obtaining the maximal current transmission 0 1 3 coefficient. However, the approach suggested allows s = 1, n = 0, p = 2 U + U + U 0 1 2 one to study corrections to the particle motion. These s = 2, n = 1, p = 3 U0 + U1 + U2 corrections are necessary for the analysis of the beam

TECHNICAL PHYSICS Vol. 46 No. 9 2001 RF FOCUSING OF ION BEAMS 1147

Ueff dynamics. For example, one must take into consider- 3 ation how the nonsynchronous harmonics affect the lon- 0.05 gitudinal motion equation, the period of the RF struc- 2 ture, and, hence, the velocity of the synchronous parti- 1 cle. In this respect, the APF modiﬁcation suggested in [3] (AAPF) can be viewed as an attempt to improve the –4 –3 –2 –1 0 1 23χ longitudinal channel acceptance through the effect of asynchronous waves on the phase motion of the beam.

–0.05 3. Accelerator without Asynchronous Harmonic –0.10 Expression (3) implies that the particle can be accelerated and, at the same time, radial stability can be pro- η vided without the synchronous harmonic as well Fig. 2. Sections obtained by cutting Ueff by the plane = 0 α for ε = (1) 0.05, (2) 0.3, and (3) 0.7. ( s = 0). The three-dimensional potential well can be

(a)

η (b)

Fig. 3. Effective potential function with two minima: (a) general view and (b) isolevel map.

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1148 MASUNOV, VINOGRADOV

χ~

0.3

0.2

0.1

–4.0 –3.2 –2.4 –1.6 –0.8 0 0.8 1.6 χ

–0.1

–0.2

–0.3

Fig. 4. Phase portrait of the system with {µ = π, s = 1, n = 2, p = 0} upon the formation of the second bunch.

V(z)cosωt

Fig. 5. Implementation of the structure.

formed by the terms U2, 3. In this case, acceleration charge can be neutralized and the beam current can takes place in the ﬁeld of the composite wave arising markedly be increased. ≠ from the combined action of two nonsynchronous har- Consider, as an example, the case U2 0 and U3 = 0 monics (the undulator mechanism of acceleration [8]). that is, the structures with {µ = 0, s = 2, n = 1, p = 3} This system has a number of intriguing features. First, and {µ = π, s = 1, n = 0, p = 2}. The rate of the acceler- modulation takes place at a doubled frequency. Second, ation is given by the particle charge enters into expressions (2) and (3) in dW 1 emaxEmax the quadratic form, which provides the chance to accel------= ---α α ------sin()2ψ . (11) dz 4 n p ()∆Ð 2 erate unlike ions in one bunch. In this case, the space sn,

TECHNICAL PHYSICS Vol. 46 No. 9 2001 RF FOCUSING OF ION BEAMS 1149

With the substitution χˆ = 2χ and ψˆ = 2ψ + π/2, the 1.2 effective potential takes the form of (9). Thus, all con- 1.0 / clusions drawn for the Hamiltonian of the system with E Emax β βmax one synchronous and one asynchronous wave remain 0.8 c/ c valid. 0.6 0.4 2ψ/π SELECTION OF ACCELERATING CHANNEL PARAMETERS 0.2 When intense beams are accelerated, the accelerat- 0 0.38 0.75 1.13 1.50 ing channel parameters must be selected with regard to z/λ the space charge field. Consider the structure with {µ = π, s = 0, n = 1} in the two-wave approximation. We sub- divide the accelerating channel into the grouping and Fig. 6. Accelerator parameters vs. longitudinal coordinate. main (accelerating) sections. In the former, the syn- ψ π chronous wave linearly decreases from = /2 to some Iout/Iin rated value and the amplitude of the RF field monoton- 0.9 ically grows. In the accelerating section, these parameters remain fixed. Such a division allows one to substantially improve the phase capture. In the grouping 0.8 section, the amplitude of the RF field as a function of the longitudinal coordinate must be defined in such a way 0.7 that the coefficient of current transmission is as high as possible. Above, the amplitudes of the RF field har- α α ≤ 0.6 monics have been defined as En(z) = nEmax(z) ( n 1). In searching for the optimal function Emax(z) for the two-wave approximation, we start with the one-dimen- 0.5 sional model. Let us represent the Hamiltonian of the 0 0.05 0.10 0.15 0.20 0.25 system at the axis (η = 0) as Iin

2 Fig. 7. Current transmission coefﬁcient vs. input current. Pϕ ()ϕ H1 = ------++V ext V c. (12) 2mϕ

Here, {Pϕ, ϕ} are the canonically conjugate variables: It can be shown that the limit current In is roughly the momentum Pϕ = W Ð W , the coordinate proportional to the longitudinal acceptance. For the s current transmission coefficient to be high, the longitu- π ϕ 2 slow()β β 2β2 ω dinal acceptance must be a nondecreasing function of ==------λ ∫dz 1/ s Ð 1/ , mϕ mc s / , the longitudinal coordinate. From this condition, we can find the function E (z). Assume that, in the group- π 3β max 2 mc s ()ϕη, ing section, V ext = ------U0 = 0 , λ () ()() In z = In 0 Fz, (15) and V is the part of the potential function that is related c where I (0) is the initial value of the limit current and to the space charge. The distribution function can be n F(z) is some nondecreasing function of the longitudinal taken in the form coordinate. ≤ ff= 0 H0 ÐH1 for H1 H0, Relationships (2), (14), and (15) allow us to relate (13) the accelerating channel parameters E (z), β (z), and > max c f = 0 for H1 H0. ψ(z). This relation involves the unknown function F(z). In this case, the indifferent equilibrium state is It must rapidly increase and, at the same time, be bounded from above. The function Emax(z) thus found established at some value In of the current (we call In the limit current of the system). This parameter is linearly can be refined by numerical optimization. related to the longitudinal coordinate: ()∝ β2() ()Φψ() IMPLEMENTATION In z c z Emax z , (14) OF THE STRUCTURE where Φ = ∆(2ψcosψ Ð sinψ) + cos(∆ Ð ψ) Ð cosψ + It is known that the amplitudes of spatial harmonics (∆2/2 Ð 2∆ψ)cosψ and ∆ is the phase width of the sep- in resonant Wideröe structures rapidly drop with aratrix. increasing harmonic number. As has been shown

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1150 MASUNOV, VINOGRADOV above, RF focusing will be efficient and the beam will 0.79, the full channel length 3 m, the rate of the accel- be accelerated if the amplitude of the first harmonic eration was 0.7 MeV/m, and the aperture radius was (n = 1) is much higher than that of the zero harmonic 0.6 cm. At the rate of the acceleration of 0.7 MeV/m, (s = 0). Such a situation is provided in the modified the current transmission coefficient is 0.79, which is a Wideröe structure, which has a complex period consist- breakthrough for ARF accelerators. The current trans- ing of drift tubes of various lengths and inner diame- mission coefficient vs. input current is plotted in Fig. 7. ters. Figure 5 demonstrates the implementation of such The averaging method was verified by numerical simu- a structure and the potential distribution along the axis. lation in both the total and the averaged fields. The The potentials applied to the adjacent electrodes are results obtained in both cases coincided up to 5Ð10%. equal in amplitude and are phase-shifted by π. The field potential as a function of z (along the axis) has two equiamplitude antinodes within the period that are CONCLUSION symmetric relative to the third one. Such a function was A new approach to describing RF focusing in the realized by properly selecting the inner diameter, axisymmetric field of an ion linac is developed. The length, and shape of the tubes. For example, the ampli- classification of RF focusing types based on the analy- tude of the negative antinode is smaller than the ampli- sis of the harmonic composition of the RF field is sug- tudes of the other two, since the inner diameter of the gested. Analyzing ARF, we selected the accelerating associated (middle) tube is larger. By properly selecting channel parameters for the Wideröe structures and the lengths of the tubes and the shapes of the roundings, obtained the high current transmission coefficient. one can almost completely eliminate higher harmonics Computer simulation of high-current ion beam dynam- in the potential distribution. The thus-obtained trans- ics in an ARF structure was performed. The validity of verse distribution of the RF field is consistent with its the averaging method was justified. In some cases, the representation in terms of the modified Bessel func- performance of the accelerator under study is close to λβ tions. The length of the period is c/2, which makes that of RFQ devices, which proves the ARF efficiency. the implementation of the structure with n ≥ 2 difficult, since five or more tubes should be arranged within the period in this case. When heavy ions are accelerated, REFERENCES the synchronous velocity drastically drops and the 1. M. L. Good, Phys. Rev. 92, 538 (1953). period shrinks. This difficulty can be obviated by oper- 2. I. B. Faœnberg, Zh. Tekh. Fiz. 29, 568 (1959) [Sov. Phys. ating at lower frequencies, i.e., by increasing λ. Tech. Phys. 4, 506 (1959)]. 3. V. V. Kushin, At. Énerg. 29 (3), 123 (1970). 4. V. S. Tkalich, Zh. Éksp. Teor. Fiz. 32, 625 (1957) [Sov. NUMERICAL SIMULATION Phys. JETP 5, 518 (1957)]. The dynamics of a high-current proton beam in the 5. V. K. Baev and S. A. Minaev, Zh. Tekh. Fiz. 51, 2310 ARF accelerator described above was numerically sim- (1981) [Sov. Phys. Tech. Phys. 26, 1360 (1981)]. ulated with the method of coarse particles using a spe- 6. V. D. Danilov and A. A. Il’in, in Theoretical and Exper- cially developed computer program. The harmonic imental Investigations of Charged-Particle Accelera- composition of the RF field was selected for the case tions (Énergoatomizdat, Moscow, 1985), pp. 93Ð97. {µ = π, s = 0, n = 1} by the above-mentioned technique. 7. H. Okamoto, Nucl. Instrum. Methods Phys. Res. A 284, The accelerating channel parameters as functions of the 233 (1989). longitudinal coordinate are represented in Fig. 6. The 8. É. S. Masunov, Zh. Tekh. Fiz. 60 (8), 152 (1990) [Sov. Phys. Tech. Phys. 35, 962 (1990)]. amplitude Emax(z) of the field in the grouping region was found with the above approach. This function can 9. E. S. Masunov, in Undulator and RF-System for Ion Lin- be refined by using special methods of numerical opti- ear Accelerators: Proceedings of the 18th International mization. In the calculation, the parameters of the Linac Conference (CERN, Geneva, 1996), Vol. 2, p. 487. accelerator were the following: the operating frequency 10. E. S. Masunov and N. E. Vinogradov, in Proceedings of α α the 1999 Particle Accelerator Conference, New York, was 150 MHz, s = 0.1, n = 1, the maximal field amplitude was 300 kV/cm, the input/output energy was 1999, Vol. 4, p. 2855. 0.1/1.22 MeV, the input/output beam current was 0.1/0.079 A, the current transmission coefficient was Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 46 No. 9 2001

ELECTRON AND ION BEAMS, ACCELERATORS

A Planar Relativistic Electron Beam in a Plasma Channel Bounded by Conducting Walls A. P. Kuryshev and V. D. Andreev Baltic State Technical University, St. Petersburg, 198005 Russia Received October 5, 2000

Abstract—The force interaction of a relativistic electron beam with a plasma in a channel bounded by plane- geometry highly conducting walls is studied. The steady-state interaction regime, ω = ku, is analyzed using the model of a cold collisional electron plasma. The formulas for the transverse component of the force acting on the beam electrons are derived for an arbitrary deviation of the beam from the symmetry plane of the channel. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION The transverse component of the Lorentz force that acts on the beam electrons moving with velocity u The problems associated with the transport of rela- along the z axis has the form tivistic electron beams (REBs) in plasma channels bounded by media with different conductivities (from ()β β u vacuum to perfect conductors) have been widely dis- FeE==x Ð By , ---. (1) cussed in the literature [1Ð10]. c It is well known that the force interaction of an REB The problem as formulated reduces to solving the with a plasma in a channel bounded by a conducting corresponding Maxwell equations under the assump- wall induces positive charges and reverse currents on tion that the linear properties of the medium are pre- the wall surfaces; moreover, the force between the scribed. In the linear model of a cold collisional elec- charges and the REB is attractive, whereas the force tron plasma, the dielectric tensor is described by the between the currents and the REB is repulsive. Clearly, familiar expression the dominance of either an attractive or a repulsive 2 force results in the preferential attraction or repulsion ω ε ()δω, k ==εω() δ1 Ð ------p , (2) between the REB and the wall. If an REB is transported ij ij ijωω()+ iν in a vacuum channel, then the induced charge dominates and the beam is attracted by the channel wall. In ω where p is the Langmuir frequency of the plasma elec- a plasma-filled transport channel, the beam charge is trons and ν is the collision frequency of the plasma partially or wholly screened. As a result, the interaction electrons. between the currents may give rise to a force that acts to stabilize the deviation of the beam from the channel We use Maxwell’s equations axis. This effect can be used to solve such problems as 1∂D 4π transverse stabilization of the trajectory of a beam curlB ==------+0,------j , divB transported through the plasma [2Ð6] and the formation c ∂t c b (3) of annular beams [7, 8]. The attraction of an REB to the ∂ 1 B πρ wall of the channel filled with a weakly ionized plasma curlE ==Ð4------∂ -, divD b was analyzed by Vladyko and Dudyak [9], who applied c t the model in which the plasma properties are described σ and the equation of state in which spatial dispersion is in terms of a constant electrical conductivity . In our neglected: earlier paper [10], we solved the problem of the transport of a relatively thin annular electron beam in a t cylindrical plasma channel. (), ε (), (), Di t r = ∫ dt' ij ttÐ ' r E j t' r . (4) A similar problem is treated in the present paper: we Ð∞ investigate the force interaction between an REB and a plasma in a channel bounded by plane-geometry highly Taking into account the equation of state (4), we conducting walls. We examine the steady-state interac- consider the projections of Eqs. (3) onto the axes of the tion regime using the model of a cold collisional elec- Cartesian coordinate system. We take the FourierÐ tron plasma and the model of a finite-duration beam Laplace transformation (in the z coordinate and time t) with constant parameters. of the electromagnetic field components, which deter-

1152 KURYSHEV, ANDREEV mine the transverse component of the Lorentz force The solution to problem (7)Ð(9) is easy to construct: acting on the beam electrons, to obtain ()κ () κ() ∂ I 2H sinh h sinh RaÐ ()κ() ik 2 E Ez = Ð------sinh Rx+ , F()ω,,kx = Ðe-----()1 Ð β ε ------z, κ2 sinh()2κR κ2 ∂x (5) 2 ()κ () κ() 2 2 ω II H cosh a cosh RhÐ ()κ κ = k Ð ------εω(). Ez = ------cosh x 2 2 cosh()κR c κ sinh()κa sinh() κ()RhÐ In order to analyze the steady-state interaction Ð ------sinh()κx Ð, 1 regime, we set ω = ku in expressions (5). As a result, the sinh()κR force component in question becomes ∂ III 2H sinh()κh sinh() κ()Ra+ i Ez E = Ð------sinh()κ()RxÐ , FFkx==(), Ðe------. (6) z 2 ()κ k ∂x κ sinh 2 R We start by investigating the case of a perfectly con- Reκ > 0. ducting channel wall.

We take the derivative of Ez with respect to x to PERFECTLY CONDUCTING WALL obtain the desired expression for the Fourier component of the sought-for force acting on the beam elec- The geometry of the problem is shown in Fig. 1. The trons that occur at the symmetry plane of the channel equations for the electric field component Ez in each of the regions of the system under investigation can ei∂ Fx()= a = Ð---- Ez readily be obtained from Maxwell’s equations: k ------∂ x xa= ∂2 (10) ᏸ ≡ Ez κ2 []η η π 2 Ez ------Ð Ez = H ()xaÐ + h Ð ()xaÐ Ð h , 4 e n i iRκsinh()2κa sinh()hκ ∂ 2 = Ð ------b ------˜j()k . x (7) R kε()ku k sinh()2κR 2 i H = 4πκ en ------j ()kx, , bkε()ku z In order to determine this force over the entire physical space, it is necessary to take the inverse Fourier η where (x) is the Heaviside step function, enbujz(k, x) transformation of expression (10) in the longitudinal is the Fourier-transformed (with respect to the z coordi- coordinate zˆ . nate) beam current density, and 2R is the width of the plasma channel. In a collisionless plasma (ν = 0), the desired expression for the force acting on the electrons of an ultrarel- For definiteness, we assume that the x component of γ ∞ the beam current density is uniform. The boundary con- ativistic ( ) beam has the form ditions for Eq. (7) have the form 2a h ()± {} sinh------sinh--- Ez x = R = 0, Ez xah= ± , 2 λ λ zˆ F = Ð2πe n λ------1 Ð.cos--- b 2R λ ∂E (8) sinh------{}B ==0 ⇔ ------z 0, λ y xah= ± ∂x ± xah= The finiteness of the relativistic factor and the colli- {}f = fx()= a± h+ 0 Ð fx()= a± hÐ 0 . (9) sions in the plasma can be incorporated in the same way as was done in [10]. As a result, expression (10) can be x represented in the form 2 4πe n i III Fk()==Ð------bG ()k G ()k jk(), G ()k ------,

J R 1 2 1 kε()ku

h ...... ˇ

2 ...... II Rκsinh()2κa sinh()hκ (11) α z G2 = i------()κ -. F k sinh 2 R R I In order to extend this expression to the entire physical space, we ﬁrst take the inverse Fourier transformation of the functions G1 and G2 and then successively ﬁnd the two Fourier convolutions that involve the ν ω Fig. 1. resulting functions. For < 2 p, the inverse Fourier

TECHNICAL PHYSICS Vol. 46 No. 9 2001 A PLANAR RELATIVISTIC ELECTRON BEAM 1153 transform of the function G1(k) is determined by the contributions of the two poles k 1 2 γ γ series k , ==±k Ð iα and k ------Ð α +--- 12 0 0 λ2β2 λ ν α λ c ==------, ω------2u p –k k that lie in the lower half-plane of the complex variable 0 0 k and describe the excitation of the electron Langmuir ν Ð------oscillations (Fig. 2): 2u ν series ν () () 1 () () Ð--- G1 k Ð G1 zˆ = ------π∫G1 k exp ikzˆ dk λ 2 γ α Ð--- γ series αzˆ η() λ = e cosk0zˆ Ð ---- sink0zˆ Ðzˆ . k0 Fig. 2. The inverse Fourier transform of the function G2(k) is determined by the contributions of the zeros of the function The contributions to the inverse Fourier transform κ ⇔ 4κR ⇔ κ sinh2 R == 0 e 1 R n G2(zˆ ) in the beam region (zˆ < 0) are determined by the collisional ν series and the first γ series, while the sec- ⇔ ()κ 2 2 = ijn R n = Ð jn, (12) ond γ series gives the corresponding contribution in the πn region ahead of the beam (zˆ > 0). It is easy to see that j = ------()n = 12,,… . n 2 the singular point k = 0 is a removable singularity and does not contribute to the function G2(zˆ ), for which we Equation (12) describes the spectrum of our prob- successively obtain lem and differs from the corresponding equation for an ∞ 3 () REB transported in a cylindrical channel [10] only in l ˆ () yn z () the numbers jn. In our analysis, we can use the solution G2 zˆ = ∑ ∑ Cnle f l zˆ , to Eq. (12) that was constructed in [10]. Equation (12), l = 1 n = 1 which can be reduced to a cubic equation, has three series of poles: one collisional series (the ν-series) 2a h n 2 sin j ------sin j --- ()1 ()1 () n n γ Ð1 jn R R kn = Ðiyn and two relativistic series (the -series) C = ------, nl 2 ()l () ()23, ()23, l R yn 2y 1 ν 1 kn = Ðiyn (Fig. 2). The poles of the collisional n ------2 + 2 () 2 series lie within the interval (0, Ðiν/u) of the negative γ λ u()y l Ð ν/u imaginary axis and are described by the formula n () η() () η() 2 2 4 f 12, zˆ ==Ðzˆ , f 3 zˆ Ð zˆ . ()1 ν 1 λ ν d R 1 y = ------1 Ð,------n d = ------. (13) n 2 2  2 3 n λ For ν ≠ 0 and for finite values of γ, we calculate the u1 + d γ u () jn n 1 + dn above two Fourier convolutions to arrive at the final ()2 expression for the force acting on the beam electrons: The first relativistic series yn lies within the interval (Ðiγ/λ, Ði∞) of the negative imaginary axis, and the π 2  ∞ 4 e nb Cn1 Cn2 Cn3 ()3 ------ F = Ð ∑ Ð ------()1- Ð ------()2- + ------()3- second relativistic series yn lies within the interval R  yn yn yn (iγ/λ, i∞) of the positive imaginary axis. The relativistic n = 1 series are determined by the formulas 2 α2 2 ∞ × k0 Ð 1 αzˆ Cn1 1 ------e sink0zˆ + ∑ ∑ ------()------2 2 α2 2 k l ()l 2 2 γ νλ () + k0 0 yn ()Ð y + α + k ()23, ± 1 + dn 1 dn dn + 2 l = 1 n = 1 n 0 yn = ------1 Ð.------3/2- (14) λ d 2u γ 2 ()l n ()1 + d () y zˆ α () n × l n zˆ l yn e + e Ðyn cosk0zˆ --- Formulas (13) and (14) imply that, in the collision- ν less limit, there is no series, while the ultrarelativistic α γ ∞ γ ()()l α  limit ( ) is free of both of the series. In the case + k0 Ð ---- Ð yn + sink0zˆ ν = 0 and γ ∞, we arrive at the above solution. k0

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()3 ()τ the beam and plasma electrons has the form C 1 ()3 Ðyn zˆ + u Ð ------n-3 ------y e --- ()3 ()3 2 2 n yn ()Ð y + α + k κ sinhκ h n 0 F ()kx, = Ð8πe2n ------2------1 [coshκ ()Rx+ 2 b 2ε 2 κ 1 k 2 sinh 2 1 R αzˆ()3 α ()3  ---- ()α  + e Ðyn cosk0zˆ + k0 Ð Ð yn + sink0zˆ . ×κ() κ ()κ()]˜() k0  sinh 1 Ra+ Ð cosh 1 RxÐ sinh 1 RaÐ j k .

Note that the total contribution of both of the relativ- We thus arrive at the sought-for expression for the istic series to the force F is proportional to 1/γ. component F2(k, a) of the force acting on the beam electrons that occur at the symmetry plane of the channel: PLASMA CHANNEL BOUNDED BY WALLS π 2 WITH FINITE CONDUCTIVITY 8 e nb F (ka, ) = Ð------2 R We consider an REB transported in a plasma channel bounded by walls with a finite conductivity. In this sinh()κ h sinh()2κ a cosh()2κ R κ situation, the walls may be either metal conductors with × R------1 1 1 ------2-˜j()k (16) σ 2 ()κ 2ε a finite conductivity or conducting media (plasma). sinh 2 1 R k 2 Let us denote the parameters of the plasma channel and of the conducting walls by the indices 1 and 2, respec- 8πe2n tively. Now, our task is to solve Eq. (7) for the electric = Ð------bG ()k G ()k ˜j()k . R 1 2 field component Ez with the nonuniform boundary conditions One can readily see that the contribution of the plasma channel and the walls to expression (16) for the ε ∂E {} {}≡ z force component F2 is multiplicative, which allows a Ez ± ==0, By ------0, (15) xR= κ2 ∂x separate treatment of media 1 and 2, whose properties xR= ± are described in terms of the inverse Fourier transforms which imply that the z and y components of the electro- of the functions Gl. The desired force component can be magnetic field should be continuous at the boundary solved for by successively calculating the convolutions between the two media. of these inverse Fourier transforms, which, of course, are to be found in advance. According to the superposition principle, we can represent the electric field component in question as First, we consider the model of an ultrarelativistic electron beam (γ ∞) transported in a channel filled () () ν E = E 1 + E 2 . with a collisionless plasma ( 1 0). In this model, z z z expression (16) takes the form ()1 2 Since the solution Ez to the inhomogeneous equa- 8πe n F ()ka, = Ð------bG ()∞ G ()k ˜j()k , tion (7) with uniform boundary conditions was con- 2 R 1 2 ()2 structed above, we only need to find the solution Ez to the homogeneous equation (7), h 2a 2R sinhλ----- sinh-----λ- cosh------λ - () ()∞ 1 1 1 ᏸ 2 G1 = ------. Ez = 0, 2 2R sinh ------λ - with the nonuniform boundary conditions (15). 1 The solution to Eq. (7) with boundary conditions (15) Thus, we have to calculate the inverse Fourier trans- κ 2ε can be constructed using a familiar approach. Inside the form of the function G2(k) = 2/(k 2) with respect to the plasma channel bounded by highly conducting walls, z coordinate. we have The physical condition that the electric field should κ κ decrease at infinity, ()2 π i b sinh 1h Ez = Ð8 enb------2 kκ ε κ 2 1 2 sinh 2 1 R Imκ = 0 κ > ⇔ 2 ×κ[ ()κ() Re 2 0  sinh 1 Rx+ sinh 1 Ra+  κ2 < Re 2 0, + sinhκ ()κRxÐ sinh ()]RaÐ ˜j()k . 1 1 ν corresponds to the cut (0, Ð 2) (which will be denoted Γ The component F2(k, x) of the force acting on both by on the imaginary axis (Fig. 3), in which case we

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λ have xzˆ/ 2 1 Ð e ()2 × ------dx + λ [()α A˜ Ð k B˜ 2 ˜ 2 2 0 2 1 v i v x Ð1λ2 x + (17) κ ==------< 0 ⇔ κ ±----- Ð.------2 Γ 2 2 Γ− 2 λ v + ν + λ v + ν α zˆ ()2 ()2 2 2 2 2 2 (()α ˜ ˜ Ð e 2 A Ð k0 B cosk0 zˆ Consequently, for ω > ν /2, the function G (k) has p2 2 2  two poles in the lower half-plane of the complex vari- ()()2 ˜ α ˜ ()2 )] + k0 A + 2 B sink0 zˆ . able k = u + iv, 

()2 ()2 1 2 k , ==Ð iα ± k , k ----- Ð,α In the collisionless limit, the expression for the con- 12 2 0 0 λ2 2 2 tribution of the component F2 to the force F acting on the REB follows from formula (17): ν ν α 2 ν i G ()∞ 2 ==-----, i ---- , ()π 2 λ zˆ 1 2 c F2 xa= = 8 e nb 2 1 Ð cos----λ------. 2 R and is characterized by the cut Γ between the poles γ ∞ ν ω ν (Fig. 3). With these properties in mind, we obtain for Under the conditions , 1 = 0, and p2 < 2/2, which imply that no oscillations are excited in the G2():zˆ plasma channel, the poles of the function G1(k) (or, α 2zˆ ()2 ()2 equivalently, the zeros of the function ε ) lie on the G = Ðe ()ηA˜ cosk zˆ + B˜ sink zˆ ()Ðzˆ 2 2 0 0 cut Γ (Fig. 4) and do not contribute to the desired λ˜ 2 xzˆ/λ expression for the force component in question. Thus, 1 λ˜ Ð x e 2 Ð ---ᏼ ------2 ------dxη()Ðzˆ , we are left with the problem of determining the contri- π ∫ 2 ˜ x x Ð1λ2 x + 0

˜ ν λ2 b ()α ν λ2 A = a 2 2 Ð ------()1 Ð 2 2 2 , k 2 0 ω ω – p2 p2 ˜ a ()α ν λ2 B = ------1 Ð 2 2 2 , ()2 Γ Γ k0 – + Γ Γ 2 1 2 1 2 − ()2 κ ()k , = -----2Ð α + ik 2α 2 12 λ2 2 0 2 2 ν – 2 ρ2 +−Φi ⇒ρκ () +−Φi − = e 2 k12, ==e aib+ ,

()2 Γ 1 2k α ∞ Φ = --- arctan------0 2 , 2 λÐ2 α2 2 Ð 2 2 Fig. 3. λ ˜ 2 c ˜ 1 λ2 ==------, zν -----, λ2 < ---. zν 2 ν 2 2 2 Γ ˜ The Fourier convolution of G1(zˆ ) with j (zˆ ) can be calculated in the same way as in section 2. As a result, s γ (1) γ (1) under the conditions – +

ν h1 ν = 0, γ∞, -----2 < ω , 1 2 p2 h we arrive at the following expression for the component 2 F2 of the sought-for force ν λ˜ 2 ()∞  2 λ˜ (), π 2 λ G1 1ᏼ 1 2 Ð x F2 zˆ a = 8 e nb 2--------- ∫ ------R π x x 0 Fig. 4.

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1156 KURYSHEV, ANDREEV ν υ the collisionless limit 1 = 0. In this case, the component F2 of the desired force can be found from the formula (), ()(), ν () ()∞ F2 zˆ a = F2 zˆ 1 = 0 G1 zˆ /G1 , (18) s ν in which the component F2(,zˆ 1 = 0) was determined above.

The inverse Fourier transform G1(zˆ ) should be calculated with allowance for the properties of the ν function G1(k, 1) both at infinity and in the vicinity of ()1 ()1 the poles of the collisional series kn = Ðiyn . One can readily single out the singularities of the function Γ Γ ν ()1 – + G1(k, 1) in the vicinities k kn of the second-order ()1 poles kn () Fig. 5. fk()1 G ()k, ν ∼ ------n , 1 1 ()()1 2 kkÐ n bution of the cut Γ, in which case the integral along the contour that envelopes the cut should be understood in where terms of the Cauchy principal value. The expression for ν 2 ()1 n + 1R F (,zˆ ν = 0) is governed by the first term in braces in fk()= ()Ð1 -----1 ---- 2 1 n c 2 formula (17). jn First, we consider the so-called σ-model, in which × 2a h 1 the wall properties are described by Ohm’s law: sinjn------sinjn------2. R R ()1 + d2 j = σE. n At infinity, the function G (k, ν ) approaches a non- The physical condition that the electric field should 1 1 decrease at infinity corresponds to the semi-infinite cut zero constant. Γ on the imaginary axis (Fig. 5). On this cut, the func- The contribution of the second-order poles is calculated from the familiar Cauchy formula: tion G2(k) has the form () κ v fk()1 eikzˆ () 2 () ± Ð (), ν 1 n G2 k Γ ==G2 iv Γ− =i------. G zˆ = ∑------dk ------+ 1 1 ∫ 2ε 4πσ 2π ()()1 2 k 2 Γ vv+ ------n C kkÐ n  n (19) c ∞ () () 1 ˆ ()1 yn zη() We successively calculate the inverse Fourier trans- = ∑ fkn zê Ðzˆ . form G2()zˆ Ð G2(k) and its convolution with the Fou- n = 1 rier-transformed beam current density. As a result, we Consequently, for ν ≠ 0, the inverse Fourier trans- arrive at the following formula for the force component 1 ν form G1(zˆ ) is described by the expression F2(,zˆ 1 = 0): () ()δ∞ () (), ν (), ≡ (), ν G1 zˆ = G1 zˆ + G1 zˆ 1 . (20) F2 zˆ a F2 zˆ 1 = 0 ∞ 2 We can easily see that the contribution of the first 8πe n z xzˆ/z b ()∞ mᏼ dxx()m term in expression (20) to the force component F2 coin- = ------G1 ----- ∫------1 Ð e , R π x2()1 Ð x cides with the contribution that was already obtained in 0 ν the collisionless limit 1 = 0. As a result, the expression ν ν 2 πσ where zm = m/c and m = c /(4 ) is the magnetic vis- for the force component F2 contains two terms: cosity. (), (), ν (), ν F2 zˆ a = F2 zˆ 1 = 0 + F2 zˆ 1 . (21) We consider an ultrarelativistic REB (γ ∞) ν ≠ ν transported through a collisional plasma ( 1 0). For The contribution F2(,zˆ 1) of collisions in the such a system, it is sufficient to determine the inverse plasma channel (medium 1) to expression (21) for the Fourier transform G (zˆ ) of the function G (k) and then desired force component is determined by substituting 2 2 ν to calculate the convolution of this inverse Fourier the formulas for F2(,zˆ 1 = 0) obtained in different transform with the corresponding function obtained in physical models of the wall substance and calculating

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Fourier convolutions. Thus, under the conditions γ  ∞ ν ≠ ω ν 1 ()1 2 zˆ ()1 zˆ , 1 0, and p2 > 2/2, we arrive at the expression Ð------()()y λ Ð1 cos----- +2y λ sin----- , ()1 2 n 2 λ n 2 λ ()λ 2 2  yn 2 + 1 2 ∞ 8πe n () F ()zˆ, ν = ------b ∑ fk()1 γ ∞ ν ≠ ω ν 2 1 R n so that, for , 1 0 and p2 > 2/2, we obtain n = 1 2 ∞ 3 λ˜ ()1 ()1 π ()λ 3 2 y zˆ ()λ ˆ 8 e nb 1 2 λ n xyÐ n 2 z F ()zˆ, ν = ------∑ fk()----- × 2ᏼ e e 2 1 R n π ----- ∫ ------() ------() Ð zˆ  π xyÐ 1 λ xyÐ 1 λ n = 1 0 n 2 n 2 λ˜ ()1 () 2 y zˆ ()1 n xyÐ n zm zˆ ()1 e e yn zˆ ()1 × ᏼ ------ ˆ λ˜ ∫ ()1 ------() Ð zˆ 1 e Ð 1 zˆ yn z 1 2 Ð x 1 + ------------Ð -----e ------xyÐ n zm xyÐ z ()1 λ ()1 λ λ x x 0 n m yn 2 yn 2 2 ()1 yn zˆ ()1 () y zˆ λ˜  1 ˆ 1 e Ð 1 zˆ n 1 2 Ð x dx dx 2 C yn z1 1 + ------------Ð -----e ------. × ------+ λ ------e Ð zˆ + ------Ð ------()1 ()1 λ 2 2 2 ()1 ()1 ()1 λ λ 2 x x λ˜ λ˜ yn 2 yn 2 x Ð12 x + x Ð12 x + yn yn yn σ γ ∞ ν ≠ In the model, for and 1 0, we ﬁnd ()1 1 yn zˆ 2 ()1 + ------e ()k D Ð ()α Ð y C zˆ--- 2 ∞ 3 ()2 ()2 0 2 n 8πe n ()zˆ ()α Ð y 1 + k 2 F ()zˆ, ν = ------b ∑ fk()1 ----m- 2 n 0 2 1 R n π n = 1 1 ()1 ()2 Ð ------(2D()α Ð y k (22) ∞ ()1 ()1 ()2 ()2 2 n 0 y zˆ ()xyÐ z zˆ ()α Ð y 1 + k 2 e n e n m 2 n 0 × ᏼ ------------Ð zˆ ∫ ()1 ()1 xyÐ n zm xyÐ z 2 0 n m ()1 2 ()2  ---+ C()()α Ð y + k 2 n 0  ()1 yn zˆ ()1 1 e Ð 1 zˆ yn zˆ xdx + ------------Ð -----e ------. α zˆ ()1 ()1 2 2 y z y z zm x ()1 Ð x e [( ()α ()1 ()2 n m n m + ------2 2D 2 Ð yn k0 ()α ()1 2 ()2 2 Ð yn + k0 Now, we proceed to a calculation of the inverse Fou-

2 rier transforms of the function G2(k), which serves to ())()α ()1 2 ()2 ()2 + C 2 Ð yn + k0 cosk0 zˆ describe the properties of the conducting wall in different models. 2 ( ()()α ()1 2 ()2 In the σ model of the wall substance, the finiteness + D 2 Ð yn Ð k0 of the relativistic factor is easy to take into account. The   physical condition that the electric field should ()α ()1 ()2 ) ()2 Γ ---Ð2C 2 Ð yn k0 sink0 zˆ]  , decrease at infinity corresponds to the two cuts i on the   imaginary axis (Fig. 6):

() () where C = α A˜ Ð k 2 B˜ , D = k 2 A˜ + α B˜ , and the s = 0 2 0 0 2  quantities A˜ and B˜ were deﬁned above.  βγ2 Γ > i :  vv0 = ------In the model in which plasma medium 2 is collision-  zm less (ν = 0), expression (22) gives  2  v < 0. π 2 λ3 ∞  (), ν 8 e nb 2 ()()1 1 Also, the function F2 zˆ 1 = ------∑ fkn Ð------R  ()1 λ n = 1 yn 2 κ κ G ()k ==------2------2 ()1 2 2 yn zˆzˆ 1 1 k ε 1 × e Ð ----- + ------Ð ------2 kk+ i------()1 ()1 β λ λ λ zm 2 yn 2 yn 2 has the ﬁrst-order pole () 2  ()1 ()1 λ 1 yn zˆ ()1 y Ð 1 + ------e Ð y zˆ + ------n 2 - 1 ()1 2 n ()1 2 v ()λ  ()λ k2 ==Ði------β Ði 2. yn 2 + 1 yn 2 + 1 zm

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1158 KURYSHEV, ANDREEV

2a h sin j ------sin j --- 2 n R n R ()()l ()n + 1 jn Γ fkn = Ð1 ------. 2 3 2 R ()l 2y 1 ν 1 n ------1 ------2 + 2 2 γ λ u ()l ν ν 1 y Ð -----1 0 n u

The expression for the force component F2(,zˆ a) can be obtained by calculating the corresponding two Fou- rier convolutions. Performing simple but rather labori- s Γ ous manipulations and taking into account ﬁnite γ val- 1 ues and collisions in the plasma channel, we arrive at ν the following formula for the force component F2 in the 2 σ model of the wall substance:

2 3 () 8πe n z  () y 3 ()zˆ + uτ (), b m 3 n F2 zˆ a = Ð------∑ f n e R  Fig. 6. n

x () τ ∞ x ----- + β ()3 zˆ + u One can readily see that the pole k2 does not contrib- 2 yn zm Ð x ------Ð 1 γ z Ðxuτ/z ute to the corresponding integral because the function × ᏼ dx------m -()1 Ð e m κ ∫ Ð1 2 ()3 2 2 has different signs on different sides of the cut. In ()β Ð x x () 0 yn zm Ð x addition, the singular point k1 = 0 is a removable singularity. As a result, the integral should again be under- x stood in terms of the Cauchy principal value, so that we ∞ x ----- + β ()3 2 () ()ˆ τ γ arrive at the following expression for the inverse Fou- 3 yn z + u Ð ∑ f n e ∫dx------Ð1 2 rier transform G2(zˆ ) of the function G2(k): ()β Ð x x n 0

x ()3 ()τ ∞ x ----- + β 1 yn zˆ + u ()3 2 × ------()1 + e ()y ()zˆ + uτ Ð 1 γ ˆ 2 n () 1ᏼ xz/zm η() ()3 G2 zˆ = Ð--- ∫------e dx Ðzˆ ()y z π ()βÐ1 Ð x x n m 0 xzˆ/z (23) m  ()()3 ()τ e yn zm Ð x zˆ + u /zm x Ð ------1 + e - ∞ x Ð ----- + β ()3 2 2 () γ yn zm Ð x 1 Ðxzˆ/zm + --- ∫ ------e dxη()zˆ . π ()βÐ1 + x x 2 x βγ ∞ x Ð ----- + β  2 ()3 zˆ + uτ ()3 γ The first and second terms on the right-hand side of × ()y z Ð x ------Ð 1  Ð ∑ f dx------Γ n m n ∫ 2 Ð1 this expression are the contributions of the cuts 1 and zm  x ()x + β Γ γ ∞ n βγ2 2, respectively (Fig. 6). For , the second term, ()3 ()τ which describes the perturbation ahead of the beam yn zˆ + u γ 1 + e ()3 front, vanishes. For finite values of , we first determine × ------()y ()zˆ + uτ Ð 1 ()3 2 n the inverse Fourier transform of the function G1(k) by ()y z using the above multiplicative property of expression (16) n m Ðxz()ˆ + uτ /z for the force component F2. The singular points where m ()3 )()τ e yn zm + x zˆ + u /zm the denominator of the function G (k) vanishes are sec- Ð ------(1 + e 1 ()()3 2 ond-order poles; specifically, these are the poles of the yn zm + x ν series and the two γ series, which were already found in Section 2. In this case, the contribution of the second- × ()()()3 ()τ order poles can be evaluated in the same way as was yn zm + x zˆ + u Ð 1 done when deriving formula (19). Thus, we successively find x β ∞ 2 ∞ x ----- + 3 () 2 () l ˆ ()l γ l yn z ᏼ G ()k Ð G ()zˆ = ∑ ∑ fk()zê f ()zˆ , + ∑∑ f n ∫dx------1 1 n l ()βÐ1 Ð x x2 l = 1 n = 1 l = 1 n 0

TECHNICAL PHYSICS Vol. 46 No. 9 2001 A PLANAR RELATIVISTIC ELECTRON BEAM 1159

() xzˆ/zm l ˆ () × 1 ()yn z()l e Ð ------e yn zˆ Ð 1 + 1 + ------Γ ()()l 2 ()()l 2 3 yn zm yn zm Ð x

() ()l () × yn zm Ð x /zm()l zˆ υ - e yn zm Ð x ----- Ð 1 + 1 2 zm

x ∞ β 2 xÐ -----2 + ()l γ + ∑∑ f n ∫ dx------()βÐ1 + x x2 l = 1 n βγ2 s

()l 1 yn zˆ ()l k × Ð------()e ()y zˆ Ð 1 + 1 2 k1 ()l 2 n Γ ()y z γ ν 1 γ n m 2 – 2 1

()τ Ðxzˆ + u /zm () ()l ˆ () e ()yn zm + x z/zm()()l + ------e yn zm + x zˆ/zm Ð 1 + 1 ()()l 2 υ yn zm + x 1 Γ x 2 ∞ β 2 xÐ -----2 + ()l γ + ∑∑ f n ∫ dx------()βÐ1 + x x2 l = 1 n 2 βγ Fig. 7.

()τ Ðxzˆ/zm Ðxzˆ + u /zm κ Φ × ()e Ð e --- for 2(k1, 2), a, b, and are somewhat different:

2 1 2 1 − ()2 2 2 +−2Φi κ ()k , = -----2Ð α 1 + ----- + i2α k β = ρ e , ()l 2 12 2 2 2 2 0 ()l λ γ ()y z + x zˆ/z ()y z + x zˆ/z Ð 1  2 × n m m n m m  e ------() 2 - . ()y l z + x  κ ()ρ+−Φi − n m 2 k12, ==e abi+ ,

Finally, we consider the model in which the proper- ()2 2 1 2α k β ties of the conducting wall are described by the disper- Φ = --- arctan------2 0 -. sion law (2), the relativistic factor is ﬁnite, and the con- 2 ()λÐ2 Ð 2α2 ()11/+ γ 2 ω ν 2 2 dition p2 > /2 holds. For this model, the singularities As a result, we obtain of the function G2(k) are illustrated in Fig. 7. These are three cuts Γ, which lie on the imaginary axis of the ()≡ () G1Ð2 zˆ Gzˆ plane of the complex variable k = s + iv, and two poles k12, 2 ()2 ()2 Ð2 2 λ α ˆ () k = ±k Ð iα , where k = ()βλ Ð α , α = 2 2z{[]α 2 ()λÐ2 α2 1, 2 0 2 0 2 2 2 = ------()e 2a 2k0 Ð b 2 Ð 2 2 ν ν ν k 2 2 /2, and 2 = 2/u: 0 × ()2 []α ()2 λÐ2 α2 ()2 }η  s = 0 cosk0 zˆ + 2b 2k0 + a()2 Ð 2 2 sink0 zˆ ().Ðzˆ  Γ ()() i : vvÐ v1 vvÐ 2 , The inverse Fourier transform G2(zˆ ) is determined i = 123,,------> 0 v Ð ν by the additive contributions of the poles and the cuts  2 Γ i. The contributions of the cuts are as follows:

2 λ˜ ν 2 2 1 2 γ ν +− λ˜ 2 v 12, = Ð --- 2 ----- +.------() 1ᏼ dx 2 Ð x 1 ()λ˜ 2 4 β2λ2 Gzˆ Γ = Ð--- ∫ ------+ ----- 2 Ð x 2 1 π x x γ 2 ω ν 0 For p2 > 2/2, the contributions of the poles to the zˆ xλ----- inverse Fourier transform G (zˆ ) of the function G (k) 2 2 2 e are evaluated in a way similar to what we did in the × ------η()Ðzˆ , 2 ˜ limit γ ∞, in which case, however, the expressions x Ð1λ2 x +

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1160 KURYSHEV, ANDREEV

∞ beam of arbitrary thickness at an arbitrary distance λ˜ () 1ᏼ dx 2 Ð x 1 ()λ˜ 2 from the symmetry plane of the channel. In the case of Gzˆ Γ = --- ∫ ------+ ----- 2 Ð x 2 π x x γ 2 a cylindrical channel, we succeeded in analyzing the x1 problem at hand only for thin beams. zˆ xλ----- e 2 × ------η()Ðzˆ , ACKNOWLEDGMENTS 2 λ˜ x Ð12 x + We would like to acknowledge the ﬁnancial support ∞ from the Integration program. λ˜ () 1 dx 2 + x 1 ()λ˜ 2 Gzˆ Γ = --- ∫ ----- Ð ------+ ----- 2 Ð x 3 π x x γ 2 x2 REFERENCES zˆ 1. A. A. Rukhadze, L. S. Bogdankevich, S. E. Rosinskiœ, Ðxλ----- e 2 and V. G. Rukhlin, Physics of High-Current Relativistic × ------η()zˆ , Electron Beams (Atomizdat, Moscow, 1980). 2 ˜ x ++λ2 x 1 2. W. E. Martin, Phys. Rev. Lett. 54, 685 (1985). where 3. P. S. Strelkov and D. K. Ul’yanov, Fiz. Plazmy 26, 329 (2000) [Plasma Phys. Rep. 26, 303 (2000)]. ν 2 ν 2 2 2 4. A. P. Kuryshev, V. B. Sokolyuk, and S. V. Chernov, Zh. x , = ------+ γ ± ------. 12 2βω 2βω Tekh. Fiz. 57, 1292 (1987) [Sov. Phys. Tech. Phys. 32, p2 p2 764 (1987)]. ω ν For p2 < 2/2, the contribution of the poles equals 5. A. P. Kuryshev and S. V. Chernov, Zh. Tekh. Fiz. 58, 2106 (1988) [Sov. Phys. Tech. Phys. 33, 1279 (1988)]. zero; i.e., the term G2(zˆ ) drops out of the expression for the inverse Fourier transform G (zˆ ). In other respects, 6. E. K. Kolesnikov and A. S. Manuœlov, Zh. Tekh. Fiz. 60 1Ð2 (3), 40 (1990) [Sov. Phys. Tech. Phys. 35, 298 (1990)]. the force component F2 is calculated in the same way as σ 7. A. G. Zelenskiœ and E. K. Kolesnikov, Zh. Tekh. Fiz. 65 in the model. (5), 188 (1995) [Tech. Phys. 40, 510 (1995)]. 8. V. P. Grigor’ev, A. V. Didenko, and G. P. Isaev, Fiz. CONCLUSION Plazmy 9, 1254 (1983) [Sov. J. Plasma Phys. 9, 723 (1983)]. The results obtained in this paper make it possible to apply different models of the media of the plasma chan- 9. V. B. Vladyko and Yu. V. Rudyak, Zh. Tekh. Fiz. 60 (8), 199 (1990) [Sov. Phys. Tech. Phys. 35, 993 (1990)]. nel and the wall and take into account a wide variety of physical circumstances in order to describe the force 10. A. P. Kuryshev and V. D. Andreev, Zh. Tekh. Fiz. 66 (8), interaction of an REB with a plasma in a channel 143 (1996) [Tech. Phys. 41, 821 (1996)]. bounded by highly conducting walls. We have completely solved the problem of the transport of a planar Translated by O. Khadin

TECHNICAL PHYSICS Vol. 46 No. 9 2001

Technical Physics, Vol. 46, No. 9, 2001, pp. 1161Ð1167. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 71, No. 9, 2001, pp. 97Ð104. Original Russian Text Copyright © 2001 by Gurin, Korsukova, Loginov, Shrednik.

SURFACES, ELECTRON AND ION EMISSION

The Properties of the Nonequilibrium LaB6 Surface Resulting from Field Evaporation V. N. Gurin, M. M. Korsukova, M. V. Loginov, and V. N. Shrednik Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia Received January 18, 2000

Abstract—The nonequilibrium surface of single-crystal lanthanum hexaboride needles and its modifications are studied with a time-of-flight atomic probe. The surface is obtained by room-temperature field evaporation. The mass spectra of field evaporation shed light on the surface composition at the needle tip immediately after tip etching, corrosion in residual gases, intense cleaning by field evaporation, and the relaxation of the nonequilibrium surface by heating to 1250 K. Conditions for the breakdown of an oxide film on the tip surface and for obtaining the mass spectra of field evaporation for stoichiometric or lanthanum-enriched pure LaB6 single crystals are discussed. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION pressure in the atomic probe chamber was maintained at a level of 10Ð9Ð10Ð10 torr with an electrical discharge Ion flows due to the room-temperature field evapo- pump. The major residual gases were CO, H2O, and ration of LaB6 have been studied in [1, 2]. Those works have been concerned with the stability of free ionized CH4. The crystal tips were attached to the support +m (anode) with either Aquadag or conducting epoxy resin clusters of the LaBn type (where n = 1Ð6 and m = 1Ð4), containing fine-grain tantalum powder. In the former which are present in abundance in the ion flow. It has case, the chamber with the tip can be heated to 200 and been shown that the flow largely contains poorly stable 1300°C, respectively, and the water content in the resid- and even metastable ions. ual gases was insignificant. In the latter case, the cham- In this work, we use the same experimental tech- ber and the tip were kept at room temperature and the amount of water was higher. The well-defined water nique as in [1, 2] but concentrate on studying the LaB6 surface resulting from the field evaporation and serving peak was used to calibrate the mass spectra. as an active source of evaporating ions. In addition, modifications of this surface due to heating or interaction with adsorbed residual gases are of no less interest. RESULTS AND DISCUSSION Moreover, it is significant to know the properties of the 1. Mass spectra of LaB6 field evaporation. Figure 1 LaB6 surface initially kept under a vacuum after it has been etched in concentrated H SO to form a fine tip shows three mass spectra within one series of runs with 2 4 the same LaB tip. In all spectra, the base, V , and the and then kept in air. All these points have not been 6 b touched upon in [1, 2] and are dealt with in this article. pulse, Vp, voltages were 13 and 5.6 kV, respectively. Finally, we will try to find conditions under which the The horizontal and vertical scales for the three spectra ion flow is enriched by lanthanum. The knowledge of are the same. The spectra differ because they have the these conditions would help to design point ion sources. different histories. The first spectrum (Fig. 1a) was taken from the as-etched LaB6 surface and contains a minor amount of the ions generated by 59 pulses of a EXPERIMENT total of 5000 pulses applied. The highest peak is that of CO+ (CO is among the residual gases). Experiments were carried out with a time-of-flight atomic probe [3] with a mass resolution M/∆M ≈ 30. It is of interest that the water peak in Fig. 1a is Recently, its detecting system has been upgraded: the absent, although the tip was fixed by epoxy resin and rate of data collection has been raised 30 times or more the instrument was not heated. Several weak peaks ris- and the time resolution has been greatly improved. ing above unit ones (the latter contain also noise peaks) Accordingly, the reliability of the mass spectra has been can be identified as corresponding to oxide ions, improved and the time of recording the mass spectra ++ + +++ ++ has been cut. The tips were made of the needles grown B2 O3 , B2O , and LaO2 , and to La2B ions. Before from the solution in aluminum melt [4]. As in [1, 2], the this spectrum, we recorded four spectra with Vb = 8, 9, temperature of the tip was kept at the room value. The and 11 kV and with the same Vp = 5.6 kV. They all con-

1162 GURIN et al. + I (a) CO +++ 2

5 ++ ++ 3 + O 2 O O LaO 2 2 B B La

25 + O 2 H

(b) 20

15 ++ ++ 5 +++ 2 ++ 10 3 LaB LaB LaB LaB +++ ++ 8 ++ 9 + ++ LaB 2 + +++ B CO LaB La 5 LaB La

0 30 +++ ++

La (c) La 25 ++ LaB ++ 5 ++ 4 +++

20 LaB LaB LaB + ++ 3 O O 2 2

15 H +++ 2 H 2 + B LaB ++ 6 ++ 8 +++ 2 ) 4 +++ 2 LaB ) 5 10 LaB (LaB 7 (LaB ++ B 2

5 La

0 0 20 40 60 80 100 120 140 160 180 200 m/q

Fig. 1. Field evaporation mass spectra for the LaB6 single crystal that were obtained with the atomic probe. The spectra were obtained for the sample after various treatments. m/q is the ion mass-to-charge ratio, and I is the number of ions. (a) The surface is covered by the oxide ﬁlm after tip etching and exposure to air, n/N = 59/5000; (b) the surface after the removal of the ﬁlm, n/N = 422/5000; and (c) the La-enriched surface, n/N = 1000/7586. N is the total number of pulses applied, n is the number of ion-generating pulses.

+ tained one well-deﬁned CO peak and the same weak ion showed up as a high peak when Vb was decreased peaks. None of the spectra contained water. by 1 kV (at Vb + Vp = 14 + 5.6 kV). Then, Vb was An increase in Vb to 15 kV resulted in a spectrum decreased to 13 kV, and the spectrum depicted in with a plenty of peaks. The spectrum was similar to that Fig. 1b was obtained. The spectra in Figs. 1a and 1b + in Fig. 1a and also did not contain the H2O ion. This radically differ: the latter contains the high well-deﬁned

TECHNICAL PHYSICS Vol. 46 No. 9 2001

THE PROPERTIES OF THE NONEQUILIBRIUM 1163

+ +m + H2O peak and many cluster ions of the LaBn type. Of observed, for example, for the B2 peak, which did these peaks, those of the doubly charged ions were the grow 2.6 times. At the same time, the La+++ and La++ highest. This spectrum does not contain excess lantha- peaks grew, respectively, seven- and fivefold. These num in large amounts, although an excess of lanthanum estimates were made from the peak heights. A sharper is typical of developed field evaporation of LaB6 [1, 2]. estimate including ions in the adjacent channels and not It was found that the developed process is established if only those in the channel of their maximal accumula- Vb is reduced for a while. In runs following those where tion (i.e., an estimate of all ions of a given sort that is the spectrum in Fig. 1b had been obtained, Vb was suc- made from the whole area of the related peak) would cessively reduced to 11 and 9 kV and then was raised most likely aggravate the differences mentioned above. by steps to 11, 12, 12.5, and 13 kV. In the last case, an However, such a sharp estimate is difficult to make interesting spectrum (Fig. 1c) was obtained. It has because of the strong overlap of the peaks at the bot- intense La+++ and La++ peaks, La-enriched LaB+++ and tom. The selective growth is not observed for the peaks +m ++ +m of LaB -type ions with n = 1Ð5 and m = 2 or 3. In the LaB peaks, and a variety of other peaks of the LaBn n type. The La+++ peak of the same height was present spectrum in Fig. 1c, these peaks grew two or three + also at V = 12 and 12.5 kV. We, however, show the times, which is not surprising. In Fig. 1c, the CO and b + spectrum in Fig. 1c, because it was recorded for the H2O peaks are noticeably weaker than in Fig. 1b same Vb + Vp = 13 + 5.6 kV as those in Figs. 1a and 1b. because of a decrease in the residual pressure as the mass spectra are recorded. What happens when Vb grows starting from 8 kV and then varies back and forth? What is the reason that 2. The reason for excess lanthanum in the mass the three spectra for the same object differ so much at spectra in Fig. 1c. The spectrum in Fig. 1c was taken the same V + V ? from the La-enriched surface. For this spectrum, the b p total content of La and B in the evaporated flow corre- The surface of an as-prepared air-exposed LaB tip 6 sponds to the LaB formula (where n = 1Ð3) rather than (to which the spectrum in Fig. 1a corresponds) is inert n + to stoichiometric LaB6. Excess lanthanum seems to be and hard. It does not adsorb water (there are no H2O present also in the spectrum in Fig. 1b but to a lesser ions in the spectrum), and the surface atoms almost do extent: here, possibly, 6 > n > 3. The reason why the not evaporate at a pulse voltage of up to 18.6 kV. Near + flow of LaB6 evaporation is La-enriched in the latter the surface, only CO ions are generated. From those case remains to be clarified. It is hardly probable that few ions that are still recorded, one can suggest that the the LaB6 single crystals are so nonuniform in composi- surface is covered by boron and lanthanum oxides. The tion that the spectra in Figs. 1b and 1c reflect the vol- oxides are broken down at a pulse voltage of as high as ume contents of the elements. The elemental composi- 20.6 kV. Water is absent in the spectrum presumably tion of the LaB needles was determined by precise because it is absent at the very end of the tip. At 15 + 6 5.6 kV, the strong base field V = 15 kV removes water gravimetric analysis, and the homogeneity of the nee- b dles follows from X-ray diffraction measurements [4]. at some distance away from the tip. When V is dimin- b η ished by 1 kV, the adsorbed water reaches the tip and, The ratio = Vp/(Vb + Vp) equals 0.3. It is unlikely + that steady-state field evaporation at V = 13 kV subsequently, the H2O ion is observed at each of Vb b from 14 to 9 kV. intensely and selectively removes boron, resulting in an The spectrum in Fig. 1b, like those for V = 15 and excess of lanthanum (especially with regard for the fact b that the previous spectra at V = 15 and 14 kV have +m b 14 kV, covers many peaks of the LaBn type, where n already been taken). Note that the spectrum was La- η ≈ for the highest peaks lies between 1 and 5 and m = 2. enriched even at Vb = 11 kV ( 0.34). Moreover, the Such products of field evaporation are typical of the practice of LaB6 field evaporation suggests that selec- case when the LaB6 lattice is broken down at room tem- tive removal is typical of lanthanum rather than boron. perature. Spectra like that depicted in Fig. 1b are invari- Hence, other reasons for excess lanthanum in the spec- ably obtained when excess lanthanum is not delivered tra should be looked for. to the pure LaB6 surface subjected to field evaporation. Once the oxide film has been removed from the end The spectrum in Fig. 1b is the spectrum of ions gen- of the tip (at Vb + Vp = 15 + 5.6 kV), the evaporation erated by n = 422 pulses of N = 5000 pulses applied. proceeds from the pure surface and generates various, Actually, however, the number of ions was greater than including cluster, ions. Being strongly nonequilibrium, n (namely, about 600), since two ions were evaporated the surface thus produced contains a certain amount of simultaneously in some cases. La atoms. The field evaporation breaks down the LaB6 For the spectrum in Fig. 1c, the number of ions is lattice, which generates surface La atoms, among them, 1600 (n = 1000, N = 7586); it might be expected that those capable of migrating over the surface at room here the number of peaks will be much greater than in temperature. These atoms are free to move over the sur- Fig. 1b. If the former qualitatively copied the latter, the face for some time (that is, they do not evaporate or peaks would be 1600/600 = 2.67 times higher. This is occupy lattice sites at once). Under the steady-state

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1164 GURIN et al. conditions, the surface concentration of these atoms were so, the water peak would be absent, since water depends on field and temperature. evaporates at a smaller field than lanthanum. In all These atoms alone do not produce an excess of lan- probability, the excess lanthanum was spent during the thanum. Basically, they can raise the local concentra- intense field evaporation at Vb = 14 and 15 kV (the pre- tion at one site through a decrease in the concentration vious spectra) and lanthanum had no time to be accu- at another if the field and the temperature provide a mulated during recording of the spectrum in Fig. 1b. considerable free path of these atoms on the surface. When Vb was reduced to 11, 10, and 9 kV, the evapora- The free path must be no less than the diameter of the tion of La was insignificant but the lanthanum was sub- evaporating area of the tip. This area is roughly three stantially redistributed by electromigration to provide orders of magnitude larger than the probing zone, subsequent enrichment of the probing zone. which is projected onto a diaphragm through which The La concentration in this zone sharply decreased ions to be analyzed escape into the drift space. In our in one of the runs at Vb = 11 kV preceding the spectrum experiments, the probing zone was at the end of the tip. shown in Fig. 1c. For this run, the curve of ion accumu- The atomic probe spectrum detects excess lanthanum lation (Fig. 2) dramatically increases after the applica- (exceeding the stoichiometric La : B = 1 : 6 proportion) tion of 2800 pulses (at Vb + Vp). Partial spectra for this if an excess concentration of La is produced in the run before and after the kink in the curve are shown in probing zone by depleting the periphery of the evapo- Fig. 3. They conclusively show that the rise in the accu- rating area. mulation rate is associated largely with the evaporation The permanently applied base field Vb produced a of the excess lanthanum. In the spectrum before the gradient of the field strength E at the curved surface. kink (Fig. 3a), the peaks are weak and scarce. It is sim- The gradient is directed toward the top of the tip. Under ilar to the previous spectra at small Vb (10 and 9 kV). the action of this gradient, the migrating electropositive The spectrum after the kink is more intense. The peaks lanthanum atoms produce a gradient of the La concen- of pure lanthanum (La+++ and La++), as well as the clus- tration C with a maximum at the very end of the tip [5], +m ter peaks of the LaB type (especially LaB+++ and i.e., in the probing zone. The field evaporation rate, as n +++ well as the rate of electromigration, depends on E, and LaB2 ), are substantially higher. Hence, the excess the flow of evaporating La atoms creates an oppositely lanthanum is not only removed in the form of mona- directed gradient of C. The balance between the inflow tomic ions but also may capture one or more boron to the tip due to electromigration and the outflow from atoms. the tip due to evaporation specifies the resulting surface concentration of La. Here, we may face different situa- As Vb increases to 12, 12.5, and then to 13 kV (the tions. spectrum in Fig. 1c), the lanthanum inflow to the probing zone remains unchanged. In the course of evapora- (i) The field evaporation from the tip at a given base tion, the tip might become blunt. Therefore, if the voltage exceeds the electromigration inflow to the tip. steady-state evaporation of La did not occur at V = Then, a boundary where the two fluxes are equal to b 13 kV (Fig. 1b), it a fortiori could not take place at V = each other is set at some distance away from the tip. b 11, 12, or 12.5 kV (Fig. 1c). Note, however, that the The region adjacent to the tip will be free of excess lan- + thanum, while the region on the other side of the bound- appreciable H2O peak was always present. In all of the ary will be La-enriched. It seems that this situation cases, the lattice broken down over the evaporating area takes place immediately after the breakdown of the generated free lanthanum in large amounts and the sufficiently high V set the La concentration gradient, oxide film at Vb = 15 kV and then at Vb = 14 kV. The b associated spectrum corresponds to the volume (sto- which made the probing zone La-enriched. At the same ichiometric) composition. time, the pulse field had no time to counterbalance the electromigration flow while such a tendency was (ii) As Vb is decreased or the tip becomes dull, this observed: the slope of the ion accumulation curve for boundary approaches the tip. When it enters the prob- the spectrum in Fig. 1c progressively declined. ing zone, the spectrum will show excess lanthanum. The boundary may completely disappear (collapse) at 3. Formation of the oxide film on the LaB6 sur- the center of the probing zone, and the amount of face exposed to a vacuum. From subsections 1 and 2, excess lanthanum in the spectrum will rise with a fur- it follows that the surface conditions of the same tip at the same V + V may vary: It may be covered by an ther decrease in Vb. b p inert oxide film (the scarce spectrum in Fig. 1a), its sur- (iii) The field evaporation at a base voltage is insig- face may have the volume composition once the film nificant. In this case, the pulsed evaporation at Vb + Vp has been broken down (various cluster peaks in the governs the La concentration in (and, generally, spectrum in Fig. 1b), and finally the surface may be La- beyond) the probing zone. enriched (the intense many-lined spectrum of field So, what is the reason for such a large difference evaporation in Fig. 1c). Note also that, once the excess between the spectra in Figs. 1b and 1c? The intense lanthanum had been accumulated on the top of the tip, evaporation of La at Vb in Fig. 1b seems unlikely. If this spectra qualitatively similar to that in Fig. 1c (perhaps

TECHNICAL PHYSICS Vol. 46 No. 9 2001 THE PROPERTIES OF THE NONEQUILIBRIUM 1165 less intense) were obtained even at Vb + Vp = 11 + J 5.6 kV (Fig. 3), i.e., at a voltage markedly smaller than 800 13 + 5.6 kV, when the oxide film almost blocked the evaporation. Such an inert film forms also upon long-term stor- age in the residual gases (at a pressure of about − 10 7 torr), especially if the gases contain moisture. For 600 example, after being exposed to such an environment, the tip (whose properties have been discussed in Sec- tions 1 and 2) showed scarce spectra for Vb between 9 and 14 kV (Vb = 5.6 kV). These resembled the spectrum in Fig. 1a with the only difference: the moderate H O+ 2 400 peak was present even at Vb = 12 kV, becoming pronounced at Vb = 13 and 14 kV. The associated film resulting from the oxidation of boron and lanthanum on the surface in the residual gases seems to be somewhat thinner than the initial film, grown after etching and exposure in air (Fig. 1a). In spite of its smaller thick- 200 ness, this film also prevents the evaporation of single- crystal LaB6. However, it may have a better conductivity and another composition, which causes water adsorption and ionization. This surface film was broken down at Vb + Vp = 15 + 5.6 kV with the formation of the developed (many-lined) La-enriched spectrum like in 0 1000 2000 3000 4000 N Fig. 1c. Unlike the film associated with Fig. 1a, which was broken down during the rise in Vb from 13 to 15 kV, Fig. 2. Ion accumulation curve for the spectrum depicted in the film being discussed was broken down during Fig. 3. Vb + Vp = 11 + 5.6 kV. For the complete spectrum, recording the spectrum, as follows from the ion accu- n/N = 500/4250. J is the number of ions collected. mulation curve. Before the breakdown, the spectrum contained boron ions and boron oxides, as well as car- siderably smaller (hours instead of days) and the exper- bon ions and adsorbed CO. After the breakdown, the imental conditions were cleaner (the absence of mois- high La+++ peaks were observed not only at 15 + 5.6 kV ture in the setup heated). Carefully increasing Vb from and 15 + 6.2 kV but also at 13 + 6.2 kV and even 12 + spectrum to spectrum, we were able to detect a break- 6.2 kV (which is much less than 14 + 5.6 kV, when the down in a spectrum. Then, analyzing the partial spectra evaporation was blocked by the stable inert film). In all before and after the breakdown, we made certain that intense spectra taken after the breakdown at Vb + Vp the former spectra contained boron oxide ions, carbon +++ ++ from 12 + 6.2 kV to 16 + 6.2 kV, high La , La , and compounds, and, possibly, lanthanum oxides in greater +m LaBn peaks were observed along with a more or less amounts. The La spectra, however, were difficult to + record, since they were superposed on the LaBn spectra. appreciable H2O peak. Thus, it could be concluded that La and LaBn clusters were not evaporated in con- 4. Relaxation of the nonequilibrium LaB6 surface siderable amounts at the base voltage; hence, the spec- by heating. In another series of experiments, a LaB6 tip tra were valid. was fixed on a tantalum arc with Aquadag and could be The tip became blunt, and the constant field applied heated by passing the current through the arc. In this series, the intense evaporation at V + V = 15 + 6.2 kV (Vb) was moderate; therefore, excess lanthanum was b p always present and spectra like in Fig. 1b (without gave a spectrum like that in Fig. 1b with high and broad +++ +++ +++ ++ ++ excess lanthanum) were not observed. By selecting Vb LaB3 , LaB4 , LaB6 , La , and LaB peaks (with not too high so as to prevent La evaporation but suffi- a minor amount of excess lanthanum). Then, Vb was cient to draw free La into the probing zone, and Vp suf- decreased to 13 kV. With this voltage, the evaporation ficient for the intense evaporation of La, one can pro- intensity was reduced but the spectrum did not change +++ vide conditions where the La peak becomes the qualitatively. At Vb + Vp = 13 + 6.2 kV, the following highest. This is of importance in designing point procedure was repeated several times. The spectrum sources of La ions or La-enriched combined sources of was obtained at room temperature, then the tip was boron and lanthanum. heated to 1250 K in a high vacuum (10Ð10 torr) in the Upon exposure to the residual gases, the inert film absence of the field, and subsequently the spectrum was formed in many other cases as well. The film was thin- taken again at room temperature. The spectra and the ner and easier to break down if the exposure was con- ion accumulation curves before and after the heating

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1166 GURIN et al.

I + +++ 3 +++ 2

O (a) 5 2 H + LaB LaB + BO CO

0 15 +++ La (b) +++ LaB +++ 10 5 ++ 4 LaB +++ 6 LaB LaB ++ +++ ++ 3 5 ++ ++ 3 LaB + LaB LaB La + 4 BO LaB CH +

5 + ++ ++ 7 3 + CO B O 2 LaB LaB H

0 0 20 40 60 80 100 120 m/q

Fig. 3. LaB6 ﬁeld evaporation mass spectrum corresponding to the ion accumulation curve in Fig. 2 (a) Before (N = 1Ð2800) and (b) after (N = 2801Ð4250) the kink in the curve. For the complete parts of the spectrum, n/N = (a) 220/2800 and (b) 280/1450. Vb + Vp = 11 + 5.6 kV.

J J 400 800 (a) (b)

300 600

200 400

100 200

0 2000 6000 10000 0 2000 6000 10000 N N

Fig. 4. Ion accumulation curves for the LaB6 ﬁeld evaporation spectra obtained at Vb + Vp = 13 + 6.2 kV from the same sample after (a) the intense evaporation at Vb = 13 and 15 kV, Vp = 6.2 kV, n/N = 202/10 000 and (b) heating to 1250 K (in the absence of the ﬁeld) and subsequent cooling to room temperature, n/N = 379/10 000. Fine straight lines are drawn for comparison.

TECHNICAL PHYSICS Vol. 46 No. 9 2001 THE PROPERTIES OF THE NONEQUILIBRIUM 1167 were compared. The basic difference was that the evap- sition of the crystal) and with excess (as compared to oration intensity increased roughly twice. The spectrum the stoichiometric ratio 1 : 6) lanthanum. The latter before the heating was obtained at the total number of spectra exhibit high La+++ and/or La++ peaks, high clus- pulses applied N = 10 000 and the number of ion-gen- ter LaB+++ and LaB++ peaks, and lower peaks of other erating pulses n = 185. For the post-heating spectrum, +m n = 315 at the same N. In another run of this series, n = cluster ions like LaBn (they are also typical of the 202 and 379, respectively, for the same N. Qualitatively, former spectra). the pre- and postheating spectra were identical: all the (4) The presence of the excess lanthanum in the peaks grew in proportion to n. In these experiments, the mass spectra in Figs. 1c and 3b is explained by its redis- tip was pointed at the dark area (as demonstrated by the tribution over the surface due to the constant electric cold-electron imaging), which had a larger work func- voltage Vb. By properly selecting Vb and Vp, one can tion. Previously, the spectra with the excess lanthanum provide the field evaporation flows substantially were obtained with the tip pointed at the readily emit- enriched by lanthanum. This might serve as a basis for ++ ting area. The peaks for the clusters from LaB3 to designing controllable point ion sources. LaB++ were present in all these spectra. The ion accu- (5) The inert film like that mentioned in item 1 2 results during the interaction of the pure LaB surface mulation curves (two of which are shown in Fig. 4) are 6 nearly linear for the field-cleaned surface (Fig. 4a) and (after the field evaporation) with the residual gases in tend to decrease their slope with increasing N (Fig. 4b) the atomic probe chamber. The longer the exposure for the evaporation from the heated and then cooled time and the more H2O in the residual gases, the more surface. readily the film is produced. At a sufficiently high Vb + Vp, the film can be broken down and evaporated. Subse- The explanation for the effect of enhanced ion col- quently, the intense field evaporation of the crystal may lection after heating (with all other things being equal) take place at voltages lower than the breakdown is straightforward. After intense field evaporation, the voltage. surface becomes atomically smooth: all asperities disappear. The remaining structure is very stable against (6) The highly nonequilibrium final surface of LaB6 field evaporation. The resulting surface is strongly non- relaxes under heating (in the absence of the field) to equilibrium and is retained only because it was formed 1250 K. The relaxed surface evaporates atoms with an at a sufficiently low (room) temperature. Heating intensity twice as much as the initial nonequilibrium allows the atoms to occupy more equilibrium sites; i.e., surface (all other things being equal). the surface relaxes. However, a number of atoms extending outward appear after relaxation. They provide the enhanced ion collection, which was detected in ACKNOWLEDGMENTS the experiment. Thus, our earlier assumptions that a This work was supported by the program “Surface part of the La atoms can migrate even at room temper- Atomic Structures” of the Ministry of Science of the ature are confirmed by the above experiments: after Russian Federation (project no. 4.12.99). room-temperature LaB6 field evaporation, we do obtain the strongly nonequilibrium “frozen” surface. REFERENCES CONCLUSIONS 1. I. Bustani, R. Byunker, G. Khirsh, et al., Pis’ma Zh. Tekh. Fiz. 25 (23), 43 (1999) [Tech. Phys. Lett. 25, 944 (1) The surface of as-etched LaB6 single crystals (1999)]. (Names as in Tech. Phys. Lett.) kept in a vacuum is covered by an oxide, presumably 2. I. Boustani, R. Buenker, V. N. Shrednik, et al., J. Chem. poorly conducting, film that is highly resistant to field Phys. (2001) (in press). evaporation. 3. M. V. Loginov, O. G. Savel’ev, and V. N. Shrednik, Zh. (2) A sufficiently high electric field E breaks down Tekh. Fiz. 64 (8), 123 (1994) [Tech. Phys. 39, 811 the oxide film; subsequently, the intense field evapora- (1994)]. tion of LaB crystals can proceed at appreciably smaller E. 6 4. M. M. Korsukova and V. N. Gurin, Usp. Khim. 56, 3 (3) According to the relationship between the base, (1987). Vb, and pulse, Vp, voltages and also to the area of ion 5. A. G. Naumovets, Ukr. Fiz. Zh. 9, 223 (1964). collection, basically two characteristic spectra of LaB6 field evaporation are observed: without excess lanthanum (this spectrum roughly reflects the volume compo- Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 46 No. 9 2001

Technical Physics, Vol. 46, No. 9, 2001, pp. 1168Ð1174. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 71, No. 9, 2001, pp. 105Ð111. Original Russian Text Copyright © 2001 by Knat’ko, Lapushkin, Paleev.

SURFACES, ELECTRON AND ION EMISSION

Surface Photoemission from Ultrathin Potassium Films Adsorbed on Tungsten M. V. Knat’ko, M. N. Lapushkin, and V. I. Paleev Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia e-mail: [email protected] Received November 23, 2000

Abstract—The initial stage of formation of ultrathin potassium ﬁlms on W(100) is studied by threshold photoemission spectroscopy using p- and s-polarized light in a photon energy range of 1.6Ð3.5 eV. It is found that the photoemission current spectrum depends on the surface coverage by the alkaline atoms. Mathemati- cally, this shows up as the dependence of the matrix elements responsible for photoemission excitation on surface coverage. The matrix elements vary because the photoelectron escape depth is small; hence, the emission comes from the surface layer under irradiation by both p- and s-polarized light. © 2001 MAIK “Nauka/Inter- periodica”.

INTRODUCTION in a submonolayer range of Cs films have been found. For the Cs/Au system [10], TPS has been applied to Systems involving alkaline adatoms on metallic study the initial formation stage of the CsAu surface substrates have been the subject of much investigation alloy. over many decades. These systems are used as a model in studying the interaction of atoms with a solid adsor- The SS spectrum for potassium-on-metal adsorption bent. In addition, alkaline adsorbates are finding wide has been taken for K/Al(111) [3, 11], K/Cu(110) [12], application in various devices. Recent investigations in and K/Cu(100) [13] systems. Below, we report TPS this field have been concerned with the generation and results on the formation of the surface electron struc- modification of surface states (SSs) induced by alkaline ture when ultrathin potassium films are applied on a adsorbates on metallic adsorbents. W(100) substrate. Experimental [1Ð3] and theoretical [4Ð7] studies suggest the following scenario of the SS density varia- 1. EXPERIMENT tion near the Fermi level EF when an alkaline adsorbate is applied. The band of intrinsic SSs (IntSSs) of the Experiments were carried out at a pressure P ≈ 5 × adsorbent lies below EF. At the initial stage of coating 10Ð10 torr. Single-crystal W(100) was used as a sub- formation, adsorbateÐsubstrate interaction shifts the strate. A total of three potassium monolayers were IntSS band toward higher binding energies and causes applied on the substrate at room temperature from an the appearance of alkaline-atom-induced SSs (IndSSs). atomically clean potassium source. The coverage ϑ was The IndSS band initially lies above EF. As the surface found from the well-known coverage dependence of coverage grows, both IntSS and IndSS bands move the work function ϕ. The minimal work function of toward higher binding energies. For the coverage at the K/W(100) system was observed at ϑ = 0.6 mono- which the work function of the adsystem is minimum, layer [7]. We recorded spectral dependences of the inte- ν ν the IndSS band goes below EF. With a further increase gral photoemission currents Ip(h ) and Is(h ) induced in the coverage, both bands experience modifications. by p- and s-polarized light, respectively, for photon The electron configuration of an alkaline metal– energies between 1.6 and 3.5 eV and various ϑ. The angle of incidence of light was γ = 45°. metallic substrate system near its EF is most conveniently studied by threshold photoemission spectros- The spectral dependences of the photocurrents copy (TPS). This technique is much more sensitive to ν ν Ip(h ) and Is(h ) for various K coatings are shown in surface states located near EF than conventionally used Figs. 1 and 2. As for the systems Cs/W(100), UV spectroscopy [1, 8]. Cs/W(111), Cs/W(110) [1], and Cs/Ag [9] studied ear- ν ν The TPS technique has been applied to studying lier, the Ip(h ) and Is(h ) curves differ in shape and ϑ ν Cs/W(100), Cs/W(111), Cs/W(110) [1], and Cs/Ag [9] intensity. In addition, when is changed, the Ip(h ) and ν systems. In all of them, near-EF SS bands induced by Is(h ) curves vary in a different way. The latter mono- ν ϑ alkaline adsorption and the modification of these bands tonically increase, except for Is(h ) at = 0.6, which

SURFACE PHOTOEMISSION FROM ULTRATHIN POTASSIUM FILMS 1169

ν ν has a convex form. The shape of the Ip(h ) curves sub- Is(h ), arb. units stantially changes with ϑ. 3 ν Obviously, the differences between the Ip(h ) and ν ϑ 3 2 Is(h ) curves for = 3.0 and those obtained for “thick” films (100 nm thick) [14] are due to the fact that K films in the former case are extremely too thin. The shape of ν 1 the Is(h ) curve for thick films is specified largely by 2 4 the narrow conduction band of K, Eband = 1.60 eV [15]. 5 Consider the curves obtained at the different light ν 6 polarizations in greater detail. The photocurrent Is(h ) near the threshold obeys the Fowler law [16] 1 ()νϕ2 Is = ah Ð , (1) where a is a constant. 0 For metals, the threshold photon energy of photoe- 23hν, eV ν ϕ mission equals the work function: h 0 = . The energy Fig. 1. Spectral dependences of the photocurrent I (hν) for range where the photocurrent obeys the Fowler law ϑ s depends on ϑ; in other words, in general, the energy = (1) 0.30, (2) 0.60, (3) 1.0, (4) 2.0, and (5) 3.0. (6) Curve ν ϑ for the thick potassium film [14]. range where Is(h ) is a quadratic function varies with . However, there are two subranges, ϑ < 0.3 and 1.1 < ν ϑ < 2.8, where this curve remains quadratic throughout Ip(h ), arb. units the excitation energy range. 14 From I (hν) curves [formula (1)] taken for various ϑ, s 12 one can find the shape of the ϕ(ϑ) curve (Fig. 3). The 5 value of ϕ = 2.30 ± 0.03 eV at ϑ = 3.0 is somewhat 10 greater than the reference data ϕ = 2.22 eV [17]. 3 For p-polarized exciting light, the photocurrent near 8 2 the threshold does not follow the Fowler law. 6 4 2. THEORETICAL CONSIDERATION 4 AND DATA PROCESSING 2 1 6 In the theory of threshold photoemission [18], the ν ν 0 photocurrents Ip(h ) and Is(h ) are given by 1.52.0 2.5 3.0 3.5 4.0 hν, eV ()ν 2()ν ν 2θ()ν ν I p h = K h Ð h 0 h Ð h 0 ν Fig. 2. Spectral dependences of the photocurrent Ip(h ). Ᏹ 2 2 γ (1)Ð(6) the same as in Fig. 1. × 0 sin ------2 ε γ ε 2 γ (2) m cos + m Ð sin Ᏹ Ᏹ electric vector components 2 and 3 that are parallel × [ 2 ε 2 2 γ 2 ε 2 γ to the surface; and K is a constant that takes into M1 m sin + M2 m Ð sin account the SS density features near EF and the features γ ()]ε ε 2 γ of electron transitions at the metalÐvacuum interface. +2sin Re M1*M2 m* m Ð sin , Note that the matrix elements M2 and M3 are theo- I ()hν = K 2()hν Ð hν 2θ()hν Ð hν retically responsible for photoemission from the inte- s 0 0 rior of the metal, i.e., for bulk photoemission, while the Ᏹ 2 2 γ 2 element M , for photoemission excited at the metalÐ 0 cos M3 (3) 1 × ------. vacuum interface, i.e., for surface photoemission. Thus, γ ε 2 γ 2 cos + m Ð sin s-polarized light causes only bulk photoemission, while ν θ ν ν p-polarized light must cause both. For isotropic materi- Here, 0 is the threshold frequency; (h Ð h 0) is the ε als, M2 = M3 [18]. Heaviside function; m is the permittivity of the metal; Ᏹ From (3), we have 0 is the amplitude of the incident light wave; M1 is the 2 matrix element of transitions due to the electric vector I cosγ + ε Ð sin2 γ component Ᏹ that is perpendicular to the surface; M K 2 M 2 = ------s m . (4) 1 2 3 ()ν ν 2θ()ν ν Ᏹ 2 2 γ and M3 are the matrix elements of transitions due to the h Ð h 0 h Ð h 0 0 cos

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1170 KNAT’KO et al.

hν ; hν , eV ϕ, eV Then, the interference term in (2) depends on the pl m0 δ δ δ 1.4 2.4 phase difference = 1 Ð 2 alone. Hence, the interference term can be rejected when the factor 1.2 1 2.3 −iδ ε ε 2 γ e m* m Ð sin approaches zero. This procedure requires special analysis. 1.0 3 2.2 | |2| |2 4 To ﬁnd the value of K M1 , we must make assump- 0.8 2.1 tions on the form of this function. It is selected for each 2 speciﬁc system. Parameters that specify the form of | |2| |2 0.6 2.0 K M1 are found by a method of minimum search. The minimum of the expression 0.4 1.9 exp  I ()hν 2 ------p Ð K θ()hν Ð hν 0.2 ∑()ν ν 0 1.8 h Ð h 0 1.7 Ᏹ 2 sin2 γ 0 132 × ------0 ϑ ε γ ε 2 γ 2 m cos + m Ð sin (5) ϕ ν ν ϑ ν ϑ 2 2 2 2 2 Fig. 3. (1) , (2) h pl and (3) h m0 vs. . (4) h pl vs. curve × [ M ε sin γ + M ε Ð sin γ calculated for the Cs/W(100) system according to [1]. 1 m 2 m 2 2 γ ()]ε ε 2 γ  ν -+2sin Re M1*M2 m* m Ð sin , h max  M 1 where Iexp (hν) are the experimentally found photocur- hν p m0 rent values, is sought. Below, results of processing experimental data by formulas (4) and (5) are presented. | |2| |2 2.1. s-polarization. All functions K M3 obtained Γ by data processing (Fig. 2) are well described by the hν equation M3 plato ν (θ()θν ν ()ν h pl KM3 = Km3 h Ð h 0 + h Ð hv plato ν (6) h 0 × ())nνn ÐnνÐn h h plato Ð 1 ,

where Km3 is the amplitude of the matrix element (its value is related to the density of states near EF). ν ≤ ν ≤ ν ν 0 EF E, eV In the range h 0 h h plato, the current Is(h ) obeys the Fowler law; i.e., it is speciﬁed by electron | |2| |2 | |2| |2 Fig. 4. Matrix elements K M3 and K M1 . states below EF. For photon energies inducing the pho- toelectric effect, their density can be considered constant. The exponent n deﬁnes the rate of fall or rise of Figure 4 shows the energy dependence of the matrix the density of states when photon energies induce pho- | |2| |2 ν ν element K M3 that was calculated by formula (4). toemission (h > h plato). ν From (2), the matrix element of surface photoemission The values of h plato, n, and Kms were found from cannot be found directly because of the interference the minimum of the expression term M1* M2. In the optical frequency range, this term  Iexp()hν for metals cannot be ignored unless additional assump- ∑------s - Ð K 2θ()hν Ð hν -- tions are made. ()ν ν 2 0 h Ð h 0 Theoretically [18], the elements M are represented (7) i 2 Ᏹ 2 cos2 γ M 2  as × 0 3  ------2 , iδ γ ε 2 γ  i cos + m Ð sin Mi = Mi e , δ exp ν where i is the matrix element phase that is independent where Is (h ) are the experimentally found photo- of hν. currents.

TECHNICAL PHYSICS Vol. 46 No. 9 2001 SURFACE PHOTOEMISSION FROM ULTRATHIN POTASSIUM FILMS 1171

ν ν ν ϑ | |2 Figure 3 plots h pl = h plato Ð h 0 vs. . Km1,3 ϑ When the coverage increases to min, the plateau 140 starts shrinking. As the potassium film thickens, the ν ϑ 120 value of h pl grows. In the range 1.2 < < 2.0, the plateau reaches its maximum length and then remains ϑ ν ϑ 100 unchanged. For > 2.0, h pl decreases. For = 3.0, ν h pl = 0.85 eV, which exceeds 0.40 eV for thick potas- 80 ν sium layers according to [14]. Note that h pl tends to ϑ decline with increasing . 60 3 In the range ϑ < 0.3, n = 0. For ϑ between 0.3 and ϑ ϑ 40 1 min, n drops to Ð3 at min. As the film grows further, n increases to 0 at ϑ = 1.0. In the range 1 < ϑ < 2.5, n = 20 × 5 const = 0. As the film grows further, n drops to Ð0.5 at 2 ϑ = 3.0. For thick potassium films, the value of n calcu- ϑ lated according to [14] was found to be n = Ð11. This 0 132 supports the tendency of n to decrease with growing | |2| |2 | |2| |2 ϑ | |2| |2 adsorbate thickness. Fig. 5. (1) K m3 and (2) K m1 vs. . (3) K m3 vs. ϑ | |2| |2 ϑ ϑ ϑ curve calculated for the Cs/W(100) system according to Figure 5 plots K m3 against . For 0.3 < < min, [1]. the matrix element grows and attains a maximum at ϑ | |2| |2 ϑ min. At thicker films, K m3 decreases. For > 2.0, 2 2 ϑ ν ϑ the value of |K| |m | remains almost constant. We did For 0 < < 2, the shape of the curve h m0( ) closely 3 ν ϑ ϑ ϑ not compare our data for |K|2|m |2 with its value for resembles that of the curve h pl( ). At small ( < 3 ν thick potassium layers, because the absolute values of 0.6), its increase causes a decrease in h m0, which drops ϑ ϑ the photocurrents were not measured. to its minimal value at min. A further increase in ν ϑ ν It is seen that the experimentally found values of the causes h m0 to increase. At 1.0 < < 1.8, h m0 remains ν | |2| |2 ϑ ϕ ϑ ϑ ν ϑ parameters h plato, n, and K m3 vary with like ( ). unchanged. For > 1.8, h m0 grows with . 2.2. p-polarization. For the K/W(100) system, the ϑ | |2| |2 The dependence of K m1 is demonstrated in matrix element KM1, which is responsible for surface | |2| |2 | |2| |2 photoemission, is best described by the Gaussian func- Fig. 5. K m1 is much less than K m3 throughout tion the range of potassium film thickness. A similar result has been obtained for photoemission from indium into ( ( () KM1 = Km1 exp Ðln 0.5 (8) an electrolyte [18]. Although KM1 is much smaller than × ()ν ν ÷ (Γ ()()ν ν Γ ))) KM3, the photoemission due to the normal component h Ð h max /2 1 + Casym h max Ð h / , Ᏹ 1 of the electric vector of p-polarized light is stronger ν Γ where h max is the position of the peak, is the half- than the photoemission due to the parallel component Ᏹ width of the peak, Km1 is the amplitude of the matrix 2 of p-polarized light. This is because in Eq. (2) for ν ε element, and Casym is the asymmetry factor. photocurrent Ip(h ), the coefficient before M1 is m The matrix element M1 is depicted in Fig. 4. The times greater than the coefficient before M2. For tung- ν ε ≈ ϑ position of the peak, h max, is slightly shifted toward sten, m 20 in the optical range [19]. The depen- | |2| |2 higher energies relative to the peak of the SS band. In dence of K m1 is similar to the same dependence of the threshold approximation, the half-width Γ of the | |2| |2 | |2| |2 ϑ K m3 . The value of K m1 grows to = 0.80 and peak of the matrix element responsible for the excita- then drops nearly to the initial value. tion of the surface band is much greater than the width of the SS band [1]. The sign of the factor Casym shows Figure 6 plots Γ against ϑ. Initially, Γ decreases and ϑ the direction of broadening of the matrix element. At attains its minimum at the coverage min. At a further positive Casym, the element broadens toward lower ener- rise in the coverage, Γ grows insignificantly; for 1.3 < | |2| |2 ϑ gies. The value of K m1 is proportional to the density < 2.5, this parameter remains constant. With the cov- of states. erage increasing still further, Γ increases. The phase difference between the matrix elements ϑ Γ ϑ δ ± The curve Casym( ) is similar to the curve ( ) M1 and M3 is = 1.0 0.1 irrespective of the film thick- ϑ δ (Fig. 6). The parameter Casym( ) is negative, except for ness. This value is somewhat larger than = 0.8, which ϑ ϑ has been found for the Cs/W system [1]. the coverage ranges 0.4 < and > 2.5. ν ϑ Figure 3 shows the dependence of h m0 on , where It is seen that the experimentally found values of the ν ν ν ν ν Γ | |2 ϑ h m0 = h max Ð h 0 is the shift of h max relative to the parameters h m0, , Km1 , and Casym vary with like ν photoemission threshold h 0. ϕ(ϑ).

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1172 KNAT’KO et al.

Γ ϑ , eV Casym matrix elements M1 and M3, with allow one to quali- 1.8 3 tatively visualize the variation of the SS density near EF. The general picture of SS modification is repre- 1.6 2 sented in Fig. 7. 3.1. Analysis of the matrix element M ; SS band 1.4 1 1 due to %1. Consider the variation of the SS⊥ density (SSs in the direction normal to the surface) near E with 1.2 F 1 0 the degree of coverage (the upper part of Fig. 7). Sur- face photoemission is known to proceed from the SS 1.0 2 –1 band. Ideally, we would have to calculate the matrix element responsible for surface photoemission with 0.8 regard for the wave functions of the adsorbate and SSs –2 of W, as well as their variation during K adsorption. 0.6 0 132 Such a computational procedure is very tedious. There- ϑ fore, we will consider on a qualitative basis a relation between the parameters M1 and the parameters of the Γ ϑ Fig. 6. (1) and (2) Casym vs. SS band (Fig. 8). For threshold photoemission, where the transition from the surface band to the continuum of Γ ρ excited states takes place, the half-width of the matrix (E), arb. units element is several times larger than its associated SS band [1]. The parameter Casyn indicates the SS band 2 ν asymmetry. The position h m0 of the element maximum is 0.1 to 0.2 eV close to the Fermi level than the maximum of the SS band [1]. The position of the surface 1 band maximum Emax relative to the Fermi level is given ≈ ν | |2| |2 by Emax Ðh m0. The value of K m1 is proportional to the SS density ρ(E). We detect the integral photocurrent and, hence, obtain the smoothed pattern that shades 0 drastic changes in ρ(E). The resulting variation of the SS band with increasing degree of coverage for the K/W(100) system is shown at the top of Fig. 7. At ϑ = 0.3, the SS band is wide, as indicated by large Γ 2 and by the low density of states (the small value of | |2| |2 K m1 ). This band peaks at high binding energies ν (h m0 is large). It is logical to assume that this surface band is the IntSS band of tungsten, as for the 1 Cs/W(100) system [1]. ϑ As the coverage increases to min, the SS band expe- ρ | |2 riences great modifications. First, (E) grows ( KM1 0 increases); second, the band shifts toward smaller bind- 0.3 ν ing energies (which the decrease in h m0 indicates); 0.6 Γ 1.0 third, the band shrinks ( decreases). These modifica- 2.0 tions can be explained by shifting the band of potas- 3.0 sium valence states below E , that is, by the formation ϑ F >100 of the IndSS band induced by potassium adsorption. 0 –1 –2 Similar behavior has been observed for Cs adsorption on E, eV various W faces [1] and for K adsorption on Al(111) [3]. Fig. 7. Variation of the SS band for the K/W(100) system For the monolayer-thick adsorbate, the SS band with the coverage ϑ. shifts downward, i.e., toward higher binding energies ν (h m0 grows), and slightly broadens. The density of | |2| |2 3. DISCUSSION states increases ( K m1 rises). The SS band presumably consists of potassium valence electron levels on The variations of the photoemission parameters, as which the W substrate has a minor effect. Our results well as of the functional shape and the value of the correlate with data in [20], where it has been shown that

TECHNICAL PHYSICS Vol. 46 No. 9 2001 SURFACE PHOTOEMISSION FROM ULTRATHIN POTASSIUM FILMS 1173 ε ε the SS band width for an isolated K monolayer is 2,3 1 ≈1.5 eV and the SS density ρ(E) has no singularities. Evac ϑ The tendencies observed are retained at 1.0 < < 0 max ν plato ν h 2.2. The surface band consists of potassium valence ν h electron levels and is affected by the W substrate only h EF insigniﬁcantly. Hence, the formation of the potassium pl

a ν

band of surface states continues. Such a situation is h 0

ϑ m

observed to = 3.0, which the variation of the parame- ν ters studied is an indication. The SS band broadens (Γ h increases). The density of states in the peak declines b | |2| |2 ( K m1 decreases). The band peak shifts toward ν higher binding energies (h m0 increases). Thus, the IntSS band of potassium continues to form. Fig. 8. SS band.

3.2. Analysis of the matrix element M3; SS band due to %3. The theory states [18] that bulk photoemis- θ where r is a constant, EF = 0, and (E Ð EF, pl) is the sion from a metal is described by the matrix element ν Heaviside function; here, the relationship Epl = h pl M3. Earlier, it was assumed that M3 must remain unchanged during the application of the three-mono- holds. layer-thick potassium films, since the depth of light The parameter n shows how much the density of penetration into the metal is ≈100 nm, so that the escape states in the band b is greater (smaller) than in band a | |2| |2 depth of low-energy photoelectrons is too large (for the (Fig. 8). The parameter K m3 is proportional to the general energy dependence of the electron escape depth density of states in the band. The associated qualitative see [21]). It should be noted here that the escape depth picture is depicted at the bottom of Fig. 7. of low-energy (less than 1 eV) photoelectrons from At ϑ = 0.3, there exists the flat band (hν = 1.2 eV) alkaline metals is very difficult to measure. Anyway, we pl below E . This band is formed by the SSs of W(100), have not found relevant data in the literature. As follows F which are symmetric in the plane parallel to the surface. from Section 2.1, the parameters of the element M3 vary The presence of the SS|| on W(100) near E is supported during potassium deposition. This might indicate that F the photoelectrons are emitted not from the bulk of by calculation [5]. ϑ tungsten, where ρ(E) and M cannot vary during potas- When the coverage rises to min, the SS band expe- 3 Ᏹ sium adsorption, but from a very narrow (several riences sharp modifications due to the action of 3. The ν atomic layer thick) surface region. Such an assumption value of Epl (or h pl) decreases. The density of states in correlates with data in [22], where it has been shown the band grows (|K|2|m |2 increases). At E < E , the den- ≈ 3 pl that the escape depth of 4-eV photoelectrons from Cs sity of states diminished (n = Ð3). The band seems ≈ is 1 nm. Consequently, one can suppose that photo- likely to include the SS band of W(100) at binding ener- electrons come from the surface band that is excited by gies above Epl and the SS band of K near EF. This result %3. In the dipole approximation, this means that the is consistent with the calculation of sodium adsorption ≠ scalar product ((m||%3) 0 (m|| is the dipole moment par- on aluminum [4]. It has been shown in [4] that SSs with allel to the surface). This is possible if the SS|| band with the preferential direction parallel to the surface exist the preferential direction parallel to the surface is even at ϑ = 0.2. formed in the surface layer. Such states in photoemis- ϑ Ᏹ sion have been detected, for example, for oxygen When the coverage increases to = 1.0, the 3-sen- adsorbed on Ni(100) (p and p states) [23] (x and y are sitive SS zone changes again. It becomes flat (n = 0), x y | |2| |2 coordinates in the surface plane). The processing of and the density of states drops ( K m3 decreases). experimental data for the Cs/W(100) system [1] also Apparently, the photoemission from the SS zone of ϑ W(100) gives way to that from the forming IndSS|| band indicates that M3 depends on (see Figs. 3, 5). of potassium. In his classical work [16], Fowler considered photo- At ϑ = 2.0, the density of states in the surface SS|| emission in the flat-band approximation. Our results indicate that this approximation is valid. Below, we will zone somewhat declines (as follows from decreasing | |2| |2 K m3 ). This means that the photoemission from relate the parameters of the element M3 in formula (6) Ᏹ W(100) is virtually absent. to the parameters of the band responsible for 3- induced photoemission. For threshold photoemission At ϑ = 3.0, the surface SS|| band changes once again. ν from the flat band (Fig. 8) in which the density of states We observe the flat band near EF up to E > Ð0.7 eV (h pl goes to zero at Epl, we have decreases) and the slight decrease in the density of states at high binding energies (n = Ð0.5) The density of ρ() ()θ()θ() | |2| |2 E = r EEÐ F Ð EEÐ pl , states near EF remains the same ( K m3 is constant).

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1174 KNAT’KO et al.

The SS band for the thick K ﬁlms narrows near EF, as REFERENCES follows from calculations according to [14], and the 1. G. V. Benemanskaya, M. N. Lapushkin, and M. I. Ur- density of state in it sharply drops at high binding ener- bakh, Zh. Éksp. Teor. Fiz. 102, 1664 (1992) [Sov. Phys. gies. JETP 75, 899 (1992)]. 2. S. A. Lindgren and L. Wallden, Solid State Commun. 28, 283 (1978). CONCLUSIONS 3. K. H. Frank, H. J. Sagner, and D. Heskett, Phys. Rev. B We have studied photoemission in the K/W(100) 40, 2767 (1989). system subjected to visible polarized light at an adsor- 4. H. Ishida, Phys. Rev. B 38, 8006 (1988). bate amount of 0.3 < ϑ < 3.0. 5. E. Wimmer, A. J. Freeman, J. R. Hiskes, et al., Phys. Rev. B 28, 3074 (1983). A method for separating the matrix elements M1 and 6. R. Q. Wu, K. L. Chen, D. S. Wang, et al., Phys. Rev. B M3, responsible for the photoemission, from the spec- 38, 3180 (1988). tral dependences of the photocurrents under s and p 7. J. Cousty, R. Riwan, and P. Soukiassian, J. Phys. (Paris) excitations has been suggested. The parameters of the 46, 1693 (1985). matrix elements have been shown to vary in accordance ϕ 8. B. Feurbacher and R. F. Wilis, J. Phys. C 9, 169 (1976). with the variation of . 9. A. Libsch, G. V. Benemanskaya, and M. N. Lapushkin, The variations of the SS density (for both the per- Surf. Sci. 302, 303 (1994). pendicular, SS⊥, and the parallel, SS||, components) 10. M. V. Knat’ko, M. N. Lapushkin, and V. I. Paleev, Pis’ma Zh. Tekh. Fiz. 24 (10), 48 (1998) [Tech. Phys. Lett. 24, near and below EF have been found. The variations of both components characterize the formation of the 390 (1998)]. potassium SS band. 11. K. Horn, A. Hohﬁeld, J. Somers, et al., Phys. Rev. Lett. 61, 2488 (1988). It has been established that, under the excitation by 12. B. Woratschek, W. Sesselmann, J. Kuppers, et al., Phys. s-polarized light, the matrix element responsible for the Rev. Lett. 55, 1231 (1985). photoemission changes even at a submonolayer cover- 13. T. Aruga, H. Toshihara, and Y. Murata, Phys. Rev. B 34, age. This means that threshold photoemission is initi- 8237 (1986). ated at a very small distance (on the order of several 14. J. Monin and G. A. Boutry, Phys. Rev. B 9, 1309 (1974). atomic layers) from the surface. 15. B. S. Itchkawitz, I. W. Lyo, and E. W. Plummer, Phys. Ᏹ Thus, both the normal, 1, component of the elec- Rev. B 41, 8075 (1990). tric vector of p-polarized light and the tangential, Ᏹ|| 16. R. H. Fowler, Phys. Rev. 1, 35 (1938). components of s- and p-polarized light cause photoe- 17. V. S. Fomenko, Emission Properties of Materials (Kiev, mission from the SSs of the alkaline metalÐmetallic Naukova Dumka, 1981). substrate adsystem. The former component causes pho- 18. A. M. Brodskiœ and M. I. Urbakh, Electrodynamics of toemission from the SSs of the adsystem that have the Metal/Electrolyte Boundary (Moscow, 1989). preferential direction normal to the surface, while the 19. J. H. Weaver, C. G. Olson, and D. V. Linch, Phys. Rev. B latter two cause photoemission from the SSs of the 12, 1293 (1975). adsystem that have the preferential direction parallel to 20. E. Wimmer, J. Phys. F 13, 2313 (1983). the surface. 21. M. P. Seah, Surf. Interface Anal. 1 (1), 1 (1979). 22. N. V. Smith and G. B. Fisher, Phys. Rev. B 3, 3662 (1971). ACKNOWLEDGMENTS 23. G. J. Lapeyre, J. Vac. Sci. Technol. 14, 384 (1977). This work was supported by the State Research Pro- gram “Surface Atomic Structures” (project no. 3.14.99). Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 46 No. 9 2001

SURFACES, ELECTRON AND ION EMISSION

Electrode Surface Erosion in a High-Frequency Discharge Plasma G. P. Strokan’ Research Institute of Physics, Rostov State University, pr. Stachki 194, Rostov-on-Don, 344104 Russia e-mail: [email protected] Received January 9, 2001

Abstract—The surface erosion of electrodes made of different materials in the plasma of a high-frequency discharge, which is used for pumping ion lasers at a frequency of 10 MHz, is investigated. It is found that the erosion is due to blistering. The effect of the electrode temperature and material, as well as of the gas type, on the erosion evolution under typical operating conditions of a gas discharge tube is studied. The concentration of blistering products (dust particles) in the discharge is estimated in the framework of geometrical optics. Ways to prevent blistering in the discharge under such conditions are suggested. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION decreases and drops to zero at some threshold concentration. The characteristics of laser radiation are greatly affected by the electrode surface condition, since the In the second case (Fig. 1b), the test tube was placed pumping of transverse high-frequency discharge outside the cavity and was subjected to a lower-inten- (THFD) lasers is affected by processes in the electrode sity irradiation. In such a configuration, losses up to region [1Ð5]. Moreover, during the operation, the elec- 100% due to light scattering by dust particles in the trode surface is exposed to intense ion irradiation with tube could be measured. The compensation measuring an ion energy comparable or equal to that of near-cath- scheme was used to detect a small absorption (about ode acceleration. As a result, the cathode surface condi- 0.2%). Using a microscope, we studied the particle tion changes and the electrode material is sputtered motion and measured the particle sizes. Selective mir- both in the atomic form and as clusters into the dis- ror 7 prevented the spontaneous radiation of the test charge volume. Eventually, the gas discharge character- tube from reaching a photodetector. Before the mea- istics [6Ð10] and, hence, lasing parameters degrade. surements, a heliumÐneon laser was heated for 3Ð4 h to exclude the influence of intrinsic fluctuations. Along with the well-known phenomenon of cathode sputtering [11], the electrodes in a gas discharge are (a) eroded, for example, by blistering, which is well known 31 2 3 in the technology of thermonuclear reactors [12]. How- ever, this process in THFD lasers has not been studied. The purpose of this paper is to study blistering in a discharge used for ion laser pumping. 45 6

EXPERIMENTAL (b) 31327 Our experimental setup (Fig. 1) was used to observe macroparticles produced in the discharge tube and to investigate their effect on lasing parameters. In the first case, the test tube was placed in a cavity with the tube of an LG-75 heliumÐneon laser (Fig. 1a). The cavity incorporated mirrors with a reflection coefficient of 5 99.9%. Because of a high intensity inside the cavity, the 8 birth of particle bunches and their subsequent localization could be observed. With this technique, we were Fig. 1. Block diagram of the experimental setup for obser- µ vation of blistering products. The test tube is (a) inside and able to observe the ejection of particles 10Ð30 m in (b) outside the cavity. 1, LG-75 laser; 2, test tube; 3, cavity size. The drawback of this technique is that, as the par- mirrors; 4, deflection plate; 5, power meter; 6, microscope; ticle concentration increases, the lasing power 7, selective mirror; and 8, compensation signal source.

1176 STROKAN’

(‡) (1) Macroscopic particles result both in helium and in heliumÐcadmium discharges, the concentration of the particles being higher in the helium discharge. In discharges of other gases (air or krypton), the particles are virtually absent. (2) At 300°C, the concentration of the particles is the highest in the discharge initiated by the stainless-steel and nickel electrodes. For the aluminum and nitride- coated electrodes, the particles are released in small amounts. The quartz and ceramic electrodes practically do not generate the particles. (3) The temperature dependence of the particle generation was studied in the helium discharge initiated by the stainless steel electrodes. At low temperatures (T < 150°C), dust formation is negligible. At T = 280°C, the generation of the macroparticles becomes intense and (b) the particles settle on the walls of the discharge tube. A further increase in the temperature reduced the dust formation rate, which was minimal at T = 700°C. It should be noted that the heating of the electrodes in the absence of the discharge did not cause appreciable powder formation. The electrodes were heated by the external resistance heater, as well as by the induction heater, to about 600°C. These observations are explained by the properties of blistering described in [12, 13]. (1) Blistering should be most pronounced in a pure- helium discharge, since helium is not bonded to the electrode material and, ﬁlling available voids, causes blistering. Fig. 2. Image of the stainless steel electrode surface in the case of (a) exfoliation and (b) blistering. Scale is 100 µm. (2) The data obtained for the electrodes made of the different materials are explained as follows. When penetrating into the electrode, helium atoms are accumu- The test tube with an electrode inside was outgassed lated in voids, exerting a high pressure in them. As a by evacuating to a pressure of about 10Ð3 Pa. Then, it result, blistering and then electrode erosion occur. If the was heated to a temperature of 300Ð400°C by an exter- surface is porous, the helium atoms emerge on the sur- nal resistance or induction-heated and evacuated again. face through pores without causing electrode erosion. If the currentÐvoltage characteristics remained This is seen most clearly in the case of the quartz elec- unchanged after several measurements, the electrode trode, which has a helium permeability as high as 7 × surface was assumed to be clean. It should be noted that 10Ð3 Pa l/cm2 at 300°C [14]. This comes into conﬂict the heating of the electrode in the absence of the dis- with the Norton rule, according to which metals are charge did not lead to the erosion of the electrode sur- impermeable to inert gases and must show blistering. face as in the case of blistering. The much lesser generation of the macroparticles in the We used metallic (St3 steel, stainless steel, nickel, discharge with the aluminum electrodes is due to the aluminum), nitride-coated (titanium nitride and chro- presence of an oxide layer whose thickness exceeds the mium nitride), and insulating (quartz and ceramic) penetration depth of helium atoms. Moreover, the alu- electrodes. The thickness of the nitride coatings was minum electrode was investigated at a temperature of about 20 µm. An HF generator with a matched load pro- 0.7T1 (T1 is the melting point), while at temperatures of vided an output of up to 1 kW at a frequency from 1 to 0.5Ð0.6T1 or higher, blistering is highly improbable 20 MHz. [12]. Similar results were obtained for the electrodes coated by ≈20-µm-thick titanium nitride. (3) The temperature dependences of the macroparti- RESULTS AND DISCUSSION cle generation are in agreement with those obtained in When studying blistering in the discharge tubes, we [12, 13] for fusion reactors. The temperature depen- took into consideration that this phenomenon depends dences of blistering are in good agreement with those of on temperature, type of the gas, and electrode material. particle generation that were observed in the helium The results obtained are as follows. discharge with the stainless steel electrodes. At temper-

TECHNICAL PHYSICS Vol. 46 No. 9 2001

ELECTRODE SURFACE EROSION 1177 atures less than 150°C, small (about several microme- K, % ters in diameter) blisters are produced on the electrode surface by the flow of helium ions. The detachment of 30 small blister caps does not lead to an appreciable dust formation, because the surface erosion rate at this temperature is much smaller (by one or two orders of mag- 20 nitude) than the rate of exfoliation [13]. At T = 0.3T1, blistering gives way to exfoliation. In this case, the erosion rate and the particle sizes grow. The surface of the 10 stainless steel electrode in the case of blistering and exfoliation is shown in Fig. 2. At an electrode temperature of about 700°C, small blisters appear again on the 0 electrode surface as a result of helium ion bombard- 5 101520 ment; i.e., exfoliation gives way to blistering. Experi- t, min mentally, this shows up as a substantial decrease in the particle concentration in the discharge volume. Fig. 3. Time dependence of the radiation absorption coefficient in the tube with the nitride-coated electrode. The electrode temperature is 300°C. ABSORPTION OF RADIATION IN LASER TUBES WITH NITRIDE-COATED ELECTRODES It has been shown above that the nitride-coated electrodes generate a small amount of macroscopic particles. However, the radiation absorption in laser tubes with such electrodes remains to be estimated. In addition, to obtain the maximum lasing power, one should know losses in the tube, for example, for appropriately selecting the electrode material. Therefore, a more detailed study of dust formation and radiation absorption in the laser tube, as well as the determination of the particle concentration due to blistering, would be of much interest. The measuring setup is shown in Fig. 1b. The measurements were carried out in an asymmetric tube with a 0.5-m-long electrode inside. The active channel has the form of a slot 3 × 2.5 mm in size. The inner surface of the channel was coated by titanium nitride. Fig. 4. Surface of the nitride-coated electrode after ten-hour operation under the THFD conditions at a temperature of After long-term heating and degassing of the test ° µ tube, the absorption coefficient was estimated at K ≈ 300 C. Scale is 10 m. 25% and remained constant for a long time (Fig. 3). In this case, the discharge volume was free of dust clouds, obtain with our method which are typical of the stainless steel electrodes. The slight absorption can be accounted for by the presence 4 100 n = 3.14× 10 ln ------. of a small amount of the particles released from the 100 Ð K electrode surface. A decrease in the transmitted light intensity due to In our case, the particle concentration is about particle-related losses in the discharge can be estimated 104 cmÐ3. in the geometrical optics approximation: Such estimations allow one to compare electrodes made of different materials in order to minimize radia- ln()J /J πr2αn = ------0 -. tion losses in the discharge tube of a THFD laser, as l well as to reduce the noise level in the low-frequency spectral range. Here, n is the average particle density in the discharge; r is the particle diameter; l is the tube length; J0 and J are the intensities at the tube inlet and outlet, respec- CONCLUSION α tively; and is the damping parameter. Thus, in both HF and dc discharges, the surface of At a particle diameter of 3 µm, which corresponds to the metallic electrodes is eroded. Depending on the the typical particle diameter in Fig. 4 and also to the electrode material and temperature, erosion may show microscopic visualization parameter α = 2 [15], we up as the blistering (the collapse of blisters with diam-

TECHNICAL PHYSICS Vol. 46 No. 9 2001

1178 STROKAN’ eters from units to tens of micrometers) or exfoliation. 5. A. A. Kuzovnikov, V. P. Savinov, and V. G. Yakunin, In the latter case, the erosion is one or two orders of Vestn. Mosk. Univ., Ser. 3: Fiz., Astron. 21 (4), 75 magnitude faster and the erosion products are several (1980). hundreds of micrometers in size. Unlike the stainless 6. V. V. Savranskii and G. P. Strokan’, J. Russ. Laser Res. steel electrodes, the aluminum and nitride-coated elec- 15 (1), 81 (1994). trodes are not prone to exfoliation during the operation 7. V. S. Mikhalevskiœ, G. P. Strokan’, M. F. Sém, et al., of a heliumÐcadmium laser, as indicated by the small Kvantovaya Élektron. (Moscow) 16, 37 (1989). concentration of dust particles. 8. G. P. Strokan’ and G. N. Tolmachev, Avtometriya, No. 1, To reduce the effect of blistering products on laser 61 (1984). performance, we suggest using electrodes coated by 9. A. V. Borodin, V. F. Kravchenko, and G. P. Strokan’, Zh. nitrides or by a material the melting point of which Tekh. Fiz. 66 (8), 44 (1996) [Tech. Phys. 41, 770 exceeds the operating temperature by no more than a (1996)]. factor of 1.5. 10. G. P. Strokan’ and G. N. Tolmachev, in TQE-85, Bucha- rest, 1985, Vol. I, p. 89. 11. N. V. Pleshivtsev, Cathode Sputtering (Atomizdat, Mos- REFERENCES cow, 1968). 1. E. L. Latush, V. S. Mikhalevskiœ, M. F. Sém, et al., 12. M. I. Guseva and Yu. V. Martynenko, Usp. Fiz. Nauk Pis’ma Zh. Éksp. Teor. Fiz. 24 (2), 81 (1976) [JETP Lett. 135, 671 (1981) [Sov. Phys. Usp. 24, 996 (1981)]. 24, 69 (1976)]. 13. K. L. Wilson, J. Plasma Phys. Thermonuclear Fusion, Special Issue, 85 (1984). 2. S. V. Aleksandrov, V. V. Elagin, and A. É. Fotiadi, Pis’ma Zh. Tekh. Fiz. 6 (3), 160 (1980) [Sov. Tech. Phys. Lett. 14. Handbook of Thin-Film Technology, Ed. by L. I. Maissel 6, 70 (1980)]. and R. Glang (McGraw-Hill, New York, 1970; Sov. Radio, Moscow, 1977), Vol. 1. 3. M. G. Dyatlov, V. G. Kas’yan, and V. G. Levin, Pis’ma Zh. Tekh. Fiz. 3, 644 (1977) [Sov. Tech. Phys. Lett. 3, 15. D. Deirmendjian, Electromagnetic Scattering on Spher- 264 (1977)]. ical Polydispersions (Elsevier, New York, 1969; Mir, Moscow, 1971). 4. A. N. Korol’kov, S. A. Rudelev, and V. A. Stepanov, Éle- ktron. Tekh., Ser. 4: Élektrovak. Gazorazryadn. Prib. 9, 12 (1977). Translated by M. Astrov

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EXPERIMENTAL INSTRUMENTS AND TECHNIQUES

The Effect of Multiple ElectronÐElectron Scattering on the Energy Distribution of Electrons in a Correlated Pair O. M. Artamonov and S. N. Samarin Research Institute of Physics, St. Petersburg State University, Ul’yanovskaya ul. 1, Petrodvorets, St. Petersburg, 198906 Russia Received May 30, 2000; in ﬁnal form, December 26, 2000

Abstract—A unit event of electronÐelectron scattering in LiF layers is studied by correlation spectroscopy of scattered electrons. The energy distribution of electrons in a correlated pair when a 15- to 55-eV free electron is scattered by a valence electron of LiF is studied. It is shown that single electronÐelectron scattering prevails and the distribution is uniform when the energy of the primary electron is below 25 eV. As the energy of the primary electron increases, the formation of correlated pairs of electrons with equal energies becomes the most probable. With the energy of the primary electron above 40 eV, the pairs with substantially different electron energies dominate. Such evolution of the energy distribution of the electrons in the pair stems from the fact that ﬁrst one and then the other electron of the pair successively takes part in electronÐelectron scattering. A phe- nomenological model for the single scattering and double scattering of primary electrons in LiF ﬁlms is considered. Results obtained indicate that the strengths of single scattering and double scattering channels become comparable at electron energies above 25 eV. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION electron is scattered by a valence electron of the solid, the energy distribution of correlated electrons depends Correlation spectroscopy of scattered electrons, or on many factors: (1) multiple electronÐelectron scatter- spectroscopy of electronÐelectron coincidences, is ing in the solid, (2) the escape of the correlated pair among the most promising methods for studying elec- from the solid to a vacuum, (3) the nonuniform distri- tronÐelectron scattering on and near the surface of sol- bution of vacant and occupied electron states in the ids. In correlation spectroscopy, a pair of correlated solid, (4) the presence of surface electron states, (5) electrons that results from a unit event of scattering of electron diffraction by the lattice, etc. It has been shown a primary electron by a valence electron of the solid is experimentally [2Ð7] that the energy distribution of detected. The idea of using correlation spectroscopy of electrons in a correlated pair may have both a minimum secondary electrons for studying a solid surface was and a maximum at equal energies of scattered electrons first suggested by us in [1]. Correlation spectroscopy of when primary electrons of energy between 10 and low-energy backscattered electrons was first applied to 50 eV are scattered by the metal surface. metal surfaces by a joint team of Russian and German researchers [2Ð7]. In recent works, the dynamics of In this work, we study the effect of multiple elec- slow electron surface scattering in the coincidence tronÐelectron collisions on the energy distribution of spectroscopy method has been treated theoretically the correlated carriers. Experiments were performed [6, 8Ð10]. with insulating LiF films. Insulators feature a well- defined energy threshold for the scattering of a non- The energy distribution of electrons in a correlated equilibrium electron by a valence one. This threshold pair is a basic parameter of electronÐelectron scatter- equals the bandgap E (11.5Ð13 eV for LiF). Thus, we ing. In the free-electron binary scattering approxima- g tion, this distribution reflects the dynamics of electron– can easily determine the energy threshold for the band- electron scattering and is adequately described by the to-band excitation of valence electrons [12, 13]. well-known Mott formula, which is the quantum- mechanical generalization of the Rutherford formula EXPERIMENT (see, e.g., [11]). The probability of generating a correlated pair as a function of the energy of one of the elec- 1. Equipment. Experiments were performed with trons has a minimum when the electrons have equal an (e, 2e) electron spectrometer configured with two energies (this is valid if their spins are not taken into time-of-flight energy analyzers (Fig. 1) at a pressure on account, i.e., when the correlated electrons are undis- the order of 10Ð11 torr. A W(100) substrate was mounted tinguishable). When a primary (especially low-energy) on a rotatable support and was current-heated to

1180 ARTAMONOV, SAMARIN

Detector 1 ration of energy-correlated electron pairs is accom- Start Delay plished by analyzing experimentally found distribu- L Stop tions. For example, if the target is metallic and only TAC those pairs of scattered electrons for which the total energy equals the energy of the primary electron minus Flight distance, the work function of the target are separated, it becomes

40° ADC Electron clear that these pairs arose when the primary particles gun Delay 40° was scattered by valence electrons on the Fermi level. W(100) Computer Coincidence

Pulse ADC 2. Experimental results. Experimental 2D time-of- generator ﬂight distributions were converted to 2D energy distributions of correlated electrons. Figure 2 shows three Stop 2D energy distribution of the correlated electrons for

Delay TAC Start incident electron energies of 23, 35, and 45 eV. The Detector 2 energy of the electron pair is plotted on the coordinate axes. The degree of blackening depends on the amount Fig. 1. (e, 2e) time-of-flight spectrometer. of pairs detected (the stronger the blackening, the larger the amount of the pairs). Any line parallel to the dashed line corresponds to pairs with a constant total energy of ° 2000 C at regular intervals for cleaning. The purity of correlated electrons Etot = E1 + E2 (where E1 and E2 are the substrate was checked by Auger spectroscopy. LiF the energies of the first and the second electron, respec- films were applied on the W(100) substrate by thermal tively) or to the constant binding energy of the valence evaporation from an electron-beam-heated molybde- electron Eb. The peaks of the 2D distributions lie on the num crucible. The thickness of the films was deter- dashed lines and correspond to certain values of the mined with a quartz microanalytical balance. An elec- total energy of the pair. In Fig. 2a, the peak is at Etot = tron gun generated a pulsed electron beam with a mean 12 eV for E = 23 eV (E is the energy of a primary elec- Ð14 p p current on the order of 10 A and a diameter of less tron), which corresponds to the binding energy of the than 3 mm. Two microchannel plates (MCPs) of diam- valence electron Eb = 11 eV excited from the valence eter 30 mm were used as detectors. The detectors and band top. In Figs. 2b (E = 22 eV at E = 35 eV) and 2c the electron gun were arranged in the plane of the nor- tot p Θ Θ (Etot = 29 eV at Ep = 45 eV), the binding energy is mal to the surface. The angular coordinates 1 and 2 ° ° greater; here, the pairs are most likely to be excited of the detectors could be varied between 30 and 80 from the center of the LiF filled gap. relative to the normal to the surface. Measurements Θ Θ ° were taken for 1 = 2 = 50 . The magnetic field of the Sections of the 2D distribution that are parallel to Earth was attenuated approximately 100-fold by means the dashed line in Fig. 2 yield energy distributions of Helmholtz coils and a shield made of mu-metal, the between the electrons in a pair at constant Etot. Figure 3 shield being placed inside the vacuum chamber. depicts the energy distribution between the electrons of a pair for several E and E . The selected values of E The energy of scattered electrons was measured p tot tot from the time of flight of the electrons from the sample correspond to the valence electron excited from the to the detector. The incident electron beam was modu- valence band top. The intensity axis shows the relative amount of correlated pairs with the electron energies E lated by pulses with a repetition rate of 2.5 × 106 pulse/s 1 at a pulse width of no more than 5 ns. A modulating and E2 = Etot Ð E1. The energy distributions in Fig. 3 pulse from a pulse generator was used as the zero time correspond to different energies of primary electrons of the time-of-flight scale and triggered two time-to- and to the same energy of the valence electron. The val- amplitude converters, TAC1 and TAC2. A correlated ues of Ep and the total energy Etot are indicated by pair was detected as follows. An incident electron is arrows. The spacing between Ep and Etot in the energy scattered by a valence one, and both scattered electrons axis gives the binding energy of the scattering valence leave the surface. They are detected separately by the electron (Fig. 3c). first MCP (MCP1) and the second MCP (MCP2) and All the distributions in Fig. 3 are symmetric about generate pulses that shut down the TAC1 and the TAC2, the midpoint E1 = E2, since the geometry of the experi- respectively. A coincidence circuit (threshold genera- ment is symmetric and the detectors are undistinguish- tor) rejected all pulses that do not coincide in time able for the scattered electrons. The energy of each of (within 200 ns). The times of flight T1 and T2 are digi- the electrons can vary between 0 and Etot. However, the tized with two ADCs and memorized. The result is a 2D range of the energies measured is somewhat narrower, distribution of the amount of electron pairs over the because only electrons of energies above 1.5Ð3.0 eV times of flight of both electrons of the pair. were detected (this range depends on the time interval Electron detection in the coincidence mode allows that is measured by the time-of-flight analyzers). The us to separate those electron pairs originating from distributions in Figs. 3a and 3b are nearly uniform inelastic scattering of one primary electron. The sepa- within the statistical spread; that is, the probability of

TECHNICAL PHYSICS Vol. 46 No. 9 2001

THE EFFECT OF MULTIPLE ELECTRONÐELECTRON SCATTERING 1181 detecting pairs with different combinations of the elec- E2, eV tron energies is constant within the available energy 12 interval. The distribution in Fig. 3c is sharper, while that in Fig. 3d has distinct steps on both sides of the (a) center. The height of the steps is roughly 0.5. In Figs. 3e and 3f, the distributions have a minimum at the center. In Fig. 3g, this minimum occupies the whole central 8 region. The intensity in the minimum is 0.3. Later, we will show that the steps and the minima can be related to double scattering.

DISCUSSION 4 1. Model of electronÐelectron scattering on the surface. The elastic scattering of a primary electron by the ion core plays an essential part in the scattering 4 8 12 kinematics and turns the total moment of a correlated pair toward a vacuum. Elastic scattering will be included in the scattering model under consideration. 20 (b) To be definite, we assume that elastic scattering pre- cedes inelastic electronÐelectron scattering (the left of Fig. 4a), although the assumption that a primary electron interacts simultaneously with the core and with a 15 valence electron would be more plausible. The energy balance for single electronÐelectron scattering is shown in the band diagram on the right of Fig. 4a. The lowest 10 energy of the primary electron that is needed for the excitation of a correlated pair into a vacuum, Eth' , can be estimated from the law of conservation of energy. If 5 the electron affinity of a material is close to zero (that is, the conduction band bottom roughly coincides with 5 10 15 20 ≅ the position of the vacuum level), then Eth' Eg, where Eg is the bandgap of the insulator. In this case, the scat- 30 (Ò) tered primary and valence electrons pass to the conduction band bottom and find themselves in a vacuum with the zero energy. With the potential barrier at the surface,

Eth' grows. Its value also becomes larger than that esti- 20 mated from the energy balance for momentum-correlated electrons. The second event of electronÐelectron scattering may take place if the energy of at least one of the corre- 10 lated electrons exceeds Eth' . This is possible when Ep > ≅ Eth'' , where Eth'' 2Eth' is the threshold of double scattering. With such an energy of the primary electrons, at 10 20 30 least one of the correlated electrons has an energy that E1, eV is sufﬁcient for the excitation of an electron from the valence band. In this case, the initial correlated pair col- Fig. 2. 2D energy distribution of the correlated electrons lapses. One would expect that the collapse is responsi- Θ Θ ° that were scattered by the LiF/W(100) layer. 1 = 2 = 45 . ble for the symmetric steps on both sides of the major The correlated pairs are near the dashed lines. peak in Fig. 3d, where the energy of the primary electrons is 35 eV. As the energy of the primary electron increases fur- threshold for both electrons is E''' ≅ 3E' . In this situ- ther, it may so happen that the energies of both elec- th th ation, the number of pairs with (E1, E2) > Eth' decreases trons in correlated pairs exceed 2Eth' ; hence, both may ≅ take part in double scattering. The double scattering further. Initially, at Ep Eth'' , the number of the pairs

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1182 ARTAMONOV, SAMARIN

Intensity, arb. units range considered. Energies falling into this range are 1 (a) smaller than Ep, since the range of the measured elec- Etot Ep tron energy is bounded from below by a value varying from 1.5 to 3.0 eV according to the energy of the pri- 0 mary electron. With these assumptions, a function that describes the energy distribution between the electrons 1 (b) in the pair can be represented as a plateau with steeply Etot Ep falling “side walls” at E1 = 3 eV and E2 = Etot Ð 3 eV (Fig. 3b). This function can be extrapolated to the entire 0 energy range of the primary electron. Actually, however, the shape of the side walls is of minor signiﬁcance 1 (c) for reasoning that follows. Etot Ep E Consider the effect of inelastic electronÐelectron b scattering on the energy distribution in a correlated pair 0 immediately after its generation in a solid. We assume 1 (d) that an electron of the pair may either escape into a vac- Etot Ep uum (and, hence, be detected as an electron of the pair) or take part in additional (second) electronÐelectron 0 scattering, which results in the collapse of the pair. One can suppose that the energy dependence of the inelastic ω 1 (e) scattering probability, ee(E1), can roughly be approxi- Etot Ep mated by the step function ω ()ω> ω ()≤ 0 ee E1 Eg ==ee, ee E1 Eg 0, (1) where E is the threshold. 1 (f) g E Function (1) characterizes the probability that an tot electron of a correlated pair loses its energy. If one of 0 the electrons loses its energy, the pair disappears. The probability P0(E1,E2) of detecting a pair of elec- 1 (g) trons is the product of the probabilities of detecting Etot either electron, P1(E1) and P2(E2); that is, P ==P ()E P ()E P ()E P ()E Ð E . (2) 0 1020 30 40 0 1 1 2 2 1 1 2 tot 1 E1, eV On the other hand, () []ω () P1 E1 = A 1 Ð ee E1 , Fig. 3. Energy distributions of the correlated electrons with (3) the total energy Etot of the pair for Ep = (a) 18, (b) 23, (c) 30, ()[]ω () P2 Etot Ð E1 = A 1 Ð ee Etot Ð E1 , (d) 35, (e) 40, (f) 45, and (g) 55 eV. Etot = (a) 6, (b) 12, (c) 16, (d) 22, (e) 26, (f) 30, and (g) 39 eV. The energy of where A is a constant speciﬁc of the given model. primary electrons and the total energy of the pair are indicated by arrows. Continuous lines in (b), (d), and (g) are Eventually, model results. (), 2[]ω ()[]ω () P0 E1 Etot = A 1 Ð ee E1 1 Ð ee Etot Ð E1 .(4) starts decreasing from the center of the energy distribu- Expression (4) is the probability of detecting a cor- tion; as the energy of the primary electron increases, related pair of electrons where one of them has an this process spans the successively wider central region energy E and the other, Etot Ð E1. If the initial energy (Fig. 3g). distribution of the correlated electrons is assumed to be time-invariable, then function (4) describes the energy 2. The effect of double scattering on the energy distribution of the electrons in a pair after some corre- distribution of correlated pairs. To estimate the effect lated pairs have disappeared as a result of inelastic elec- of multiple electronÐelectron scattering on the energy tronÐelectron scattering. distribution of correlated electrons, one should make an 3. Comparison of the model with experimental assumption of this distribution without multiple scatter- data. To compare experimental distributions with the ing. The distributions in Figs. 3a and 3b refer to single model ones, we normalized the former to their maxima. electronÐelectron scattering, since the energy of the As adjustable parameters, we used Eg (its value affects ' ω electrons in the pair is below Eth . Within the statistical the position of the steps in the model curves) and ee spread of data points, both distributions can be approx- (the probability of electronÐelectron scattering affects imated by a constant value of the intensity in the energy the relative height of the steps). The best agreement was

TECHNICAL PHYSICS Vol. 46 No. 9 2001 THE EFFECT OF MULTIPLE ELECTRONÐELECTRON SCATTERING 1183

dependence ω (E) has a more complex form than a (a) ee Incident unit step, which was used in the calculation. Also, the electron Incident fine structure may be attributed to the dynamics of scat- electron Correlated tering of a primary electron by a valence one. Θ Θ pair 2 1 Single Vacuum scattering E1 E Solid 2 Band E gap vac CONCLUSION Valence Electron–electron electron scattering Valence electron It has been shown that, when the energy of incident Elastic electrons is below 26 eV, correlated pairs in LiF are Ion core scattering excited with the highest probability via single electronÐ electron collisions. Thus, for such energies, the contri- (b) Incident bution of multiple collisions is negligible. For single Incident electron electron electronÐelectron scattering, we can estimate the bind- First Correlated scattering ing energy of the valence electron involved in the pro- Θ Θ pair 2 1 Second cess. Accordingly, it has been demonstrated that the Vacuum scattering E1 E2 valence electrons are excited largely from the valence Solid Band E gap vac band top. At energies between 26 and 40 eV, one of the Valence Electron–electron electron scattering electrons in the pair has an energy sufficient for the Valence valence electron to be excited. At energies above 40 eV, Elastic electron both correlated electrons acquire such a capability. Ion core scattering Because of the additional scattering of the correlated electrons, the number of pairs generated by single scat- Fig. 4. Electron scattering model: (a) single and (b) double tering events decreases. A comparison between the scattering. The left-hand and right-hand sides of the figure depict scattering geometry and band diagram, respectively. model and the experimental data indicates that the probability of a correlated pair escaping from a solid and the probability that the electrons of the pair will found for Eg = 12.5 eV (for Ep = 35 eV) and 12 eV (for take part in further electronÐelectron scattering events ω may equal to each other. In the latter case, the function Ep = 55 eV) and ee = 0.5 and 0.65, respectively (continuous lines in Figs. 3d, 3g). It should be noted that of energy distribution between the correlated electrons may change. The modified function reflects both the these values of Eg are close to the literature data [12, 13]. primary scattering of the primary electron by the valence one and subsequent scattering of the correlated The results of fitting the model to the experimental electrons in the resulting pair. data are in good agreement with the above qualitative The method of (e, 2e) spectroscopy turns out to be explanation of the changes that are observed in the an efficient tool for studying the scattering of slow elec- (e, 2e) spectrum as Ep grows. When the energy of the trons by insulating surfaces. It is believed that double incident electrons rises, so does the total energy of the (or multiple) scattering of incident electrons is also pairs at the ridge of the 2D distribution. For a given Etot, essential for semiconductors and metals. Thus, multiple the energy E1 of one of the electrons takes the highest scattering plays a major part in (e, 2e) spectroscopy of value if the energy E2 of the other electron is the lowest solid surface. and vice versa, since E2 = Etot Ð E1. Pairs with such combinations of the electron energies lie at both edges of the function of energy distribution among the elec- ACKNOWLEDGMENTS trons. These pairs disappear as soon as the energy of This work was supported by the State Program one of the electrons exceeds the excitation threshold of “Integration” [project no. A 0151 (0326.37)] and by the the valence electron. In this case, the steps at the edges Russian Foundation for Basic Research (project of the distribution function arise (Fig. 3d). When the no. 99-02-16769). energies of both electrons (E1 and E2 = Etot Ð E1) exceed E , the dip at the center of the distribution appears The authors thank the administration of the Max g Planck Institute (Halle, Germany) for permission to (Fig. 3g). This dip is due to the fact that both electrons publish experimental results obtained at the institute. have an energy that is sufficient for the excitation of the valence electron through the energy gap. The probability of this pair disappearing from the distribution may REFERENCES increase twice compared with the case when only one electron has a high energy. It is worth noting that the 1. O. M. Artamonov, Zh. Tekh. Fiz. 55, 1190 (1985) [Sov. energy distribution functions for electrons in a pair Phys. Tech. Phys. 30, 681 (1985)]. have the fine structure, which is most pronounced in 2. J. Kirschner, O. M. Artamonov, and S. N. Samarin, Phys. Fig. 3d. A reason for this structure may be that the Rev. Lett. 75, 2424 (1995).

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1184 ARTAMONOV, SAMARIN

3. O. M. Artamonov, S. N. Samarin, and J. Kirschner, Phys. 9. J. Berakdar and M. P. Das, Phys. Rev. A 56, 1403 (1997). Rev. B 51, 2491 (1995). 10. J. Berakdar, S. Samarin, H. Herrmann, and J. Kirschner, 4. O. M. Artamonov, S. N. Samarin, and J. Kirschner, Appl. Phys. Rev. Lett. 81, 3535 (1998). Phys. A: Mater. Sci. Process. A65, 535 (1997). 11. L. D. Landau and E. M. Lifshitz, Course of Theoretical 5. S. Samarin, G. Herrmann, H. Schwabe, and O. Arta- Physics, Vol. 3: Quantum Mechanics: Non-Relativistic monov, J. Electron Spectrosc. Relat. Phenom. 96, 61 Theory (Fizmatgiz, Moscow, 1963; Pergamon, New (1998). York, 1977). 6. R. Feder, H. Gollisch, D. Meinert, et al., Phys. Rev. B 58, 12. S. C. Erwin and C. C. Lin, J. Phys. C 21, 4285 (1988). 16 418 (1998). 13. W. Pong, D. Pondyal, and D. Brandt, J. Electron Spec- 7. S. Samarin, J. Berakdar, R. Herrmann, et al., J. Phys. IV trosc. Relat. Phenom. 21, 261 (1980). 9 (Pr6), 137 (1999). 8. H. Gollisch, D. Meinert, Yi. Xiao, and R. Feder, Solid State Commun. 102, 317 (1997). Translated by V. Isaakyan

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EXPERIMENTAL INSTRUMENTS AND TECHNIQUES

High Electrical Conductivity of Trotyl Detonation Products S. D. Gilev and A. M. Trubachev Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, pr. Akademika Lavrent’eva 15, Novosibirsk, 630090 Russia e-mail: [email protected] Received January 9, 2001

Abstract—A new measurement scheme makes it possible to study the conductivity of detonation products of condensed explosives with a time resolution of about 10 ns. Experiments with cast trotyl show that conduction under detonation is a complex phenomenon associated with the chemical reaction zone and the expansion of the reaction products. The time variation of the electrical conductivity has a sharp peak (≈250 ΩÐ1 cmÐ1) and a plateau (≈35 ΩÐ1 cmÐ1). The peak corresponds to the highest conductivity value that has been ever observed for the products of chemical explosive detonation. The results support the validity of the contact method for measuring the detonation conductivity of trotyl. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION contacts and connects the conducting area. Such measurements pose some difficulties: (1) The time resolu- The electrical conductivity of detonation products τ ≈ of condensed explosives has been studied over the past tion of the measuring system is L/R (where L is the 40 years [1Ð13]. Interest in detonation conduction is inductance and R is the resistance of the shuntÐexplo- dictated by the fact that the material in a detonation sive circuit), which makes the measurement of very wave takes a specific state, which is a strongly imper- high conductivity difficult [16]; (2) at the initial time fect low-temperature plasma (pressure ≈20 GPa, mass instant, the extended conducting area with the a priori velocity ≈2 km/s, temperature ≈3 × 103 K, density unknown current field configuration is connected to the ≈ 3 ≈ × 22 Ð3 contacts; and (3) the contact resistances and the metalÐ 2 g/cm , and molecule concentration 3 10 cm ). plasma transition layers may affect the measurements. Available theoretical approaches to describing such a complex object are inadequate; therefore, experimental It was the aim of our work to study the conductivity techniques prevail. It has been shown in many experi- of trotyl detonation products using a scheme with a ments that the conductivity of detonation products high time resolution. We used a measuring cell [17], depends on the nature of an explosive and its parame- which to a great extent is free of the disadvantages ters (density, etc.). For many explosives, the conductiv- listed above. ity typically equals σ ≈ 1 ΩÐ1 cmÐ1. A value of σ as high With this cell (Fig. 1), we detected insulator(semi- as 100 ΩÐ1 cmÐ1 has been obtained only once for liquid conductor)Ðmetal transitions in shock waves [17Ð19]. trotyl [4]. This result has not been confirmed in later The shunt, a thin metal foil, was applied on the sample. publications, and the values of σ for solid trotyl have The shock enters the sample through the shunt. The turned out to be lower by more than one order [1, 2, 8, voltage is detected by electrodes connected to the plane 10Ð12]. Subsequently, however, it has been found that of the shunt. Such a measuring scheme provides the early experiments gave conservative conductivity val- highest time resolution and offers a well-defined cur- ues for trotyl detonation products and σ ≈ 25 ΩÐ1 cmÐ1 rent line configuration. In addition, the effect of the has been obtained [13]. In [13], the time resolution was poor, about 0.5 µs, and the reaction zone was not resolved (the accuracy dropped with increasing conductivity). Interest in detonation conduction has sharp- I ened over the last decade because of the discovery of detonation-produced diamonds [14, 15]. The electrical conductivity method was applied for testing the dia- 1 mond phase [11, 13]. 3 2 In the conventional contact method for measuring V 5 high (σ > 1 ΩÐ1 cmÐ1) electrical conductivity [1], the dc 4 mode is employed. Direct current passes in a circuit Fig. 1. Cell for measuring the conductivity of detonation consisting of a current source, a shunt, and an explosive products of condensed explosives: 1, shunt (metal foil); parallel-connected to the shunt. A detonation wave 2, explosive; 3, voltage electrodes; 4, oscilloscope; and propagating through the explosive reaches the electrical 5, insulator.

1186 GILEV, TRUBACHEV

V V0/V – 1 generated with a plane wave generator of diameter 5 (a) 2.5 75 mm. Its blasting cartridge made of cast trotyl was V0 743 60 mm thick. For this generator, the time spread of 4 2.0 wave appearance over a circle of diameter 50 mm was no more than 40 ns. The plane wave generated propa- 3 V 1.5 gated through the 5-mm-thick insulating plate and 1 entered the trotyl. The electrode spacing was 10 mm. t1 2 t0 1.0 The current value was up to 400 A. The signals were recorded by an S9-27 oscilloscope with a sampling 1 0.5 interval of 10 ns. Figure 2a shows oscillograms for the experiments 0 0 0123 with cast trotyl. The instant the shock enters the trotyl is marked by a small peak due to electromagnetic tran- σ, Ω–1 cm–1 sients in the foil. This peak is a reliable temporal 300 (b) marker and specifies the zero time in the oscillograms. It was found in several experiments that a shock- 743 induced change in the constantan conductivity is negli- 250 gible; hence, it was not taken into account. Estimations of the shock parameters made from the shock adiabat [22] yielded the pressure P = 22.5 GPa. The pressure of stationary detonation in the trotyl measured with a 50 manganin sensor in special experiments was found to be P ≈ 18 GPa. Thus, when the shock wave entered the explosive, it was in the somewhat overcompressed state 0 compared with stationary detonation. 0123 t, µs As follows from the oscillograms, at the instant the shock enters the trotyl, the voltage V starts to decrease. Fig. 2. (a) Experimental voltage vs. time curve for trotyl This means that the conducting state appears in the (thick line, left ordinate) and the same curve after integral sample without any delay. After the early sharp processing (thin line, right ordinate); (b) conductivity profile. decrease, the rate of fall of the voltage changes and the new rate is retained until the shock wave reaches the contact resistances in the electrode region is weak. The insulating plate (high-intensity oscillations in the oscil- time resolution of the circuit is restricted by electro- logram). Figure 2a also shows the processed curve in magnetic transients in the shuntÐconducting medium the (V0/V Ð 1, t) coordinates (V0 is the initial voltage). If system. This restriction, however, can be considerably the conductivity of the material behind the detonation smoothed by analyzing the dynamic skin effect in the front is constant and the electromagnetic transients are shock [20, 21] and using the procedure of restoring the absent, the curve processed must represent a straight conductivity of the conducting region based on infor- line [17]. However, the conductivity of the detonation mation coming from its boundary [19]. In this work, the products is seen to be variable. The early drop of the conductivity measurement method previously used for voltage correlates with the onset of the high conductiv- shock waves is applied to detonation processes. ity state, which exists for the time t0. Then, the conductivity noticeably decreases. At t > t1, the curve processed is linear; hence, the conductivity remains con- EXPERIMENT stant in this time region. Generally, two basic portions Trotyl filled a groove (width 20 mm, depth about where the conductivities greatly differ can be distin- 20 mm, and length 80 mm) in a textolite plate. A 0.1- guished in the curve. mm thick constantan foil was used as a shunt. Its width Experimental results on the conductivity of the cast varied from run to run. An initiating shock wave was trotyl detonation products are summarized in the table.

Table 3 µ µ σ ΩÐ1 Ð1 σ ΩÐ1 Ð1 Run no. as, mm D, 10 , m/s t0, s t1, s 1, cm 2, cm 705 5.5 6.55 ± 0.1 0.14 0.52 180 35 726 2.85 6.69 ± 0.12 ≈0.1 0.42 ≈300 33 727 2.3 6.59 ± 0.1 0.30 0.67 200 56 743 2.9 6.64 ± 0.1 0.15 0.54 240 31

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HIGH ELECTRICAL CONDUCTIVITY OF TROTYL DETONATION PRODUCTS 1187

Here, as is the shunt width, D is the detonation rate, t0 V is the instant of sharp drop of the voltage, t1 is the full 5 742 time of recording the initial portion of the oscillogram σ 4 (Fig. 2a), 1 is the mean conductivity in the range 0Ðt1, 1 and σ is the mean conductivity for t > t . 2 1 3 2 The detonation rate D was found from the instants 3 the initiating shock entered the explosive (the first peak 2 in the oscillogram) and left it, i.e., touched the insulat- 4 ing plate (deviation from the straight line, the onset of 1 3 the oscillations). For a trotyl density of 1.58 g/cm , the 0 rate of stationary detonation has been estimated at 4657 ≈6.87 km/s [22]. The measured value of D was close to t, µs this figure. The discrepancy between the experimentally found and predicted values was the least for run Fig. 3. Time variation of the voltage for the oppositely no. 726 (Ð2.6%) and the greatest for run no. 705 directed detonation wave (continuous line) and the results − of electromagnetic simulation for a system having two ( 4.7%). The delay times of initiation that were deter- regions of various constant conductivities (dashed lines). mined from the above rates are 70 and 120 ns, respec- The conductivity of the detonation products is (1) 10, (2) 25, tively. Since the time delay is small, we can assume that (3) 50, and (4) 100 ΩÐ1 cmÐ1. near-stationary detonation starts in the trotyl when the initiating shock enters the explosive. The graph of (2) is depicted in Fig. 2b. The conduc- The conductivity of the detonation products was tivity profile is intricate: initially, the conductivity is found under the assumption that the expanding con- very high; then, it sharply drops and remains constant ducting zone is connected in parallel to the shunt. In the until the detonation front comes to the insulating wall. electrotechnical approximation, the mean conductivi- σ ≈ × 2 ties for the two time ranges can be determined from the Typical values for the two portions are 1 2.5 10 σ ≈ ΩÐ1 Ð1 formulas and 2 30 cm , respectively. δ Formulas (1) and (2) are valid if two conditions are σ as s 1V 0 met: (1) the skin effect in the detonation products is 1 = ---- ρ------()---------1Ð , a s DuÐ t V negligible and (2) the conductivity profile is stationary. (1) The role of the skin effect can be estimated from the a δ σ s s 1V 0 V 0 ratio of the magnetic field diffusion time in a conductor 2 = ---- ρ------()---------Ð ------, a s DuÐ t V V 1 to the time of wave propagation. For shock waves, the µ σ 2 σ δ parameter R = 0 (D Ð u) t is estimated (where is the where a is the width of the explosive, s is the shunt ρ electrical conductivity of the conductor) [20]. If R < 1, thickness, s is the shunt resistivity, and u is the mass the magnetic field changes due to diffusion and the skin velocity at the ChapmanÐJouguet point. In (1), V1 is the effect is insignificant. Such an approach can naturally voltage at the time instant t1, from which the second be extended for detonation waves (if the mass velocity portion of the conductivity curve starts. It is assumed in a Taylor wave is not too high). In our runs, R < 0.4; that the resistance of the first conducting zone remains therefore, we can ignore the electromagnetic nonuni- unchanged (which holds for the stationary conductivity formity in the first approximation and use the electro- profile). technical model. In run no. 727, the pressure of the initiating shock in A number of factors indicate that the conductivity the trotyl was 16.3 GPa, which is lower than the pres- profile is stationary. First, the experimentally found det- sure at the ChapmanÐJouguet point for stationary deto- onation rate is close to the expected one. Second, the nation. At the initial time instant, such a detonation is voltage taken from the electrodes monotonically undercompressed. As follows from the table, the values decreases as the detonation wave propagates. If the σ of t0 and t1 are larger than in the other runs, while D, 1, peak conductivity were due to transients during the for- σ and 2 are close to the associated values for the other mation of the detonation wave, the voltage records runs. This implies that the detonation wave approaches would have had a number of singularities. For example, the stationary mode. the disappearance of the high-conductivity portion would have led to an increase in the voltage. Third, The conductivity profile for the detonation products experiments with various strengths of the initiating is obtained by using the differential processing tech- shock wave (the pressure is smaller or greater than the nique [23]; in our case, it yields ChapmanÐJouguet value) give similar results. The con- a δ V dV ductivity profile for run no. 727 is similar to that shown σ()t = Ð----s ------s ------0 ------. (2) in Fig. 2b (the only difference is the more extended ρ ()2 a s DuÐ V dt region of high conductivity in the former case). Fourth,

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1188 GILEV, TRUBACHEV the constancy of the conductivity in the second time DISCUSSION interval agrees with the results obtained for greater- diameter coaxial cells [13]. Our experiments with the The use of the new measuring scheme to study det- 38-mm-diameter coaxial (the total diameter of the onation processes has lead to several intriguing results. explosive charge is 50 mm) have shown that, in the First, the detonation conductivity in the trotyl exhibits absence of side unloading, the conductivity of the trotyl complex behavior. The initial sharp peak changes to the detonation product is constant and close to the maximal lower plateau. Then, the detonation conductivity turned value obtained in [13]. out to be considerably higher than that obtained previously. The value σ ≈ 2.5 × 102 ΩÐ1 cmÐ1 appears to be Certainly, the above considerations are indirect. 1 Experimental methods for profiling the trotyl conduc- the highest among those which have ever been obtained tivity in a strictly stationary detonation wave are lack- for explosive detonation products. Our data agree with ing. Here, difficulties are of a fundamental character, early results for liquid trotyl [4], where the conductivity σ ≈ 2 ΩÐ1 Ð1 because, in experiments, the requirement of detonation was an increasing function ( max 10 cm ) and stationarity comes in conflict with the requirement that the measurement period was no more than 20 ns. In our edge effects of current spread be absent. For detonation experiments, the observation period was much longer, to be strictly stationary, the explosive charge must be which allowed us to record the entire conductivity pro- σ very long. As such a wave propagates, the extended file. The conductivity value on the plateau, 2, exceeds conducting zone is connected to the contacts at some data in [1, 2, 8, 10Ð12] roughly by one order of magni- time instant. The current field configuration depends on tude and coincides (by one order of magnitude) with the the conductivity profile, which is a priori unknown. maximal value in [13]. The cell employed in [13] did not allow the detection of the conductivity peak near Yet, we gained qualitative information on the conductivity profile in the case of stationary detonation. To the detonation front because of the worse time resolu- this end, we used a measuring cell similar to that tion and the influence of the edge effects. depicted in Fig. 1 where a detonation wave moved in The reason for the high conductivity of trotyl the opposite direction (see the oscillogram in Fig. 3). remains to be clarified. First of all, the fact stands out The wave propagated in the trotyl and reached the shunt that the time of peak existence (≈150 ns) roughly equals contacting with the plastic insulator (Getinaks). At this the time of chemical reaction in trotyl (100Ð200 ns) time instant, the current starts to diffuse from the shunt [22]. The high conductivity in the peak seems to reflect to the conducting region of a finite thickness. The tran- physical processes taking place in the chemical reac- sient process takes some time that depends on the thick- tion zone. In particular, it indicates that the reaction ness of this region and the conductivity of the detona- produces highly conducting particles that become tion products. If the conductivity of the material behind bonded or spatially separated after a short time. These the detonation front is constant, the voltage must monotonically decrease with time. Figure 3 shows a family of might be particles of carbon heated to high tempera- model curves obtained from the analysis of electromag- tures that is released during an exothermal reaction. netic diffusion in a shuntÐdetonation product system Thermodynamic analysis of the detonation products having two regions of different conductivities. The has shown that the amount of free carbon in them is curves were taken under the assumption that the con- about 0.5 g/cm3 [24]. The carbon particles grow in the ductivity behind the detonation front is constant. As reaction zone, forming porous fractal structures. The soon as the detonation wave leaves the shunt (arrow in mechanism of macroconduction may be contact [4]. Fig. 3), the voltage is much smaller than the expected The resulting network of conducting carbon particles is one but then its variation qualitatively agrees with the responsible for the high conductivity in the peak. The model dependence. Such a behavior can be explained expansion of the detonation products in the reaction as follows. First, the narrow high-conductivity zone zone (which is accompanied by additional transforma- related to the detonation front is connected to the shunt. tions and cooling of the particles) causes the conductiv- Subsequently, this zone disappears and the voltage is ity to drop. Other mechanisms that have been invoked controlled by current diffusion into the extended con- to explain detonation conduction (thermal ionization, stant-conductivity region. The truth of the detonation chemical ionization, thermionic emission, and dissoci- product conductivity value thus obtained is open to ation of resulting water [1, 7Ð9]) cannot explain such question (the physical state of the products reflected high values of the conductivity. from the wall may change, and the accuracy of conductivity recovery is poor). Yet, the shape of the curves The total resistance of the trotyl detonation products points to the presence of the high-conductivity zone is ≈0.01 Ω, which makes possible the use of detonation that spatially contacts with the detonation front. Thus, conduction in high-current physical experiments. the conductivity peak moves in space with the detona- Because of their high density, the detonation products tion front rather than resulting from transients at the offer good dielectric strength, which is retained during insulatorÐexplosive interface. a time specified by rarefaction gas dynamics.

TECHNICAL PHYSICS Vol. 46 No. 9 2001 HIGH ELECTRICAL CONDUCTIVITY OF TROTYL DETONATION PRODUCTS 1189

CONCLUSION 7. V. V. Yakushev and A. N. Dremin, Dokl. Akad. Nauk SSSR 221, 1143 (1975). A new conductivity measuring scheme improves the 8. A. G. Antipenko, A. N. Dremin, and V. V. Yakushev, time resolution roughly by one order of magnitude, Dokl. Akad. Nauk SSSR 225, 1086 (1975). thus enabling the detection of new phenomena related to detonation conduction in the widely used explosive. 9. A. P. Ershov, Fiz. Goreniya Vzryva 11, 938 (1975). The conductivity of the detonation products is a com- 10. K. Tanaka, Report on 5th International Colloquium of Gasdynamics of Explosions and Reactive Systems plex function of time, which reflects the combination of (Bourges, France, 1975). various processes taking place in the chemical reaction 11. A. M. Staver, A. P. Ershov, and A. I. Lyamkin, Fiz. zone. The high sensitivity of the new method can be Goreniya Vzryva 20 (3), 79 (1984). useful in studying fast physicochemical conversions. 12. A. I. El’kind and F. N. Gusar, Fiz. Goreniya Vzryva 22 The effect of high conductivity concentrated in a nar- (5), 144 (1986). row layer and transferred with a detonation rate may 13. A. P. Ershov, N. P. Satonkina, O. A. Dibirov, et al., Fiz. have a number of high-energy applications. Goreniya Vzryva 36 (5), 97 (2000). 14. A. I. Lyamkin, E. A. Petrov, A. P. Ershov, et al., Dokl. ACKNOWLEDGMENTS Akad. Nauk SSSR 302, 611 (1988) [Sov. Phys. Dokl. 33, 705 (1988)]. The authors thank A.P. Ershov for the valuable dis- 15. Greiner N. Roy, D. S. Phillips, J. D. Johnson, et al., cussion. Nature 333, 440 (1988). This work was supported by the Russian Foundation 16. V. V. Yakushev, Fiz. Goreniya Vzryva 14 (2), 3 (1978). for Basic Research (grant no. 99-02-16807). 17. S. D. Gilev and A. M. Trubachev, Prikl. Mekh. Tekh. Fiz., No. 6, 61 (1988). 18. S. D. Gilev and A. M. Trubachev, Phys. Status Solidi B REFERENCES 211, 379 (1999). 1. A. A. Brish, M. S. Tarasov, and V. A. Tsukerman, Zh. 19. S. D. Gilev and T. Yu. Mikhaœlova, J. Phys. IV 5, C3-211 Éksp. Teor. Fiz. 37, 1543 (1960) [Sov. Phys. JETP 10, (1997). 1095 (1960)]. 20. S. D. Gilev and T. Yu. Mikhaœlova, Zh. Tekh. Fiz. 66 (5), 2. R. Shall and K. Vollrath, in Les ondes de d’etonation 1 (1996) [Tech. Phys. 41, 407 (1996)]. (Centre National de la Recherche Scientifique, Paris, 21. S. D. Gilev and T. Yu. Mikhaœlova, Zh. Tekh. Fiz. 66 (10), 1962), pp. 127Ð136. 109 (1996) [Tech. Phys. 41, 1029 (1996)]. 3. R. L. Jameson, S. J. Lukasik, and B. J. Pernick, J. Appl. 22. L. V. Al’tshuler, G. S. Doronin, and V. S. Zhuchenko, Fiz. Phys., Part 1 35, 714 (1964). Goreniya Vzryva 25 (2), 84 (1989). 4. B. Hayes, in Proceedings of the 4th International Sym- 23. L. V. Kuleshova, Fiz. Tverd. Tela (Leningrad) 11, 1085 posium on Detonation (Office of Naval Research, Wash- (1969) [Sov. Phys. Solid State 11, 886 (1969)]. ington, 1967), ACR-126, p. 595. 24. Ch. L. Mader, Numerical Modeling of Detonations 5. A. D. Zinchenko, V. N. Smirnov, and A. A. Chvileva, Fiz. (Univ. of California Press, Berkeley, 1979; Mir, Mos- Goreniya Vzryva 7, 422 (1971). cow, 1985). 6. A. P. Ershov, P. I. Zubkov, and L. A. Luk’yanchikov, Fiz. Goreniya Vzryva 10, 864 (1974). Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 46 No. 9 2001

EXPERIMENTAL INSTRUMENTS AND TECHNIQUES

Device Structures Based on Resonant Tunneling Diodes: A Theoretical Consideration I. I. Abramov and A. V. Korolev Belarussian State University of Informatics and Radioelectronics, ul. Brovki 6, Minsk, 220027 Belarus e-mail: [email protected] Received January 9, 2001

Abstract—Simple device structures incorporating resonant tunneling diodes (RTDs) are considered in terms of electrical models and the ECÐRTSÐNANODEV software suite. It is shown that the structures can be used in multilevel logic, frequency converters, and generators of harmonic, relaxation, and chaotic signals. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION Electrical models for resonant tunneling structures have been considered in detail elsewhere [12Ð16]; we Advanced nanodevices, offering extended function- will only briefly outline them. Within these models, the ality, have been reduced to the point where quantum- IÐV characteristic of an RTD is approximated as size effects become essential [1Ð4]. To date, a number () ()[]()()2 of device structures that integrate several resonant tun- IV = IP V/V P exp a0 ++a1 V/V P a2 V/V P neling diodes (RTDs) [5], several resonant tunneling (){ []()()2 (1) transistors [6], RTDs and heterojunction bipolar tran- + a3 IP exp a4 exp a5 V/V P + a6 V/V P sistors [7], RTDs and selectively doped heterojunction []}() FETs [8], RTDs and CMOS elements [9], etc. have Ð exp a7 V/V P , been created. These functionally integrated structures where I is current, V is voltage, IP is the RTD maximal feature unique properties; however, their physics is tunnel current, VP is the external bias at which the tun- much more complicated compared with the physics of nel current is maximal, and a Ð a are coefficients. the individual constituents. This is because various 0 7 regions and elements may interact with each other, Each of the terms in (1) contributes to the current qualitatively changing the principle of operation. The through the RTD; hence, they can be considered as volt- theoretical analysis of such structures, while difficult, is age-controlled current sources. Thus, the equivalent necessary, since the industry is now facing the onset of circuit for the RTD steady-state operation involves nanoelectronic quantum-effect ICs. three diodes (Fig. 1a). The parameter RN includes the resistance of the RTD passive parts, as well as other The aim of this paper is to theoretically treat several parasitic (under the steady-state conditions) compo- simple RTD-based structures and to illustrate their nents. The RTD equivalent circuit to study transients is wide functionality. given in Fig. 1b. Expression (1) is valid throughout the range of the applied voltage, while RP, LP, and CP describe the RTD dynamic properties at large signals. MODELS Note that RP may differ from RN, although this differ- Experience shows that integrated structures includ- ence is ignored for the small-signal case [15]. The ing quantum-effect devices are today the most conve- parameter CP includes the overall capacitance of the nient to study in terms of electrical models. More vig- RTD active part and the parasitic components. The orous models, for example, those based on a self-con- inductance LP describes dynamic processes in the quan- sistent numerical solution of the Schrödinger and tum well and also includes the parasitic components. Poisson equations, as well as of the kinetic equation for These electrical models are based on present-day the Wigner function [10], are still under development. physical concepts of RTD operation [13, 15, 16]. For Moreover, they are, as a rule, complex and can be simple devices considered in our work, such an applied only to simple quantum-effect devices. Such a approach seems to be more appropriate than using mac- situation radically differs from integrated devices romodels. It has been shown [17] that the latter are the whose operation can be described in terms of the clas- most useful in simulating complex quantum-effect sical diffusionÐdrift models [11]. Therefore, we structures, where physical processes taking place in the dwelled on electrical models in our analysis. individual element are of lesser concern. That the elec-

DEVICE STRUCTURES BASED ON RESONANT TUNNELING DIODES 1191

(a) (b)

RP LP

I(U) EB RN EB

Fig. 1. RTD equivalent circuits for the (a) steady-state and (b) transient cases.

(a) (b)

RN1 CP1 E RTD B1 LP1 RP1 Vout

RS D1 IT R EB out L Rout S D2 EB2 LP2 RP2 Vin

RN2 C E0 P2

Fig. 2. Equivalent circuits for the (a) single-RTD structure and (b) integrated structure with two parallel-connected RTDs. trical models proposed ﬁt available experimental data dently, Kawano et al. [25, 26] have demonstrated that for RTDs more adequately than other known models similar generators can be implemented with an RTD, a has been demonstrated in [12, 13, 15]. For example, the capacitor, and an inductor provided that the diode is fed maximal error in simulating the IÐV characteristic is by a harmonic voltage with the dc component. This cir- typically 10Ð15%, which is a quite reasonable value in cuit is simple but have the inductor; therefore, the view of the strong nonlinearity of the RTD IÐV curve. monolithic (integrated) implementation of the genera- The associated technique for the identiﬁcation of the tor is hardly possible. That is why the discrete version model parameters from experimental data has been of the generator was discussed in [25, 26]. reported in [18, 19]. Let us demonstrate the possibility of implementing In our study, we use the ESÐRTSÐNANODEV [19Ð signal generators of various types with a simple device 22] software suite for analyzing resonant tunneling structure incorporating only an RTD (i.e., without a devices and circuits. This suite, employing both the capacitor and an inductor). Its equivalent circuit is well-known and new electrical models, is a part of the shown in Fig. 2a. A dc bias EB is applied to the input of NANODEV suite for simulating single-electron, reso- the structure, and the output voltage Vout is taken from nant-tunneling, and quantum-interference nanodevices the resistor R . The equivalent circuit of the RTD [23, 24]. out

ANALYSIS Values of the equivalent circuit parameters Ω Ω We have reported [15, 16] that an RTD may serve as EB, V RP, LP, nH CP, pF Rout, a core for generators of harmonic, relaxation, gate relaxation, etc. signals. In these applications, a dc volt- Figs. 3a, 3c 0.575 0.17 0.05 15 0.1 age is applied to the diode. Concurrently and indepen- Figs. 3b, 3d 0.575 0.17 5 2 0.1

TECHNICAL PHYSICS Vol. 46 No. 9 2001

1192 ABRAMOV, KOROLEV

0.20 (a) 0.10 (b) 0.15 0.08 p

V 0.10 / 0.05 0.06 out V 0 0.04 –0.05 –0.10 0.02 0246810 0 20406080100 τ τ

–2 (c) –2 (d) –4 –4

), dB –6 τ

(1/ –6 S –8 –10 –8 0 0.5 1.0 1.5 2.0 2.5 0 0.1 0.2 0.3 1/τ 1/τ

τ Ð10 Fig. 3. Results of simulation of the simple single-RTD circuit ( = t/t0, t0 = 10 s): (a, b) harmonic and relaxation oscillations, respectively; (c, d) their respective spectra. alone is depicted in Fig. 1b. For the IÐV characteristic most considerable difference is the coupling character. approximated by model (1), we used experimental data In [5], two RTDs are connected through a resistance for structure A in [27]. This was done intentionally to only. In the circuit shown in Fig. 2b, the diodes are con- improve the validity of the theoretical consideration. nected via the resistor RS and inductor LS. Such a con- The results of simulation of the structure in Fig. 2a nection can be viewed as interaction of two parallel- are presented in Fig. 3 for almost harmonic (Fig. 3a) connected resonant tunneling elements in the integrated and relaxation (Fig. 3b) output signals. The associated structure through their active regions. Next, our values spectral characteristics of the signals are illustrated in of RS are much lower than in [5]. The presence of RN1 Figs. 3c and 3d. The table lists the values of the param- and RN2 does not considerably affect the operation of eters of the equivalent circuit for the signals generated. the circuit. The spread of the parameter values is seen to be small. As follows from the table, the generators of various The analytical IÐV curve of the integrated structure signals can be implemented with the simple structure (Fig. 4) establishes that the structure can be used as a including an RTD and a load resistor. The parameters of multilevel logic gate. Numerical experiments indicate that the shape of this double-peak curve strongly the equivalent circuit, namely, RP, LP, CP, and Rout, can be varied by varying the geometry of the passive part of depends on the bias EB2. This is because the partial IÐV the integrated structure, by additionally doping some of its regions, by varying the thickness of the structure layers, etc., since the design, process, and physical param- I, A eters of the RTD specify the values of the circuit param- 0.16 eters listed above. The device performance can easily 0.14 be improved by slightly modifying the fabrication tech- 0.12 nology. One can use empirical formulas for RP, LP, CP, 0.10 and Rout that are derived at the design stage. It should be noted that the parameter values listed in the table are 0.08 typical of actual RTDs [27Ð32]. 0.06 Consider a simple integrated structure that includes 0.04 two RTDs and is described by the equivalent circuit in Fig. 2b. We will show that this structure can be used in 0.02 multilevel logic, as a frequency converter, or as a signal generator, depending on applied biases and equivalent 0 0.2 0.4 0.6 0.8 circuit parameters. V, V First of all, we note that this structure is similar to Fig. 4. IÐV curve for the logic gate consisting of two paral- that experimentally studied in [5] at T = 100 K. The lel-connected RTDs.

TECHNICAL PHYSICS Vol. 46 No. 9 2001 DEVICE STRUCTURES BASED ON RESONANT TUNNELING DIODES 1193

I(t), A since these parameters specify feedback between the 0 (a) RTDs. The values of the parameters used in the numerical calculations were EB1 = 0, EB2 = 0.35 V, RP1 = RP2 = –0.02 Ω 0.17 , LP1 = LP2 = 1.01 nH, CP1 = CP2 = 2.96 pF, RS = 1.0 Ω, and L = 1.01 nH. The partial IÐV curves are sim- –0.04 S ilar to that of structure A in [27]. The resistances RN1 –0.06 and RN2 were not taken into consideration in the calculations. –0.08 The structure under study can also serve as a fre- –0.10 quency converter. Figure 5 demonstrates the results of simulation for the parameter values listed above when 0 50 100 150 200 a harmonic signal µ t, s () V in = Aftsin , (2) S( f ), dB 0 was applied to the input. Here A = 0.35 V and f = (b) 12.5 kHz. E0 was taken to be equal to 0.3 V. The current shown in Fig. 5a passes through the –1 resistor Rout. The corresponding spectrum of the output signal is shown in Fig. 5b. It is obvious that this struc- –2 ture is promising for a frequency multiplier. Finally, it remains to demonstrate that, with a dc –3 bias (Vin = 0), this integrated structure can be used as a signal generator if the parameters of the RTDs slightly –4 differ. In this case, Rout = 0 and the current IT (Fig. 2b) 0 1020304050 is taken as an output. The IÐV curve of one RTD corre- f, kHz sponds to the structure A in [27], and the IÐV curve of the other differs by approximately 2%. Figure 6 shows Fig. 5. Results of simulation of the double-RTD structure: the results of simulation for the circuit in Fig. 2b with (a) output current and (b) its spectrum. Ω Ω RP1 = 0.17 , RP2 = 0.167 , LP1 = 0.01 nH, LP2 = Ω 0.0098 nH, CP1 = 0.5 pF, CP2 = 0.49 pF, RS = 8.012 , characteristics of the RTDs virtually superpose when and LS = 0.1 nH. EB2 is below some critical value. In addition, the overall From Fig. 6a, it follows that the output signal is IÐV characteristic of the structure depends on RS and LS, nearly harmonic in this case (its spectrum is depicted in

1.4 (a) 1.5 (c)

) 1.0 1.2 τ ( 0.9

max 0.6 0.6 I / T

I 0.3 0.2 0 –0.2 –0.3 0 0.2 0.40.6 0.8 1.0 0123 4 τ τ

–2 (b) –2.5 (d) –4 –3.5 –4.5 ), dB

τ –6 –5.5 (1/

S –8 –6.5 –7.5 –10 –8.5 0 5 10 15 20 0 48 12 16 20 1/τ 1/τ

τ Ð10 ≈ Fig. 6. Results of simulation of the double-RTD structure for = t/t0, t0 = 10 s, Imax 0.2 A, E0 = 0.55 V, and EB1 = 0. (a, c) Oscillations and (b, d) their spectra. EB2 = (a, b) 0 and (c, d) 0.1 V.

TECHNICAL PHYSICS Vol. 46 No. 9 2001 1194 ABRAMOV, KOROLEV

Fig. 6b). Figure 6c shows the almost chaotic output sig- 4. Nanostructure Physics and Fabrications, Ed. by nal (its associated spectral characteristic is given in M. A. Reed and W. P. Kirk (Academic, Boston, 1989). Fig. 6d). Such a large discrepancy between the signals 5. S. Sen, F. Capasso, A. Y. Cho, and D. Sivco, IEEE Trans. was obtained with a slight difference in only one Electron Devices 34, 2185 (1987). parameter: in Fig. 6a, EB2 = 0, while in Fig. 6c, EB2 = 6. F. Capasso, S. Sen, F. Beltram, et al., IEEE Trans. Elec- 0.1 V. tron Devices 36, 2065 (1989). It is worth noting that the equivalent circuit in 7. C. E. Chang, P. M. Asbeck, K.-C. Wang, and E. R. Brown, IEEE Trans. Electron Devices 40, 685 Fig. 2b can be used as a basis for electrical models for (1993). a number of new structures [33Ð38] that exploit the 8. M. Kawashima, H. Hayashi, H. Fukuyama, et al., Jpn. J. principle of coherent charge carrier transport with self- Appl. Phys. 39, 2468 (2000). organization [39]. These structures have been theoretically treated in [33Ð39], and early attempts to analyze 9. J. I. Bergman, J. Chang, Y. Joo, et al., IEEE Electron Device Lett. 20 (3), 119 (1999). them in terms of electrical models have been made in [14, 40]. In these structures, the partial IÐV curves of 10. B. A. Biegel and J. D. Plummer, Phys. Rev. B 54, 8070 (1996). the diodes in Fig. 2b may differ much greater. Further- more, the resistances R , R and the inductances L , 11. I. I. Abramov, Modeling of Physical Processes in Silicon P1 P2 P1 Integrated Circuit Elements (Beloruss. Gos. Univ., LP2 may also substantially differ, so that there is no need Minsk, 1999). for the voltage sources EB1 and EB2. The extended func- 12. I. I. Abramov, A. L. Danilyuk, A. V. Korolev, and tionality of the new devices is due to more complex E. A. Patent, in Proceedings of the 8th International physical processes in the nanostructure rather than to Crimean Microwave Conference, Sevastopol, 1998, the spatial separation of two resonant tunneling regions p. 599. as in the structure considered before. 13. I. I. Abramov, A. L. Danilyuk, and A. V. Korolev, Izv. Beloruss. Inzh. Akad., No. 2 (6), 43 (1998). 14. I. I. Abramov, A. L. Danilyuk, and A. V. Korolev, Izv. CONCLUSION Beloruss. Inzh. Akad., No. 1 (7), 119 (1999). Using original electrical models of an RTD and the 15. I. I. Abramov, A. L. Danilyuk, and A. V. Korolev, Izv. ECÐRTSÐNANODEV software suite, we analyzed sev- Vyssh. Uchebn. Zaved., Radioélektron. 43 (3), 59 eral simple RTD-based integrated devices. In spite of (2000). the simplicity of the devices, they were shown to be 16. I. I. Abramov, A. L. Danilyuk, and A. V. Korolev, Vestsi capable of performing a variety of complex functions, Akad. Navuk Belarusi, Ser. Fiz.-Tékh. Navuk, No. 2, 75 depending on the applied (usually dc) biases and the (2000). equivalent circuit parameters. Specifically, they can 17. S. Mohan, J. P. Sun, P. Mazumber, and G. I. Haddad, serve as multilevel logic gates; frequency converters; IEEE Trans. Comput.-Aided Des. 14, 653 (1995). and generators of harmonic, relaxation, and chaotic sig- 18. I. I. Abramov and A. V. Korolev, in Proceedings of the nals. With a slight difference in the values of the equiv- 7th International Scientific and Technical Conference alent circuit parameters, the integrated nanostructures “Topical Problems of Solid-State Electronics and Micro- may provide a basis for analog and digital circuits. The electronics,” Divnomorskoe, 2000, Vol. 2, p. 16. feasibility of such an approach is also supported by the 19. I. I. Abramov, I. A. Goncharenko, and A. V. Korolev, in fact that the parameter values used in this study are typ- Proceedings of the 10th International Crimean Micro- ical of actual RTDs [27Ð32]. wave Conference, Sevastopol, 2000, p. 418. 20. I. I. Abramov, I. A. Goncharenko, A. L. Danilyuk, and A. V. Korolev, in Proceedings of the 9th International ACKNOWLEDGMENTS Crimean Microwave Conference, Sevastopol, 1999, p. 296. This work was partially supported by the Belarus- 21. I. I. Abramov, Yu. A. Berashevich, I. V. Sheremet, and sian Research Programs “Informatics,” “Low-Dimen- I. A. Yakubovskiœ, Izv. Vyssh. Uchebn. Zaved., sional Structures,” and “Nanoelectronics.” Radioélektron. 42 (2), 46 (1999). 22. A. V. Korolev and I. I. Abramov, in Proceedings of the 7th International Scientific and Technical Conference REFERENCES “Topical Problems of Solid-State Electronics and Micro- 1. Zh. I. Alferov, Fiz. Tekh. Poluprovodn. (St. Petersburg) electronics,” Divnomorskoe, 2000, Vol. 2, p. 13. 32, 3 (1998) [Semiconductors 32, 1 (1998)]. 23. I. I. Abramov, I. A. Goncharenko, E. G. Novik, and 2. Resonant Tunneling in Semiconductors: Physics and I. V. Sheremet, in Proceedings of the 6th International Applications, Ed. by L. L. Chang, E. E. Mendez, and Crimean Microwave Conference, Sevastopol, 1996, C. Tejedor (Plenum, New York, 1991), NATO ASI Ser., p. 294. Ser. B 277 (1991). 24. I. I. Abramov and E. G. Novik, Numerical Simulation of 3. Single Charge Tunneling: Coulomb Blockade Phenom- Metallic Single-Electron Transistors (Bestprint, Minsk, ena in Nanostructures, Ed. by H. Grabert and M. H. De- 2000). voret (Plenum, New York, 1992), NATO ASI Ser., Ser. B 25. Y. Kawano, Sh. Kishimoto, K. Maezawa, and 294 (1992). T. Mizutani, Jpn. J. Appl. Phys. 38, L1321 (1999).

TECHNICAL PHYSICS Vol. 46 No. 9 2001 DEVICE STRUCTURES BASED ON RESONANT TUNNELING DIODES 1195

26. Y. Kawano, Sh. Kishimoto, K. Maezawa, and 35. I. I. Abramov and A. L. Danilyuk, Vestsi Akad. Navuk T. Mizutani, Jpn. J. Appl. Phys. 39, 3334 (2000). Belarusi, Ser. Fiz.-Tékh. Navuk, No. 3, 64 (1997). 27. J. M. Gering, D. A. Crim, D. G. Morgan, et al., J. Appl. 36. I. I. Abramov and A. L. Danilyuk, Zh. Tekh. Fiz. 68 (12), Phys. 61, 271 (1987). 93 (1998) [Tech. Phys. 43, 1485 (1998)]. 28. E. R. Brown, C. D. Parker, and T. C. L. G. Sollner, Appl. 37. I. I. Abramov and A. L. Danilyuk, Dokl. Akad. Nauk Phys. Lett. 54, 934 (1989). Belarusi 42 (5), 55 (1998). 29. C. Y. Huang, J. E. Morris, and Y. K. Su, J. Appl. Phys. 82, 38. I. I. Abramov and A. L. Danilyuk, in Proceedings of the 2690 (1997). 8th International Crimean Microwave Conference, Sev- 30. K. J. Gan, Y. K. Su, and R. L. Wang, J. Appl. Phys. 81, astopol, 1998, p. 602. 6825 (1997). 39. I. I. Abramov and A. L. Danilyuk, in Proceedings of the 31. K. J. Gan and Y. K. Su, J. Appl. Phys. 82, 5822 (1997). 6th International Crimean Microwave Conference, Sev- 32. K. J. Gan and Y. K. Su, Jpn. J. Appl. Phys. 36, 6280 astopol, 1996, p. 45. (1997). 40. I. I. Abramov, A. L. Danilyuk, and A. V. Korolev, Vestsi 33. I. I. Abramov and A. L. Danilyuk, Appl. Phys. Lett. 71, Akad. Navuk Belarusi, Ser. Fiz.-Tékh. Navuk, No. 1 665 (1997). (2001). 34. I. I. Abramov and A. L. Danilyuk, in Proceedings of the 7th International Crimean Microwave Conference, Sev- astopol, 1997, p. 379. Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 46 No. 9 2001

Technical Physics, Vol. 46, No. 9, 2001, pp. 1196Ð1198. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 71, No. 9, 2001, pp. 134Ð136. Original Russian Text Copyright © 2001 by Volkolupov, Dovbnya, Zakutin, Krasnogolovets, Reshetnyak, Romas’ko.

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Fast Generation of an Electron Beam in a Magnetron Gun with a Secondary-Emission Metallic Cathode Yu. Ya. Volkolupov, A. N. Dovbnya, V. V. Zakutin, M. A. Krasnogolovets, N. G. Reshetnyak, and V. P. Romas’ko Kharkov State Technical University of Radio Electronics, Kharkov, 61726 Ukraine Received October 30, 2000

Abstract—The formation of an electron layer and the generation of an electron beam in magnetron guns where secondary emission is triggered by nanosecond pulses are studied. In the guns with small cross sizes, hollow electron beams with an outer diameter of 3Ð6 mm are generated. The beam current is 1Ð2 A, and the cathode voltage is 5Ð7 kV. Results obtained indicate that the generation of nanosecond beam-current pulses is a possibility. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION beam size was measured from its image on an X-ray Recent years have seen extensive research on cold- film and on a molybdenum foil. The cathode and the cathode electron sources that employ secondary emis- anode of the magnetron gun were made of copper and sion in crossed electric and magnetic fields [1–4]. stainless steel, respectively. The anode length was Among their merits are long operating time, high cur- 120 mm. The gun was placed in vacuum chamber 3, rent density, simple design, etc. Therefore, they could which was evacuated to a pressure of ≤10Ð6 torr. make emitters for long-lived high-power microwave generators [5] or fast high-voltage devices [6]. In this When secondary emission is initiated by nanosec- work, we studied the fast (within 1Ð10 ns) formation of ond pulses, one must reckon with distortions of their the space charge and the temporal stability of beam shapes in transmitting lines. In our setup, the initiating generation in magnetron guns. pulses pass through the coax, high-voltage insulator, vacuum transmission line, and the anode fixture. To determine the actual shape of an initiating nanopulse, EXPERIMENTAL SETUP we measured its parameters between the anode and the Experiments to generate electron beams were con- cathode with regard for parasitic inductances and ducted with the setup shown in Fig. 1. A magnetron gun capacitances. Figure 2 shows the time variation of the is driven by modulator 1, which produces 4- to 100-kV pulse shape for pulse durations of ≈2 and ≈6 ns. The µ negative pulses Uc of width 2Ð10 s and repetition rate waveforms were recorded by an I2-7 oscilloscope 10Ð50 Hz. The pulses are applied to cathode 5, whereas (transmission band ≈3000 MHz). It is seen that the anode 6 is grounded via resistor R3. Secondary emis- shorter pulse almost retains its shape (the fall time is sion takes place during the trailing edge of a voltage ≈2 ns), while the longer one is distorted: its fall time pulse. Two techniques to initiate secondary emission increased to ≈11 ns. were employed. In the first case, two pulse generators 2 were used: the former with a voltage pulse amplitude U = 2Ð15 kV and a trailing edge duration τ ≈ 70 ns; the 5 6 Ω 4 latter with U = 3.5 kV across a 50- load resistor and a 7 duration of the leading and trailing edges of ≈1 ns. In 3 the other case, secondary emission was initiated during the 0.6-µs-wide trailing edge of a specially generated overshoot [5]. This allowed us to vary the trailing edge R4 duration between 2 and 600 ns and the voltage pulse steepness, between 20 and 1200 kV/µs. A magnetic R1 R2 field of strength H ≤ 3000 Oe was generated by solenoid 4. The beam current and size were measured R3 180 mm away from the anode plane with Faraday cup 7 2 ≈ 1 ( 40-cm-long coax) and resistor R4 of value equal to the wave impedance of the coax (18 Ω). The cathode voltage was measured with voltage divider R1R2. The Fig. 1. Experimental setup.

FAST GENERATION OF AN ELECTRON BEAM 1197

RESULTS AND DISCUSSION 0 6 12 18 t, ns Secondary emission was triggered under various conditions. With the fall time of the initiating pulses equal to ≈0.6 µs and the pulse steepness to 20Ð 2 50 kV/µs, secondary emission and beam generation 1 2 started 100Ð500 ns (depending on experimental conditions) after the beginning of the voltage pulse fall. For such a small steepness, the number of primary electrons U, kV must be signiﬁcant, since only a minor part of them acquires the energy necessary to initiate secondary Fig. 2. Waveforms of the initiating pulse between the cath- emission. Therefore, the accumulation of primary elec- ode and the anode. The pulse duration is (1) 2 and (2) 6 ns. trons with the necessary energy takes a long time and is of statistical character. The time spread may be as high as several tens of nanoseconds. Accordingly, the onset U, kV 3 of current pulse generation varies in time and the elec- 0 trons in the beam are distributed in energy. On the other hand, the duration of the leading edge of a beam pulse –2 depends on the steepness of an initiating pulse; in our experiments, the duration of the leading edge of a beam –4 pulse was several tens of nanoseconds. To improve the temporal stability, cut the beam pulse duration, and –6 2 decrease the energy spread of the beam electrons, one 1 should make the initiating pulse steeper and shorter. –8 With this in mind, we performed experiments where the initiating pulses had either a nanosecond fall time I, A (Fig. 2) or a steepness of greater than 300 kV/µs. Under 3 these conditions, the electron beam arose within 1.5Ð 2.0 ns in the former case and ≈10 ns in the latter case 2 after the pulse amplitude had decayed. This is also 4 observed in Fig. 3, where typical waveforms of cathode 1 voltage pulses (with and without the current beam) and beam current pulses from the Faraday cup are presented 0 5 10 15 (cathode and anode diameters are 2 and 10 mm, respec- µ tively). The temporal instability of the onset of the t, s beam current pulse does not exceed the initiating pulse fall time and is about several nanoseconds. Fig. 3. Waveforms of (4) the beam current pulse from the Faraday cup and the cathode voltage (2) with and (1) with- The results obtained are listed in the table. Here, dc out the beam current. Arrow 3 indicates the instant the initi- and Da are the diameters of the gun cathode and anode, ating pulse is applied. respectively; Uc is the cathode voltage; I is the beam current; U and τ are the amplitude and the fall time of the initiating pulse; and H is the magnetic ﬁeld. During ary emission and beam generation are possible. For the measurements, the cathode voltage amplitude did example, the energy of primary electrons bombarding not exceed the breakdown value for the interelectrode the copper cathode should lie in the range of 0.4Ð spacing (depending on the gun geometry, the maximal 0.6 keV. With such energies, the secondary emission amplitude was between 15 and 25 kV). The table lists factor reaches its maximum value and the process of the minimal amplitude values at which beam genera- secondary emission is very intense. In this case, the tion still takes place. This value is of practical interest, energy of primary electrons ranges up to ≈10% of the because it sets the lower voltage limit at which second- electron energy at the exit from the gun. This is of inter-

Table τ Gun no. dc, mm Da, mm Uc, kV I, A H, Oe U, kV , ns 1 2 7 7 1.9 3000 2.4 2 2 2 10 5 0.8 1900 4 13 3 2 10 7 1.6 2100 3 11 4 3 14 8 2.3 1400 3 14

TECHNICAL PHYSICS Vol. 46 No. 9 2001

1198 VOLKOLUPOV et al. est for the formation of both a stable electron layer and CONCLUSION an electron beam with a considerable energy spread. Thus, our experiments have demonstrated the possi- The study of beam generation vs. pulse steepness bility of forming the space charge and generating an showed that this dependence has a threshold. For beam electron beam in a magnetron gun with a secondary generation to be stable at nanosecond fall times, the emission cathode within ≈2 ns. This enables the syn- steepness must be much greater than at larger times. For chronization of the beam current pulses with a nanosec- example, beam generation in gun no. 1 occurred at a ond accuracy. With a cathode voltage of 5Ð7 kV and a steepness of more than 1000 kV/µs and in gun no. 2, at magnetic ﬁeld strength of 1900–3000 Oe, hollow elec- a steepness above 300 kV/µs. These values are more tron beams with a current of 1Ð2 A and an outer diam- than one order of magnitude higher than those neces- eter of 3Ð6 mm have been obtained. sary for generating the beam at a fall time of 0.6 µs. The same is true for gun no. 3 and gun no. 4. REFERENCES When the gun is triggered by a voltage pulse with a 1. V. M. Lomakin and L. V. Panchenko, Élektron. Tekh., nanosecond fall time, the energy spectrum of the beam Ser. 1, No. 2, 33 (1970). is improved because of a decrease in the amount of fast 2. J. F. Skowron, Proc. IEEE 61 (3), 69 (1973). electrons that are generated during the trailing edge of 3. S. A. Cherenshchikov, Élektron. Tekh., Ser. 1, No. 6, 20 a small-amplitude pulse. (1973). Our experimental data are in satisfactory agreement 4. A. N. Dovbnya, V. V. Zakutin, N. G. Reshetnyak, et al., with results obtained by numerically simulating the for- in Proceedings of the 5th European Particle Accelerator Conference, Ed. by Myers A. Pacheco, R. Rascual, et al. mation of the electron layer in the crossed ﬁelds. Our (Inst. of Physics Publ., Bristol, 1996), Vol. 2, p. 1508. computations, as well as those in [7, 8], indicate that, 5. V. V. Zakutin, A. N. Dovbnya, N. G. Reshetnyak, et al., during the fall time of the initiating voltage pulse (1 or in Proceedings of the 1997 Particle Accelerator Confer- 2 ns), the electrons may gain energy large enough for ence, Vancouver, 1997, Ed. by M. Comyn, M. K. Crad- secondary emission to be initiated. With such short fall dok, M. Reiser, and J. Thomson, Vol. 3, p. 2820. times, the amount of primary electrons is small; how- 6. A. I. Vishnevskiœ, A. I. Soldatenko, and A. I. Shendakov, ever, because of the very high steepness, they have a Izv. Vyssh. Uchebn. Zaved., Radioélektron. 11, 555 chance of gaining the necessary energy for a small (1968). number of gyroperiods, so that intense secondary emis- 7. A. V. Agafonov, V. P. Tarakanov, and V. M. Fedorov, sion takes place. Vopr. At. Nauki Tekh., Ser. Yad.-Fiz. Issled. 1 (2Ð3), 134 In both cases, the cross section of the beams was a (1997). ring with the uniform azimuth distribution of the inten- 8. A. V. Agafonov, V. P. Tarakanov, and V. M. Fedorov, sity. The inner diameter was found to be roughly equal Vopr. At. Nauki Tekh., Ser. Yad.-Fiz. Issled., No. 4, 11 (1999). to the cathode diameter, and the “wall” thickness was 1Ð1.5 mm. For example, for gun no. 2, the beam has outer and inner diameters of 4 and 2 mm, respectively, Translated by V. Isaakyan 180 mm away from the anode plane.

TECHNICAL PHYSICS Vol. 46 No. 9 2001

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Twinning in Bismuth Single Crystals Induced by Thermal Cycling O. M. Ostrikov Sukhoœ State Technical University, Gomel, 246746 Belarus Received November 1, 1999; in ﬁnal form, May 5, 2000

Abstract—The role of twinning in bismuth single crystal plastic deformation under thermal cycling (cooling to 77 K and heating to 373 K) is studied. It is found that, under these conditions, the twinning of the crystals proceeds in several stages. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION surface can be examined metallographically without additional treatment. Under operating conditions, engineering materials are often subjected to thermal cycling. For instance, the The specimens were stressed by thermal cycling: rubbing metallic parts of an internal combustion engine cooling to 77 K in liquid nitrogen and subsequent heat- heat up at its start and cool down when it stops. Thermal ing to 373 K in boiling water. A cooling/heating cycle stresses that occur in the material under these condi- lasted 40 s. tions change its dislocation structure and may cause the Using a PMT-3 optical microscope, we measured breakdown of the material. That is why the study of the the twin dimensions and detected the instant of their effect of thermal cycling and, in particular, the effect of nucleation. the resulting thermal ﬂows on the mechanical properties of solid crystals is of great physical interest. RESULTS AND DISCUSSION In crystals, internal stress relaxation caused by tem- 〈 〉 perature variation does not always happen through dis- The {110} 001 -type twin density on the (111) bis- location glide. In a number of important engineering muth single crystal surface vs. the number n of thermal materials (α-Fe, Fe + 3.5% Si, Ti, AlTi, etc.), the relax- cycles is presented in Fig. 1. We have ation may be due to twinning. Twinning is often associ- N ated with crack nucleation in a crystal [1Ð3]; therefore, ρ = ----, (1) studying twin nucleation under thermal cycling would S help to extend the service life of the materials. where N is the number of twins on the area S of the The object of our work is to elaborate a physical (111) bismuth single crystal surface. model of thermal stress relaxation in bismuth single As is seen from Fig. 1, the curve ρ = ρ(n) has three crystals and to discover the role of twinning in this pro- stages: (1) the incubation stage (0 < n < 25), (2) the cess. stage where the number of the twins grow (25 < n < 200), and (3) the saturation stage (n > 200). EXPERIMENTAL ρ × 105, cm–1 Bismuth is a convenient model material [4Ð6] that is prone to twinning and does not require a large temper- 2 ature drop to initiate twin nucleation. Also, the twinning laws found in this material can be extended to other metals, since twinning dislocations are similar in all crystals: they are Shockley partial dislocations and 1 differ only by magnitude and sense of the Burgers vector. The bismuth single crystals were grown by the Bridgman method from a 5N-purity raw material. × × Specimens measuring 4 5 10 mm were obtained by 050100 150 n cleaving the ingot along the (111) cleavage plane. Because of the layer structure and of the crystals and Fig. 1. Twin density ρ on the (111) single-crystal bismuth pronounced cleavage properties in them, the as-cleaved surface vs. the number n of thermal cycles.

1200 OSTRIKOV

S1/S Thus, at the stage of twin growth in bismuth single crystals subjected to thermal cycling, we can separate 0.2 3 three characteristic steps (Fig. 2).

1. Nucleation of Twins 0.1 2 At this step, the {111 } perfect dislocations are split into partial twinning ones with the formation of the twinÐmother crystal interfaces. As a rule, this process 1 takes place owing to {111 } dislocation clusters formed during the incubation period. Because of the thermal 050100 150 n stresses and difficult dislocation glide along the {111 } Fig. 2. Relative area occupied by twins on the (111) surface system, the stresses in the clusters relax either through vs. the number of thermal cycles. (1Ð3) Twinning steps. the basal slip along (111) or through the generation of {110}〈001〉 twinning dislocations. The latter process is energetically more favorable due to the greater Schmidt At the incubation stage, dislocations collect on the factor and the superposition of the thermal stresses and {111 } planes becomes difficult because the defects those caused by the {111 } dislocations. already formed cause stresses that block the sources of dislocations. This makes the twinning process, the Schmidt factor of which is close to that of dislocation 2. Increasing the Number of Twing and Their Growth glide on the {111 } planes, energetically more favorable. At this step, the number of the twins increases not only because of the thermal and dislocation stresses but Note that twinning takes place for the most part also because of the stresses produced by incoherent when dislocation glide is difficult or impossible. In gen- twin boundaries in the crystal. The twins expand mostly eral, the dislocations cannot glide because of the sym- by the translation of the twinning dislocations along the metry selection rule, the low-temperature limitations, existent twin boundaries. This results in the displace- and limitations due to impurities. In our case, twinning ment of the twin edges and causes the twin growth in is associated with energetically favorable dislocation the direction normal to the boundary. glide on the {111 } planes.

The ratio S1/S, where S1 is the surface area of twins 3. Twinning Saturation and S is the area on the (111) surface, vs. the number n At this step, the twinning pattern on the (111) sur- of thermal cycles is presented in Fig. 2. The value of S1 face stabilizes and the curve S1/S = f(n) saturates was deﬁned as the sum of the products of the lengths because of the pinning of the twin boundaries by Frank and widths of individual twins. The dimensions were dislocations generated in the vicinity of the boundaries. measured on the (111) single-crystal bismuth surface. According to [12], Frank dislocations appear by the ρ ρ The curves S1/S = f(n) and = (n) are of a similar reaction shape, but the interpretation of the results shown in Fig. 2 is different. During the incubation period, the []101 0.92 + []011 3.08 []110 3.08. (2) twins are not detected. The growth of S1 after 25 cycles Here, the subscripts are the ratios of the Burgers vector is accounted for largely by twins of length to 10 µm and µ squared to the lattice parameter squared (that is, the rel- width to 2Ð3 m. Then, the ratio S1/S (Fig. 2) rises ative dislocation energy). In this case, the interacting because the number of the twins grows and they increase in size. The change in their size is due to alter- dislocations lie in the (111) and (111 ) crystallographic nating-sign stresses generated by thermal cycling. The planes and the resultant dislocation, in (001). effect of the alternating-sign stresses on the twinning Thus, like dislocation glide along {111 }, the twin- kinetics in bismuth single crystals has been described in ning process saturates (Fig. 1). Basal slip along (111), [7Ð11]. In those papers, the Bauschinger effect and which is observed at the third step of twinning as well, twin-boundary pinning have been studied. According to succeeds twinning. The thermal stresses here have only [10, 11], the twin boundaries stop moving in the direc- an indirect effect because of the symmetry selection tion normal to them after a number of loading/unload- rule, suppressing the plastic deformation process con- ing cycles. A similar situation occurs upon thermal sidered. In this case, as at the last step of twinning, cycling. The twins with at a length of 20 µm (sometimes 40 µm) and a width of 5 µm cease to grow. This basal slip is initiated by the stresses from the {111 } happens after 200 thermal cycles. dislocation clusters and by the twin boundaries. The

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TWINNING IN BISMUTH SINGLE CRYSTALS 1201

h improves the material plasticity and extends the time to cracking.

CONCLUSION 221 We showed that both twinning and plastic deforma- (111) tion in bismuth single crystals subjected to thermal α cycling develop in stages. L g REFERENCES 1. V. M. Finkel’, A. P. Korolev, A. M. Savel’ev, and V. A. Fedorov, Fiz. Met. Metalloved. 48, 415 (1979). 2. V. A. Fedorov, V. M. Finkel’, and V. P. Plotnikov, in Interaction of Lattice Defects and Properties of Metals Fig. 3. Direction of the elastic forces g of a twin boundary (Tul’sk. Politekh. Inst., Tula, 1979), pp. 143Ð148. in relation to the twin boundary and to the (111) cleavage plane: (1) twin of length L and width h and (2) mother 3. V. M. Finkel’, Physics of Destruction (Metallurgiya, crystal. Moscow, 1970). 4. O. M. Ostrikov, Fiz. Met. Metalloved. 86 (6), 106 (1998). direction of the elastic forces g of a twin boundary in 5. O. M. Ostrikov, Fiz. Met. Metalloved. 87 (1), 94 (1999). reference to the boundary and to the (111) cleavage 6. O. M. Ostrikov, Inzh.-Fiz. Zh. 72, 592 (1999). plane is shown in Fig. 3. The angle α depends on the ° 7. V. I. Bashmakov and V. P. Soldatov, Fiz. Met. Metall- twin length L and width h and is about 10 ; that is, for oved. 16, 768 (1963). the basal slip initiated by the twin boundary, the 8. V. I. Bashmakov and N. G. Yakovenko, Fiz. Met. Metall- Schmidt factor roughly equals 0.2. oved. 26, 606 (1968). It is obvious that the basal slip is also saturated after 9. V. I. Bashmakov and N. G. Yakovenko, Izv. Vyssh. a number of cycles. This process is favored by the twin Uchebn. Zaved., Fiz., No. 1, 48 (1969). boundaries, which serve as stoppers for the basal slip. 10. V. I. Bashmakov, M. E. Bosin, F. F. Lavrent’ev, and This sets conditions for microcracking. I. I. Papirov, Probl. Prochn., No. 1, 80 (1974). Notice that, in materials that are not prone to twin- 11. V. I. Bashmakov, M. E. Bosin, and P. L. Pachomov, Phys. ning, cracking at thermal cycling may appear earlier Status Solidi A 9, 69 (1972). than in those prone to twinning, because the energy is 12. F. F. Lavrent’ev, Fiz. Met. Metalloved. 18, 428 (1964). not spent on twin nucleation and growth. Hence, the tendency to twinning can be considered as a factor that Translated by B. Malyukov

TECHNICAL PHYSICS Vol. 46 No. 9 2001

Technical Physics, Vol. 46, No. 9, 2001, pp. 1202Ð1204. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 71, No. 9, 2001, pp. 140Ð142. Original Russian Text Copyright © 2001 by Volkolupov, Krasnogolovets, Ostrizhnoœ, Chumakov.

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Experiments with Pulsed High-Voltage Discharge in the Atmosphere Yu. Ya. Volkolupov, M. A. Krasnogolovets, M. A. Ostrizhnoœ, and V. I. Chumakov Kharkov State Technical University of Radio Electronics, Kharkov, 61726 Ukraine Received October 10, 2000

Abstract—The characteristics of a spark atmosphere discharge initiated at an interelectrode voltage U = 500 kV and a discharge initiated with a thermally bursting wire are studied. © 2001 MAIK “Nauka/Interperi- odica”.

A high-voltage atmosphere discharge excited with a Figure 1 shows the waveforms of the PVG current. high-voltage (several hundred kilovolts) pulse genera- When the spark is initiated between the point elec- tor produces high-power radiation whose spectrum lies trodes, damped oscillations with a frequency of in the rf and optical bands. The visualization of the dis- ≈800 Hz are observed. The loss ∆ in the equivalent charge helps to determine the characteristics of the oscillatory circuit of the discharge is small, hence, the radiation, to indirectly find the features of the dis- slow damping of the current amplitude oscillations. charge, and to evaluate the optimal parameters of Such a behavior is typical of PVG-initiated discharges experimental facilities. This paper reports the charac- [1, 2]. teristics of a high-voltage atmosphere discharge initiated in the spark gap of the generator at U = 500 kV and The time behavior of the current in the case of the a discharge initiated by a generator-terminating ther- BW is significantly different (Figs. 1b, 1c). The initial mally bursting wire (BW). BW pulses longer than 1 µs pulse of the current is higher, and the oscillations decay were obtained, and the discharge current waveform as a noticeably faster. This behavior is inconsistent with the function of load, as well as the radiation energy density linear model of oscillatory circuit, can be described as a function of BW material, were studied. only in terms of an equivalent circuit including nonlinear resistor and capacitor (see Fig. 1). The effect of the In the experiment, we used a column-type Ark- BW parameters on the current is seen in Figs. 1b and adyevÐMarx pulse voltage generator (PVG) with a 1c. Using a 0.04-mm-diameter tungsten BW instead of 10-stage voltage multiplier [1]. The maximum output a copper one results in a still shorter current pulse and voltage was 600 kV at an initially accumulated energy increases the pulse amplitude. In addition, the dis- of 3.8 kJ. The atmosphere discharge was initiated charge becomes aperiodic. between the point electrodes of the PVG (the breakdown conditions) or by means of the BW with the elec- The PVG discharge is illustrated in Fig. 2. The spark trodes removed. The discharge current was measured develops as a typical single- or multichannel process with a Rogowski loop. The optical radiation of the dis- whose shape and sizes depend on electrode configura- charge was detected in a wide range of 0.3Ð10.2 µm by tion, gap width, and applied voltage [3]. The use of the an IMO-2N device and in individual spectral bands BW stabilizes the shape of the discharge area and with a set of optical filters. makes it possible to significantly extend the discharge

(a) (b) (c)

Fig. 1. Waveforms of the current generated by the PVG discharge: (a) point electrodes, (b) copper BW, and (c) tungsten BW. Vertical and horizontal scale divisions are (a) 2 kA and 10 µs, (b) 5 kA and 5 µs, and (c) 10 kA and 5 µs, respectively.

EXPERIMENTS WITH PULSED HIGH-VOLTAGE DISCHARGE 1203 gap. The interelectrode spacing in Figs. 2a and 2b is ence in the energy of the discharge radiation in the case 25 cm. In Figs. 2c and 2d, the wire is 124 cm long. of BWs having different diameters and made of differ- Photography using optical filters allows one to ent materials shows that parameters of the plasma in the observe the spatial temperature distribution in the dis- discharge channel depend predominantly on the wire charge channel and to estimate the sizes of individual rather than on the atmosphere. temperature regions [4]. As a rule, four temperature The BW energy introduced into the discharge chan- regions of the discharge are distinguished. The highest- nel is estimated from the Joule law temperature region of a diameter no more than 1 mm is ∂Q 1 2 at the center (Figs. 2a, 2c). The plasma coat adjacent to ------∂ = σ--- j , (1) the high-temperature region has a transverse size on the t order of the electrode diameter. The outer tubular shell where Q is the specific energy being released in the coaxial with the high-temperature channel is about 1 wire, σ is the wire conductivity, and j is the current den- mm thick. Some regions of the discharge channel are sity in the wire. surrounded by asymmetric low-temperature shells. The temperature dependence of the metal conduc- When viewed through a low-pass filter, the discharge tivity at temperatures up to the temperature of vaporiza- region appears uniform and the regions with different tion can be approximated by the expression [5] temperatures are not distinguished (Fig. 2d). σ Table 1 lists the linear energy densities of the dis- σ = ------0 -, (2) charge radiation in various frequency bands. The differ- 1 + βQ

(‡) (b)

(e) (f)

Fig. 2. Discharge as observed through the optical filters: (a) spark discharge, ZhZS-17 filter; (b) spark discharge, UFS-8 filter; (c) spark discharge, TS-10 filter; (d) bursting copper wire, KS-17 filter; (e) bursting tungsten wire, NS-10 filter; and (f) bursting tungsten wire, UFS-8 filter.

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1204 VOLKOLUPOV et al.

Table 1 Linear energy density of radiation, mJ/cm PVG Gap width δ, cm without filter UFS-6 filter FS-6 filter KS-17 filter Interelectrode spacing1 25 3.5 1.5 0 0 BW/Cu2 124 4.0 0 0 0 BW/W2 124 7.5 0 1.0 1.5 1 Data taken at a distance of 0.75 m. 2 Data taken at a distance of 1.35 m.

Table 2 Heat of sub- Temperature Integral of Wire length, Wire radius, Conductivity, Energy released Material limation, coefﬁcient current, 1017 m µm 6 Ω Ð1 in the wire, J 10 ( m) 105 J/kg ×109 A2 s/m4 Al 1.25 50 39.2 105 2.15 1.09 1801 Cu 1.25 50 63.3 58.2 1.31 1.95 4164 W 1.25 20 18.2 Ð Ð Ð Ð

σ where 0 is the metal conductivity at the zero tempera- gaps or by a burst of a metal wire is developed. Our ture and β is the temperature coefﬁcient of conduc- results can be used for shortening high-voltage pulses tance. that generate short-pulse radiation and in charged parti- Substituting (2) into (1), we obtain cle accelerators that excite high-power ultrashort pulses. t β β 2 1 + Q = exp σ----- ∫ j dt . (3) 0 REFERENCES 0 With (3), the energy being released in the wire 1. S. M. Smirnov and P. V. Terent’ev, High-Voltage Pulse becomes Generators (Énergiya, Moscow, 1964). 2. A. M. Asner, Stosspannungs-Mestechnik (Springer-Ver- π 2 β π 2 r l1  lag, Berlin, 1974; Énergiya, Moscow, 1979). W1 ==Q r l1 ------β - exp-----σ I Ð, 1 (4) 0 3. J. M. Meek and J. D. Craggs, Electrical Breakdown of Gases (Clarendon, Oxford, 1953; Inostrannaya Liter- where r is the wire radius, l1 is the wire length, and I = atura, Moscow, 1960). ∫t j2 dt is the integral of the current (see, e.g., [5]). 4. V. I. Chumakov, M. A. Ostrizhnoœ, Yu. A. Volkolupov, 0 et al., Radiotekhnika 115 (2000). Table 2 lists the energies introduced into the discharge according to (4). It is seen that the conditions for 5. H. Knoepfel, Pulsed High Magnetic Fields (North-Hol- ≅ land, Amsterdam, 1970; Mir, Moscow, 1972). the wire thermal burst are met: W1 (2Ð3)W2, where W2 is the heat of sublimation of the material [6]. 6. N. N. Stolovich, V. G. Maksimov, and N. S. Minitskaya, Zh. Tekh. Fiz. 44, 2132 (1974) [Sov. Phys. Tech. Phys. Thus, the technique for visualizing high-current 19, 1321 (1975)]. optical discharges can be applied to high-voltage discharges in the atmosphere. A benchmark for investigating an atmospheric plasma generated by a spark in wide Translated by A. Khzmalyan

TECHNICAL PHYSICS Vol. 46 No. 9 2001