∫– the Contributions of Polarizabilities to Four Basis

THEORETICAL AND MATHEMATICAL PHYSICS

The Contributions of Polarizabilities to Four Basis Polarizations of Electromagnetic Media E. N. Bukina and V. M. Dubovik Joint Institute for Nuclear Research, Dubna, Moscow oblast, 141980 Russia e-mail: [email protected] Received February 8, 2000

Abstract—The contributions to four basis densities of polarization distribution for an arbitrary electromagnetic medium are determined. Mixed polarizabilities given by tensors of up to the fourth rank are considered. Speciﬁc physical examples are discussed. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION of a cyclic fragment involved in a complex molecule of a benzene ring C6 (Fig. 1). This has made it possible to Electromagnetic properties of some substances give insight into the interaction between microcrystals (e.g., crystals [1]) are described predominantly by the of aromatic compounds and an alternating magnetic distributions of dipoles. Multipole expansion of the field [14]. Such an extraordinary type of electromag- density of such a distribution differs from expansion of netic induction is known as aromagnetism. the current of charges and leads to unusual interaction of dipole systems with external fields. Comprehensive These advances, as well as the inadequacy of the analysis of such expansions is not only of academic theory of the AharonovÐBohm effect [15] and the dis- interest. It is known [2, 3] that multipole expansion of covery of similar effects (see, for example, [16]), current (charge) density within the framework of the necessitate a more systematic approach to multipole MaxwellÐLorentz electrodynamics involves three representation of various configurations that can arise, types of basis dipoles: electric Q, magnetic M, and for example, in dipole media and to selecting parame- (polar) toroidal T. An electromagnetic medium without ters for the description of electromagnetic properties of free and bound charges (ρ ≡ 0), as well as unclosed cur- quantum objects like complicated molecules. rents (divj = 0), is described only by magnetic and tor- In this work, we found complete sets of basis oidal polarizations [4]. When describing the distribu- dipoles for dipole media of most physical importance tion of magnetization in a nucleus, Blatt and Weisskopf [9]. Based on the mathematical fact of their existence, (see [5], where they cite earlier works) were apparently we demonstrate possible unusual responses of con- the first to introduce magnetic contributions to toroidal densed matter to uniform and/or dynamic electric and polarizations by the name “induced electric transverse magnetic fields. This paper is an extension of [12, 17]. moments.” In classifying point groups of magnetic symmetry of crystals, several authors [6Ð9] have pointed to the fact that a toroidal dipole exhibits sym- CONSTRUCTION OF MULTIPOLE MOMENTS metry of a special type. Magnetic substances allowing FOR SPIN (DIPOLE) MEDIA AND THEIR toroidal ordering have been referred to as toroics. The INTERACTION WITH EXTERNAL FIELDS references cited give examples of such magnetic crys- Let us consider a system of continuously distributed tals. Recent experimental studies of toroidal polariza- elementary electric dipoles that can be described by a tion have been made on magnetic piezoelectric function d : r ∈ V limited in space and, in the general Ga2 − xFexO3 [10] and magnetoelectric Cr2O3 [11]. case, time-dependent. Assume that internal interactions In 1986, a family of axial toroidal moments was in this system or applied external fields lead to the for- included in the apparatus of the electrodynamics of mation and ordering of variously configured aggre- continuous media [12].1 A short time later, these gates, each consisting of a finite number of elementary moments found prominent application in molecular dipoles. We will describe the properties of the aggre- physics. In 1991, it was demonstrated [13] that crystal- gates by multipole parameters. The first three terms in lization of some aromatic compounds (e.g., anthracene, the expansion of the energy of their interaction with phenanthrene, or pentacene) may produce an axial tor- external fields are oidal microdipole by ordering electric dipoles of atoms 3 12 ()d W ==Ðdr()Er()dr --- r divEQETÐ Ð curlE 1 Historically [8], the basis dipole of this family is sometimes des- ∫ 60 ignated as G. V

134 BUKINA, DUBOVIK

(a) Q (b) electric moment of the system, T(d) = (1/2)∫(r × d)d3r T(d) is the dipole axial toroidal moment, and P(d) = (1/10)∫[ r(rd) Ð 2r2d]d3r is the dipole polar poloidal moment (see Figs. 1aÐ1c). These three vectors form a basis allowing one to represent the density of distribution of a given type. In 2 3 (2) addition, (1) contains r0 = ∫rEd r, a scalar moment (c) (d) r0 P(d) corresponding to the time derivative of the rms charge radius in the expansion of the current (i.e., to the double 2 2 2 3 spherical layer, see Fig. 1d); r0 = ∫[ 2r(rE) + r E]d r, the rms radius of the longitudinal component of the dipole moment Q, generated by the double cylindrical (d) layer of the dipoles (see Fig. 1e); and Qij and T ij , the electric and toroidal quadrupole moments determined by conventional rules. In (1), all the interactions of the (2) (e) r1 (f) multipole characteristics with the ﬁeld or its derivatives are local and written for the “center” of the system. For the interaction energy between magnetic dipoles and an external nonuniform magnetic ﬁeld, mH is substituted for dE and we have (µ) W = Ð∫m(r)H(r)d3r = Ð MH Ð T curlH

(µ) 1 Ð P curlcurlH Ð --M (∇ H + ∇ H ) 2 ij i j j i (µ)∇ ( ) … (µ) ú Ð Tij i curlH j + Ð MH Ð T E (2) Fig. 1. (a) Model in which electric microdipoles add up to total dipole moment Q; (b) model of a dipole axial toroidal (µ) (µ) Ð P (curlj Ð Húú ) Ð MH Ð T curlH moment T(d) realized in a toroid-shaped dielectric when (µ) electric microdipoles are oriented along the azimuthal direc- + P ∆Hfree + … . tions; (c) model of a dipole poloidal moment P(d) realized in a toroid-shaped dielectric when electric microdipoles are It is seen that, according to electromagnetism dual- oriented along the meridional directions; (d) centrosymmet- ity, the dipole moment Q in Eq. (1) is replaced by the ric polarization of a spherical dielectric layer having the dipole magnetic moment M (characterizing an equiva- (2) (d) (µ) symmetry of a spherical capacitor; scalar moment r0 is lent current loop); T , by the polar toroidal dipole T ; the main multipole characteristic of this object; (e) model of and the vector poloidal moment P(d), by the dipole axial the longitudinal rms radius of the electric dipole moment poloidal moment P(µ). If the magnetic moments are car- (2) ried by spin particles (e.g., neutrons), then the multi- r1 (double cylindrical layer); (f) equatorial section of a thin-walled torus with magnetic microdipoles on the merid- pole expansion of the distribution of their magnetic µ ians; poloidal dipole moment P( ) is the multipole charac- moments (in a nucleus or a neutron star) acquires the teristic of this object upon its contraction to a point. primary sense. Zheludev [6] was the first to predict toroidal ordering of spin magnetic moments in perovskites within the theory of condensed matter. For ultrathin (d) 1 (d) films, ordering of magnetic particles into closed config- Ð P curlcurlE Ð --Q (∇ E + ∇ E ) Ð T ∇ (curlE) 2 ij i j j i ij i j urations that can be described by toroidal moments seems to be a rule rather than an exception [18, 19]. 1 (2) (d) + -----r —divE + … ÐQE Ð T curlE Note that exchange interactions between coplanar spins 10 d (1) give rise to the Kosterlitz–Thouless lattice of “antitor- oidally” arranged vortexes. (d) (d) + P (—ρ Ð ∆E) + … ÐQE + T Hú It is evident that expansions like (1) and (2) can be ( ) obtained for the remaining two fundamental (toroidal) + P d (jú Ð Eúú) + …, 2 In the specific case of zero inner radius, such a configuration is where Q = ∫d (r)d3r is the conventional total dipole sometimes called “hedgehog.”

TECHNICAL PHYSICS Vol. 46 No. 2 2001 THE CONTRIBUTIONS OF POLARIZABILITIES TO FOUR BASIS POLARIZATIONS 135 dipole distributions t(r) and g(r). It is unlikely that the which is nonlinear in field in this case. Eventually, we resulting high-order vortexes are realized in crystals. come to a qualitatively new property of the system of They may form in exotic media where the elementary dipoles. It may be concluded that the external field Hi carriers initially possess, for example, toroidal dipole induces an additional magnetic dipole polarization of moments. Recent experimental data indicate that neu- α(m) trinos are apparently Majorano particles forming dense the medium Mk = ik Hi, which can interact with the 15 Ð3 α(m) media (10 cm ). It is known [20, 21] that neutrinos field Hk. Here, ik is the quadratic magnetic suscepti- have only toroidal dipole moments. That coherent exci- bility of the medium. tations in such media, in particular, hadron gravita- tional excitations, are a possibility was first indicated Thus, we introduced a nonlinear interaction in [22]. accounting for the nonlinear response of the medium to a magnetic field. Basically, the tensor α may have an arbitrary number of subscripts. In the expression for VECTOR PARAMETERS energy, it must be convolved with the corresponding AND NONLINEAR MEDIA symmetric tensor for the vector of the external magnetic field strength. A system consisting of electric The above discussion is related to so-called linear dipoles may naturally experience deformations similar electrodynamics. In its terms, elementary objects (e.g., to those described above. If the electric susceptibility dipoles) are controlled by a field or a current of a given αe is substituted for the magnetic one, the expression type upon contact interaction so that the parameters of for the energy of interaction with an external electric the objects remain unchanged. One can also consider field will be completely identical to Eq. (5). In real sys- more complex objects consisting of dipoles: quadru- tems, atoms and molecules are in continuous motion poles, octupoles, etc. An example of associated macro- due to external and internal factors. Therefore, interre- scopic constructions is two or three current loops that lation between the sources via the fields may give rise are closely spaced or form a single circuit. Consider a to mixed polarizabilities. For example, in the case of wire ring shaped like a propeller. In each of the loops, electromagnetic polarizability, the interaction energy is the electric current generates a magnetic field and the given by magnetic properties can approximately be described by ( ) magnetic moments. The energy of local interactions W = β em E H H . (6) between the two loops and an external magnetic field of NL ijk i j k arbitrary configuration is then given by Bearing in mind the organization of computer mem- µ { } µ { } ory, we introduce the polarizability of the third rank, W L = i Hi 1 + k Hk 2 . (3) corresponding to the Hall effect symmetry, and the Here, “locality” means that both terms in the above for- respective interaction energy mula are written for the centers of the loops. In addition 1 (em) to the vectors µ and µ , which relate to the different W = --α E H J , (7) i k NL c ikl i k l loops and are specified at different points {1} and {2} in the space, we introduce a quadrupole moment of the where J is the external current passing through the sys- µ l system as a symmetric tensor ik of the second rank. As tem. in the previous section, the parameter µ takes into ik In particular, this current may be a displacement account the contribution to the energy from the gradient current when the system is inside a capacitor. Assume of the external magnetic field (i.e., due to field nonuni- α(em) formity between the centers of the loops). Then Eq. (3) that the tensor ikl is asymmetric with respect to sub- can be approximately recast as scripts i and k. Then, for a given experimental configu- µ { } µ { } µ ∇ { } ration, the asymmetric part of the tensor EiHk can be W L = i Hi 0 + k Hk 0 + ik i Hk 0 . (4) related to the Poynting vector of light passing through Now all the terms are written for some middle point a crystal with appropriate nonlinear properties. Follow- (hereafter, the center of the system). Such a representa- ing the preceding considerations, we may introduce the tion reflects the essence of multipole expansion. induced toroidal polarization Assume that the loops are made of a flexible wire. 1 (em) (em) T = --α[ ] E H ≡ α S , (8) Then, the magnetic field orienting the magnetic l c ik l i k jl j moments can deform the loops. We take this perturbation into account by introducing the quadrupole polar- responding to the current Jl passing through the crystal. izability α(m) of the medium. The additional energy of Indeed, an electromagnetic wave may reorient the local interaction of the system with a magnetic field is polarization of a ferromagnetic dielectric, turning it given by the expression into a weak ferromagnet with a toroidal polarization. Such behavior is typical of perovskites, manganites αm { } W NL = ik Hi Hk 0 , (5) [23], and other complex compounds.

TECHNICAL PHYSICS Vol. 46 No. 2 2001 136 BUKINA, DUBOVIK In experiments, approaches to measuring the toroi- Here, dots denote the number of contracted subscripts dal polarization are simpler. For example, Popov et al. in the corresponding tensor, Jm := 1/c(j + Dú ) ≡ curlH, [10] studied the response of FeÐGa oxide single crys- e tals to a magnetic field. It is known that such materials J := curlE, and subscripts at the polarizabilities denote combine the ferromagnetic and piezoelectric properties their tensor rank. One can pass to irreducible tensors by α(em) (by analogy with elastic loops). The authors of [10] introducing half-sums and half-differences for 3 measured the electric polarization induced in the crys- ( ) β me tal by a magnetic field H: and 3 , etc. For magnetic polarization, we obtain α(em) α(m) ⋅ α(m) α(m) Ӈ … Pi = ij H j, (9) M = ˆ 2 H + ˆ 3 : HH + ˆ 4 HHH + ( ) α(em) α(em) α(em) Ӈ … βˆ e ⋅ where ij is the tensor of electromagnetic suscepti- + ˆ 3 : HE + ˆ 4 HEE + + 2 E bility. (e) (e) (me) + βˆ : EE + βˆ Ӈ EEE + … + βˆ : EH α(em) 3 4 3 The asymmetric part of the tensor ij defines the (me) (τ) (m) (τ) m m dipole toroidal moment in accordance with the duality + βˆ 4 Ӈ EEH + … + γˆ ⋅ J + γˆ : J J 2 3 (12) relation (τ) ( τ) ( τ) + γˆ Ӈ Jm Jm Jm + … + γˆ e : HJm + γˆ m : EJm α(em) α(em) ⑀ 4 3 3 ij Ð ji = ijkT k, (10) (meτ) m ˆ (τ) e ˆ (τ) e e ⑀ + … + γˆ Ӈ HEJ + … + δ3 J + δ3 : J J where ijk is the Levi-Civita symmetric unit tensor. 4 The polarization per unit volume of the oxide gives ˆ (τ) e e e ˆ (eτ) e | | = µ + δ4 Ӈ J J J + … + δ3 : EJ the dipole toroidal moment T 24 B Å. Similar results were obtained for Cr O magnetoelectric sub- (mτ) e (emτ) e 2 3 + δˆ 3 : HJ + … + δˆ 4 Ӈ EHJ + …; jected to a strong magnetic field (exceeding the field of spin flop transition). for toroidal polarization of magnetic nature, α(τ) ⋅ m α(τ) m m α(τ) Ӈ m m m … TM = 2 J + 3 : J J + 4 J J J + CONTRIBUTIONS OF THE POLARIZABILITIES α(τe) m α(τe) Ӈ m … β(e) ⋅ TO THE VECTOR POLARIZATIONS AND THEIR + 3 : J E + 4 J EE + + 2 E MULTIPOLE EXPANSIONS (e) (e) (eτ) m In this section, we develop a comprehensive formal- + β3 : EE + β4 Ӈ EEE + … + β3 : EJ ism of contributions of the highest (mixed) polarizabil- β(eτ) Ӈ m … γ (m) ⋅ γ (m) ities to four basis densities of polarization distribution + 4 EEJ + + 2 H + 3 : HH (µ) (d) (13) for an arbitrary medium: P, M, T , and T . Further γ (m) Ӈ … γ (τm) m γ (τe) m multipole expansion of the energy of their interaction + 4 HHH + + 3 : J H + 3 : J E with external polarizations would generalize the above (τ) (τ) … γ (τme) Ӈ m … δ ⋅ e δ e e formulas for the case of nonlinear media. Based on the + + 4 J HE + + 2 J + 3 : J J fundamental interactions of each of the polarizations, (τ) e e e (eτ) e we introduce polarizabilities that take into account all + δ4 Ӈ J J J + … + δ3 : EJ possible mechanisms whereby external fields and cur- (mτ) e (meτ) e rents induce a given polarization. The most general + δ3 : HJ + … + δ4 Ӈ EHJ + …; expression for electric polarization up to contributions and for toroidal polarization of electric nature, of the fourth order is given by (τ) e (τ) e e (τ) e e e ( ) ( ) ( ) α˜ ⋅ α˜ α˜ Ӈ … α e ⋅ α e α 3 Ӈ T p = 2 J + 3 : J J + 4 J J J + P = 2 E + 3 : EE + 4 EEE τ (τ ) (e) ( ) ( ) ( ) α e e α e Ӈ e … β˜ ⋅ … α em α em Ӈ … β m ⋅ + ˜ 3 : J E + ˜ 4 J EE + + 2 E + + 3 : EH + 4 EEH + + 2 H (e) (e) (eτ) (m) (m) (me) ˜ ˜ ˜ e β β Ӈ … β + β3 : EE + β4 Ӈ EEE + … + β3 : EJ + 3 : HH + 4 HHH + + 3 : HE (eτ) ( ) ( ) (me) (τ) (m) (τ) m m β˜ Ӈ e … γ m ⋅ γ m + β Ӈ HHE + … +γ ⋅ J + γ : J J + 4 EEJ + + ˜ 2 H + ˜ 3 : HH 4 2 3 (14) (11) ( ) (τ ) (τ ) γ (τ) Ӈ m m m … γ (eτ) m γ (mτ) m + γ˜ m Ӈ HHH + … + γ˜ m : Je H + γ˜ e : JeE + 4 J J J + + 3 : EJ + 3 : HJ 4 3 3 (τ) (τ) … γ (emτ) Ӈ m … δ(τ) ⋅ e δ(τ) e e … γ˜ (τme) Ӈ e … δ˜ ⋅ m δ˜ m m + + 4 EHJ + + 2 J + 3 : J J + + 4 J HE + + 2 J + 3 : J J δ(τ) Ӈ e e e … δ(eτ) e ˜ (τ) m m m ˜ (eτ) m + 4 J J J + + 3 : EJ + δ4 Ӈ J J J + … + δ3 : EJ δ(mτ) e … δ(emτ) Ӈ e … ˜ (mτ) m ˜ (meτ) m + 3 : HJ + + 4 EHJ + . + δ3 : HJ + … + δ4 Ӈ EHJ + … .

TECHNICAL PHYSICS Vol. 46 No. 2 2001 THE CONTRIBUTIONS OF POLARIZABILITIES TO FOUR BASIS POLARIZATIONS 137 β(m) βˆ (e) With allowance for dual symmetries, 2 = 2 , must take into account sum and difference frequencies, (τ) (τ) which may strongly influence the amount of the effect. δ δ˜ 2 = 2 , etc. All the contributions to the polariza- Using expression (11), one can easily derive a tions containing vectors Jm and Je are called nonlocal. formula for Q where all the nonlinear local contribu- In several recent experiments, they were separated from tions are included. If in (11)Ð(14) we capitalize the let- local effects [24]. In these formulas, some of the field ters denoting the nonlocal polarizabilities, the contribu- strength vectors can be replaced by their derivatives. tion, for example, from the nonlocal polarizability of The rank of the polarizabilities will increase in this the second rank to the dipole moment will be repre- case, and these contributions to the corresponding sented as polarizations will become nonlocal. A set of nonlinear cross terms may include not only magnetoelectric ones, (e) (e) ∇ 3 Q3 = A3 : ∫ Ed r. (18) but also mechanomagnetic, thermoelectric, and others. Let us proceed to multipole expansion of the vector The tensor in the integrand can be separated into polarizations with regard for the properties of the polar- irreducible parts. Then its trace, equal to divE, will be izabilities. For this purpose, we use formulas (1) and the external charge density distribution. The symmetric (2) for the interaction energy between these polariza- and antisymmetric parts are responsible for interaction tions and external fields. We start from the analysis of with the quadrupole and axial components of an exter- electric polarization. The presence of the electric field nal field, respectively. vector in the associated expression modifies the definition of the multipole moments. The dipole moment Q = CONCLUSION P (r)d3r acquires additional terms due to the polariz- ∫ A comprehensive formalism of such multipole ( ) ( ) ( τ) abilities α e , α e , …, γ em , … . For example, the con- expansions is easy to develop. The parametrization of 2 3 4 the expression for the interaction energy between a α(e) tribution related to 3 is polarization and an external field will lead to a cubic lattice infinitely occupying the space, for example, in (e) α(e) 3 Q3 = 3 : ∫EEd r. (15) the positive directions of the X, Y, and Z axes. On the X (principal) axis, we will plot the values of the moments Such a contribution can be considerable if the describing a complex electromagnetic system in order ( ) medium exhibits strong nonlinear properties and α e is of ascending l polarity. On the Y axis, we will plot the 3 contributions from the radii (to the power 2n) of the dis- large. This addition also depends on the integral. In tributions for each of the moments in order of ascend- spherical coordinates, it can be rewritten as ing power n. Recall that multipole analysis is mathe- E E d3r = E2r2dr n n dΩ, (16) matically based on the theory of generalized functions. ∫ i k ∫ ∫ i k Therefore, in spite of the erroneous views regarding the where n = E/|E|. domination of the moments, the values of the radii may appear to be much more important for the description Such integrals are common in electrodynamics and of a particular system. Specifically, an ellipsoidal dou- statistical mechanics. They are found from the general ble layer with a small eccentricity is more appropriately formula described by the rms charge radius than by the electric n n n …n quadrupole moment. From the geometrical point of ∫ i j k l view, the radii describe the deviation of each of the   L    (17) l-pole moments from its value specified at the point 1 {δ δ …δ δ δ …δ …} = ----- ij k …l Ð ik j …l + . with actually zero dimensions [25]. As follows from L! [25], both moments and radii could be corrected for These integrals become difficult to take if divE = 0 nonlinearity in the interaction of a given system with an or divH = 0. Fields applied to crystals in physical external field. The corrections can be plotted vertically experiment may be of different natures (in particular, of in parallel to the Z axis (in order of ascending number different frequencies). Specifically, the Pockels effect, of subscripts in the polarizability tensors) over each usually considered as linear, is actually a second-order position of the matrix fixing the values of the moments effect, since it involves the simultaneous action of a or radii. low-frequency electric field and a light wave where the Note that the majority of experiments employ either electric component usually prevails. The well-known static or harmonic fields. It should be noted that, to Kerr effect is believed to be quadratic, because the reveal first-order vortex structures (toroidal dipoles), refractive index of a crystal is proportional to the square the “true” response function (i.e., free of other multi- of the low-frequency refractive field. However, judging pole interactions) is obtained by applying fields linearly from the expression for interaction energy, this effect is increasing with time. The contributions of the poloidal generally of the third order. One should also distinguish dipoles (second-order vortex structures or two-tori between static and dynamic effects. In the latter, one [26]) can be revealed by applying a magnetic field that

TECHNICAL PHYSICS Vol. 46 No. 2 2001 138 BUKINA, DUBOVIK grows quadratically with time. In general, the power in 10. Yu. F. Popov, A. K. Zvezdin, A. M. Kadomtseva, et al., the time dependence of the field must equal the order of Zh. Éksp. Teor. Fiz. 114, 263 (1998) [JETP 87, 146 an n-torus. It is natural that the response amplitude will (1998)]. decrease sharply with increasing n owing to the relativ- 11. Yu. F. Popov, A. M. Kadomtseva, A. K. Zvezdin, et al., istic factor (1/c)n in the expression for the energy of Pis’ma Zh. Éksp. Teor. Fiz. 69 (4), 302 (1999) [JETP interaction of an n-torus with an external field. Lett. 69, 330 (1999)]. Note that the separation of the vector contributions 12. V. M. Dubovik, L. A. Tosunyan, and V. V. Tugushev, Zh. Éksp. Teor. Fiz. 90, 590 (1986) [Sov. Phys. JETP 63, 344 is most convenient in analyzing, for example, small- (1986)]. size crystal objects. The advantages of the method become even more obvious in studying the electromag- 13. M. A. Martsenyuk and N. M. Martsenyuk, Pis’ma Zh. Éksp. Teor. Fiz. 53 (5), 229 (1991) [JETP Lett. 53, 243 netic responses from low-dimension structures used in (1991)]. nanotechnology. 14. N. A. Tolstoœ and A. A. Spartakov, Pis’ma Zh. Éksp. Teor. Fiz. 52 (3), 796 (1990) [JETP Lett. 52, 161 (1990)]. ACKNOWLEDGMENTS 15. G. N. Afanas’ev, Fiz. Élem. Chastits At. Yadra 21, 172 This work was supported by the Russian Interdisci- (1990) [Sov. J. Part. Nucl. 21, 74 (1990)]. plinary Scientific and Technical Program “Fundamen- 16. X. G. He and B. H. J. McKellar, Phys. Rev. A 47, 3424 tal Metrology,” project no. 2.51. (1993); G. N. Afanas’ev, Fiz. Élem. Chastits At. Yadra 24, 512 (1993) [Phys. Part. Nucl. 24, 219 (1993)]. 17. V. M. Dubovik, M. A. Martsenyuk, and N. M. Mar- REFERENCES tsenyuk, Fiz. Élem. Chastits At. Yadra 24, 1056 (1993) 1. Yu. N. Venevtsev, V. V. Gagulin, and V. N. Lyubimov, [Phys. Part. Nucl. 24, 453 (1993)]. Ferroelectric Magnets (Nauka, Moscow, 1982). 18. B. N. Filippov, L. G. Korzunin, et al., Fiz. Met. Metall- 2. V. M. Dubovik and A. A. Cheshkov, Fiz. Élem. Chastits oved. 87 (6), 17 (1999). At. Yadra 5, 791 (1974) [Sov. J. Part. Nucl. 5, 318 19. N. A. Usov, J. Magn. Magn. Mater. 203, 277 (1999). (1974)]. 20. V. M. Dubovik and V. E. Kuznetsov, J. Mod. Phys. A 3. V. M. Dubovik and L. A. Tosunyan, Fiz. Élem. Chastits 143, 5257 (1998). At. Yadra 14, 1193 (1983) [Sov. J. Part. Nucl. 14, 504 21. E. N. Bukina, V. M. Dubovik, and V. E. Kuznetsov, Phys. (1983)]. Lett. B 435, 134 (1998). 4. V. M. Dubovik, in Proceedings of the 3rd Seminar 22. V. M. Koryukin, Izv. Vyssh. Uchebn. Zaved., Fiz., “Group-Theoretical Methods in Physics (Nauka, Mos- No. 10, 119 (1996). cow, 1986), Vol. 2, pp. 356Ð362. 23. L. M. Sandratskii and J. Kübler, Phys. Rev. Lett. 76, 5. J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear 4963 (1996). Physics (Wiley, New York, 1952; Inostrannaya Lite- ratura, Moscow, 1954). 24. V. P. Drachev, S. V. Perminov, S. G. Rautian, and V. P. Safonov, Pis’ma Zh. Éksp. Teor. Fiz. 68 (8), 618 6. I. S. Zheludev, Izv. Akad. Nauk SSSR, Ser. Fiz. 33, 204 (1998) [JETP Lett. 68, 651 (1998)]. (1969). 25. V. M. Dubovik and V. V. Tugushev, Phys. Rep. 187 (4), 7. H. Schmid, Int. J. Magn. 4, 337 (1974). 145 (1990). 8. E. Ascher, in Magnetoelectric Interaction Phenomena in 26. G. N. Afanasiev and V. M. Dubovik, Fiz. Élem. Chastits Crystals, Ed. by A. Freeman and H. Schmid (Gordon and At. Yadra 29, 891 (1998) [Phys. Part. Nucl. 29, 366 Breach, New York, 1975), p. 69. (1998)]. 9. V. M. Dubovik, S. S. Krotov, and V. V. Tugushev, Kristal- lografiya 32, 540 (1987) [Sov. Phys. Crystallogr. 32, 314 (1987)]. Translated by A. Chikishev

TECHNICAL PHYSICS Vol. 46 No. 2 2001

THEORETICAL AND MATHEMATICAL PHYSICS

On the Theory of Classic Mesodiffusion V. V. Uchaikin and V. V. Saenko Ul’yanovsk State University, Institute for Theoretical Physics, Ul’yanovsk, 432700 Russia e-mail: [email protected] Received March 27, 2000

Abstract—A simple model of the classical random walk of particles with a constant speed and anisotropic angular distribution is used to study the characteristic features of mesodiffusion, that is, of an intermediate stage between the ballistic regime (short times) and ordinary diffusion (long times). In the extreme case of anisotropy, namely, walking along a straight line, the process can be described by the telegraph equation, whose solution contains δ-functions accounting for the ballistic component. As the anisotropy becomes less pronounced, the δ-singularity transforms into a frontal burst (the quasi-ballistic component), beyond which the distribution can be satisfactorily described by the telegraph approximation. In the other extreme case of isotropic walking, the frontal burst disappears and the telegraph approximation, contrary to general belief, proves to be cruder than the diffusion approximation. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION the particles Research into the preparation of high-purity semi- ∂ρ ∂ 2 ρ ∂ 2 ρ ------+ Θ ------= D ------, conductor materials with perfect crystal structure gave ∂t ∂ 2 ∂ 2 birth to a new branch in condensed matter physics, the t x study of mesoscopic systems [1Ð3]. Mesoscopic nano- which is called the telegraph equation. Its solution for a structures present a unique possibility to experimen- short-time (for example, instantaneous) source repre- tally study the transfer process in a medium with a well- sents a diffusion front, beyond which there are no dif- defined potential field not perturbed by random impuri- fusing particles, and in the vicinity of which they travel ties and other defects. The existence of three temporal in the ballistic regime. At long times, the ballistic com- intervals is essential to this process in which the trans- ponent decays and the remaining part of the solution for fer mechanisms are different: an interval (0, t1) where large samples transforms into a gaussian packet that ∞ the ballistic transfer is dominant; an interval of (t2, ); satisfies an ordinary diffusion equation. an interval (t2 > t1) of ordinary (gaussian) diffusion, and The results obtained in [4] apply, however, to a an intermediate interval (t1, t2) of a transfer mechanism rather artificial model of a particle walking along a called mesoscopic diffusion [4], which we will call fixed x-axis with zero transverse velocity component mesodiffusion. and an alternating longitudinal component (the one- The quantum-mechanical analysis of the one dimensional walk). This model does not distinguish dimensional problem carried out in [4] has shown that between the nonscattered particles and those multiply the distinctive feature of mesodiffusion is a deviation scattered in one (say, positive) direction of the x-axis, from Fick’s law thus eliminating the differences between the telegraph and kinetic equations, on the one hand, and creating the ∂ρ specific δ-singularity at the diffusion wave front that is jD= Ð ------, ∂x lacking in the real physical process on the other. The other extreme case is considered to be the one- which is replaced in this region by the MaxwellÐCatta- velocity model of isotropic scattering, a time-indepen- neo relationship dent version of which was thoroughly studied in connection with its use in neutron physics and nuclear ∂ρ ∂ j jD= Ð ------Ð Θ ----- , Θ > 0. reactor problems [5Ð7]. The time-dependent version of ∂x ∂ t the model is considered in relatively few papers (see reference in [8Ð10]) in which the telegraph equation In combination with the continuity equation appears as an approximation. However, the quantitative analysis of its accuracy has not still been carried out. ∂ρ ∂ j ------= Ð ------, The present study considers the class of the mesod- ∂t ∂ x iffusion models relying on a parameter ν ∈ [1/3 , 1]. it gives an equation for the distribution density ρ(x, t) of It is assumed that the distribution of the cosine of the

140 UCHAIKIN, SAENKO angle ω between the x-axis and the direction of particle Note the following properties of this solution: movement after scattering is independent of the direc- ( , ) > ≤ ν tion of movement before scattering and has the density f ν x t 0, x t, (4) ( , ) > ν α ν2 f ν x t = 0, x t, (ω) + 1 ω α ω ≤ α 3 Ð 1 ≥ W ν = ------, 1, = ------0.(1) ν 2 1 Ð ν2 t f ν(x, t)dx = 1, (5) The initial direction of a particle which starts to ∫ move from the origin of the coordinates at the time t = 0 Ðνt has the same density distribution. The parameter νt 2 2 1 1/2 x f ν(x, t)dx = 2ν [t + exp{Ðt} Ð 1], (6) α 1/2 ∫ ν ω2 (ω) ω  + 1 ≥ 1 = ∫ W ν d = ------ ------Ðνt α + 3 3 Ð1 1  x2  f ν(x, t) ∼ gν(x, t) = ------expÐ------, is the root mean square cosine, ν = 1/ 3 for isotropic 2 4πν2t  4ν t  (7) walking and ν = 1 for one-dimensional walking. The smooth dependence of the scattering indicatrix (1) on t ∞. the parameter ν enables the evolution of distributions with the variation of ν to be followed and a conclusion These turned out to be just the properties needed for about the relationship between the solutions of the dif- the description of a quite different process—symmetri- fusion, the telegraph, and the kinetic equations, to be cal random walk along a straight line [12Ð15], in which made. fν(x, t) has the meaning of the probability density at time t and the parameter ν is the velocity of free move- (0) ment of a particle. The term f ν (x, t) describes the dis- THE TELEGRAPH EQUATION AND ONE-DIMENSIONAL MESODIFFUSION tribution of the particles which did not change their direction of movement during time t. They are found at The telegraph equation, derived (as pointed out in points x = νt and x = Ðνt and form the front of the dif- [11]) by Lord Kelvin in connection with the laying of fusion packet which occupies the interval [Ðνt, νt]; the the first transatlantic cable, has the form (in dimension- probability to find a particle outside this interval is less time units t) equal to zero. At t ∞, the function fν(x, t) tends to 2 2 2 the normal distribution gν(x, t) with the variance 2ν t ∂ f ν ∂ f ν ∂ f ν ------+ ------= ν2------. (2) [13] and satisfies the ordinary diffusion equation ∂t2 ∂t ∂x2 ∂ ∂2 gν ν2 gν Its solution fν(x, t) has the meaning of a current at a ------= ------∂t ∂ 2 point x of the conductor at t. The parameter ν is related x to the self-induction and the resistance of a conductor with the initial condition gν(x, 0) = δ(x). Because of its of unit length. Under the initial conditions approximate (asymptotic) nature, the function does not f ν(x, 0) = δ(x), [∂ f ν(x, t)/∂t] = 0, contain information about the diffusion front, but t = 0 describes only the large central part of the diffusion it consists of two terms packet (Fig. 1).

(0) (s) f ν(x, t) = f ν (x, t) + f ν (x, t). (3) ISOTROPIC MESODIFFUSION The first term describes two instant pulses moving ν: From the physical point of view, however, the one- away from the origin of the coordinates with a velocity dimensional model considered in [4] proves to be rather (0) 1 artificial, since, as a result of collisions, the velocity of f ν (x, t) = --[δ(x Ð νt) + δ(x + νt)]exp{Ðt/2}. 2 a real particle may take other directions as well, not just the ones colinear to the x-axis. We chose the unit of The second term gives a continuous part of the solu- length in such a way that the absolute value of the par- tion that fills the interval between these pulses (Ðνt < ticle velocity between collisions (assumed to be con- x < νt): stant) would be equal to 1. In the case of one-dimensional walking of such a particle along the x-axis, its (s)( , ) 1 [ ( ( 2 2 ν2) ) distribution will be described by the density f (x, t). If f ν x t = ------I0 t Ð x / /4 1 4ν the particle acquires a new direction independent of the previous one and possessing the same distribution as ( ( 2 2 ν2) ) 2 2 ν2 ] Ðt/2 + tI1 t Ð x / /4 / t Ð x / e . the initial one after a collision then the density ρ(x, t) of

TECHNICAL PHYSICS Vol. 46 No. 2 2001 ON THE THEORY OF CLASSIC MESODIFFUSION 141 the distribution of its x-coordinate at time t satisfies the ψ integral kinetic equation [10, 16] 101 ( ) ρ(x, t) = ρ 0 (x, t) 100 t t' = 3 ( ) (8) t + ∫dt' ∫ dx'ρ 0 (x', t')ρ(x Ð x', t Ð t'), –1 0 Ðt' 10 t = 10 where ρ(0)(x, t) is the distribution density of the x-coor- t = 30 –2 dinate of the nonscattered particle. 10 t = 100 From elementary probabilistic considerations, it can be found that this random coordinate is uniformly dis- 10–3 tributed between Ðt and t for an isotropic source (we recall that the velocity is equal to 1) 10–4 Ð1 (0) (2t) exp{Ðt}, x < t ρ (x, t) =  (9) 0, x > t. 10–5 0 0.2 0.4 0.6 0.8 1.0 ξ Let ψ(x, ω, t) be the joint distribution density of the x-coordinate and the cosine of the angle between the Fig. 1. The distributions ψ(ξ, t) = tf (ξt, t) (solid curve) and particle velocity and the x-axis at time t: ψ ξ ξ 1 '( , t) = tg1( t, t) (dashed curve) for one-dimensional ran- 1 dom walking. ∫ ψ(x, ω, t)dω = ρ(x, t). ψ Ð1 Eliminating 1 from this equation, we obtain an ψ ≡ ρ It is known [16] that the integral equation (8) with equation for 0 in the form kernel (9) is equivalent to the integrodifferential Boltz- ∂2ρ ∂ρ 1∂2ρ mann equation ------+ ------= ------, ∂ 2 ∂t 3∂ 2 1 t x ∂ ∂ 1 ---- + ω----- + 1 ψ(x, ω, t) = -- ψ(x, ω, t)dω (10) so that ∂t ∂x 2 ∫ Ð1 ρ( , ) ( , ) x t = f 1/ 3 x t . (11) with the initial condition The presence of ν = 1/ 3 is clear since the root 1 ψ(x, ω, t) = --δ(x). mean square projection of the isotropically distributed 2 unit vector of the velocity onto one of the axes is con- Expansion in terms of Legendre polynomials sidered. However, the solution of Eq. (11) according to the properties (4)Ð(7) is nonzero only in the interval ∞ − δ ψ( , ω, ) ( ) (ω)ψ ( , ) [ t/ 3 , t/ 3] and has -singularities at the boundary x t = ∑ l + 1/2 Pl l x t , points, whereas the exact solution covers the interval l = 0 [−t, t] and at the boundary points has finite discontinu- 1 ities equal to (2t)Ð1exp{Ðt} due to nonscattered radia- ψ (x, t) = P (ω)ψ(x, ω, t)dω, ψ (x, t) ≡ ρ(x, t) tion (9). Figure 2 shows that in this problem, the tele- l ∫ l 0 graph approximation describes the transfer process less Ð1 accurately than the diffusion approximation. This transforms Eq. (10) to a chain of equations result, unexpected at first glance, calls for care regarding the conclusions of studies in which the time-depen- ∂ψ ∂ψ ∂ψ l 1 l + 1 l Ð 1 dent diffusion approximation is refined with the use of ------+ ------(l + 1)------+ l------+ (1 Ð δ )ψ = 0. ∂t 2l + 1 ∂x ∂x l0 l the telegraph equation [17, 18]. Retaining the first L + 1 equations and discarding in ∂ψ ∂ the last of them the term L + 1/ x, we obtain a PL ANISOTROPIC MESODIFFUSION approximation that is well known in neutron physics As is evident from the foregoing, the relations [7Ð9]. In particular, in the P1 approximation, between the diffusion and telegraph equations and the kinetic equation in the considered extreme cases are ∂ψ ∂ψ ∂ψ 1∂ψ ------0 + ------1 = 0, ------1 + ------0 + ψ = 0. opposite in a certain sense; namely, in the one-dimen- ∂t ∂x ∂t 3 ∂x 1 sional case, the diffusion solution is a poorer approxi-

TECHNICAL PHYSICS Vol. 46 No. 2 2001 142 UCHAIKIN, SAENKO

ψ ∞ α + 1 (Ð1)n  k  2n 101 = ------∑ ------. λ + 1 2n + α + 1λ + 1 n = 0 100 At k 0, this transform has the asymptotics

(0) λ + 1 ρ˜ (k, λ) ∼ ------. 10–1 t = 3 (λ + 1)2 + ν2k2 t = 10 Substituting it into Eq. (13) and executing simple –2 transformations, we obtain 10 t = 30 t = 100 2 2 2 [λ + λ + ν k ]ρ˜ (k, λ) = λ + 1. –3 10 This relationship is nothing else than the FourierÐ Laplace transform of the telegraph equation 10–4 ∂2ρ ∂ρ ∂2ρ ------+ ------= ν2------, t > 0 (14) ∂t2 ∂t ∂x2 10–5 0 0.2 0.4 0.6 0.8 1.0 with the initial conditions ρ(x, 0) = δ(x) and ξ ∂ρ | [ (x, t)/dt] t = 0 = 0. A comparison of solution (3) above to Eq. (14) with Fig. 2. The distributions ψ(ξ, t) = tρ(ξt, t) for isotropic random walking. Dots are the results of the Monte Carlo simu- the results of numerical solution of the integral equa- lation, solid curves are the telegraph approximation, and tion (8) with kernel (12) by Monte Carlo simulation dashed curves are the diffusion approximation. (see the Appendix) is shown in Fig. 3 in variables ξ = x/t and mation, while in the three-dimensional isotropic case, ψ(ξ, t) = tρ(ξt, t). conversely, the telegraph approximation is less accurate than the diffusion approximation. A smooth transition In the top part of the ﬁgure, it is seen that even at ν2 = 1/2, where the angle distribution linearly depends from one case to the other can be conveniently achieved ω ξ ν by taking the angle distribution of a particle after exit on , in the region < the telegraph solution is closer from the source and after each scattering in form (1). to the exact solution than the diffusion one. As the anisotropy is further increased, a frontal burst appears The telegraph approximation in this scheme is easy in the exact solution, which describes the distribution to obtain using a FourierÐLaplace transform of the of the particles that always move with a positive x-pro- integral equation (8) with the distribution of nonscat- jection of the velocity. In the limit ν2 1, this burst tered particles corresponding to the indicatrix of scat- transforms to the limit δ-function of the telegraph solu- tering (1): tion. At ν2 ≥ 0.7 outside the frontal burst, the telegraph (0) Ð1 approximation is in good agreement with the exact ρ (x, t) = t W ν(x/t)exp{Ðt}, x < t. (12) solution. As it follows from Eq. (8), the transform

∞ t FRONTAL BURST ρ˜ (k, λ) = ∫dt∫ dxexp{ikx Ð λt}ρ(x, t) An analytical description of the frontal burst can be 0 Ðt conveniently obtained using the so-called method of moments [19, 20]. Let z = t Ð x be a lag of the particle satisﬁes the equation from the extreme point of the front t and ρ+(z, t) the dis- ( ) tribution over z at time t of particles moving all the time ρ( , λ) ρ 0 ( , λ)[ ρ( , λ)] ˜ k = ˜ k 1 + ˜ k , (13) in the positive x-axis direction, more speciﬁcally, those where invariably having a positive projection of the velocity on this axis. For this density, an equation similar to ∞ t Eq. (8) will also be valid (corrected for the nonnegativ- ( ) ( ) ρ˜ 0 (k, λ) = ∫dt∫ dxexp{ikx Ð λt}ρ 0 (x, t) ity of z): ( ) 0 Ðt ρ+(z, t) = ρ 0 (z, t)

1 α t t' ω dω (0) + (15) = (α + 1)(λ + 1)∫------+ ∫dt'∫dz'ρ (z', t')ρ (z Ð z', t Ð t'), (λ + 1)2 + (ωk)2 0 0 0

TECHNICAL PHYSICS Vol. 46 No. 2 2001 ON THE THEORY OF CLASSIC MESODIFFUSION 143 where ψ

(0) Ð1 0 ρ (z, t) ≡ t W ν(1 Ð z/t)exp{Ðt}, 0 < z < t. (16) 10 It is convenient now to perform the Laplace transform over both variables, z and t: v2 = 1/3 ∞ ∞ ρ˜ (s, λ) = ∫dt∫dzexp{Ð λt Ð sz}ρ+(z, t), –1 0 0 10 giving the formula ( ) ρ˜ 0 ( , λ) 0 ρ+( , λ) s 10 ˜ s = ------(----)------. (17) 1 Ð ρ˜ 0 (s, λ) Obviously, 1/2 [∂nρ+( , λ) ∂ n] ( )n (λ) ˜ s / s s = 0 = Ð1 m˜ n , where 10–1 ∞ (λ) { λ } ( ) m˜ n = ∫ exp Ð t mn t dt 0 100 is the Laplace transform of the nth moment of the distribution ρ+(z, t) 0.80 ∞ ( ) nρ+( , ) mn t = ∫z z t dz. 0 10–1 For nonscattered radiation,

1 α (0) α + 1 ω dω 0 ρ˜ (s, λ) = ------, 10 2 ∫1 + λ + (1 Ð ω)s 0 from whence 0.90

(0)(λ) 1 (0)(λ) 1 m˜ 0 = ------, m˜ 1 = ------, 2(1 + λ) 2(1 + λ)2(α + 2) 10–1 (0)(λ) 2 m˜ 2 = ------, (1 + λ)3(α + 2)(α + 3) 0 and so forth. Differentiating Eq. (17) with respect to s, 10 we obtain the corresponding expressions for the moments of the distribution of all particles forming the frontal burst: 0.95 1 m˜ (λ) = ------, 0 1 + 2λ 10–1 0 0.2 0.4 0.6 0.8 1.0 (λ) 2 ξ m˜ 1 = ------, (α + 2)(2λ + 1)2 ( λ )(α ) (α ) (λ) 2 2 + 1 + 2 + + 3 ψ ξ m˜ 2 = 4------, Fig. 3. The distributions ( , t) for t = 5 and the speciﬁed (α + 2)2(α + 3)(λ + 1)(2λ + 1)3 values of ν2. The top plot corresponds to isotropic walk. The notations are the same as in Fig. 2. The δ-singularities in the and so forth. The inverse Laplace transform by the res- telegraph approximation are denoted by vertical arrows.

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ψ+ at ν2 close to the limiting value of 1 is only slightly less than the velocity of the front formed by the nonscat- 0.5 v2 = 0.80 tered particles (16) moving parallel to the x-axis. The dispersion of the considered packet σ2 σ2 ( ) ( ) [ ( ) ( )]2 0 x = z = m2 t /m0 t Ð m1 t /m0 t

1.0 ∞ α α 2 α 0.90 increases at t as 4( + 1)t/[( + 2) ( + 3)], so that its relative width

0.5 δ( ) σ ( ) ( ) ∼ 2 t = x t /x t ------(α + 1)(α + 3)t

Ð1/2 0 tends to zero with time as t . The propagation regime of such particles will be referred to as quasiballistic. It 1.5 also includes particles in the ballistic regime; that is, the 0.95 nonscattered particles whose distribution is given by density (16) and characterized by a ﬁnite surge 1.0 (0) (0) α + 1 ∆ρ = ρ (0, t) = ------exp{Ðt}. (20) 2t 0.5 The fraction of ballistic particles in a quasi-ballistic packet is η( ) (0)( ) ( ) { } 0 t = m0 t /m0 t = exp Ðt/2 . 0.8 0.9 1.0 ξ Except for step (20), the solution of Eq. (15) can be approximated by the formula Fig. 4. Distributions ψ+(ξ, t) = tρ+(ξt, t) of particles moving in the positive x-axis direction. Solid circles are the results 2 + γ µ z γ of Monte Carlo simulation; solid curves are approximations ρ (z, t) = ------exp{Ð t/2 Ð (µz) }. by Eq. (21) and for particles which changed their x-compo- 2Γ(2/γ ) nent sign of the velocity at least once. Open circles are the results of the Monte Carlo simulation. It meets the normalizing condition

∞ idue method gives ∫ρ+(z, t)dz = (1/2)exp{Ðt/2} ( ) ( ) { } m0 t = 1/2 exp Ðt/2 , 0 (18) ( ) t { } and has the moments m1 t = ----(--α------) exp Ðt/2 , 2 + 2 ∞ n + Γ((n + 2)/γ ) 4 Ð 2t + t2/2 4(t Ð 2) z ρ (z, t)dz = ------exp{Ðt/2}. m (t) = ------+ ------exp{Ðt/2} ∫ µnΓ( γ ) 2 2 (α )(α ) 2 2/ (α + 2) + 2 + 3 0 (19) 4(α + 1) Equating the ﬁrst and second moments to those + ------exp{Ðt}. (α + 2)2(α + 3) given by Eqs. (18) and (19), respectively, we obtain a system of equations to determine parameters γ and µ as Using the obtained relationships for the moments, it functions of t and ν2: is easy to ﬁnd the center of gravity of the considered ( ) particle distribution: 2Γ(4/γ )Γ(2/γ ) m2 t ------= ------exp{Ðt/2}, [Γ( γ )]2 [ ( )]2 α + 1 2t 3/ m1 t x = t Ð z = t Ð m (t)/m (t) = ------t = ------, 1 0 α + 2 Ð2 1 + ν Γ( γ ) µ 3/ { } = ------(----)---Γ---(------γ----) exp Ðt/2 . whose velocity 2m1 t 2/ ú 2 The results of the numerical calculations of γ(t, ν2) x = ------Ð---2 1 + ν and µ(t, ν2) are presented in the table. In Fig. 4, the

TECHNICAL PHYSICS Vol. 46 No. 2 2001 ON THE THEORY OF CLASSIC MESODIFFUSION 145

The parameters γ and µ in formula (21) ν2

t 0.80 0.90 0.95 0.99 γ µ γ µ γ µ γ µ

2 0.854 12.69 0.765 35.73 0.729 84.69 0.703 484.8 4 1.103 3.777 0.978 9.908 0.928 22.70 0.892 126.5 6 1.401 1.808 1.229 4.505 1.161 10.07 1.111 55.01 8 1.756 1.093 1.521 2.622 1.428 5.752 1.362 30.98 10 2.184 0.758 1.861 1.767 1.737 3.822 1.650 20.36 12 2.709 0.573 2.262 1.308 2.096 2.799 1.981 14.78 14 3.378 0.459 2.747 1.032 2.521 2.189 2.368 11.48 16 4.280 0.384 0.348 0.851 3.035 1.793 2.828 9.359 18 5.609 0.332 4.129 0.726 3.678 1.521 3.391 7.900 20 7.914 0.294 5.213 0.635 4.524 1.324 4.107 6.849 approximation At ν2 > 0.7, the continuous part of the telegraph + + equation is in satisfactory agreement with the solution ψ (ξ, t) = tρ ((1 Ð ξ)t, t) of the kinetic equation (if ξ is not too close to 1) and, 2 2 (21) together with the above-mentioned approximation of t γµ (1 Ð ξ) γ = ------exp{Ð t/2 Ð (µt(1 Ð ξ)) } the frontal burst, can be used as an approximation of the 2Γ(2/γ ) exact solution. is compared with the results of the Monte Carlo simu- In the case of weak anisotropy, the telegraph lation of the distribution of particles moving all the time approximation approximates the exact solution less in the positive x-axis direction. Good agreement accurately than the diffusion approximation. between these results enables the solution of the kinetic equation to be presented in the form of a sum of the slowly varying component of the solution of the tele- ACKNOWLEDGMENTS graph equation and two bursts ψ+(ξ, t) and ψÐ(ξ, t), which are symmetric about the origin of coordinates. This work was supported in part by the Russian Foundation for Basic Research, project nos. 000100284 and 000217507. CONCLUSIONS The main conclusions of the study are as follows. The spatial distribution of particles emitted by an APPENDIX instantaneous point source in the case of mesodiffusion differs from the ordinary diffusion distribution by the SOLUTIONS OF EQUATIONS (8) AND (15) occurrence of a front, beyond which the density is zero. USING MONTE CARLO SIMULATION The front moves away from the source with the velocity The analog simulation of the considered process of a freely moving particle. presents no special problems; however, the obtained In the case of one-dimensional mesodiffusion, there histograms are not very convenient for describing the are δ-singularities at both fronts accounting for the bal- density in the region of high gradient. Indeed, in the listic regime. Together with the continuous part of the analog simulations, the estimation of the density ρ at solution of the telegraph equation, they give an exact point x* at time t* is obtained by building an elemen- solution of Boltzmann’s kinetic equation. tary section (layer) of length ∆x* in the vicinity of the Taking into account the angle distribution of the par- point x* and calculating the quantity ticles results in the disappearance of the singularities N(∆x*, t*) and the emergence instead of well pronounced frontal ρˆ (∆x*, t*) = ------, (A1) 2 ∆ bursts in the case of strong anisotropy when ν > 0.8, N0 x* which merge into the background of the smooth com- 2 ponent at ν < 0.7. The shape of the surges correspond- where N0 is the total number of statistical sampling tra- ing to the ballistic regime is well approximated by for- jectories; N(∆x*, t*) is the number of trajectories which mula (21). are found in the layer ∆x* at the moment of time t*.

TECHNICAL PHYSICS Vol. 46 No. 2 2001 146 UCHAIKIN, SAENKO

The estimator (A1) is not unbiased and its mathe- collisions of nonscattered particles, then estimator (A4) matical expectation can be considered as a sort of local estimation [21]. 1 Mρˆ (∆x*, t*) = ------ρ(x, t*)dx ∆x* ∫ REFERENCES ∆x* 1. I. M. Lifshits, S. A. Gredeskul, and L. A. Pastur, Intro- is equal to the function ρ(x, t*) averaged over ∆x* duction to the Theory of Disordered Systems (Nauka, rather than its sought-for value ρ(x*, t*). According to Moscow, 1982; Wiley, New York, 1988). the mean value theorem, 2. V. F. Gantmakher and M. V. Feœgel’son, Usp. Fiz. Nauk 168, 113 (1998) [Phys. Usp. 41, 105 (1998)]. 1 3. Direction in Condensed Matter Physics. Physics of ------ρ(x, t*) = ρ(x', t*), ∆x* ∫ Mesoscopic Systems, Ed. by G. Grinstein and G. Ma- ∆x* zenko (World Scientific, Singapore, 1986). where x' ∈ ∆x*, but the exact position of x' is not 4. S. Godoy and L. S. García-Colin, Physica A (Amster- known. dam) 258, 414 (1998). 5. B. Davison, Neutron Transport Theory (Clarendon, This uncertainty is the source of a horizontal error. Oxford, 1957; Atomizdat, Moscow, 1960). Estimator (A1) itself contains a statistical “vertical” 6. G. I. Marchuk, Computing Methods of Nuclear Reactors error. (Gosatomizdat, Moscow, 1961). Obviously, the horizontal error component can be 7. K. M. Case and P. F. Zweifel, Linear Transport Theory reduced by decreasing ∆x*, which in turn results in an (Addison-Wesley, Reading, Mass., 1967; Mir, Moscow, increase in the vertical error. It is possible to completely 1972). eliminate the horizontal error from the results by tran- 8. A. M. Weinberg and E. P. Wigner, The Physical Theory sition from the analog scheme to the following modifi- of Neutron Chaim Reactors (Univ. of Chicago Press, cation of the method. Let Chicago, 1959; Inostrannaya Literatura, Moscow, 1961). t* 9. K. H. Beckurts and K. Wirtz, Neutron Physics (Springer- ( ) ( , )ρ( , ) Verlag, Berlin, 1964; Atomizdat, Moscow, 1968). Jh t* = ∫ dt∫dxh x t x t , (A2) 10. R. García-Pelayo, Physica A (Amsterdam) 258, 365 0 (1998). and X1, T1, X2, T2, …, XN(t*), TN(t*) be random coordi- 11. R. García-Pelayo, Physica A (Amsterdam) 216, 299 nates and instants of time of the collisions of the parti- (1995). cle in a given trajectory; N(t*) is a random number of 12. P. Frank and R. von Mises, Die Differential- und Integ- collisions in the interval (0, t*). It is obvious that ralgleichungen der Mechanik und Physik (Vieweg, Brunswick, 1930, 1935; ONTI, Moscow, 1937), Vols. 1, 2. N(t*) ˆ ( ) ≡ ( , ) 13. A. S. Monin, Izv. Akad. Nauk, Ser. Geofiz., No. 3, 234 Jh t* ∑ h Xi T i (A3) (1995). i = 1 14. M. Kac, Some Stochastic Problems in Physics and Math- is the unbiased estimate of the functional (A2) ematics (Magnolia Petrolium Co., Dallas, 1956; Nauka, Moscow, 1967). ˆ ( ) ( ) MJ h t* = Jh t* . 15. A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics (MIT, Cambridge, 1971, 1975; Gidrome- Now represent Eq. (8) in the form teoizdat, St. Petersburg, 1992). ρ( , ) ρ ( , ) ( ) 16. V. V. Uchaikin, Physica A (Amsterdam) 255, 65 (1998). x* t* = 0 x* t* + Jh t* , 17. S. L. Sobolev, Usp. Fiz. Nauk 167, 1095 (1997) [Phys. where Usp. 40, 1043 (1997)]. ( , ) ρ ( ) 18. V. A. Lykov and V. G. Nikolaev, Zh. Vychisl. Mat. Mat. h x t = 0 x* Ð x, t* Ð t . Fiz. 38, 1907 (1998). According to Eqs. (A2) and (A3), the unbiased esti- 19. A. Ya. Énder and I. A. Énder, Zh. Tekh. Fiz. 69 (6), 22 (1999) [Tech. Phys. 44, 628 (1999)]. mator of the functional Jh(t*) is 20. A. Ya. Énder and I. A. Énder, Zh. Tekh. Fiz. 69 (9), 6 N(t*) (1999) [Tech. Phys. 44, 1005 (1999)]. ˆ ρ ( , ) Jh = ∑ 0 x* Ð Xi t* Ð T i . (A4) 21. S. M. Ermakov and G. A. Mikhaœlov, Course of Statisti- i = 1 cal Modeling (Nauka, Moscow, 1976). ρ Since 0(x* Ð Xi, t* Ð Ti) in the adopted system of units describes the density of the flux and the density of Translated by M. Lebedev

TECHNICAL PHYSICS Vol. 46 No. 2 2001

THEORETICAL AND MATHEMATICAL PHYSICS

On the Nature of the AharonovÐBohm Effect A. G. Chirkov* and A. N. Ageev** *St. Petersburg State Technical University, Politekhnicheskaya ul. 29, St. Petersburg, 195251 Russia **Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia Received March 31, 2000

Abstract—It is shown that zero-field potentials may be responsible for the AharonovÐBohm effect. A magnetic f f 0 field B = curlA has a physical (gauge-invariant) meaning for field potentials A , whereas a circulation °∫C A dr has a physical meaning for zero-field potentials A0. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION sibility of the quasi-ABE occurring in the presence of solely field potentials was considered. In [10−12], a The AharonovÐBohm effect, an intriguing quantum- variable magnetic flux was specified but corresponding mechanical phenomenon, was predicted in 1939 and fields and potentials were not discussed. Nevertheless, 1949 [1, 2] and then discovered and theoretically the origin and structure of the fields are of fundamental treated in 1959 [3]. Its essence is that a charged quan- importance for the ABE. Indeed, as was shown (see, tum particle moving in a region where a stationary e.g., [13]), in the stationary case, the ABE is due to magnetic (or electric) field is absent but the vector (or zero-field potentials, changing only the phase of the scalar) potential is nonzero is subjected to electromag- wave function. The necessary condition for the ABE is netic influence. A deeper insight into the nature of the the presence of zero-field potentials that cannot be AharonovÐBohm effect (ABE) has been given in the eliminated by gauge transformation. course of long-standing discussion [4, 5]. However, inexact and sometimes incorrect assertions on this The notion of zero-field (redundant) potentials was problem are still appearing in the physics literature. first introduced for solving boundary problems in elec- The necessary condition for the ABE is the presence of trodynamics of anisotropic media [14Ð16] and is very gauge-invariant potentials that do not produce electro- seldom encountered in the physics literature. The magnetic fields. Since such potentials (zero-field poten- authors of [17] argue that “electric and magnetic field tials) must have the form vectors cannot be expressed in terms of vector potentials” in anisotropic media. The conventional approach 0 0 1∂Ψ to such problems was to use electric and magnetic field A ==gradΨ, ϕ Ð ------, c ∂t strengths or, for zero scalar potentials, vector potentials proportional to them as unknown functions (Coulomb the function Ψ was, generally speaking, mistaken for gauge). Such an approach turned out to exclude the the function of gradient transformation in the vast possibility of satisfying boundary conditions in aniso- majority of works. This resulted in a paradox in both tropic media due to the intrinsic structure of Maxwell’s classical and quantum cases: “…, we may retain the equations (see Section 2). present local theory and, instead, we may try to give a The use of electromagnetic potentials with nonzero further new interpretation to the potentials. In other scalar potential in order to regularly satisfy boundary words, we are led to regard A(r) as a physical variable. conditions was first proposed by academician This means that we must be able to define the physical Tikhonov in 1959 [18]. The evolution of these ideas has difference between two quantum states which differ given rise to a general method for solving boundary only by gauge transformation” [3]. In papers devoted to problems in electrodynamics of anisotropic media, the ABE, the necessary condition for its existence has which is referred to as the method of redundant poten- virtually not been discussed. In the stationary case, it tials [14Ð16, 19]. Its basic idea is the use of potentials was generally assumed that “a nontrivial topology of that do not generate an electromagnetic field to regu- the region of charged particle propagation” is necessary larly satisfy boundary conditions. The main results of for the ABE [6]. this method are reported in Sections 1 and 2. However, After the cogent experiments of Tonomura et al. [7], even its authors state that “these potentials make no the potentialities of studying the stationary ABE had physical sense, since, being a direct consequence of apparently been exhausted and physicists began to gauge invariance, they generate zero electromagnetic study the nonstationary ABE [8Ð12]. In [8, 9], the pos- fields” [19].

148 CHIRKOV, AGEEV As is shown in Section 5, zero-field potentials, gen- find the equations of differential constraints imposed erally speaking, differ from gauge transformation on the unknown functions: despite the same form. The fact that two equivalence ∂ relations for vector potentials A and A' (curlA = curlA' ∇ ϕ ∂------LL2A = Ð LL2 , (3) and A Ð A' = gradχ) do not coincide in the general case x0 was apparently first mentioned in [20] and rigorously where x = ct and the operators L and L have the form proven in the cohomology theory [21]. 0 2 ∂2 ∂2 ∂2 ∂2 L = ------+ ------+ ------Ð µε------, (4a) ∂ 2 ∂ 2 ∂ 2 ∂ 2 1. ZERO-FIELD POTENTIALS IN A UNIAXIAL x y z x0 DIELECTRIC MEDIUM ∂2 ∂2 ∂2 ∂2 Consider an anisotropic dielectric medium with a L = ------+ ------+ ------Ð µε ------, (4b) ε ε ε ε 1 ∂ 2 ∂ 2 ∂ 2 1∂ 2 permittivity tensor ˆ = diag( , , 1), i.e., a uniaxial x y z x0 medium, which is the simplest special case of a homo- ε 2 geneous anisotropic medium.  1 ∂ L2 = L1 Ð 1 Ð ---- ------The free-field Maxwell equations for electromag- ε ∂z2 netic potentials have the form (in any gauge) (4c) ∂2 ∂2 ε ∂2 ∂2 2 1 µε µ ∂ µ ∂ = ------+ ------+ ------Ð 1------. graddivA Ð ∆A + ------εÂ + ------εˆgradϕ = 0, (1a) ∂x2 ∂y2 ε ∂z2 ∂x2 c2 ∂t2 c ∂t 0 Relation (3) implies that the general potentials A 1∂A  ϕ f 0 div εˆ ------+ gradϕ = 0, (1b) and are representable as the sums A = A + A and c ∂t  ϕ = ϕf + ϕ0. The field-producing terms Af and ϕ f sat- δ isfy the basic (dispersion) relation (irrespective of the where is the Laplacian operator. Projecting these gauge) equations on the principal axes of anisotropy, we obtain f ϕ f ∂2 ∂2 ∂2 LL2A = 0, LL2 = 0. (5) Ax Ax µε Ax Ð ------Ð ------+ ------0 ϕ0 ∂y2 ∂z2 c2 ∂t2 The terms A and , which identically satisfy sys- (2a) tem (2) but do not satisfy (5), are related by the condi- ∂2 A ∂2 A µε ∂2ϕ tion + ------y + ------z + ------= 0, ∂x∂y ∂x∂z c ∂t∂x ∂ 0 A ϕ0 -∂------+ grad = 0; ∂2 A ∂2 A ∂2 A x0 ------x Ð ------y Ð ------y ∂y∂x ∂x2 ∂z2 i.e., they do not generate fields. (2b) 2 2 In quantum problems, just as in the stationary ABE, µε∂ A ∂ A µε ∂2ϕ allowing for zero-field potentials causes only a change + ------y + ------z + ------= 0, c2 ∂t2 ∂y∂z c ∂t∂y in the phase of the wave function. Therefore, the presence of zero-field potentials is the necessary condition ∂2 ∂2 ∂2 for the ABE. Ax Ay Az ------+ ------Ð ------In the isotropic case (ε = ε), each of the Cartesian ∂z∂x ∂z∂y ∂x2 1 (2c) coordinates of the vector potential is an independent ∂2 A µε ∂2 A µε ∂2ϕ solution of the wave equation. In order to satisfy the Ð ------z + ------1 ------z + ------1 ------= 0, boundary conditions, three independent solutions are 2 2 2 ∂ ∂ ∂y c ∂t c t z required [9]. Hence, the system of boundary conditions is closed. For an anisotropic medium, this is not the ∂2 ∂2 ε ∂2 ε Ax ε Ay 1 Az case: the introduction of zero-field potentials becomes ------+ ------+ ------c∂x∂t c∂y∂t c ∂z∂t a must to satisfy the boundary conditions. This asser- (2d) tion will be clarified in the next section. ∂2ϕ ∂2ϕ ∂2ϕ + ε------+ ε------+ ε ------= 0. ∂ 2 ∂ 2 1 ∂ 2 x y z 2. GENERAL REPRESENTATIONS FOR A VECTOR Let us use the Chetaev method [16, 19] to study sys- POTENTIAL IN A UNIAXIAL MEDIUM tem (2). Its determinant, involving the operator coeffi- To complete the definition of system (2), we replace cients, is identically zero. This means that the equations its last equation by a linear equation in the general form in system (2) are linearly dependent. In this case, one of [19] the equations, e.g., the last one, may be eliminated. ϕ Solving the obtained system by the Cramer rule, we Lx Ax + Ly Ay + Lz Az + L0 = 0, (6)

TECHNICAL PHYSICS Vol. 46 No. 2 2001 ON THE NATURE OF THE AHARONOVÐBOHM EFFECT 149 which is the most general gauge condition (Lk are linear f f f  ε  ∂  A A  operators of differentiation with respect to coordinates L A = 1 Ð ----1 ------x + -----y . (11c) 2 z  ε  ∂z∂x ∂y xk). Calculating the determinant of the resulting system, we ﬁnd that the potentials have to satisfy the equations From these equations, LL SA = 0, LL Sϕ = 0 (7) f (2) f (2) f f (1) f (2) 2 2 Ax = 0, Ay = 0, Az = Az + Az (12) with the operator f (1) f (2) with L Az = 0 and L2 Az = 0 in accordance with (8). ∂ ∂ ∂ ∂ From the last equations in (11) and (12), we come to the S = ∂-----Lx + ∂-----Ly + ∂-----Lz Ð ∂------L0. x y z x0 equation f (1) f (1) Thus, the general representation of the potentials f (1)  ε  ∂ ∂A ∂A  L A = 1 Ð ----1 ------x------+ ------y------. (13) f f 0 2 z  ε  ∂  ∂ ∂  has the form A = A(1) + A(2) + A , where z x y

f f 0 In view of the Chetaev gauge condition, Eq. (13) LA(1) = 0, A(2) = 0, SA = 0 (8) yields ( ) and the potentials must satisfy the additional condition ∂A f 1 ∂ϕ f (1) ------z------+ ------= 0; (14) 0 ∂ ∂ ∂A 0 x0 z ------+ gradϕ = 0; ∂ ( ) x0 f 1 ϕ f(1) i.e., Az and coincide with the zero-field poten- 0 Ψ ϕ0 ∂Ψ ∂ Ψ f (1) i.e., A = grad and = Ð / x0, where is any dif- tials. Therefore, A = 0, which results in ferentiable function. z ( ) ( ) The last equation in (8) is referred to as redundant ∂A f 1 ∂A f 1 ------x------+ ------y------= 0. (15) [19]. ∂x ∂y ∂ ∂ α β Let us put Lα = eαβ / xβ ( and = x, y, z) and L0 = µ∂ ∂ Finally, for the Chetaev gauge condition, the general / x0. Then, Eq. (6) may be represented in the form of representation of the vector potential in a uniaxial the generalized Lorentz condition ϕ iωt medium (for Ak, ~ e ) has the form [19] f ( ) f ∂ϕ ∂ϕ 1 µ f (1) 1 diveÂ + -∂------= 0, (9) A = A + ------0---, (16a) x0 x x ∂ ik0 x and the operator of the redundant equation is given by ∂ϕ(1) f (1) 1 0 2 A = A + ------, (16b) ∂ y y ik ∂y S = diveˆgrad Ð µ------. (10) 0 ∂ 2 x0 (1) f (1) 1 ∂ϕ Az = Zz + ------∂------, (16c) Choosing the tensor eˆ in the form ik0 z 1 1 1  ω ω(1) eˆ = diag --, --, ---- , where k0 = /c and 0 satisfy the redundant equation   ( ) e e e1 ϕ 1 L 0 = 0. we obtain The general representation for the potentials (16) makes it possible to solve boundary problems in elec- ∂2 ∂2 e ∂2 ∂2 S = ------+ ------+ ------Ð µe------. (10a) trodynamics of uniaxial media. Unlike the case of an ∂ 2 ∂ 2 e ∂ 2 ∂ 2 isotropic medium, here there appears additional condi- x y 1 z x0 f (1) f (1) ε tion (15), relating Ax to Ay . However, the bound- For e = e1 = , the redundant operator S coincides ε ε ary conditions cannot be satisfied using solely the com- with L; for e = 1 and e1 = , S = L2. The former case ponents of the field potentials. Indeed, let it be required coincides with the Chetaev optimal gauge condition to solve the FockÐSommerfeld problem of a dipole [19]; and the latter, with the Tikhonov condition [18]. field generalized for an anisotropic medium. In this In these and only in these cases, the redundant equation case, it is necessary to determine seven functions appears, showing up as one of the second-order equa- 0 0 tions into which the general equation decomposes [19]. (bounded at infinity): three components (Ax , Ay , and ε For e = e1 = , general system (2) takes the form 0 Az ) of the vector potential in a vacuum (z < 0), three f f f (1) f (1) f (1) LAx = 0, LAy = 0, (11a, 11b) components ( Ax , Ay , and Az ) of the vector

TECHNICAL PHYSICS Vol. 46 No. 2 2001 150 CHIRKOV, AGEEV ϕ(1) potential, and the function 0 in the anisotropic half- true in a simply connected domain), since this is the space (z > 0). Thus, to satisfy the boundary conditions, only case when the circulations of A and A' coincide: we have 14 constants. They can be determined from the seven conditions of boundedness at infinity. Three °∫Adl = °∫Adl. (20) more equations result from continuity conditions for We have come to the conclusion that the equalities the vector potential components at z = 0. The continuity χ conditions for the normal derivatives of the tangential curlA = curlA' and A' = A + grad are equivalent only components of the vector potential yield two additional in a simply connected domain [21]. In contrast to dif- equations. One equation results from relation (15) ferential equations describing local properties of the f (1) f (1) vector fields A and A', condition (20) describes their between Ax and Ay . Finally, we should take into global properties. account the Tikhonov field excitation condition (the Thus, even if the redundant equation for zero-field jump of the normal derivative of the vector potential potentials coincides with that for the function of gauge component that is parallel to the direction of current), transformation, their solutions [in view of (20)] may be which allows for the presence of a dipole. different in multiply connected domains. It is this case Thus, one can uniquely solve this problem only by that takes place in the ABE [3]. introducing zero-field potentials. This also means that, along with field potentials, zero-field potentials are also uniquely determined in a given gauge. 4. THE STRUCTURE OF THE VECTOR POTENTIAL IN THE CASE OF MAGNETOSTATIC ABE In the initial version of the magnetostatic ABE [3], 3. RELATION BETWEEN ZERO-FIELD the influence of the region outside an infinite solenoid POTENTIALS AND GAUGE TRANSFORMATION with constant current on electron motion was studied. Let initial potentials A and ϕ be related to each other Let us consider the structure of the vector potential in by general gauge relation (6); that is, this problem on the basis of the above-developed the- 3 3 ory. It is worth mentioning that, if the solenoid is con- Ð1 sidered as a smooth thin-walled continuous cylinder, L A + L ϕ = 0, ϕ = ÐL L A , (17) ∑ k k 0 0 ∑ k k we can take the entire space to be a homogeneous k = 1 k = 1 anisotropic medium with a conductivity that is nonzero ϕ and new potentials A' and ' are related as ρ ρ 2 2 only in the ea direction at = R ( = x + y and the 3 3 z axis is aligned with the axis of solenoid symmetry). ϕ ϕ ( )Ð1 ∑ Lk' Ak' + L0' ' = 0, ' = Ð L0' ∑ Lk' Ak' . (18) The thinner the wire and the less the winding pitch, the k = 1 k = 1 more accurate such a treatment. Therefore, the potential must have the form A = Af + A0, where Af is the Here, L , L' , L , and L' are any linear operators with k k 0 0 field-generating potential and A0 = gradΨ is the zero- constant coefficients. field potential. It is well known that electromagnetic fields E and B In the magnetostatic case (ϕ = 0 and µ = ε = 1), L = are gauge-invariable: ∆ L2 = . The equations of differential constraints (3) will 1∂χ take the form A' = A + gradχ, ϕ' = ϕ Ð ------. c ∂t ∂ ∂ -----∆A = -----∆A , (21a) Substituting these expressions into Eq. (18), we ∂z x ∂x z arrive at ∂ ∂ 3 3 -----∆A = -----∆A . (21b) ∂χ ∂χ ∂z y ∂y z 1 ( Ð1 ) ∑ Lk' ∂------Ð --L0' -∂----- = Ð ∑ Lk' Ð L0' L0 Lk Ak. (19) xk c t Hence, either ∆A = ∆A = ∆A = 0 or k = 1 k = 1 x y z The last equation represents the condition that must ∂A ∂A ------x Ð ------z = B = 0, be satisfied by the function χ of gauge transformation if ∂z ∂x y the potentials are related to each other by gauge condi- ∂ ∂ tions (17) and (18). Generally, this equation differs Az Ay Ψ ------Ð ------= Bx = 0. from the redundant equation S = 0. The only special ∂y ∂z case where Eq. (19) coincides with the redundant one is Completing the definition of this system by operator (6) the restricted gauge transformation Lk = Lk' and ∂ ∂ ∂ ∂ ∂ ∂ with Lx = / x, Ly = / y, Lz = / z, and L0 = 0, i.e., by L0 = L0' . the condition divA = 0, we obtain Eq. (7) in the form Finally, we should take into account that the func- ∆2A = 0. This means that the operator of the redundant tion χ in gradχ must be single-valued (which is always equation is S = ∆ and coincides with L. By virtue of the

TECHNICAL PHYSICS Vol. 46 No. 2 2001 ON THE NATURE OF THE AHARONOVÐBOHM EFFECT 151 symmetry of the problem, one can put Aρ = Az = 0 and tically zero. Thus, the potentials A1α and A2α are defined Aα = Aα(ρ). In view of the boundedness of the solution uniquely (in the gauge divA = 0). at the origin, its form in the range 0 ≤ ρ < R may be ρ f 0 A1α = c1 ( A1 = A1α and A1 = 0). In the outer region ρ ρ 5. THE STRUCTURE OF THE VECTOR ( > R), the general solution has the form A2α = c3 + ρ POTENTIAL FOR AN AC SOLENOID c2/ . However, one cannot require the potential to be bounded at infinity, since the system is infinite. Physi- Consider nonstationary excitation of an infinite cally, it is clear that the magnetic field outside the sole- cylindrical solenoid with infinitely thin walls to illus- noid is equal to zero ( A f = 0). Therefore, in the outer trate the existence of nonstationary zero-field potentials 2 as a generalization of the stationary magnetostatic region, the redundant equation ∆A0 = 0 subject to ρ α 0 ABE. Choosing a cylindrical coordinate frame ( , , z) curlA = 0 should be solved. This condition selects the with the z axis directed along the solenoid axis, we can ρ proper solution A2α = c2/ . This fact was not found out represent the volume distribution of current as earlier due to the coincidence of the redundant and general operators in this problem. In this case, the (ρ, α, ) δ(ρ ) ( α ω ) jα z = I0 Ð R expi Ð n + t , Tikhonov condition for field excitation is given by (22) jρ = jz = 0, ∂A α ∂ 2I ------1--- Ð A2α = ------, ∂ρ ---∂----ρ---- cR where R is the solenoid radius and ω is the circular fre- ρ = R quency of the current. where I is the current per unit solenoid length. This The nonzero components of the vector potential, Aρ condition is the same as that of the jump of the mag- and Aα, have the form [17] (further, we will omit the netic field tangential component in the usual statement factor showing the harmonic dependence on time) of the problem. Using the Tikhonov condition and the continuity ρ ( ) (α α ) ( , ) condition for the potential at = R, we obtain the final Ar = ∫ jα r' sin Ð ' G r r' dV', (23a) solution: A α = Iρ/cR (0 ≤ ρ < R) and A α = IR/cρ (R < 1 2 V ρ < ∞). The constants in these formulas can be expressed in terms of the magnetic field flux inside the Φ π 2 ρ Φ πρ solenoid: A1α = ( /2 R ) and A2α = /2 . ( ) (α α ) ( , ) Aα = ∫ jα r' cos Ð ' G r r' dV', (23b) The potential in the outer region is a zero-field Ψ V potential, so that A2 = grad . However, since this region (R < ρ < ∞) is doubly connected, the function Ψ iπ (2) where G(r, r') = Ð---- H (k|r Ð r') is the Green func- ≠ c 0 is multiple-valued and °∫A2 dr 0 despite the fact that ≡ (2) curlA2 0. Here, the nonequivalence mentioned in Sec- tion for the Helmholtz equation [17], H0 is the Han- tion 3 takes place. This means that the Stokes theorem kel function, and k = ω/c. (in the form °∫A dr = °∫B ds, where the contour C The integrals appearing in (23) are easily evaluated encloses the solenoid in the outer region and S is the using the addition formula for the Hankel functions area inside the contour C) is inapplicable [21]. Never- [17]: theless, this formula is incorrectly used in all the papers devoted to the ABE. (2)( ρ2 2 ρ (α α )) H0 k + R Ð 2 Rcos Ð ' (24) Outside the solenoid, the Stokes theorem applies to +∞ (2) (α α )H (kR)J (kρ), ρ < R the region between two closed contours C1 and C2 that Ðim Ð '  m m = ∑ e ( ) enclose the solenoid. A gap between the contours J (kR)H 2 (kρ), ρ > R. makes the region simply connected. In this case, m = Ð∞ m m ω ω ∫ A1 dr = ∫ A2 = 1, where 1 is the cyclic constant Eventually, we obtain °C1 °C2 [21] that equals the flux Φ = 2πIR/c = πR2B in our case, π2 where B is the field inside the solenoid. For a restricted i I0 R Ðinα Aα = Ð------e gauge transformation, the function χ of gauge transfor- c mation (the same both inside and outside the solenoid) ( ) ( ) (25a) has to satisfy the Laplace equation ∆χ = 0 (0 ≤ ρ < ∞). H 2 (kR)J (kρ) + H 2 (kR)J (kρ), ρ < R ×  n + 1 n + 1 n Ð 1 n Ð 1 It is well known [22, 23] that the solution of the Laplace ( ) ( ) χ|  2 ρ 2 ρ ρ > equation in the entire space subject to ρ = ∞ = 0 is iden- Jn + 1(kR)Hn + 1(k ) + Jn Ð 1(kR)Hn Ð 1(k ), R,

TECHNICAL PHYSICS Vol. 46 No. 2 2001 152 CHIRKOV, AGEEV

π2 where I0 R Ðinα Aρ = Ð------e 2πI RJ (kR) 2c W = ------0------1------( ) ( ) (25b) c H 2 (kR)J (kρ) Ð H 2 (kR)J (kρ), ρ < R × n + 1 n + 1 n Ð 1 n Ð 1  and Y1 is the Neumann function.  (2) ρ (2) ρ ρ > Jn + 1(kR)Hn + 1(k ) Ð Jn Ð 1(kR)Hn Ð 1(k ), R. f 0 In the stationary case, Aα = 0 and Aα = JR/cρ. From In the case n = 0, i.e., in the absence of angle modu- the condition of absence of an electric field, one can lation, formula (11) yields obtain the expression for the scalar zero-field potential: ( ) π 2 2 ( ) ( ρ) ρ < 0 4 iI0 R 2iπ I RH1 kR J1 k , R ϕ = Ð------J (kR)α, (31) 0  1 Aα = Ð------( ) c c J (kR)H 2 (kρ), ρ > R, (26) 1 1 where α is the azimuthal angle. In the stationary case, 0 Aρ = 0. ϕ = 0. Note that there exists a relation between the sole- In the stationary case (ω 0), these relations result ρ ρ noid radius and the electromagnetic field wavelength. in the well-known formulas Aα = J /cR and Aα = JR/c This relation is defined by the roots of the equation (ρ > R), where J = 2πRI is the current per unit length 0 J1(kR) = 0 and yields the zero field outside the solenoid, in the solenoid wall. as follows from (15). However, in this nonstationary The only nonzero magnetic-field component corre- case, zero-field potentials also vanish; here, we are sponding to potentials (26) has the form dealing with a closed waveguide. Zero-field potentials are equal to zero for all n ≠ 0 as well. One can easily 2 (2)( ) ( ρ) ρ < 2iπ I RkH1 kR J0 k , R evaluate the flux inside the solenoid: 0  Bz = Ð------( ) (27) R 2π c J (kR)H 2 (kρ), ρ > R, 1 0 Φ (ρ)ρ ρ α = ∫ ∫ B1z d d for alternating current or 0 0 (32) 3 2 4πI i4π R I ( ) B = ------0 (ρ < R), B = 0 (ρ > R) (28) = Ð ------0 J (k R)H 2 (k R). z c z c 1 0 1 0 in the stationary case. The circulation of the vector A0 on any contour In order to divide the vector potential outside the enclosing the solenoid (cyclic constant) is given by solenoid into two terms according to (1), we separate π2 out the term whose curl is equal to zero in (12). This can ω 8 I0 R ( ) 1 = °∫A0dr = ------J1 k0 R . (33) be done uniquely as follows: ck0 C π2 ω ≠ Φ 2i I0 R (2) Obviously, 1 in the general (nonstationary) Aα = Ð------J (kR)H˜ (kρ) c 1 1 case. Therefore, their coincidence in the stationary case (29) (k 0) should be considered as accidental. 4πI RJ (kR) 0 0 1 ≡ f 0 Generally speaking, this fact completely changes + ------ρ------Aα + Aα, our understanding of the ABE. The reason for the ABE f f 0 is zero-field potentials. A magnetic field B = curlA has where Aα is the component of the field potential, Aα is a physical (gauge-invariant) meaning for field poten- the component of the zero-field potential, and 0 tials Af, whereas a circulation A dr has a physical (2) (2) °∫C H˜ (kρ) = H (kρ) Ð 2i/πkρ. 1 1 meaning for zero-field potentials A0. Let us write the real parts of the potential components in (29): REFERENCES f ReAα 1. W. Franz, Verh. Dtsch. Phys. Ges. 20, 65 (1939). 2. W. Ehrenberg and R. E. Siday, Proc. Phys. Soc. London,   (30a) Sect. B 62, 8 (1949). π ( ρ) ω 2 π ( ρ) ω = W J1 k sin t Ð ----- + Y1 k cos t,  kρ  3. V. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959). 4. S. Olariu and I. I. Popesku, Rev. Mod. Phys. 57, 339 (1985). 0 2 ReAα = W----- cosωt, (30b) 5. M. Peskin and A. Tonomura, Lect. Notes Phys. 340, 115 kρ (1989).

TECHNICAL PHYSICS Vol. 46 No. 2 2001 ON THE NATURE OF THE AHARONOVÐBOHM EFFECT 153

6. V. G. Bagrov, D. N. Gitman, and V. D. Skarzhinskiœ, Tr. 16. D. N. Chetaev, Dokl. Akad. Nauk SSSR 174, 775 (1967) Fiz. Inst. Akad. Nauk SSSR 176, 151 (1986). [Sov. Phys. Dokl. 12, 555 (1967)]. 7. A. Tonomura, N. Osakabe, T. Matsuda, et al., Phys. Rev. 17. G. T. Markov and A. F. Chaplin, Excitation of Electro- Lett. 56, 792 (1986). magnetic Waves (Moscow, 1967). 8. B. Lee, E. Yin, T. K. Gustafson, and R. Chiao, Phys. Rev. 18. A. N. Tikhonov, Dokl. Akad. Nauk SSSR 126, 967 A 45, 4319 (1992). (1959) [Sov. Phys. Dokl. 4, 566 (1959)]. 9. A. N. Ageev, S. Yu. Davydov, and A. G. Chirkov, Pis’ma 19. M. G. Savin, Problems of Lorentz Gauge in an Anisotro- Zh. Tekh. Fiz. 26 (9), 70 (2000) [Tech. Phys. Lett. 26, pic Medium (Moscow, 1979). 392 (2000)]. 20. F. Valz-Gris and C. Magni, J. Math. Phys. 36, 177 10. A. Vourdas, Europhys. Lett. 32, 289 (1995). (1995). 11. A. Vourdas, Phys. Rev. A 56, 2408 (1997). 21. B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Mod- 12. G. Afanasiev and M. Nelhiebel, Phys. Scr. 54, 417 ern Geometry: Methods of Homology Theory (Nauka, (1996). Moscow, 1984). 13. D. H. Kobe, Ann. Phys. 123, 381 (1979). 22. V. I. Smirnov, A Course of Higher Mathematics (Nauka, Moscow, 1968; Addison-Wesley, Reading, 1964), Vol. 2. 14. D. N. Chetaev, Zh. Tekh. Fiz. 32, 1342 (1962) [Sov. Phys. Tech. Phys. 7, 991 (1963)]. 23. N. E. Kochin, Vector Calculus (Moscow, 1967). 15. D. N. Chetaev, Izv. Akad. Nauk SSSR, Fiz. Zemli, No. 10, 45 (1966). Translated by M. Fofanov

TECHNICAL PHYSICS Vol. 46 No. 2 2001

THEORETICAL AND MATHEMATICAL PHYSICS

Transformation Properties of Wave Functions of a Two-Dimensional Harmonic Oscillator I. E. Mazets Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Received May 16, 2000

Abstract—A transformation formula for wave functions of type ψ (x )(ψ x), where ψ is the eigenfunction n1 1 n2 2 n of a harmonic oscillator for the nth energy level, was found for an arbitrary rotation of the (x1, x2) Cartesian system. By way of example, matrix elements for contact interaction of two particles in an external harmonic potential ﬁeld were calculated. © 2001 MAIK “Nauka/Interperiodica”.

STATEMENT OF THE PROBLEM words, how the wave functions transform in going to Let us have an isotropic harmonic oscillator of mass the new (primed) coordinates m and fundamental frequency ω. Its complete Hamilto- ' χ χ ' χ ξ nian can be represented as the sum of one-dimensional x1 ==cos x1 + sin x2, x2 Ðsin x1 + cos x2. (6) Hamiltonians describing oscillation along two orthogonal axes x and x : As far as we know, such a transformation has not 1 2 been considered in the literature. This is associated Hˆ = Hˆ + Hˆ , largely with the fact that it is natural to represent the 1 2 wave function of an axially symmetric harmonic oscil- 2 2 ប ∂ 1 2 2 (1) lator as the product of the radial part, which depends on Hˆ ==Ð12.------+ --- m ω x , j , j 2m∂ 2 2 j the polar radius alone and hence is invariant under rota- x j tions, and a function like exp(iMϕ), where ϕ is an angu- The properties of the eigenfunctions of operators lar variable and M is an integer. Moreover, the addition ˆ theorem for Hermitian polynomials, or, which is the Hj are well known [1]. We will take advantage of the same, for eigenfunctions of a harmonic oscillator, has expression for the eigenfunction for the nth energy also not been covered in the mathematical literature level (n = 0, 1, 2, …) normalized to unity: concerning special functions (see, e.g., the comprehensive monograph [2]). ψ() 1()† nψ() nxj=------aˆ j 0xj, (2) n! Before calculating the coefﬁcients in the transform where n1'n2' 2 ψ()ψx()x =∑K ()ψχ ()ψx' ()x',(7) ω1/4 mωx n11n2 2 n1n2 n1' 1 n2' 2 m j , ψ()x = ------exp Ð ------(3) n1'n2' 0 j πប 2ប we will point to their obvious properties. First, when is the wave function of the ground state. The quantum the coordinate axes are rotated, only degenerate states production operator is explicitly written as belonging to the same eigenvalue of the complete

† mω ប ∂ Hamiltonian Hˆ mix. Hence, the necessary condition aˆ j = ------x j Ð ------, (4) ប ω ∂ n1'n2' 2 2 m x j for K (χ) to be other than zero is the fulﬁllment of n1n2 and its conjugate (annihilation operator), as the equality ω ប ∂ m n+n=n'+n'. (8) aˆ j = ------ប x j + ------ω- ∂------. (5) 1 2 1 2 2 2 m x j Second, since the basis functions are real, so are the Our goal is to see how the wave functions of type coefﬁcients of the transform; for them, the unitary ψ (x )(ψ x) transform when the coordinate system n 1 1 n 2 2 property changes to the property of orthogonality. χ rotates through an angle in the plane (x1, x2); in other Thus, the inverse transform (in going from the new

1063-7842/01/4602-0154 $21.00 © 2001 MAIK “Nauka/Interperiodica” TRANSFORMATION PROPERTIES OF WAVE FUNCTIONS 155 coordinates to the old) has the form primed quantities are their values in the new (rotated) coordinate system. In essence, relation (13) defines the n' n' Wigner functions. ψ (x' )ψ (x' ) = ∑ K 1 2(χ)ψ (x )ψ (x ). (9) n1' 1 n2' 2 n1n2 n1 1 n2 2 We omit a routine procedure of directly computing n , n 1 2 n n the transform coefficients K 1' 2' (χ), the more so as n1n2 CALCULATION OF THE TRANSFORM expressions (11) and (12) can be obtained by another, COEFFICIENTS less trivial but faster, way. It uses isospin formalism, widely employed in the theory of two-level atom n n Direct calculation of the coefficients K 1' 2' (χ) ensembles [4]. Let us introduce the operators of isospin n1n2 projections onto three orthogonal axes in an auxiliary implies the definition of the quantum production oper- (isospin) space: '† '† ators aˆ 1 and aˆ 2 in the new coordinates. This is done 1 † † i † † Iˆ = --(aˆ aˆ + aˆ aˆ ), Iˆ = --(aˆ aˆ Ð aˆ aˆ ), by replacing x in (4) by x' . With (6), we find 1 2 1 2 1 2 2 2 1 2 1 2 j j (14) † † † 1 † † χ ' χ ' Iˆ3 = --(aˆ aˆ Ð aˆ aˆ ). aˆ 1 = cos aˆ 1 Ð sin aˆ 2 , 1 1 2 2 (10) 2 † χ '† χ '† It is easy to check that these operators satisfy the aˆ 2 = sin aˆ 1 + cos aˆ 2 . usual commutation relations for angular momentum These expressions are substituted into the definitions of ψ (x ) and ψ (x ). Removing the parentheses [Î1, Î2] = iÎ3; (15) n1 1 n2 2 in the ψ (x )ψ (x ) product, collecting similar terms, two other similar relations are obtained by circularly n1 1 n2 2 ψ ψ ψ ψ permutating subscripts 1, 2, and 3. Also, each of the and bearing in mind that 0(x1) 0(x2) = 0(x1' ) 0(x2' ), operators Iˆk (k = 1, 2, or 3) is commutatively related to we find the coefficient preceding ψ (x' )ψ (x' ) in n1' 1 n2' 2 the spin square operator: the form of a finite sum. After cumbersome computa- 3 tion using various identities known from the angular ˆ2 ˆ(ˆ ) momentum theory [3], we eventually come to ∑ Ik = I I + 1 , (16) k = 1 1( ) -- n1 + n2 n1' n2' (χ) 2 ( χ) where Kn n = d1 1 2 . (11) 1 2 --(n' Ð n' )--(n Ð n ) 2 2 1 2 2 1 1 † † Iˆ = --(aˆ aˆ + aˆ aˆ ). (17) In view of the known property [3] 2 1 1 2 2 dJ (β) = dJ (β), Thus, the wave function ψ (x )ψ (x ) is the eigen- MM' Ð M' Ð M n1 1 n2 2 formula (11) can be recast as function of the operators Iˆ and Iˆ3 , and the eigenvalues

1( ) of the operators are 1/2(n1 + n2) and 1/2(n1 Ð n2), -- n1 + n2 n1' n2' (χ) 2 ( χ) respectively. Kn n = d1 1 2 . (12) 1 2 --(n Ð n )--(n' Ð n' ) 2 1 2 2 1 2 Now consider a rotation through an angle χ in the J (x1, x2) plane. It is evident that Here, dMM' is the function whereby the Wigner D func-  ∂  tion that corresponds to a rotation through Eulerian ψ (x' )ψ (x' ) = exp Ðχ------ψ (x )ψ (x ), (18) angles (α, β, γ) is expressed n1' 1 n2' 2  ∂φ n1 1 n2 2 J (α, β, γ ) ( α) J (β) ( γ ) φ DMM' = exp ÐiM dMM' exp ÐiM' . where the variable is a polar angle reckoned from the x axis toward the x axis. In addition [1, 3], Under rotation, the wave functions with certain total 1 2 angular momentum J and its projection M onto the z ∂ ∂ ∂ axis transform as ∂----φ-- = x1∂------Ð x2∂------. x2 x1 Ψ (ϑ , ϕ , σ ) J M' ' ' ' Using deﬁnitions (4) and (14), one can show that J J (13) ψ ( )ψ ( ) ( χˆ )ψ ( )ψ ( ) Ψ (ϑ, ϕ, σ) (α, β, γ ) n' x1' n' x2' = exp Ð2i I2 n x1 n x2 ; (19) = ∑ JM DMM' . 1 2 1 2 M = ÐJ in other words, a rotation through an angle χ in the χ The argument of the wave function involves two (x1, x2) plane is reduced to a rotation through an angle 2 angular variables ϑ and ϕ and the spin variable σ; the about the x2 axis in the isospin plane. Thus, formula (12)

TECHNICAL PHYSICS Vol. 46 No. 2 2001 156 MAZETS and its equivalent (11) immediately follow from (19) in of-mass motion of two particles and their relative view of (13). motion:

1 ( ) 1 ( ) x1' = ------x1 + x2 , x2' = ------x2 Ð x1 . (23) MATRIX ELEMENTS OF CONTACT 2 2 INTERACTION Formally, this transformation is a rotation through By way of example, the relations obtained will be an angle of π/4 in the (x , x ) “plane.” With (7), we ﬁnd used in calculating matrix elements of contact interac- 1 2 tion. As is known [5], in ensembles of ultracold atoms, ν ν ν ν 1 1 2 1' 2' where s-wave isotropic scattering of length a dominates T = ------∑ K (π/2)K (π/2) n2' n1' n1n2 n1n2 n1' n2' because of the smallness of the collision energy, atomic 2ν , ν , ν , ν (24) interaction can be approximated by the contact pseudo- 1 2 1' 2' × δν ν ψν (0)ψν (0) potential 1 1' 2' 2'

(3) ( ) δ ( ) (δν ν is Kronecker’s symbol). Substituting the eigen- U r Ð r' = g r Ð r' , (20) 1 1' functions of a harmonic oscillator in the explicit form where r and r' are the positions of colliding atoms (µ is (i.e., expressing them through Hermitian polynomials) their reduced mass), g = 2πប2µÐ1a is the effective inter- [1] and using the basic result of this work [expression action constant, and δ(3) is the three-dimensional Dirac (11)], one can represent (24) in the form delta function. J' Ð J mω To refine the existing theory of degenerate atomic T = (Ð1) ------n2' n1' n1n2 πប gas in a magnetic trap [5], it seems of interest to calcu- 2 late matrix elements of contact interaction using the basis of eigenfunctions of a harmonic oscillation. Con- × ∑dJ (π/2)dJ (π/2) (25) µ M2s Ð J M' + 2s + J' Ð 2J sider two atoms of equal mass m ( = m/2) that had s k quantum numbers n j before collision (the subscript j = (2s)!(2s + 2J' Ð 2J)! 1, 2 numbers the atoms, while the superscript k = x, y, z × ------, stands for the Cartesian axes directed along the mag- s!(s + J' Ð J)!22s + J' Ð J netic axes of the trap). The quantum numbers after collision will be primed. In the general case, all three fun- where s under the summation sign runs over all integers damental frequencies of oscillation along the axes dif- for which the arguments of the factorials in (25) are fer; we, however, will omit the super- and subscripts at nonnegative. Hereafter, we introduce the designations ω to make the mathematical notation simpler. The desired matrix element is defined as 1 1 J = --(n + n ), J' = --(n' + n' ), 2 1 2 2 1 2 z 〈 k' k〉 n j U n j = g∏T k' k' k k , (21) 1 1 n2 n1 n1n2 ( ) ( ) M = -- n2 Ð n1 , M' = -- n2' Ð n1' . k = x 2 2 where After tedious routine rearrangements that follow from the angular momentum theory [3], (25) is reduced +∞ +∞ to a form that is free of the Wigner functions but con- T = dx dx ψ (x )ψ (x) jm n2' n1' n1n2 ∫ 1 ∫ 2 n2' 2 n1' tains ClebschÐGordan coefficients C (such a (22) j1m1 j2m2 Ð∞ Ð∞ form is very convenient when the computation is per- × δ(x Ð x )ψ (x )ψ (x ) 1 2 n1 1 n2 2 formed with the Mathematica 3.0 program package, where the ClebschÐGordan coefficients are used as is the matrix element from the one-dimensional delta standard functions): function. Two properties of T immediately fol- n2' n1' n1n2 J + J' J + J' J3 + l mω low from definition (22). First, this element remains T = ζ ------∑ ∑ ∑ invariable for any permutation of its four subscripts. n2' n1' n2 2(J + J') 2πប , l = J Ð J' J3 = J Ð J' J4 = max( J3 Ð l M Ð M' ) Second, it is other than zero only if the sum n1 + n2 + ' ' (J + M' Ð M + J + J' Ð l)/2 n1 + n2 is even. × ( ) 4 ζ Ð1 l Ð J' + J (26) To compute matrix element (22), we will transform J M Ð M' J M Ð M' J J Ð J' J 0 × ζ C 3 C 4 C 3 C 4 the coordinates in such a way as to separate the center- J4 Ð M + M' JMJ' Ð M' J3 M Ð M'l0 Jl Ð J'J'J Ð l J3 J Ð J'lJ' Ð J

TECHNICAL PHYSICS Vol. 46 No. 2 2001 TRANSFORMATION PROPERTIES OF WAVE FUNCTIONS 157

( ) ( ) REFERENCES J4 Ð M + M' ! J4 + M Ð M' ! × ------. 1. A. S. Davydov, Quantum Mechanics (Nauka, Moscow, J4[( ) ] [ ) ] 2 J4 Ð M + M' /2 ! J4 + M Ð M' /2 ! 1973; Pergamon, Oxford, 1976); L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 3: 1 Quantum Mechanics: Non-Relativistic Theory (Nauka, Here, ζ = -- [(Ð1)s + 1] (i.e., ζ = 1 or 0 if s is even or s 2 s Moscow, 1989, 4th ed.; Pergamon, New York, 1977, 3rd odd, respectively). Matrix elements for powers of the ed.). difference between the coordinates of two atoms (and, 2. V. T. Erofeenko, Theory of Addition (Nauka i Tekhnika, hence, for the energy of their interaction in the form of Minsk, 1989). a power series other than pseudopotential (20)) can be 3. D. A. Varshalovich, A. N. Moskalev, and V. K. Kherson- skiœ, Quantum Theory of Angular Momentum (Nauka, calculated in a similar manner. Leningrad, 1975; World Scientiﬁc, Singapore, 1988). 4. A. V. Andreev, V. I. Emel’yanov, and Yu. A. Il’inskiœ, ACKNOWLEDGMENTS Cooperative Phenomena in Optics (Nauka, Moscow, 1988); M. M. Al’perin, Ya. D. Klubis, and A. I. Khizh- This work was supported by the Russian Foundation nyak, Introduction to the Physics of Two-Level Atoms for Basic Research (grant no. 99-02-17 076) and (Naukova Dumka, Kiev, 1987). the Program “Universities of Russia” (grant 5. L. P. Pitaevskiœ, Usp. Fiz. Nauk 168 (6), 641 (1998) no. 015.01.01.04). [Phys. Usp. 41, 569 (1998)]. The author thanks D.A. Varshalovich for valuable discussions. Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 46 No. 2 2001

GASES AND LIQUIDS

Nonlinear Capillary Vibrations and Stability of a Highly Charged Drop under Single-Mode Initial Deformation of Large Amplitude S. O. Shiryaeva Demidov State University, Sovetskaya ul. 14, Yaroslavl, 150000 Russia e-mail: [email protected] Received April 12, 2000

Abstract—Nonlinear vibrations of a charged drop that are caused by a virtual initial perturbation of the equilibrium spherical shape of the drop were considered. The perturbation can be proportional to any of the free vibration modes. The spectrum and stability of the vibrations were studied correct to second order of smallness. © 2001 MAIK “Nauka/Interperiodica”.

(1) Charged drop stability is of considerable interest In what follows, we will use the dimensionless vari- in geophysical applications, scientific instrumentation, ables for which R = ρ = σ = 1. technical physics, and engineering [1]. However, most Since the initial perturbation of the drop is axisym- of the theoretical investigations into the problem were metric and small, we introduce the following simplifi- based on a linear approximation. The nonlinearity of cations. The shape of the drop is assumed to be axisym- the phenomenon has been taken into account in analyt- metric both at the initial and at each subsequent time ical and numerical studies only recently [2Ð10], instant, and the equation for the shape of the drop writ- although some results of the nonlinear approach were ten in polar coordinates with the origin at the center of also obtained within simpler linear approximations the drop has the form [11, 12]. r()Θ, t = 1+ ξΘ(),t,ξ Ӷ 1. (1) We will consider a drop whose electric charge is smaller than critical in terms of Rayleigh instability. The motion of the liquid inside the drop is assumed Let its surface undergo vibrations of large amplitude. to be potential. This means that the field of the liquid Our aim is to calculate the shape of the drop and the velocities V(r, t) is wholly defined by the potential parameters of surface vibrations. We formulate the function ψ(r, t); i.e., V(r, t) = —ψ(r, t). problem in a more general nonlinear form than in − Under the above conditions, the evolution of the [2 10] and solve it by the multiple scale method up to drop is described by the Laplace equations for velocity terms of second order of smallness in amplitude of the potential ψ(r, t) and electrostatic potential Φ(r, t): initial disturbance of the drop surface. We use a different method of satisfying initial conditions than in ∆ψ()r, t ==0, ∆Φ()r, t 0, (2), (3) [2−10]. It allows us to obtain the solution up to second- order terms when the initial perturbation is associated where ∆ is the Laplacian, with the boundary conditions with the excitation of an arbitrary vibrational mode. r0: ψ()r, t0, (4) (2) Consider a drop of an ideal incompressible perfectly conducting liquid with a density ρ and a surface r∞: Φ()r, t 0, (5) tension coefficient σ and study the time evolution of its ∂ξ ∂ψ 1 ∂ξ ∂ψ shape. We assume that the drop is in a vacuum, its r ==1 + ξΘ(),t: ------Ð ------, (6) charge equals Q, and its volume equals the volume of a ∂t ∂r r 2 ∂Θ∂Θ sphere of radius R. At the initial time instant t = 0, the equilibrium spherical shape of the drop undergoes a ∂ψ 1 21 2 ∆pÐ------Ð--- ()Ð — ψ +------()— Φ =—n, (7) virtual axisymmetric perturbation of fixed amplitude ∂t 2 8π that is much smaller than the drop radius and is propor- Φ()Φ() tional to one of the Legendre polynomials. We aim at r, t=gt. (8) calculating the spectrum of drop capillary vibrations occurring under such conditions, i.e., at finding the To make system (2)Ð(8) closed, we formulate addi- shape of the drop at each subsequent time instant t > 0. tional conditions:

NONLINEAR CAPILLARY VIBRATIONS AND STABILITY 159 conservation of drop charge The time differentiation is performed with respect to each time scale by the rule r = 1 + ξ(Θ, t) 1 ∂ ∂ ∂ ∂ Ð------(n—Φ)dS = Q, where S = 0 ≤ Θ ≤ π (9) ε ε2 (ε2) π°∫ ∂---- = -∂------+ -∂------+ -∂------+ O . 4 t T0 T1 T 2 S 0 ≤ φ ≤ 2π, Substituting (14) into boundary problem (2)Ð(10) and conservation of drop volume and equating terms of the same order of smallness to each other in each of the equations, we easily obtain a 2 4 r drsinΘdΘdφ = --π, set of boundary problems from which one can succes- ∫ 3 sively determine the unknown functions ξ(m), ψ(m), and V Φ(m), where m = 1, 2, 3, … . Since Eqs. (2)Ð(5) are lin- where ear, each of the functions ψ(m) and Φ(m) from (14) must obey them. Therefore, these functions, as solutions to 0 ≤ r ≤ 1 + ξ(Θ, t) (2)Ð(5), may be given by

V = 0 ≤ Θ ≤ π (10) ( ) ψ m (r, Θ, T , T , T , …) 0 ≤ φ ≤ 2π. 0 1 2 ∞ The initial conditions for the problem will be set by (m)( , , , …) n (µ) taking the initial axisymmetric perturbation of the equi- = ∑ Dn T 0 T 1 T 2 r Pn , (15) librium spherical surface of the drop in the form n = 1 Φ(m)( , Θ, , , , …) t = 0: ξ(Θ) = ξ + εP (µ) (k ≥ 2) r T 0 T 1 T 2 0 k (11) µ ≡ cosΘ ∞ (m)( , , , …) Ð(n + 1) (µ) and by equating the initial speed of the surface to zero: = ∑ Fn T 0 T 1 T 2 r Pn . n = 0 ∂ξ(Θ) t = 0: ------= 0. (12) ξ(m) ∂t Successive corrections to the expression for drop shape will be sought as a series in Legendre poly- In (6)Ð(12), ∆p is the difference between constant nomials: pressures inside and outside the drop in equilibrium, n Φ ξ(m)( , Θ, , , , …) is the unit normal vector to surface (1), g(t) is the posi- r T 0 T 1 T 2 tion-independent potential on the drop surface, ε is the ∞ (16) amplitude of the initial perturbation of the drop surface, (m)( , , , …) (µ) µ ξ = ∑ Mn T 0 T 1 T 2 Pn . Pk( ) is the kth-order Legendre polynomial, and 0 is a constant determined from (10) up to terms of second n = 1 order of smallness: (4) Up to the zeroth order in ε, the solution for the equilibrium system is ξ ε2 1 (ε2) 0 = Ð ------+ O . (13) (2k + 1) ξ(0)( , Θ, , , , …) r T 0 T 1 T 2 = 0; (3) We will take advantage of the well-known mul- ψ(0)( , Θ, , , , …) tiple scale method to solve the problem up to quadratic r T 0 T 1 T 2 = 0; (17) ε terms in the small parameter [13]. To do this, we rep- (0) Q ξ Θ ψ Φ Φ (r, Θ, T , T , T , …) = --- , resent the required functions ( , t), (r, t), and (r, t) 0 1 2 r in the form of a series in ε and assume that the functions depend not merely on time but on various time scales as follows from (2)Ð(10). T ≡ εmt; that is, m (5) Up to the ﬁrst order in ε (m = 1), boundary con- ξ(Θ, ) εm, ξ(m)(Θ, , , , …) ditions (6)Ð(10) for solutions (15) and (16) to deter- t = ∑ T 0 T 1 T 2 , ( ) ( ) ( ) mine D m , F m and M m are transformed to m = 0 n n n ψ( , ) εm, ψ(m)(Θ, , , , …) ∂ξ(1) ∂ψ(1) r t = ∑ T 0 T 1 T 2 , (14) r = 1: --∂------= ----∂------, m = 0 T 0 r ( ) ( ) ( ) m (m) 1 Φ 0 ∂Φ 1 Φ(r, t) = ∑ ε , Φ (Θ, T , T , T , …). ∂ψ 1 d  0 1 2 Ð --∂------+ ----π----------∂------m = 0 T 0 4 dr r

TECHNICAL PHYSICS Vol. 46 No. 2 2001 160 SHIRYAEVA

Φ(0) O(1); the error in this case is of the order of ~ε2. How- d ξ(1) ( ∆ )ξ(1) + ------ = Ð 2 + Ω , ever, (23) can also be used in the time interval t ≤ O(εÐ1) ∂r2 (18) to analyze surface motion if the first-order solution is Φ(0) comparable to the initial perturbation. A more detailed Φ(1) d ξ(1) Φ(1) + ------Ð = g , estimate of the applicability of (23) can be obtained in dr the next (second) order of smallness in ε. π ∂Φ(1) 2Φ(0) Φ(0) Substituting (23) into initial conditions (11)Ð(13) d d  ξ(1) µ ∫ ------+ ------+ 2------ d = 0, with regard for (16) and (19)Ð(22) and equating the ∂r dr2 dr terms of the same order of smallness to each other, we 0 π easily find ∂ ∂ ξ(1) µ ∆ ( µ2)  ∫ d = 0, Ω = ------1 Ð ------. 1δ ( ≥ ) ∂µ ∂µ aˆ n = -- nk, bn = 0 n 1 , 0 2 δ δ From (18), we easily obtain first-order corrections where nk is the Kronecker symbol ( nk = 1, if n = k, to the coefficients of series (15) and (16): and 0 if n ≠ k). ( ) M 1 (T , T , T , …) = 0, Eventually, the function describing the surface evo- 0 0 1 2 lution in the linear approximation in ε has the form ( ) M 1 (T , T , T , …) 0 0 1 2 (19) ξ(Θ, t) = εcos(ω t)P (µ) + O(ε2). (24) ( , , …) ( ω ) k k = An T 1 T 2 exp i nT 0 + c.c. ( ≥ ) It follows from this solution that, in the first-order n 1 , approximation in the perturbation amplitude ε, the drop ∂ (1)( , , ) surface executes harmonic vibrations around the equi- (1)( , , , …) 1 Mn T 0 T 1 T 2 ( ≥ ) librium sphere that are associated with the kth (initially Dn T 0 T 1 T 2 = ------∂------n 1 , n T0 excited) mode. (1)( , , , …) (1)( , , , …) In the same approximation, the velocity and electro- Fn T 0 T 1 T 2 = QMn T 0 T 1 T 2 (20) static potentials have the form (n ≥ 0), ω ψ(r, t) = Ð ε-----k sin(ω t)P (µ) + O(ε2), A (T , T ) ≡ a (T , T , …)exp[ib (T , T , …)], k k k n 1 2 n 1 2 n 1 2 (25) 2 (21) Q Q 2 (1) 2 Q Φ(r, t) = --- + εcos(ω t)----P (µ) + O(ε ). Φ = 0, ω ≡ n(n Ð 1)[(n + 2) Ð W], W ≡ ------, k 3 k g n 4π r r where c.c. means complex conjugate terms. The func- (6) To find the functions ξ(2)(Θ, t), ψ(2)(r, t), and tions an and bn are determined up to higher orders of Φ(2)(r, t) (that is, second-order corrections to the solu- smallness. To complete the first-order problem, an and tions), we first derive a set of equations that is obtained 2 bn are set to be constants and derived from initial con- from (6)Ð(10) by equating terms proportional to ~ε : ditions (11) and (12). It is of interest to find the error of ( ) the first-order approximation of the solution and also ∂ξ(2) ∂ξ 1 r = 1: ------+ ------the time intervals where such an approximation is valid. ∂T ∂T It is clear that 0 1 (2) 2 (1) (1) (1) ( , , …) ≈ (ε ) ∂ψ ∂ Ψ (1) ∂ξ ∂ψ an T 1 T 2 aˆ n + O t ; = ------+ ------ξ Ð ------, (22) ∂r ∂r2 ∂Θ ∂Θ ( , , …) ≈ ˆ (ε ) bn T 1 T 2 bn + O t , (2) (1) 2 (1) ∂ψ ∂ψ ∂ ψ (1) where aˆ and bˆ are constants. Ð ------Ð ------Ð ------ξ n n ∂T ∂T ∂r∂T ε 0 1 0 From (14), we have (in the linear approximation in ) ( ) ( ) ( ) ( ) 1 ∂ψ 1  2 ∂ψ 1  2 1 ∂Φ 0 ∂Φ 2 ξ(Θ, ) ≈ εξ(1)[Θ, , ( , …) ( , …)] (ε2) Ð ------+ ------+ ------2------t T 0 an T 1 , bn T 1 + O 2  ∂r   ∂Θ  8π ∂r ∂r or, in view of (22), (1) 2 (1) 2 (0) 2 (0) ∂Φ  ∂Φ  dΦ d Φ (2) (1) + ------+ ------+ 2------ξ ξ(Θ, t) ≈ εξ (Θ, t, aˆ , bˆ ) + εO(εt). (23)     2 n n ∂r ∂Θ dr dr The error in this expansion is of the order of the first (0) 2 (1) 2 (0) (1) dΦ ∂ Φ d Φ ∂Φ  (1) term if t ≈ O(εÐ1). The expansion is not valid for t ≥ + 2 ------+ ------ξ  2 2 ∂  O(ε−1). Thus, (23) is applicable in the time interval t ≤ dr ∂r dr r

TECHNICAL PHYSICS Vol. 46 No. 2 2001 NONLINEAR CAPILLARY VIBRATIONS AND STABILITY 161

(0) 3 (0) 2 (0) 2 × [ (ω ω ) ] } dΦ d Φ d Φ  (1) 2 exp i m Ð l T0 + c.c. , + ------+ ------(ξ ) (26)  dr 3  2  γ ± ω ω η dr dr λ(+) mln m l mln mln = ------, ω2 Ð (ω ± ω )2 (2) (1) (1) n m l + (2 + ∆Ω)ξ Ð 2ξ (1 + ∆Ω)ξ = 0, (27) (0) (1) 2 (0) 2 (2) dΦ (2) ∂Φ (1) 1d Φ (1) 2 (2) ω [ ] ( Φ ξ ξ (ξ ) Φ γ ≡ m(n Ð m + 1) + 2n l(l + 1) Ð 1 + [l m + 1) + ------+ ------+ ------= g , mln Kmln dr ∂r 2 dr2

π W 1 2 W ( ) ( ) ( ) ( ) ( ) ] ---- α --- ω ---- Φ 2 ∂Φ 1 ∂Φ 1 ∂Φ 0 ∂Φ 0 Ð m(2m Ð 2n + 7) + 3 n + mln m + n , d   ξ(1)   2 m 2 ∫ ------------+ 2------ + ------+ 2------ dr ∂r2 ∂r ∂r2 ∂r 0 η ≡ n  α 1  n  mln Kmln-- Ð m + 1 + mln--- 1 + ---- , 3Φ(0) Φ(0) Φ(0) 2 m 2l × ξ(2) 1d d d  + ------+ 2------+ ------ 2 3 2 dr ≡ [ 000 ]2 α ≡ 000 Ð110 dr dr Kmln Cmln , mln Ð m(m + 1)l(l + 1)CmlnCmln , ∂Φ(1) ∂ξ(1) × (ξ(1))2 µ 0 if m + l + n = 2g + 1, Ð ---∂---Θ------∂----Θ----- d = 0, where g is an integer, π g Ð n ( ) ( ) 2 ( ) [ξ 2 (Θ, t) + (ξ 1 (Θ, t)) ]dµ = 0. Ð1 2n + 1g! ∫ 000 ≡ -(------)----(------)-----(------)--- Cmln g Ð m ! g Ð l ! g Ð n ! 0 (2g Ð 2m)!(2g Ð 2l)!(2g Ð 2n)! 1/2 × ------Now, substituting (15) and (16) (m = 2), along with (2g + 1)! solutions (17) and (19)Ð(21), into boundary conditions (26), one obtains differential equations for the coefficients if m + l + n = 2g(g is an integer), (2) Mn (T0, T1, T2, …). The solutions to this system do not Ð110 ≡ have secular terms if Cmln 2n + 1n! ( ) ( ) 1/2 ∂a ∂b × m + l Ð n !m m + 1 n n (------)-----(------)----(------)------(------)- ∂------= 0, ∂------= 0. n + m Ð l ! n Ð m + l ! m + l + n + 1 !l l + 1 T 1 T 1 ( )m + 1 + z( ) ( ) Hence, an and bn do not depend on time T1. Their × Ð1 m + z Ð 1 ! n + l Ð z + 1 ! ∑------(------)-----(------)----(------)----. dependence on the “slower” time scales T2, T3, etc. can z! m Ð z + 1 ! n Ð z ! l Ð n + z Ð 1 ! be found in higher orders of approximation. Finally, the z solution to the equations derived from (26) has the form In the last series, the summation is performed over all integer z for which the expressions under the facto- ∞ rial sign are nonnegative. The bar over An in (27) means (2)( , , …) 1 { ( , …) M0 T 0 T 2 = Ð ∑ (------)- An T 2 the complex conjugate. The ClebschÐGordan coeffi- 2n + 1 000 110 n = 1 cients Cmln and Cmln are other than zero only if their × ( , …) [ ( , …)]2 ( ω ) } An T 2 + An T 2 exp i2 nT 0 + c.c. , subscripts satisfy the relations [14] m Ð l ≤ n ≤ (m + l),  (28) (2) c (T , …) ( ) ( , , …)  n 2 ( , …) m + l + n = 2g g is an integer . M0 T 0 T 2 = ------(------,---…------)- + idn T 2  an T 2 The coefficients in series (15) for ψ(m)(r, t) and Φ(m)  (r, t) are related to solutions (19) and (27) as fol- × ( , …) ( ω ) ( , , …) lows: - An T 2 exp i nT 0 + c.c.  + Nn T 0 T 2 ,  ∂ (2)( , , …) (2) 1 M T T D (T , T , …) = --------n------0------2------∞ ∞ n 0 2 n ∂T ( ) ( )  0 N 2 (T , T , …) ≡ {λ + A (T , …)A (T , …) n 0 2 ∑ ∑ mln m 2 l 2 ∞ ∞ n = 1 l = 1 1 × ∑ ∑ [m(m Ð 1)K Ð α ]--- × [ (ω ω ) ] λ(Ð) ( , …) ( , …) mln mln m exp i m + l T0 + mln Am T 2 Al T 2 m = 1 l = 1

TECHNICAL PHYSICS Vol. 46 No. 2 2001 162 SHIRYAEVA

(1) ships: ∂M (T , T , …) ( )  × ------m------0------2------M 1 (T , T , …) , δ ∂ l 0 2 (1) (2) n0 T 0  t = 0: M = δ , M = Ð------, n nk n 2k + 1 (2) ( , , …) ( ) (2) F0 T 0 T 2 = 0, ∂ 1 ∂ Mn Mn ( , , , …) ------= 0, -----∂------= 0 n = 0 1 2 . ( ) ( ) ∂t t F 2 (T , T , …) = Q{M 2 (T , T , …) n 0 2 n 0 2 From these expressions and using (19), (20), and ∞ ∞ ˆ ˆ , , … (27), we determine the constants aˆ n , bn , cˆ n , and dn : + ∑ ∑ mKmln(T 0 T 2 ) m = 1 l = 1 1 aˆ = --δ , bˆ = 0, × (1) , , … (1) , , … } n 2 nk n Mn (T 0 T 2 )Ml (T 0 T 2 ) . 1 ( ) ˆ ( , , , …) cˆ n = Ð--Nn t = 0 , dn = 0 n = 0 1 2 . The coefficients cn(T2, …) and dn(T2, …) in (27) are 2 unknown functions of time. Like the functions an and (m) bn, they do not depend on the time scales T0 and T1. By Then, the coefficients Mn (t) from (19), (20), and analogy with the linear approximation, we complete (27) take the final form the study of the problem in the second-order approxi- (1)( ) δ (ω ) ε Mn t = nk cos nt , mation in by assuming that the unknown functions cn, dn, an, and bn are constants determined by initial condi- (2)( ) 1 [ ( ω )] M0 t = Ð----(------)- 1 + cos 2 kt , tions (11)Ð(13). Then, series (14) for drop surface per- 2 2k + 1 (30) turbation takes the form (2)( ) ( ) (ω ) ( ) ( ≥ ) Mn t = Ð Nn 0 cos nt + Nn t n 1 , ( ) ξ(Θ, t) ≈ εξ 1 (Θ, T , aˆ , bˆ ) 0 n n ( ) 1(λ(Ð) λ(+) ( ω )) (29) Nn t = -- kkn + kkn cos 2 kt . ε2ξ(2)(Θ, , , ˆ , , ˆ ) ε (ε2 ) 2 + T 0 aˆ n bn cˆ n dn + O t , ( ) It can be seen that the coefficients M 2 (t) are pro- ˆ ˆ n where aˆ n , bn , cˆ n , and dn are constants. λ(±) portional to the quantities kkn . Due to (27), the latter Approximation (29) is valid in the time interval t ≤ are proportional to the ClebschÐGordan coefficients ε3 ≤ 000 Ð110 O(1) with an error ~ . In the time interval O(1) < t Ckkn and Ckkn and, according to (28), are nonzero O(εÐ1), the error is comparable to the second term in only if n = 2j, where j = 0, …, k. (29), which is of second order of smallness. Therefore, Substituting (30) into (29), we find that the evolu- only the first term, associated with the linear approxi- tion of the drop surface is described up to terms of sec- mation, remains valid. Thus, approximate linear solu- ond order of smallness (in the time interval t ≤ O(εÐ1) tions (24) and (25) are strictly applicable (uniformly by the function suitable) in the time interval t ≤ O(εÐ1). The entire  expansion (29) can be used in the time interval t ≤ O(εÐ1) ξ(Θ, ) ≈ ε (ω ) (µ) ε21 1 t cos kt Pk Ð --------provided that the term quadratic in ε is a small correc- 2(2k + 1) tion to the linear term. k × [ ( ω )] [(λ(Ð) λ(+) ) (ω ) The characteristic time scale for the dimensionless 1 + cos 2 kt + ∑ k, k, 2 j + k, k, 2 j cos 2 jt (31) 3 1/2 variables adopted is t∗ = (R ρ/σ) . It is about 0.004 s j = 1 for a water drop of radius 1 mm and about 0.12 s when  (λ(Ð) λ(+) ( ω )) ] (µ) (ε3 ) R = 1 cm. Hence, (29) adequately describes time evolu- --Ð k, k, 2 j + k, k, 2 j cos 2 kt P2 j  + O t . tion of the water drop shape at times t ≤ 0.01 s for R =  1 mm and t ≤ 0.3 s for R = 1 cm. If the perturbation (7) It follows from (31) that the initial perturbation amplitude ε is about 0.1, solution (24) remains valid up of any even or odd kth single mode of capillary vibra- to times one order of magnitude greater than those tions results in the excitation (in the second order of given above. smallness) of only even modes with numbers from the Substituting (29) in view of (16) into initial condi- interval [0, 2k]. This is illustrated in Figs. 1aÐ1d. These figures show the time dependences of the amplitudes tions (11)Ð(13) and equating the terms of the same (2) order of smallness, we obtain the following relation- Mn of various second-order modes of charged drop

TECHNICAL PHYSICS Vol. 46 No. 2 2001 NONLINEAR CAPILLARY VIBRATIONS AND STABILITY 163

(2) (2) Mn Mn (a) (b) 2 2 0.04

0.02 0.02 3 4 4 3 0.01 3 0 1 1 2

0 1 2 3 –0.02 1

–0.01 –0.04

4 0.03 0.02 5 4 5 3 3 5 5 6 4 6 3 3 4 0 1 1 2 t 0 1 1 t

–0.03 –0.02

(2) Fig. 1. Dimensionless time dependence of the dimensionless amplitudes Mn of the (1) zeroth, (2) second (principal), (3) fourth, (4) sixth, (5) eighth, and (6) tenth modes of charged drop capillary vibrations at W = 3.9. The modes are excited as a result of second- order coupling. The initial deformation of the equilibrium spherical shape is due to the perturbation of the (a) second, (b) third, (c) fourth, and (d) fifth modes. The dimensionless amplitude of the perturbation is ε. capillary vibrations excited as a result of coupling [see tative agreement with data from [8] in the second order (31)]. The initial perturbation of the equilibrium spher- of smallness. ical shape is specified by the excitation of modes 2Ð5 It follows from Fig. 1 that the rate of increase in the (Figs. 1aÐ1d, respectively) at W = 3.9. (A spherical principal mode amplitude grows with the number of the drop becomes unstable when the parameter W attains mode that specifies the initial deformation. As the num- the critical value W∗ = 4.) It is seen that a drop with a ber of this mode increases, so does the total number of charge somewhat smaller than critical may exhibit its modes of capillary vibrations due to coupling. Odd charge instability through a fast increase in the princi- modes are excited only in the third-order approxima- pal mode amplitude (n = 2). Such behavior is indepen- tion as a result of coupling between the excited odd kth dent of the initial perturbation of the equilibrium spher- mode and second-order even modes produced by the ical shape. This result is in conflict with the predictions kth one. from the linear theory but is in qualitative agreement Figure 2 illustrates drop shapes calculated using (1) with [8], where nonlinear vibrations of a charged drop and (31) at various dimensionless time instants with W were numerically evaluated. When the initial perturba- values close to critical. In this case, the initial deformation is associated with the fifth mode, the time depen- tion of the equilibrium spherical shape is specified by dences of the excitation amplitudes are also in quanti- the perturbation of the third (Fig. 2a) and fourth

TECHNICAL PHYSICS Vol. 46 No. 2 2001 164 SHIRYAEVA

(a) (b) 1.0 1 4 2 1 3 1.0 4 2 3 5 5 0.5 0.5 3 4 5 1 2 4 1 2 3 5 5 4 2 3 1 1 3 2 4 5 –1.0 –0.5 0 0.5 1.0 1.5 –1.5 –1.0 –0.5 0 0.5 1.0 1.5

–0.5 –0.5

–1.0 –1.0

Fig. 2. Shapes taken by the drop at various time instants tε when the initial deformation ε = 0.3 of the equilibrium spherical shape is due to the virtual perturbation of various modes at W = 3.8. In both ﬁgures, curves 1 depict the equilibrium spherical shape and ε µ ε curves 2, the shape at the initial time instant [the spherical shape deformed by a perturbation of type Pn( )]. (a) n = 3 and t = (3) 0.065, (4) 0.26, and (5) 0.585; (b) n = 4 and tε = (3) 0.24, (4) 0.36, and (5) 0.57.

(Fig. 2b) modes. The axis of symmetry is horizontal. It initial perturbation. Therefore, an increase in the prin- should be noted that, according to perturbation theory, cipal mode amplitude to a value of the order of ε means formula (31) is uniformly suitable when t < εÐ1. It fol- an extension of the drop into a spheroid with the eccen- lows from the figures that the restriction on time t is tricity squared, e2 ≈ 3ε Ð 5.25ε2 [15]. Even small values actually even more stringent. Curves 4 in all the figures of ε ~ 0.1 will lead to a considerable extension of the were drawn at the boundary of the domain of uniform drop and, according to [16], to a decrease in the W value expansion applicability. This follows from a compari- critical for the onset of charge instability of the drop. In son of the resulting deviation of the drop shape the approximation linear in e2, this critical value for a (curves 4) from that at the initial time instant (curves 2). spheroidal drop has the form Moreover, the volume of the drop associated with 2 ( 2 ) ≈ [ ( ε ε2) ] curves 4 is apparently different from the initial one. Yet, W*(e ) = 4 1 Ð 2e /7 4 1 Ð 2 3 Ð 4.25 /7 .(32) it is seen that, when the initial deformation of the equi- Thus, if the Rayleigh parameter W is close to the librium shape is specified by even Legendre polynomi- critical value, the drop may become unstable. Let the als, its time evolution is also described by even Legendre Rayleigh parameter of the drop W = W+ be slightly polynomials and the shape remains symmetric about the smaller than its critical value W = 4. Then, from (32), origin. For large enough t (at the boundary of the domain ∗ where the solution is uniformly suitable), the drop tends one can find the current dimensionless amplitude a2 of to break down into two equal parts. If the initial deforma- the principal mode when the drop becomes unstable: tion is due to odd Legendre polynomials, the drop shape ≈ 1 { [ ( )]1/2} at each subsequent time instant is asymmetric about the a2 ------1 Ð 1 Ð 8.17 1 Ð 0.25W+ . origin even though only even modes are excited by sec- 3.5 ond-order mode coupling. At large t, such drops tend to If, for example, W+ = 3.6, the drop becomes unstable break down asymmetrically. when the dimensionless amplitude of the principal ≈ It is clear from physical reasoning that viscosity, mode reaches a value of a2 0.16. Being unstable, the which has not been taken into account in our study, drop loses some amount of charge by emitting many would lead to damping of all the modes. However, the fine, heavily charged daughter droplets [1, 17]. damping decrement for higher order modes is larger (8) It is seen from Fig. 1 that the amplitudes of all than for lower order ones. If a time interval is large second-order modes oscillate around some midlines enough, the amplitude of an initially excited higher above the abscissa. This means that drop vibrations order odd mode may go to zero faster than the ampli- occur not around a sphere but around a lower symmetry tude of lower even modes excited by it. Then, further body. Time-independent terms in (31) specify this vibrations of the drop and its possible breakdown into body: two parts will be symmetric.  (Θ) ≈ ε21 1 It follows from the above discussion and from the r 1 Ð --(------) figures that the amplitude of the principal mode of cap- 2 2k + 1 illary vibrations increases fastest, irrespective of the (33) k form of the initial deformation. The calculations used ( )  λ Ð (µ) (ε3 ) in our study are valid as long as the amplitudes of sec- Ð ∑ k, k, 2 jP2 j  + O t ,  ond-order modes are smaller than the amplitude of the j = 1

TECHNICAL PHYSICS Vol. 46 No. 2 2001 NONLINEAR CAPILLARY VIBRATIONS AND STABILITY 165

1.0 M6

1 0.5 0.04 1 2 3 4 5 –1.5 –1.0 –0.5 0 0.5 1.0 1.5 2

–0.5 0 1 2 3 4 t –1.0

Fig. 3. Bodies around which the drop nonlinearly vibrates at –0.04 W = 3 when the initial deformation of amplitude ε = 0.3 is due to the virtual excitation of various modes. Curve 1 is the equilibrium spherical shape, and curves 2Ð5 correspond to initially excited modes with numbers 2Ð5. where k is the number of an initially excited mode. Fig. 4. Time dependence of the dimensionless amplitude of the 6th mode when the 4th mode is initially excited with ε = It is easy to check that the body around which sec- 0.1 (1) under resonance and (2) far from it. ond-order vibrations occur is composed only of even modes independently of the type of first-order modes excited. Hence, this body is a prolate spheroid, not a resonance, W = 3.9 (curve 2), when the fourth mode sphere as predicted in the Rayleigh linear analysis [1]. was initially excited. The time interval in Fig. 4 is larger It is interesting that the parameters of such a spheroid than in Fig. 1c. The run of the curves is obviously dif- depend on the drop charge, i.e., on the parameter W, ferent. Resonance coupling takes place when terms λ± ω2 2 ω2 Ð1 through the coefficients k, k, 2 j . The shapes of the bod- with the factors ~( m Ð j n ) appear in (31). ies discussed above and calculated with (33) for various k are shown in Fig. 3. It is seen that, as the number of CONCLUSION an initially excited mode increases, the stretching of the body around which vibrations take place grows. The instability of a drop bearing a charge that is slightly smaller than critical was studied in the qua- (9) The solutions obtained also describe the reso- dratic approximation with respect to the amplitude of nance coupling between individual modes of drop the initial deformation. Due to coupling between vari- vibrations that was analyzed in detail in [3]. Such cou- ous modes of capillary vibrations, this instability can be pling arises when second-order modes oscillate with initiated by virtual excitation of not only the principal ω2 2 ω2 frequencies satisfying the condition m = j n for mode but of any mode, no matter whether it is odd or some value of charge Q, where j is an integer and m ≠ n. even. The drop exhibits nonlinear vibrations around a As a result, the amplitude of one of the modes increases prolate spheroid, not a sphere as follows from a linear in time periodically and infinitely (within our approxi- analysis. In contrast to aperiodic instability of such a mation). In [3], the resonance between the initially drop, which is a consequence of energy transfer excited fourth and sixth modes at W = 2.67, i.e., when between excited adjacent modes, vibrational instability ω2 ω2 may result from resonance coupling of nonadjacent 0 = 4 4 , was studied. The same resonance takes modes. place in our study when only the fourth mode is excited. The time dependence of the amplitude of the sixth mode excited by the initial generation of the REFERENCES fourth mode for W = 2.67 and small time intervals is 1. A. I. Grigor’ev and S. O. Shiryaeva, Izv. Ross. Akad. similar to curve 4 in Fig. 1c. However, Fig. 1c demon- Nauk, Mekh. Zhidk. Gaza, No. 3, 3 (1994). strates the results of calculation at W = 3.9; here, the 2. J. A. Tsamopoulos and R. A. Brown, J. Fluid Mech. 127, sixth mode amplitude increases because of energy 519 (1983). transfer between modes coupled in the second order. In 3. J. A. Tsamopoulos and R. A. Brown, J. Fluid Mech. 147, a time interval larger than that used in Fig. 1c, the 373 (1984). amplitude of the sixth mode decreases with time. The 4. J. A. Tsamopoulos, T. R. Akulas, and R. A. Brown, Proc. time dependence of the sixth mode amplitude is shown R. Soc. London, Ser. A 401, 67 (1985). in Fig. 4 for comparison. The graphs are constructed for 5. R. Natarajan and R. A. Brown, Proc. R. Soc. London, resonance conditions, W ≈ 2.667 (curve 1), and far from Ser. A 410, 209 (1987).

TECHNICAL PHYSICS Vol. 46 No. 2 2001 166 SHIRYAEVA

6. N. A. Pelekasis, J. A. Tsamopoulos, and G. D. Manolis, 13. A.-H. Nayfeh, Perturbation Methods (Wiley, New York, Phys. Fluids A 2 (8), 1328 (1990). 1973; Mir, Moscow, 1976). 7. T. G. Wang, A. V. Anilkumar, and C. P. Lee, J. Fluid 14. D. A. Varshalovich, A. N. Moskalev, and V. K. Kherson- Mech. 308, 1 (1996). skiœ, Quantum Theory of Angular Momentum (Nauka, 8. Z. Feng, J. Fluid Mech. 333, 1 (1997). Leningrad, 1975; World Scientiﬁc, Singapore, 1988). 9. D. F. Belonozhko and A. I. Grigor’ev, Pis’ma Zh. Tekh. 15. S. O. Shiryaeva, A. I. Grigor’ev, and I. D. Grigor’eva, Fiz. 25 (15), 41 (1999) [Tech. Phys. Lett. 25, 610 Zh. Tekh. Fiz. 65 (9), 39 (1995) [Tech. Phys. 40, 885 (1999)]. (1995)]. 10. O. A. Basaran and L. E. Scriven, Phys. Fluids A 1 (5), 16. S. O. Shiryaeva and A. I. Grigor’ev, Zh. Tekh. Fiz. 66 (9), 795 (1989). 12 (1996) [Tech. Phys. 41, 865 (1996)]. 11. A. I. Grigor’ev and S. O. Shiryaeva, Zh. Tekh. Fiz. 69 17. A. I. Grigor’ev and S. O. Shiryaeva, Zh. Tekh. Fiz. 61 (7), 10 (1999) [Tech. Phys. 44, 745 (1999)]. (3), 19 (1991) [Sov. Phys. Tech. Phys. 36, 258 (1991)]. 12. S. I. Shchukin and A. I. Grigor’ev, Zh. Tekh. Fiz. 68 (11), 48 (1998) [Tech. Phys. 43, 1314 (1998)]. Translated by V. Gursky

TECHNICAL PHYSICS Vol. 46 No. 2 2001

GAS DISCHARGES, PLASMA

Influence of the Ion Charge Sign on the Stimulation of Plasmochemical Etching of Silicon S. N. Pavlov Scientiﬁc Center Institute for Nuclear Research, National Academy of Sciences of Ukraine, Kiev, 03028 Ukraine e-mail: [email protected] Received February 1, 2000

Abstract—The influence of ions with different charge signs on the stimulation of silicon etching under plasma conditions is studied. Fluorine radicals are produced in a glow discharge with a nonuniform pressure. A beam of positive or negative ions is created using a Penning ion source. The flow of fluorine radicals and the ion beam are superposed on a silicon surface placed in a high vacuum. Positive ions may be converted into fast neutrals via resonance charge exchange in the parent gas. It is shown that fast neutrals have the highest catalytic effect. The catalytic effect of positive ions is about two times less. Negative ions occupy the intermediate position. For the first time, it is found that some kinds of ions (e.g., molecular oxygen) do not accelerate, but rather decelerate the etching process; i.e., they behave as inhibitors. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION The process of plasmochemical etching can be divided into the following main stages: (i) delivery of The phenomenon of ion-stimulated etching of sili- the working gas molecules to the plasma discharge con was discovered more than 20 years ago [1, 2] and, region, (ii) transformation of the working gas mole- at present, is widely used in microelectronics [3, 4]. cules into chemically active particles via dissociation The phenomenon, in essence, is as follows. When a sil- and ionization, (iii) delivery of the chemically active icon surface is bombarded by ions, e.g., argon ions with particles to the etched surface, (iv) physical and chem- energies of 500Ð1000 eV, the solid surface is sputtered. ical adsorption of the chemically active particles on the Under the above conditions, the sputtering yield is surface, (v) the chemical reaction itself, (vi) desorption approximately unity. In microelectronics, this process of the reaction products from the surface, and (vii) is referred to as ion-beam etching and is characterized removal of the reaction products from the plasma by the rate Vph = dh/dt, where h is the plate thickness. region and vacuum chamber. This comprehensive clas- Another limiting case is etching by halogen radicals sification is given according to [4]. Stages (i), (iii), and (F*, Cl*, Br*, or I*, fluorine being most often used). (vii) are important for designing plasma chemical reac- This is a heterogeneous reaction of the gasÐsolid type. tors but do not directly affect ion stimulation itself. Silicon is removed as a result of a chemical reaction Stage (ii) may have only indirect influence via the proceeding according to the generalized formula Si + chemical composition of the reacting system. It worth 4F SiF . Silicon tetrafluoride is a gas that sponta- noting that mass spectroscopy and optical studies show 4 [7, 9] that all of the neutral fragments of the molecules neously leaves the surface of the treated material. In of the initial gas (as a rule, CF4 or SF6), as well as their this case, the plasma is only used to obtain fluorine rad- positive and negative ions, have been recorded in an RF icals from the molecules of more stable initial gases discharge plasma. However, more than 75% of dissoci- (CF , SF , etc.). Etching proceeds at a rate of V , 4 6 ch * which depends on the halogen concentration and the ating CF4 molecules decompose into CF3 and F* rad- solid temperature. If the silicon surface is exposed to icals [10]. the simultaneous action of an ion flow and fluorine rad- When analyzing the mechanism for ion stimulation, icals, etching proceeds at a rate of Vi, which may be most authors restrict themselves to considering stages several times higher than the sum Vph + Vch. The process (iv)Ð(vi). Let us consider the possible consequences of is synergistic in character [5, 6] and, hence, is of gen- ion bombardment of a silicon surface. In stage (iv), ions eral scientific interest. It can be observed not only in sil- may (a) enhance nondissociative chemisorption due to iconÐhalogen systems, but also in a number of gasifica- the breaking of SiÐSi bonds in the crystal lattice, which tion reactions, such as carbon–oxygen or tungsten–flu- results in the formation of free valence silicon bonds, orine reactions. Keeping in mind the general character and (b) cause the dissociation of complex radicals (like of the results, we nevertheless will focus on the SiÐF CF3* or SF5* ) absorbed on the surface with subsequent system, because it is the most widely used in practice chemisorption of their fragments. In stage (v), the reac- and, hence, the best studied. tion rate may increase due to surface loosening, which

168 PAVLOV leads to both the breaking of SiÐSi bonds and the for- ters constant. The design and features of the device are mation of channels in the treated material, along which described in the next section. chemical reagents can penetrate into deeper layers. In Sulfur hexafluoride and oxygen were used as work- stage (vi), ion bombardment may cause forced desorp- ing gases because they form a very compatible pair. To tion of (a) the intermediate reaction products (usually study the problem in question, it is necessary to pro- volatile SiF2), (b) the final product SiF4, and (c) the duce beams of positive ions, negative ions, and fast nonvolatile products. neutrals of the same gas with approximately the same There are arguments in favor of each of the effects intensities. Fluorine is less appropriate for such pur- listed above. The problem is that the results of each poses. Our investigations showed that in a Penning dis- experiment are usually treated as supporting one of the charge, which is often used as an ion source, the inten- mechanisms, but as discarding the other ones. For sity of the beam of negative fluorine ions is almost one example, the authors of [5, 11] conclude that dissocia- order of magnitude higher than that of F+ ions. Due to tive chemisorption plays a major role, assuming that inevitable loss during charge exchange, we could not surface loosening is of minor importance. The results obtain fast neutrals in the necessary amounts. As will be of [12Ð14] contradict this opinion. In 1978, Mauer et shown below, oxygen is free of these drawbacks. On al. [15] showed that ion bombardment may cause the other hand, the silicon etching rate is constant over induced desorption of the intermediate reaction compo- a wide range of the O2 relative concentration in a mix- nent SiF2. This concept has been well developed in ture of sulfur hexafluoride with oxygen, which consid- recent years. The results of ion-stimulated etching erably facilitates the interpretation of the results. [16, 17] show that surface bombardment by heavy par- Finally, the yield of Si atoms per incident ion is much ticles may result in the complete rearrangement of higher for etching in SF6 in comparison with that in chemical bonds in the SF6ÐSi system, which makes the CF4, which facilitates performing the experiments. situation even more intricate. As reference beams, we used beams of argon ions Thus, a unified opinion on the mechanism for ion and fast argon atoms, because argon has been used in a stimulation of plasmochemical etching of silicon is still great number of experiments and its properties are well lacking; therefore, the problem calls for further analy- studied. sis. Taking into account the previous long-term and comprehensive investigations, one can expect that only new approaches may lead to considerable progress. An EXPERIMENTAL SETUP attempt at such an approach, as well as the first experi- As was mentioned above, the prototype of our mental results, is described below. experimental device was that used in [5]. The idea of the experiment is to use two independent sources, one of which produces an ion flow and the other produces a FORMULATION OF THE PROBLEM flow of radicals. Both flows are superposed on a silicon The following fact is worthy of attention. Etching of surface placed in a high vacuum. In this case, the sys- silicon in halogen-containing plasma can be acceler- tem parameters can be varied independently over a ated not only by ion flows, but also by electron flows. wide range. The experiments were carried out in a cap- The difference is that ion stimulation occurs in almost type device at a background pressure of ~2 × 10Ð3 Pa. any medium, whereas electron stimulation can occur A diagram of the device is shown in Fig. 1. only in some specific media. Thus, electron beams with 2 We used a Penning discharge with a cold cathode as densities of 2Ð4 mA/cm increase the etching rate by a an ion source. The source design was similar to that factor of four to six when CF4 or CF3Cl is used; how- described in [20]. The source chamber was sealed and ever, the effect is not observed in XeF2 and SF6 plasmas connected to a high-vacuum chamber through a [11, 18]. This means that it is expedient to investigate 1.5-mm-diameter aperture. This allowed us to have a the influence of the ion charge sign on the stimulation working pressure in the source of 2Ð10 Pa at a pressure of silicon etching. This problem is of especial interest, in the high-vacuum chamber no higher than 10Ð2 Pa. because negative ions are proposed to be used to A cell was placed in the gap of a magnetic core assem- improve the treatment of semiconductors (see, e.g., bled from permanent magnets. The magnetic induction [19]). in the center of the source was ~0.07 T. The character- Finally, when comparing positive and negative ions, istic discharge current was 100Ð120 mA at a discharge it would also be reasonable to investigate the role of fast voltage of 600Ð750 V (the cathode diameter was 2 cm, neutrals, which can be regarded as ions with zero and the anode length was 2.2 cm). The working gases charge. were argon and oxygen. The ions were extracted across The prototype of our experimental device was that the magnetic field. The extractor voltage was 2.5 kV. used in [5]. Such a system allows one to imitate plasma The beam was focused by a single lens and then decel- conditions, which offers wide experimental possibili- erated to an energy of 1 keV. ties, because in a real plasma, it is impossible to change As is known [21], when an ion beam is extracted only one of the parameters and hold the other parame- from the source into a vacuum, intense ion charge

TECHNICAL PHYSICS Vol. 46 No. 2 2001

INFLUENCE OF THE ION CHARGE SIGN 169 exchange occurs in the parent gas near the extraction 7 15 S9 10 aperture. Under our conditions, the conversion coeffi- 5 4 cient due to this effect can attain 30%. To prevent fast 3 neutrals from reaching the silicon surface, the beam 2 x 1 i was turned through an angle of ~12° by a cylindrical electrostatic capacitor and then passed through a Winn filter, which was assembled from permanent magnets in y an armored case. The transit base was ~6.5 cm. The 8 magnetic field was almost uniform in the ~4-cm-long 6 central region, where the magnetic induction attained 15 14 0.16 T. Outside of this region, the magnetic induction dropped steeply. The distance between the plates of the 13 electrostatic capacitor in the filter was 8 mm. The plates 15 12 were 14 mm high. The inlet and outlet slits of the filter 11 were rectangular in shape (5 × 12 and 3 × 12 mm in size, respectively). About 2 cm behind the filter, the Fig. 1. Diagram of the experimental device: (1) ion source, strip beam reached a grounded receiver (Fig. 2). The (2) extractor, (3) single lens, (4) decelerating electrode, upper part of the beam was used to treat the silicon sur- (5) electrostatic turning capacitor, (6, 7) magnet yoke and face. A 2.5-mm-diameter aperture 3 (Fig. 2) in the main electrostatic capacitor of the Winn filter, (8) inlet and outlet plate of the receiver was used to monitor the beam filter slits, (S9) auxiliary electrode, (10) receiving unit, (11) cathode, (12) quartz retort, (13) glass tube, (14) auxil- parameters. The system was supplied with an addi- iary anode, (15) gas inlet, and (i) trajectory of the ion beam. tional electrode ES, which either suppressed secondary electron emission or extracted secondary electrons from collector 2, depending on the voltage applied. 5 4 1 When operating with positive ions, electrode ES was used to suppress secondary emission. In the Winn filter, the ion beam was separated either z + + Ð Ð + by mass (O2 and O ; O2 and O ) or by charge (Ar and Ar2+). Figure 3 shows that the source produced negative i and positive oxygen ions in comparable amounts in y both atomic and molecular form. The separation degree was 70Ð90%. The beam of argon ions almost com- + pletely consisted of Ar . 3 0 0 0 ES Fast neutrals (Ar , O2 , and O ) with an energy of 2 1 keV were produced from positive ions via resonant Fig. 2. Diagram of the receiving unit: (1) grounded case, + + charge exchange in the parent gas (Ar in Ar, O2 in O2, (2) collector, (3) aperture for monitoring the beam parame- + ters, (4) treated silicon plate, (5) shielding case, (i) ion and O in O2). For this purpose, the sides of the electro- beam, (ES) electrode for suppression of secondary emission static capacitor of the Winn filter were sealed to obtain or extraction of secondary electrons from the collector. The a 64-mm-long tube of a rectangular cross section (8 × x- and y-axes in Fig. 1 and the z- and y-axes in Fig. 2 make 14 mm). In the middle part of the tube, there were aper- the same left-hand Cartesian triple. tures for measuring the pressure and the gas flow rate. Such a design, together with a high-vacuum space range in which kinetic ejection prevails over potential under the cap of the device, formed a charge-exchange ejection. chamber with differential pumping. The inlet and outlet The measurement procedure was as follows. In the slits of the Winn filter increased the pressure gradient. beginning, the gas inlet into the recharging chamber At a gas pressure of ~0.4 Pa in the center of the charge- was shut, an auxiliary electrode S9 (Fig. 1) was exchange chamber, the pressure in the space under the Ð2 grounded, and the beam losses were almost absent. The cap was no higher than 10 Pa. Winn filter was tuned to a certain type of ions, e.g., Ar+. The degree of beam charge exchange was measured By varying the voltage polarity on electrode ES, we according to the technique described in [22]. The tech- could either suppress secondary electron emission or nique is based on the assumption that ions and fast neu- extract all secondary electrons from the collector. In the trals of the same elements eject electrons from the first case, we obtained the ion current Ii, whereas in the metal surface with the same efficiency; i.e., their coef- second case, we obtained the collector current Ik = Ii + ficients of secondary emission are the same (ki Ð e = Ie (where Ie is the secondary electron current). Then, we kn − e). Strictly speaking, this is true only in the energy have ki Ð e = Ie/Ii = (Ik Ð Ii)/Ii. It turned out that this

TECHNICAL PHYSICS Vol. 46 No. 2 2001 170 PAVLOV

Ii, nA Then, the gas was supplied to the charge-exchange chamber. A part of the ions was recharged, while the 600 (a) other part passed through the chamber unchanged. + + O2 O When a sufficiently high positive voltage was applied to electrode S9, all the recharged ions were reflected 300 from it and were not able to leave the Winn filter. This can be easily verified by measuring the collector current in the regime of suppression of secondary emission. The measurements showed that the total reflection was observed at US9 higher than 1100 V. Let us now 600 (b) change the polarity of electrode ES to positive and mea- O– sure the current of secondary electrons. Since the ions – do not reach the collector, the current I may be caused O2 e 300 only by the flow of fast neutrals. According to [22], we have In = Ie/ki Ð e. Here, the quantity In is the electric equivalent of the flow of neutrals, i.e., the current to the collector if every fast atom had a charge equal to the electron charge. 600 (c) When the gas inlet to the charge-exchange chamber is open, electrode S9 is grounded, and secondary emis- Ar+ sion from the collector is suppressed, one more param- 300 eter Iir (the current of unrecharged ions) can be measured. This current was measured only in auxiliary Ar2+ experiments. 0 100 200 The dependence of Iir, In, and their sum Is on the gas U, V pressure in the central part of the charge-exchange chamber is shown in Fig. 4. The working gas is argon. Fig. 3. Mass spectra of the ion beam: Ii is the beam current It is seen that In reaches its peak value at P ~ 0.5 Pa and recorded by the collector of the receiving unit, U is the voltage on the electrostatic capacitor of the Winn filter, the pres- then decreases slowly. The latter is associated with the sure in the source is (a, b) ~6 and (c) ~4 Pa, and the dis- scattering of fast neutrals by gas atoms. A similar charge current is (a, b) 120 and (c) 80 mA. dependence is observed for molecular and atomic oxygen, but at a pressure two to three times higher. I, nA We draw attention to two important facts. First, Is is 400 equal to the initial current of the ion beam (before the gas was supplied to the charge-exchange chamber) with a high accuracy in the range P ≤ 0.2 Pa, which supports I the validity of this measurement technique. Second, if s we assume that the measurement error is high, the intensity of the actual flow of neutrals cannot be higher I than a preset value, because the intensity of the total 200 n flow of neutrals and unrecharged ions cannot exceed that of the initial positive ion beam. A glow discharge in a gas flow with a highly nonuni- Iir form pressure served as a source of fluorine radicals (Fig. 1). A duralumin cathode 11 with an area of 2 cm2 was placed into quartz bulb 12 with a volume of 3 0 ~10 cm , into which SF6 gas was input. The bulb was 0.4 0.8 connected to a vacuum chamber through a 22-cm-long P, Pa glass tube 13 with an inner diameter of 8 mm. There Fig. 4. Unrecharged ion current I , electric equivalent of the was a side branch at a distance of 6 cm from the tube ir end, where the auxiliary anode 14 was placed. At a flow of fast neutrals In, and their sum Is = Iir + In as functions 3 of the pressure in the charge-exchange chamber; the work- standard SF6 flow rate of 900 cm Pa/s, the gas pressure ing gas is argon. near the cathode was 25Ð30 Pa and, in the space under the cap, it was ~7 × 10Ð3 Pa. The auxiliary anode was parameter varied in a rather wide range; hence, it was used to help initiate the discharge with a highly nonuni- measured more than once at certain time intervals dur- form pressure. In the normal regime, the voltage across ing every experiment. the discharge was 2.5Ð3 kV and the current was 4 mA.

TECHNICAL PHYSICS Vol. 46 No. 2 2001 INFLUENCE OF THE ION CHARGE SIGN 171

The current was about equally shared between the aux- h, µm iliary anode and the grounded elements of the cap. 10 Fluorine atoms were obtained due to dissociation of (a) sulfur hexafluoride molecules in the positive column of the discharge. The measurements showed that, outside of the glass tube, the etching rate of Si, which is deter- 5 mined by the concentration P*, is described well by the 2 dependence Vet ~ 1/R , where R is the distance from the tube end. The axis of the radical source made an angle 0 ° of ~20 with the plane of the receiver plate and with the 10 silicon surface. The distance between the tube end and d (b) the center of the ion beam (Fig. 1) was 2 cm. A permanent magnet was placed behind the receiver plate to ∆ create a magnetic field (~7 × 10Ð3 T) directed along the 5 h y-axis (Fig. 1). This field prevented electrons of the glow discharge from reaching the region where the e beam ions and fluorine radicals interacted with the 0 treated surface. 10 (c) MEASUREMENT TECHNIQUE AND EXPERIMENTAL RESULTS 5 The etching rate was determined by direct measurements of the etch depth. For this purpose, Si sample 4 was placed on receiving plate 1 (Fig. 2). Its upper part 0 was protected from the influence of fluorine and the ion 1 2 3 beam by casing 5. The interferometer microscope mea- x, cm sured the height of the step between treated and untreated parts of the sample; usually, it was about sev- Fig. 5. Qualitative dependence of the silicon etch depth on eral microns. As was mentioned in the previous section, the x coordinate (the point x = 0 is at the end of the radical 2 source tube; see Fig. 1): (a) the ion source is switched off, the rate of radical etching followed the law Vet ~ 1/R . and only the radical source operates; (b) etching is stimu- For this reason, when the ion beam was switched off, lated by the ion flow; and (c) the case when the process is the profile of the etch depth along the x-axis had the decelerated by some external factors. shape shown in Fig. 5a. The origin of the x-axis was at the end of the glass tube. When the silicon surface was the value I∆t, its integral value Φ = ∫I dt was measured. simultaneously exposed to the radical and ion flows, the etching was accelerated in the region of their combined To determine it, an integrating scheme was installed in action and a step appeared in the profile of the etch the collector circuit. To find the yield of silicon atoms per incident fast neutral atom, we measured the value of depth (Fig. 5b). It is easy to extrapolate curves d and e Φ and to obtain the value of ∆h. The distance R was taken , in which the current I was replaced with the equiva- ∆ ≅ lent current In described above. As a result, we obtained equal to 2 cm to obtain h h, where h is the depth of ∆ ρ µΦ silicon etching by fluorine radicals without ion stimula- Y = h NaeS/ . For each type of stimulating particle, tion. a series of not less than five measurements was performed. The averaged data are given below. The yield of silicon atoms per incident ion Y (the main measured quantity) was determined as follows. It is known [13] that the yield of silicon atoms Let the increment of the etch depth due to ion bombard- strongly depends not only on the ion mass and ion ment during the time ∆t be ∆h. The number of atoms energy, but also on the experimental conditions. The removed from the surface is determined by the expres- yield is typically from 4 [5] to 20 [23] for Ar+ ions with ∆ ρ µ ρ µ an energy of 1 keV. In our experiments, we obtained sion NSi = h Na/ , where and are the density and Y(Ar+) = 5.7 ± 1.3 atom/ion. Since this value was closer molar weight of silicon, respectively, and Na is Avogadro’s number. The number of ions incident per to the results of [5], we used it as a reference value. ∆ ∆ 0 ± unit area is Ni = j t/e = I t/eS, where j is the ion current For fast argon atoms, we obtain Y(Ar ) = 10.6 density, I is the collector current, S is the collector area, 1.5 atom/neutral, i.e., Y(Ar0)/Y(Ar+) ≅ 1.85. This result and e is the electron charge. Since the beam current is unexpected for two reasons. First, as mentioned in density was about several microamperes per square the Introduction, the stimulating influence of ion bom- centimeter, the value ∆t was typically several hours. It bardment is mainly associated with the breaking of was impossible to keep the parameters of the source Si−Si bonds, loosening of the silicon surface, and unchanged over such a long period. Hence, instead of induced desorption of the intermediate products of the

TECHNICAL PHYSICS Vol. 46 No. 2 2001 172 PAVLOV

SiFx reaction (x = 1, 2, or 3). The existence of these cess of etching, although the chemical element is the effects and their substantial contribution to the etching same. We cannot exclude the possibility that the reason process is undoubted. However, this influence is only is the difference between the ion masses and, hence, the mechanical; it cannot depend on the states of the upper ion momentum at the same energy. However, it is electron shells (recall that the ion energy is 1 keV, while unlikely because, as shown in [23], the yield of silicon the electron transitions at the upper shells correspond to atoms per incident ion is a very weak function of the ion energies of one to two tens of electronvolts). This energy, at least in the range from 500 to 3000 eV. Solv- makes us assume the existence of an additional etching ing this problem requires further investigations. channel, which has not been previously accounted for. Second, it is known [24, 25] that when an ion approaches the surface, it is usually neutralized before CONCLUSIONS colliding due to Auger or tunnel processes; i.e., almost The first results of investigations of the influence of always, it is fast neutrals that reach the surface. How- the ion charge sign on the stimulation of plasmochem- ever, the place, time, and mechanism of neutralization ical etching of silicon show the following. are different. For this reason, we doubt the validity of (1) Among the particles that accelerate the etching the measurement technique for In. However, as was process, fast neutrals have the maximum catalytic shown in the previous section, even if we assume the effect. The catalytic influence of positive ions is about presence of a substantial error, the actual flow of fast two times less. Negative ions occupy an intermediate neutrals cannot exceed a preset value, because this position. would contradict the current balance. On the other (2) It is found that some kinds of ions do not accel- hand, Y ~ 1/j and, consequently, the obtained value of erate, but decelerate the etching process; i.e., they Y(Ar0) is actually the lower limit. Therefore, it is behave as inhibitors. Among such particles (at energies proved that stimulating influence of ion bombardment + Ð on plasmochemical etching of silicon depends not only of 1 keV) are all kinds of molecular oxygen (O2 , O2 , 0 on the ion mass and ion energy, but also on the popula- and O2 ). tion of its upper electron levels. (3) The results obtained allow us to conclude that The experiments showed that atomic oxygen also the effects of loosening the silicon surface and forced + accelerates the etching process. In this case, Y(O ) = desorption of the intermediate reaction products 1.9 ± 0.2 atom/ion, Y(O0) = 3.4 ± 0.5 atom/neutral, and incompletely describe the influence of ion bombard- Y(OÐ) = 2.1 ± 0.5 atom/ion. Fast neutrals have the stron- ment. There should be an additional channel of silicon gest stimulating effect: Y(O0)/Y(O+) ≅ 1.8. Negative etching, whose efficiency depends not only on the ions take an intermediate position. energy and mass of the ion, but also on the population Unlike the particles mentioned above, the ions of of its upper electron levels. molecular oxygen do not accelerate, but decelerate the etching process. As a result, a hole (rather than a step) REFERENCES is formed in the etch depth profile in the region exposed to the beam action (Fig. 5c). For convenience, this pro- 1. N. Hosokawa, R. Matsuzaki, and T. Asamaki, Jpn. J. cess can be regarded as negative catalysis, whereas its Appl. Phys., Suppl. 2, 435 (1974). intensity can be characterized by negative values of Y, 2. L. Holland and S. M. Ojha, Vacuum 26 (1), 53 (1976). because ∆h is negative. It was found that all kinds of 3. Plasma Processing for VLSI, Ed. by N. Einspruch and + Ð 0 D. Brown (Academic, New York, 1984; Mir, Moscow, molecular oxygen (O2 , O2 , and O2 ) exhibit properties 1987). of inhibitors with Y(O+ ) = Ð2.9 ± 0.3 atom/ion, 4. B. S. Danilin and V. S. Kireev, Applications of Low Tem- 2 perature Plasmas for Etching and Purification of Mate- 0 ± Ð ± Y(O2 ) = Ð2.3 0.2 atom/neutral, and Y(O2 ) = Ð2.6 rials (Énergoatomizdat, Moscow, 1987). 0.3 atom/ion. Negative ions again occupy an intermedi- 5. J. W. Coburn and H. W. Winters, J. Appl. Phys. 50, 3189 ate position between positive ions and fast neutrals. The (1979). observed decelerating action cannot be directly associ- 6. D. L. Flamm and V. M. Donnelly, J. Vac. Sci. Technol. B 1, 23 (1983). ated with the formation of SiO2, which, as is known, is etched by fluorine at a much slower rate than pure sili- 7. M. J. Pabst, H. S. Tan, J. L. Franklin, et al., Int. J. Mass Spectrom. Ion Phys. 20 (1), 191 (1976). con. Such effects are observed in the mixture of SF6 and oxygen only when the relative concentration of O is 8. R. F. Williams, in Plasma Processing of Semiconduc- 2 tors, Ed. by R. F. Williams (Kluwer, Dordrecht, 1996) higher than 50%. In our case, the flow of SF6 molecules [NATO ASI Ser., Ser. E 336, 321 (1996)]. onto the target is more than two orders of magnitude 9. J. Perrin, in Plasma Processing of Semiconductors, higher than that of molecular ions. Ed. by R. F. Williams (Kluwer, Dordrecht, 1996) [NATO The qualitative difference between the influence of ASI Ser., Ser. E 336, 397 (1996)]. atomic and molecular oxygen is noteworthy. The 10. D. I. Slovetskiœ, in Plasma Chemistry (Énergoizdat, former accelerates, while the latter decelerates the pro- Moscow, 1983), Vol. 10, p. 108.

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11. H. F. Winters and J. W. Coburn, J. Vac. Sci. Technol. B 18. V. Yu. Kireev and M. A. Kremerov, Élektron. Tekh., 3, 1376 (1985). Ser. 3: Mikroélektron., No. 1, 3 (1985). 12. V. M. Donnelly, N. Layadi, J. T. C. Lee, et al., in Plasma 19. J. H. Keller, Plasma Phys. Controlled Fusion 39, A437 Processing of Semiconductors, Ed. by R. F. Williams (1997). (Kluwer, Dordrecht, 1996) [NATO ASI Ser., Ser. E 336, 20. T. I. Danilina, E. V. Ivanova, Yu. E. Kreœndel’, and 243 (1996)]. L. A. Levshchuk, Prib. Tekh. Éksp., No. 3, 158 (1968). 13. H. F. Winters and J. W. Coburn, Surf. Sci. Rep. 14 (4Ð6), 21. J. B. Hasted, Physics of Atomic Collisions (Butterworths, 161 (1992). London, 1964; Mir, Moscow, 1965). 22. M. S. Ioffe, R. I. Sobolev, V. G. Tel’kovskiœ, and 14. D. L. Flamm, in Plasma Processing of Semiconductors, E. E. Yushmanov, Zh. Éksp. Teor. Fiz. 39, 1602 (1960) Ed. by R. F. Williams (Kluwer, Dordrecht, 1996) [NATO [Sov. Phys. JETP 12, 1117 (1960)]. ASI Ser., Ser. E 336, 23 (1996)]. 23. U. Gerlach-Meyer and J. W. Coburn, Surf. Sci. 103, 177 15. J. L. Mauer, J. S. Logan, L. B. Zielinski, and (1981). G. C. Schwartz, J. Vac. Sci. Technol. 15, 1734 (1978). 24. H. D. Hagstrum, Phys. Rev. 104, 672 (1956). 16. V. Yu. Kireev, D. A. Nazarov, and V. I. Kuznetsov, Élek- 25. H. D. Hagstrum and Y. Tekeishi, Phys. Rev. 139, 526 tron. Obrab. Mater., No. 6, 37 (1986). (1965). 17. D. J. Oostra, A. Haring, A. E. de Vries, et al., Nucl. Instrum. Methods Phys. Res. B 13, 556 (1986). Translated by M. Astrov

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Technical Physics, Vol. 46, No. 2, 2001, pp. 174Ð178. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 71, No. 2, 2001, pp. 43Ð47. Original Russian Text Copyright © 2001 by Boguslavsky, Khaœnatsky, Shcherbak.

GAS DISCHARGES, PLASMA

Optical Studies of the PlasmaÐLiquid Transition Layer in Pulsed Corona Discharges in Strong Water Electrolytes L. Z. Boguslavsky, S. A. Khaœnatsky, and A. N. Shcherbak Institute of Pulse Research and Engineering, National Academy of Sciences of Ukraine, Oktyabrskiœ pr. 43a, Nikolaev, 54018 Ukraine e-mail: [email protected] Received March 17, 2000

Abstract—The properties of corona discharge in strong water electrolytes are studied experimentally. An analysis of the measured time-integrated spectrum of the corona plasma emission shows the presence of a thin vapor layer between the plasma and liquid. The time dependences of the layer thickness are derived from the discharge shadowgraphs. The role of the transition layer in the energy conversion process in a corona discharge in strong electrolytes is determined. © 2001 MAIK “Nauka/Interperiodica”.

Studies of the electric and hydrodynamic character- techniques, it is impossible to divide the total energy istics of pulsed corona discharges in water electrolytes dissipated in the discharge gap into equivalent compo- [1, 2] showed that, under certain conditions, the nents. In this case, the transition from plasma to liquid branched spray corona transforms into a continuous is regarded as a jump. However, studies of the underwa- plasma formation, in which the level of hydrodynamic ter-spark channel show that, between the plasma and perturbations is comparable to that in spark underwater liquid, there is a thin transition layer in which the ther- ρ discharges [3]. At initial electric fields in the range E0 = modynamic parameters T and vary gradually [6, 7]. 106Ð108 V/m, the key condition for such a transforma- The existence of such a layer is explained by the facts tion is the high electrical conductivity of electrolytes that the characteristic pressure in the plasma channel (on the order of 10 S/m). In this case, if the surface of attains 108Ð109 Pa and the water parameters are higher the point electrode provides an electric-current density than the critical ones, so that the channel cannot have a and electric-field strength high enough to ignite a dis- sharp plasmaÐliquid boundary. charge over the entire surface of the point, then a Studies of a pulsed diaphragm discharge [8, 9] plasma piston with any given configuration, depending (which is similar to a pulsed corona discharge because on the geometry of the point, can be formed in a liquid it occurs in the same media and also leads to the forma- [2]. These factors, along with the high stability of the tion of a plasma bunch) demonstrate the presence of a electric and hydrodynamic characteristics, stimulate transitional gas layer which plays a decisive role in the interest in corona discharges in strong electrolytes. discharge evolution because of the high current density Such discharges possess a high technical potential and near the plasma bunch. have advantages over other types of electric explosions The facts listed above show that it is necessary to in liquids. They can be realized in very small volumes answer the question of whether a plasmaÐliquid transi- and are characterized by stable pressure levels, elevated tion layer arises in a corona discharge in a strong elec- resources of the electrode systems, and a wider range of trolyte when a continuous plasma region is formed, and electric conductivity of the media in which the electric- to study the role of the transition layer in the discharge explosion energy conversion occurs. evolution. A further investigation of corona discharges in strong electrolytes requires a detailed explanation of EXPERIMENTAL SETUP the mechanism for this phenomenon. In the existing electrodynamic models of corona discharge in a strong The experimental setup consisted of electrical and electrolyte [4, 5], the equivalent circuit for the dis- optical parts. The electrophysical part was the discharge gap is represented by two series-connected non- charge circuit of a capacitor bank with the parameters µ µ linear resistors formed by the expanding plasma piston U0 = 10Ð30 kV, C = 3Ð6 F, and L = 2.4 H. The dis- and liquid. The total resistance varies due to variations charge current and the voltage across the discharge gap in the plasma-piston radius and nonlinear electric-con- were recorded by an S8-17 oscilloscope with the help ductivity distribution, which is related to the tempera- of a coaxial shunt and capacitive voltage divider. The ture gradient caused by Joule heating. This model is optical part of the experimental setup (Fig. 1, top view) based on the fact that, using conventional experimental consisted of the spectral and shadowgraph sections.

OPTICAL STUDIES OF THE PLASMAÐLIQUID TRANSITION LAYER 175

A discharge was ignited in a discharge chamber, which 3 4 5 6 7 was a parallelepiped 240 × 480 × 390 mm in size. The 2 chamber had three windows made of highly transparent 1 8 9 polished Plexiglas. The chamber material was 10 Kh18N9T stainless steel. A brass rod electrode (with a curvature radius of the point of r0 = 1.5 mm) insulated with polyethylene was mounted at the flange of the lateral wall of the chamber. The discharge chamber wall served as the negative electrode. As an electrolyte, we used an optically transparent distilled water solution of 11 NaCl, which was preliminarily settled and filtered. The 16 15 14 13 12 salt concentration in the electrolyte varied depending on the required electrolyte conductivity. In this experi- Fig. 1. Optical part of the experimental setup: (1) photore- σ ment, it was 0 = 10 S/m. One of the windows was corder, (2) visualizing screen, (3) recording objective, mounted perpendicular to the rod electrode to prevent (4) receiving objective, (5) visualizing diaphragm, (6) electrode, (7) discharge chamber, (8) collimator, (9) pulsed defocusing of the optical scheme during the spectral laser, (10) adjusting laser, (11) diffraction-grating spec- measurements of the plasma evolution. With the help of trograph, (12) intermediate lens, (13) removable deflecting a deflecting mirror and an intermediate achromatic mirror, (14) standard spectrum source, (15) intermediate quartz lens (f = 75 mm), the optical plasma radiation lens, and (16) deflecting mirror. was focused at the entrance slit of a DFS-452 spectrograph. The spectrograph recorded the emission spectra over the range 190Ð1100 nm with a high resolution. light spot. A visualizing screen made of opal optical In our experiments, we used a built-in 600-line/mm glass was placed behind the recording objective. Using grating with maximum reflectivity at a wavelength of this screen, we could visually observe the image when 500 nm. In this case, a film covers the wavelength range adjusting the system or preparing to film and could up to 360 nm, which is convenient when investigating check the quality of the device adjustment. For high- the overview spectra of little-studied objects. To iden- speed photography, we used a VFU-1 photorecorder tify the spectrum under investigation, we used an with a magnifying lens (5- and 10-fold magnification), IVS-28 standard spectrum source in whose discharge which operated as a streak camera. The optical system chamber an aluminum emission with a well-known as a whole permitted a resolution of 10Ð5Ð10Ð4 m. As a spectrum was generated. With the help of a deflecting visualizing diaphragm, a set of grids with a step of 0.05 mirror and an intermediate lens, the aluminum emis- to 1.0 mm was used. The diaphragm was positioned sion was focused at the entrance slit of the spectrograph directly behind the object, because, in this case, light and was recorded on a free part of the film parallel to rays are deflected at large angles in the transition layer. the spectrum under investigation. The mutual position All components of the optical system were mounted of the spectrum under investigation and the reference on a 3750-mm-long optical bench to prevent an adverse spectrum was adjusted by a special film-carrier mecha- vibration effect. The frame of the optical system lay on nism. 100-mm-thick rubber pads. The discharge chamber The shadowgraph section, which was developed for was installed on an individual platform fastened rigidly the visual identification of a thin plasmaÐliquid transi- to the laboratory floor and was mechanically isolated tion layer, operated on the principle of the defocused from the optical system to prevent misalignment of the diaphragm [10]. A GOR-100M pulsed ruby laser was latter. used as a light source. Using a plane-parallel light In the first stage of adjusting the optical system, we beam, we could partially prevent the lens effect of the also carried out an experiment to determine the shape shock front of the corona discharge. To enlarge the size of the plasma bunch at given parameters of the electric of the light spot in which optical nonhomogeneities circuit, point radius, and electrolyte. In this case, the were studied, a collimator broadening the light beam to time evolution of the plasma corona was recorded by a 50 mm was placed between the laser and the discharge VFU-1 streak camera in the time loupe regime with a chamber. A receiving Yupiter-36B objective (f = time resolution of 5 × 10Ð7 s. An IFK-2000 pulsed lamp 250 mm) was placed on the optical axis behind the dis- was used as the illuminator. charge chamber. In the recording part of the optical system, a Kaleinar-3B objective (f = 150 mm) was used to produce a sharp image of an object (which was posi- EXPERIMENTAL RESULTS tioned in the object plane) on the visualizing screen or Frame-by frame photography of the discharge demon the photorecorder film. The use of the above objec- onstrated that, in the chosen range of electric-circuit tives in the optical system was necessary because of parameters and σ = 10 S/m, a plasma bunch produced fairly long (≥74 mm) working segments, which is with a hemispherical point is shaped like a hemispher- important for the defocused diaphragm method with a ical layer. Investigations of the time-integrated emis- high resolution (up to 45 line/mm) in the center of the sion spectrum of the continuous plasma of a corona dis-

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176 BOGUSLAVSKY et al.

1.0 The total absorption corresponding to the Na resonance 1 doublet (λ = 589Ð589.6 nm) is most pronounced when the initial voltage U0 is increased to 15 kV. This indi- 2 cates that, between the plasma and electrolyte, there is 3 a boundary gas layer in which the temperature is nearly one order of magnitude lower than the plasma temper- 0.5 ature and the transition from the plasma to the gas layer may occur abruptly. The fact that the recorded spectrum does not contain additional absorption lines corresponding to the transitions from the resonant level is evidence that the emitted radiation passes through a fairly narrow, weakly absorbing vapor layer in the visi- 0 ble region. 580 584 588 592 596 λ, nm From the time-integrated emission spectrum obtained in our experiments, we could not accurately Fig. 2. Time evolution of the Na doublet profile in the evaluate the size of the observed transition layer. The plasma emission spectrum of the corona discharge at differ- thickness of the transition layer was estimated by µ ent values of U0: (1) 10, (2) 15, and (3) 20 kV for C = 3 F. assuming that it is inversely proportional to the coefficient of continuous absorption in the spectral region near λ = 589 nm. Using the data on the absorption coefficient calculated for a low-temperature oxygenÐhydro- χ ≈ × 4 Ð1 gen plasma [11], we found that, for 589 4 10 mm , the half-thickness of the gas layer is equal to δ ≈ 1/χ ≅ 2.5 × 10Ð2 mm. Figure 3 shows the shadowgraphs of a corona discharge. The inflection of the diaphragm-grid shadows is seen against the background of the time evolution of the plasma bunch. The shadow bands are inflected in three regions: (i) the compression-wave region, (ii) the region where the electrolyte is heated, and (iii) the transition-layer region. At high initial capacitive-storage (‡) voltages, a shock is observed in the compression-wave region; as the initial voltage decreases, the shock becomes diffuse. The electrolyte is heated more inten- sively when the electric parameters ensure a more intense energy release in the discharge. By the transition-layer region, we mean the region adjacent to the plasma where the inflection angle of the grid shadows becomes negative. From the shadowgraphs of the discharge, the time dependences of the thickness of the layer in which the inflection of shadow bands was observed were calculated for different discharge-circuit parameters (Fig. 4). The dependences were calculated taking into account (b) the inflection of shadows due to the lens effect in the compression-wave region [7]. The layer thickness increases rapidly only during the first microseconds Fig. 3. Shadowgraphs of the corona discharge at different µ values of U : (a) 29 and (b) 15 kV for C = 6 µF. (to 2 s) of the discharge. For most of operating condi- 0 tions, the transition layer thickness reaches 0.05 mm and increases only by the end of the energy release in charge have revealed that, against the background of an the discharge gap. However, we also observed dis- intense continuum, a wide absorption band correspond- charges with high stored energies (U0 = 29 kV, C = ing to the sodium resonance doublet is observed in all 6 µF) in which the transition layer thickness reached a of the discharge regimes under study. The band broad- fairly large value of 0.2Ð0.3 mm. ens with increasing initial voltage. Figure 2 shows the On the whole, the experimental data on the transi- time evolution of the Na doublet profile in the emission tion layer thickness that were obtained using the shad- spectrum of the corona-discharge plasma, measured owgraph technique agree with the above estimate with reference to the known emission spectrum of Al. obtained from spectral measurements. This fact is evi-

TECHNICAL PHYSICS Vol. 46 No. 2 2001

OPTICAL STUDIES OF THE PLASMAÐLIQUID TRANSITION LAYER 177 dence that the plasmaÐliquid transition layer is cor- δ, mm rectly identified in the shadowgraphs. 0.30 1 0.25 DISCUSSION 0.20 The experimental results obtained allow us to estimate the role of the transition layer in the time evolu- 0.15 2 tion of a pulsed corona and the influence of the processes of energy conversion in this layer on the total 0.10 3 energy balance in a discharge. During the evolution of 0.05 the plasma bunch in a corona discharge, the intensity of 4 surface evaporation of the liquid is governed by radia- 0 2 4 6 8 10 tion, heat conduction, and Joule heating. To determine µ the energy that is spent on the formation of the transi- t, s tion layer and is dissipated in it, we consider the condi- Fig. 4. Thickness of the visualized transition layer at U = tions at the plasmaÐliquid interface, i.e., at the phase- µ 0 transition boundary. The temperature T of the liquid (1, 3) 29 and (2, 4) 22 kV for C = (1, 2) 6 and (3, 4) 3 F. layer adjacent to the transition layer grows due to Joule heating by the current I flowing through the layer. The α mechanism in such systems and for which we take d = time-dependent distribution of T can be determined 1.4 W/(m deg). For the maximum attainable tempera- from the one-dimensional heat conduction equation ture gradient, this flow can be estimated as ∂ ∂T ∂ ∂  T l l  T α∆ 7 2 ρ cν ------+ ν ------= ---- α------+ w(t), (1) T × l l  ∂ r ∂  ∂  ∂  ------= 5.46 10 W/m , t r r r δ 4 T = 2 × 10 K where ρ is the mass density, cν is the specific heat, α l l α ∆ d T × 8 2 is the thermal conductivity of water, ν is the radial ------= 1.12 10 W/m . r δ 4 velocity, and w(t) is the heat-source power density. T = 2 × 10 K In our case, we have In addition, the gas can be heated due to radiant heat σ 4 × 7 × 9 2 2( ) transfer with g = ST = 3.5 10 Ð9 10 W/m , where ( ) J t σ is the StefanÐBoltzmann constant. Joule heating in w t = 2-σ------(------(-----,-----)--)-. (2) S 1 T l r t the transition layer can be estimated from Eq. (2) as

The label l stands for the liquid parameters. Similar 2 I δ 16 δ 2 relations hold for both the transition-layer region and ------≈ 4 × 10 ----- W/m . 2 4 δ the coronal plasma. The continuity conditions for the 4π σ r tl current density J, mass and energy flows, potential ϕ, tl p σ and temperature T are satisfied at the phase-transition Even assuming that tl is on the order of the electric boundary. A mathematical description of this time- conductivity of water, the power of Joule heating in the dependent process with nonlinear boundary conditions transition layer is 3Ð4 orders of magnitude higher than at the initially unknown phase-transition boundaries, the heat power transferred from the plasma. In fact, the even with some model simplifications, is rather compli- temperature and pressure dependences of the electric cated. For this reason, we restrict ourselves to some conductivity of the vapor of a water solution of NaCl estimates in our analysis. We will use the following [12] are such that, at temperatures on the order of estimation parameters: the transition layer thickness 400°C, the conductivity reaches its maximum (nearly δ × Ð4 × 4 σ ° = 2.5 10 m, the current I = 2 10 A, the temper- ten times 0 for vapor at 100 C) and then decreases × 3 × 4 σ ° ature T = 5 10 Ð2 10 K, the plasma-bunch radius below 0 at temperatures on the order of 600 C at × Ð3 8 8 (including the electrode radius) rp = 4 10 m, and the 10 Pa or 800°C at 2 × 10 Pa. This means that the elec- σ initial electrical conductivity of the liquid 0 = 10 S/m. tric conductivity of the gas in the transition layer is This parameters correspond to the regime with high maximum at both boundaries of the layer and is mini- σ ° stored energy. The induced convection of liquid by a mum (below 0 for vapor at 100 C) in the center of the plasma piston may be ignored. The reason is that, in layer. From here, we estimate the electric conductivity spite of rather high flow velocities (ν can attain 1/3 of averaged over the layer to be on the order of two to four r σ ° the speed of sound), convection does not redistribute times 0 for 100 C, which increases the power released the temperature in the local regions of interest. Heat can in the layer by one order of magnitude. In this case, the be transferred through the transition layer from the energy released in the transitional layer for 10 µs is esti- plasma to the liquid by convective heat conduction (α = mated to be on the order of 400 J. Evidently, this is the 0.683 W/(m deg)) and dissociative heat conduction, upper estimate and the actual value may be several which, according to [8], is the main heat-conduction times lower; however, this estimate shows that the tran-

TECHNICAL PHYSICS Vol. 46 No. 2 2001 178 BOGUSLAVSKY et al.

Table µ δ δ δ δ δ t, s I, A rp, m , m d / d / , calculation 1.75 17 027 0.0070 2.1 × 10Ð4 Ð 0.155 3.50 18 611 0.0080 2.3 × 10Ð4 0.087 0.108 5.25 17 027 0.0085 2.5 × 10Ð4 0.080 0.071 7.00 13 859 0.0085 2.7 × 10Ð4 0.074 0.047 sition layer plays an important role in the process of stage of the corona evolution, its propagation deep energy conversion in corona discharges in electrolytes. inside the gap may be largely related to heating and a Finally, we estimate the role that Joule heating of phase transition, probably a transition of the second water adjacent to the transition layer plays in the layer kind (over the lability line), rather than hydrodynamical formation. We consider a water layer with an arbitrary or other processes. thickness x; the only requirement is that x be much less Thus, optical studies of the plasmaÐliquid transition than rp. The density of the heat ﬂux due to evaporation layer in a pulsed corona discharge in a strong water from a spherical surface of radius R is electrolyte have shown that this layer plays a signiﬁcant (or even decisive) role in the formation and evolution of dR q = ρ ------[c (T Ð T ) + r ]. the corona. l dt p 0 s

Here, rs is the specific heat of evaporation and cp is the REFERENCES specific heat of water. We assume that this flux is com- 1. G. A. Ostroumov, Interactions of Electric and Hydrody- pletely provided by Joule heating at the boundary of the namic Fields (Nauka, Moscow, 1979). layer. Taking into account that, in our case, the radius is R = r + δ (where only the transition layer thickness 2. L. Z. Boguslavsky, V. V. Kucherenko, and E. V. Kri- p vitskiœ, Preprint No. 22, IIPT (Institute of Pulse Research changes due to evaporation from the surface) and x and δ and Engineering, National Academy of Sciences of are small, using Eqs. (1) and (2) we can write Ukraine, Nikolaev, 1993). dδ I2dt 3. E. V. Krivitskiœ, Zh. Tekh. Fiz. 61 (1), 9 (1991) [Sov. ----- = ------. Phys. Tech. Phys. 36, 4 (1991)]. x π2 4 σ ρ [ ( ) ] 4 r p l l c p T Ð T0 + rs 4. V. A. Pozdeev, N. M. Beskaravaœnyœ, and V. K. Sholom, Élektron. Obrab. Mater., No. 3, 33 (1990). Since x is chosen arbitrarily and the main scale length of the problem is δ, we can substitute x with δ 5. L. Z. Boguslavsky and E. V. Krivitskiœ, Teor., Éksp., and integrate this expression using the experimental Prakt. Élektrorazryad. Tekhnol. 1, 4 (1993). time dependences of the current and corona radius. 6. V. V. Arsent’ev, Prikl. Mekh. Tekh. Fiz., No. 5, 51 Since the water temperature distribution across the (1965). layer and its time dependence, as well as the phase- 7. A. L. Kupershtokh, Prikl. Mekh. Tekh. Fiz., No. 6, 64 transition dynamics, are unknown, we consider only (1980). the limiting case (water heated to the critical tempera- 8. É. M. Drobyshevskiœ, B. G. Zhukov, and B. I. Reznikov, ture). Then, we obtain Zh. Tekh. Fiz. 47, 256 (1977) [Sov. Phys. Tech. Phys. 22, 148 (1977)]. dδ I2dt 9. V. M. Sokolov, Zh. Tekh. Fiz. 54, 1890 (1984) [Sov. ----- ≈ ------4 , Phys. Tech. Phys. 29, 1112 (1984)]. δ σ r Q l p 10. M. M. Skotnikov, Quantitative Shadowgraphy Methods π2ρ in Gas Dynamics (Nauka, Moscow, 1976). where Q = 4 c[cp(Tc Ð T0) + rs]. Here, the label c stands for the parameters at the critical point. The val- 11. A. N. Shcherbak, Physics and Technology of Electro- ues of dδ/δ calculated using this expression and the pulse Processing of Materials (Naukova Dumka, Kiev, experimental values of the transition layer thickness for 1984), p. 27. one of the regimes with a high stored energy are given 12. A. S. Quist and W. L. Marshall, J. Phys. Chem. 72, 684 in the table. (1968). A rather close agreement between the calculated and experimental values confirms that, in the active Translated by N. Larionova

TECHNICAL PHYSICS Vol. 46 No. 2 2001

Technical Physics, Vol. 46, No. 2, 2001, pp. 179Ð181. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 71, No. 2, 2001, pp. 48Ð50. Original Russian Text Copyright © 2001 by Burachevskiœ, Burdovitsin, Mytnikov, Oks.

GAS DISCHARGES, PLASMA

Limiting Operating Pressure in a Plasma Source of Electrons with Hollow-Cathode Discharge Yu. A. Burachevskiœ, V. A. Burdovitsin, A. V. Mytnikov, and E. M. Oks Tomsk State University of Control Systems and Radioelectronics, Tomsk, 634050 Russia e-mail: oks@fet.tusur.ru Received March 23, 2000

Abstract—Results of an experimental study of the correlation between the accelerating gap parameters in a plasma source of electrons and the limiting magnitudes of the gas pressure and the voltage applied to the gap are presented. It has been found that the electron beam increases the electrical strength of the gap. © 2001 MAIK “Nauka/Interperiodica”.

The importance of the problem of beam plasma gen- working chamber of the setup. This produced equal eration in the fore-vacuum range of gas pressures for pressures in the gas-discharge chamber and in the some technological applications, such as annealing and region of beam acceleration and transport. The design melting of materials, electron-beam surface treatment, of the source and its parameters have been detailed in and initiating beam-plasma discharges in plasmochem- [4, 7]. istry [1], has stimulated a demand for reliable and effi- The limiting magnitudes of the gas pressure p and cient electron sources capable of producing an electron m beam at pressures of about 100 mtorr. This problem can voltage Um were registered at the moment immediately be solved with plasma sources of electrons utilizing a preceding a big increase in current Ie (breakdown) in cold (unheated) cathode discharge [2, 3]. It is the capa- the circuit of the accelerating voltage source. As could bility of producing intense electron beams at elevated be expected, the values of pm and Um varied in inverse pressures that makes plasma electron emitters superior proportion. The experiments showed that both quanti- to hot-cathode systems whose lifetime at such pres- ties increased with decreasing mesh dimension h and sures is very short. the distance d between the anode and the extractor, as illustrated in Figs. 2 and 3. The variation of the break- Earlier [4Ð7], we developed a plasma source of elec- down voltage with pressure (Fig. 3) is, in fact, the Pas- trons on the basis of a hollow-cathode discharge capa- chen curve for a constant distance between the elec- ble of producing a stable electron beam with a current up to 1 A and energy of the order of 10 keV. _ The results of experimental investigations aimed at Id 1 U determining the maximum pressures at which electron + d beam generation is still possible are presented in this paper. Basic factors limiting the operating pressure are discussed as well. The plasma source of electrons used in our experiments is shown schematically in Fig. 1. The electron 2 Ie _ emitting plasma is formed in the discharge chamber, Ue which consists of a hollow copper cathode 1 and a flat 3 + hollow anode 2 with an axial emission hole 3 16 mm in diameter. To stabilize the plasma boundary and screen 4 the accelerating field in the discharge system, a fine metal grid was placed in the anode hole. The mesh dimension varied from 0.25 × 0.25 to 1.0 × 1.0 mm. The geometric transparency of the grid was close to 70%. In some cases, the grid was replaced with a perforated electrode with hole sizes close to the grid mesh dimensions. A beam of electrons was extracted through the Ib emitting anode hole by applying a voltage across the accelerating gap between anode 2 and extractor 4. Pres- sure was increased by admitting a gas (air) into the Fig. 1. Diagram of the plasma source of electrons.

180 BURACHEVSKIŒ et al.

pm, mtorr trodes. It was expected that the presence of an electron beam in the accelerating gap would facilitate discharge 100 1 initiation (breakdown) in the gap due to effective ion- 2 ization of the residual gas by the electron beam. In the 80 experiment, however, the breakdown voltage in the 3 presence of the electron beam in the accelerating gap 60 turned out to be higher compared to the conventional Paschen’s breakdown (Fig. 4). 40 As noted in some papers [8, 9], the basic distinctive feature of the fore-vacuum operating regime of a 20 plasma source of electrons is the high ionization rate of the gas in the accelerating gap producing a backflow of 0.2 0.4 0.6 0.8 1.0 1.2 1.4 ions, which affects the parameters of the electron emit- h, mm ting plasma in such a way that the plasma density becomes higher. This, in turn, causes an increase in the Fig. 2. Variation of the maximum operational pressure with emission current and further growth of the gas ioniza- the mesh dimensions of the anode grid. Extracting voltage tion rate. Under certain conditions, an avalanche ion- Ue, kV: (1) 4, (2) 7, (3) 12; discharge current Id = 500 mA. ization develops that switches the discharge over from the anode to the extractor. Following this, the accelerat- Um, kV ing gap voltage drops down to a few tens of volts and 16 the beam collapses. 1 14 Growth of the plasma density in the vicinity of the 2 emission hole also shifts the plasma boundary toward 12 the extractor, reducing the extent of the space charge 10 3 layer in the accelerating gap and, consequently, the 4 8 effective length of the accelerating gap (the distance 5 between the emission boundary in the plasma and the 6 accelerating electrode). The avalanche growth of the 4 current and the shift of the plasma boundary are pre- 2 vented from occurring by the grid, which performs its stabilizing function for as long as an increase in the 0 emission current is compensated by the plasma poten- 60 80 100 120 140 160 p, mtorr tial growth and corresponding decrease in the emission surface area of the plasma due to expansion of the space charge layer separating the plasma from the grid elec- Fig. 3. Variation of the maximum extracting voltage with the trode [10]. Thus, p is determined by the combined gas pressure. Distance d between the anode and the extrac- m tor, mm: (1) 5, (2) 15, (3) 25, (4) 50, (5) 75. The discharge action of the two processes. current is 1 A; h = 0.45 mm. We assume that the basic relaxation mechanism of plasma ions in the accelerating gap is their recharge by gas molecules and that the mechanism of ion escape is Um, kV diffusion [9]. Then, taking into account the condition of 16 1 plasma quasi-neutrality and confining ourselves to the 14 2 one-dimensional case, we obtain a relation between the 12 3 plasma density n(0) in the emission hole and the current density ji due to ions exiting the accelerating gap, 10 π 8 ( ) 3 jiQi Mi n 0 = n0 + ------, (1) 6 eQn 8kT i 4 where n0 is the plasma density in the absence of ion 2 flow; Qi is the total interaction cross section of slow 0 ions in the plasma; Qn is the cross section for the 60 80 100 120 140 160 recharging of fast ions; and Mi and Ti are the ion mass p, mtorr and the plasma ion component temperature, respectively.

Fig. 4. Same as in Fig. 3 at different values of the discharge Taking into account known relations for a chaotic current Id, A: (1) 1.0, (2) 0.5, (3) 0. electron current from the plasma and for the efﬁciency

TECHNICAL PHYSICS Vol. 46 No. 2 2001

LIMITING OPERATING PRESSURE IN A PLASMA SOURCE 181 of ionization by fast-moving electrons, we obtain Ue, we conclude that it is necessary to reduce h to provide the plasma boundary stability. ( )ν en(0) 8kT en 0 e Enhancement of the electrical strength of the accel- je = ------= ------π------, (2) 4 4 m erating gap due to the electron beam observed in the Z = n Q d, (3) experiment (Fig. 4) may be caused by local heating of n e the gas in the beam region. At constant pressure, heat- where je is the electron current density at the plasma ing of the gas reduces the gas density and, conse- boundary; ν is the mean thermal velocity of plasma quently, the ion backflow [11]. Another possible reason electrons; Te and m are the electron component temper- is overcompensation of the electron beam, which ature and the electron mass, respectively; Z is the num- causes the formation of a positive space charge in the ber of ions produced by an electron in the accelerating accelerating gap [12]. In this case, a substantially nonuniform distribution of the accelerating field potential gap; nn is the concentration of neutral molecules in the is realized, which is equivalent to a decrease in the accelerating gap; Qe is the ionization cross section of the gas molecules for interaction with fast electrons; accelerating gap effective length. Under experimental and d is the accelerating gap length. conditions, the operating point in the space of the accelerating gap parameters is on the left branch of Pas- Then, we can write for the ion current density chen’s curve, and the corresponding decrease in the gas en(0) 8kT density and the accelerating gap length due to the flow e of electrons make the electric strength of the gap ji = nnQed------π------. (4) 4 m higher. Substituting Eq. (4) in Eq. (1) and solving the Thus, our investigations have demonstrated the pos- obtained expression with respect to n(0), we get sibility of producing an electron beam in a system with Ð1 a plasma cathode in the fore-vacuum range up to ( )  3 Qi MTe 100 mtorr. n 0 = n01 Ð nnQed------ . (5) 4Qn mT i Let us determine the thickness of a layer separating REFERENCES the plasma from the grid within a grid opening, assum- 1. A. A. Ivanov and V. A. Nikiforov, in Plasma Chemistry ing that the anode current in the operating regime of the (Atomizdat, Moscow, 1978), Vol. 5, p. 148. electron source is zero. Under such conditions, the thickness l of the layer is 2. Yu. E. Kreœndel’, Plasma Sources of Electrons (Éner- l goatomizdat, Moscow, 1977). ε 3/2( ( ) )Ð1 3. E. M. Oks and P. M. Schanin, Phys. Plasmas 7 (5), 1649 ll = 2 0Ul n 0 ekT e , (6) (1999). where Ul is the potential drop across the layer, which 4. A. V. Mytnikov, E. M. Oks, and A. A. Chagin, Prib. Tekh. depends on the ratio of the mean thermal velocities of Éksp., No. 1, 1 (1998). the plasma ions and electrons. 5. V. A. Burdovitsin, E. M. Oks, and A. A. Serov, in Pro- Substitution of Eq. (5) into Eq. (6) gives an expres- ceedings of the 12th International Conference on High- sion convenient for qualitative analysis: Power Particle Bearns, Haifa, 1998, p. 412. 6. V. A. Burdovitsin, A. V. Mytnikov, E. M. Oks, and 3/2 Ð1 3 Qi MT  A. A. Serov, in Proceedings of the 14th International l = 2 ε U (n ekT ) 1 Ð --n Q d------e . (7) Symposium on Plasma Chemistry, Prague, 1999, p. 595. l 0 l 0 e  4 n e Q mT  n i 7. V. A. Burdovitsin and E. M. Oks, Rev. Sci. Instrum. 70 ≈ If an approximate equality ll h is adopted as a cri- (7), 2975 (1999). terion of plasma boundary stability, then in the case 8. Yu. E. Kreœndel’ and V. A. Nikitinskiœ, Zh. Tekh. Fiz. 41 where p = nnkT and the dependence of Ui on pressure is (11), 2378 (1971) [Sov. Phys. Tech. Phys. 16, 1888 (1971)]. weak, the relationships between pm, h, and d become clear. Increasing either the gas pressure p or distance d 9. V. A. Gruzdev and Yu. E. Kreœndel’, Development and causes, according to Eq. (7), a reduction of the layer Application of Intense Electron Beam Sources (Nauka, Novosibirsk, 1976), pp. 130Ð135. thickness ll and, consequently, the plasma boundary stability can be ensured with smaller dimensions h of 10. V. L. Galanskiœ, Yu. E. Kreœndel’, E. M. Oks, et al., Zh. the grid opening. Tekh. Fiz. 57 (5), 877 (1987) [Sov. Phys. Tech. Phys. 32, To analyze the role of voltage U across the acceler- 533 (1987)]. e 11. A. Hershcovitch, J. Appl. Phys. 78 (9), 5283 (1995). ating gap, let us turn to expression (1) and note that ji grows with U due to better focusing of the electron and 12. V. Ya. Martens, Pis’ma Zh. Tekh. Fiz. 12 (13), 769 e (1986) [Sov. Tech. Phys. Lett. 12, 317 (1986)]. ion beams. Thus, as Ue increases, the plasma density near its boundary increases as well, with all the ensuing consequences. From the decrease in ll with increasing Translated by N. Mende

TECHNICAL PHYSICS Vol. 46 No. 2 2001

GAS DISCHARGES, PLASMA

Spark Discharges in Condensed Media N. I. Kuskova Institute of Pulsed Processes and Technologies, National Academy of Sciences of Ukraine, Nikolaev, 54018 Ukraine e-mail: [email protected] Received March 13, 2000

Abstract—The formalism of phase transition waves is shown to be applicable to the description of both fast and slow discharges in condensed media. The deﬁnitions of a streamer and leader are reﬁned, and the mechanisms for their formation and propagation are described. The discharge propagation velocities in low- and high- resistivity media are estimated. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION varies, and v is the velocity of thermal expansion or flow of the medium. An analysis of the available experimental data on different types of discharges in condensed media When describing the phase transition waves, it is showed that they can be divided into fast and slow dis- important to know the mechanism for expelling the charges. The discharges in low-resistivity media propa- field from the region where the phase transition occurs gate with velocities no higher than the speed of sound and to find the width of the wave front, i.e., the charac- and should be classified as slow discharges. Fast dis- teristic length of the region in which the field is nonuni- charges are formed in high-resistivity media and prop- form. agate with supersonic speeds. In this paper, we present a unified approach to STREAMER DISCHARGE describing spark discharges as phase transition waves. IN A CONDENSED MEDIUM The fact that plasma channels in dielectrics and PHASE TRANSITION WAVES semiconductors propagate with supersonic velocities exceeding the drift velocities of the current carriers The action of strong electric fields on low-conduc- point to the wave nature of the processes in a streamer tivity or dielectric condensed media can result in phase discharge. transitions such as melting and/or evaporation during the electric or laser breakdown of solid or liquid low- A plasma channel that is formed in a high-resistivity conductivity media such as dielectricÐsemiconductors medium in the region with a strong electric field due to or semiconductorÐmetals in dielectrics or high-resistiv- impact ionization, photoionization, or generation of ity semiconductors. Under certain conditions, the current carriers by the electric field and that propagates above phase transitions propagate in the form of waves toward the opposite electrode in the form of an ioniza- [1Ð3]. tion wave will be referred to as a streamer. Let us consider phase transition waves whose for- Streamer propagation in high-resistivity condensed mation and propagation require both the spatial inho- media exhibits some general features. Discharge is ini- mogeneity of the electric field E and mechanisms pro- tiated in regions where the electric field is highly non- δ viding its propagation in a medium. Let E be the char- uniform, e.g., at the interfaces with dielectric inclu- acteristic length of the region in which the electric field sions, rough electrode surfaces, and electrodes with is nonuniform and in which a phase transition occurs, small radii. A random distribution of different types of and some parameter f of a medium (e.g., the internal inhomogeneities on the electrode surfaces and in the energy, density, electric conductivity, or density of cur- medium itself results in the stochastic nature of rent carriers) varies steeply. Then, the velocity of the streamer formation and propagation. phase transition wave can be estimated from the In the region where the electric field is strong, the expression [2] density of current carriers sharply increases. The one- wE()δ dimensional equation for the current carrier density in u ≅ ------E+ v , (1) Cartesian coordinates has the form ∆f ∂n ∂ ()nE ∂ 2 n n where ∆f = |f Ð f |, f is the initial distribution of the ------Ð µ ------Ð D ------= wE()Ð ---- , (2) max 0 0 ∂ ∂ 2 τ parameter f, w(E) is the rate at which this parameter t x ∂ x r

SPARK DISCHARGES IN CONDENSED MEDIA 183 where w(E) is the rate of electronÐhole pair production; plasma channel, the field is not fully expelled from it. τ r is the recombination time; D is the diffusion coeffi- Hence, a necessary condition for streamer self-propa- cient; and µ and n are the electron mobility and density, gation is the formation of a space charge in its head. respectively. Let us consider the processes occurring in the anode From Eq. (2), we can obtain a rough estimate for the region. The electrons are able to leave for the electrode; ionization wave velocity, this effect is related to the so-called field ionization [4]. The higher the electronic work function of the metal w  ≅ 0 1 δ D v and the electric field near the anode, the higher the field u ----- Ð -τ-- E + δ----- + 0, (3) n0 r E ionization probability. The departure of electrons µ results in the formation of a positive space charge. Dur- where w0 = w(Emax), v0 = Emax, and n0 = nmax. ing the time τ, the space charge will shift by a distance In a strong electric field, the ionization rate substan- r = (µ + µ+)Eτ, where µ+ is the mobility of the positive tially exceeds the recombination rate and the drift current carriers (ions or holes). The space-charge elec- velocity is much higher than the diffusion velocity. tric field E' is determined from Poisson’s equation Thus, the ionization wave velocity is determined by the divE ' = ep/ε, where p is the density of the positive cur- expression rent carriers. The streamer starts propagating if the δ space-charge electric field reaches the threshold value ≅ w0 E E* (E ' ~ E*). The time necessary for satisfying this u ------+ v 0. (4) n0 condition can be determined from the above relations by taking into account the rough equality p ~ ω τ. Thus, ӷ τÐ1 0 The inequality w(E)/n0 r determines the thresh- the time during which the streamer is formed is esti- old electric field E* at which the wave is formed and the mated as δ length E of the region in which the nonuniform electric 1/2 τ  e  field at a given voltage is E > E*. The ionization wave = ------ . (6) velocity is always higher than the sum of the drift and e(µ + µ+)w(E*) diffusion velocities of carriers. In the case of intense ionization, it is much higher than this sum. Hence, if a Since no melting or evaporation occurs in a streamer strong electric field is applied to a high-resistivity discharge, the slope of the phase trajectory ∆P/∆T on medium, a streamer propagates with supersonic speed. the PÐT phase diagram must be larger than the derivative dP/dT taken along the solidÐliquid (or liquidÐgas The increase in the electron density at the wave front for liquid media) phase equilibrium curve: is limited in time by the Maxwellian relaxation time τ ε σ ε σ m = / , where is the permittivity and is the electric ∆P dP ------< ------. (7) conductivity. Then, the maximum electron density is ∆T dT determined by the expression

1/2 In phase transitions in strong electric fields, the elec- εw  ε 2 n = ------0 , (5) tric field pressure P = E /2 plays a decisive role. The 0  eµ  area on the phase diagram that corresponds to the conditions in the streamer head is characterized by the where e is the electron charge. electric field pressure and the temperature, which Correspondingly, the maximum conductivity in the increases due to Joule heating. The high local electric σ ≅ εµ 1/2 9 streamer head is 0 (e w0) . During the same time field in the streamer head (E > 10 V/m) corresponds to τ 7 m, the electric field is expelled to the region in front of a pressure of P > 10 Pa. The rates at which the electric the streamer head. Since the conductivity in the field pressure and the temperature vary are determined streamer is limited and the strong electric field at the by the following expressions: streamer front exists for a very short time, the streamer ∂ ∂ ε ∂ ∂ ≅ ε 2 τ is a weakly ionized low-temperature plasma channel. P/ t = E E/ t E / f , (8) Consequently, the lattices of solid dielectrics or semiconductors do not melt during the streamer discharge. ∂T/∂t = σE2(ρc), (9) In gas discharge physics, the low degree of ionization τ and low temperature of a streamer are used for its defi- where f is the time during which the field increases to nition. the amplitude value, ρ is the mass density, and c is the An important property of a streamer is its self-prop- heat capacity. agation ability, which is associated with the mechanism Since ∆P/∆T Ӎ (∂P/∂t)/(∂T/∂t), we can find an for expelling a strong electric field from the streamer upper limit for the time during which the electric field head. However, the voltage drop along the plasma increases to the threshold value E*: channel dictates a rapid increase in the applied voltage ερcdT as the distance passed by the streamer in the divergent τ < ------, (10) field increases. Since the streamer is a weakly ionized f σ dP

TECHNICAL PHYSICS Vol. 46 No. 2 2001 184 KUSKOVA where the derivative dT/dP characterizes how the phase mined by the dependence of the ionization rate on the transition temperature varies with pressure along the temperature. phase equilibrium curve. A plasma channel that is formed in a low-conductiv- From the above considerations, it follows that, for a ity medium due to nonuniform Joule heating of a streamer discharge to form, the rise time of the applied plasma and propagates toward the opposite electrode in voltage pulse must satisfy inequality (10). the form of ionization wave will be referred to as a leader. Hence, the leader may propagate along the One of the most interesting features of a streamer already formed streamer channel. The leader velocity is discharge is the crystallographic orientation of plasma also higher than the drift velocity of the current carriers. channels, whose description sometimes requires rather The leader is a high-temperature plasma channel. sophisticated hypotheses. As an example, it was shown that in the most thoroughly studied crystals of zinc As an illustration, let us consider the breakdown in selenide and cadmium sulfide, the directions of water in the nanosecond range [9]. For a positive elec- streamer discharges coincide with the calculated direc- trode potential, the breakdown in distilled water in the tions of synchronization of microwave and light waves nanosecond range proceeds as follows. Dissociation, [5, 6]. The microwaves are generated during the propa- which leads to streamer formation within t < 10 ns, gation of the streamer head as a result of the competi- starts in the local regions with a strong electric field, tion among different processes, which leads to periodic which appear near microinhomogeneities on the elec- changes (oscillations) in the electric field and plasma trode surface. An increase in the temperature in the density at the streamer front [7]. streamer results in a sharp increase in the probability of thermal ionization w(T); consequently, conductivity Although modeling streamer propagation with continues to rise and the streamer channel transforms allowance for the emission of electromagnetic waves is into the leader channel. rather complicated, the parameters of a streamer prop- ≅ agating in cadmium sulfide at different voltages can be The minimum time for the streamer formation is t estimated using formulas (5) and (6). According to the 4 ns. There is a narrow peak of the field in the ionization experimental data [1], the characteristic size of the wave propagating from the anode to cathode. This peak δ × Ð6 is caused by a positive space charge at the streamer region of a strong electric field is E = 5 10 m. Then, ≅ × 8 δ × head, as well as by its elongated shape. As a result, the the threshold field is E* 3.2 10 V/m. For E = 2 field in front of the streamer increases to E ≅ 2 × Ð6 ≅ × 8 10 m, we have E* 3.5 10 V/m. 109 V/m for U = 100 kV, interelectrode spacing l = −2 σ × The electron drift velocity in cadmium sulfide at E > 10 m, and initial conductivity of water 0 = 2.76 108 V/m is saturated and equal to v = 105 m/s [1]. For 106 S/m. The field inside the streamer decreases to E ≅ ≅ Ð1 2 δ × Ð6 5 D 10 m /s and E = 2 10 m, the calculated min- 3 × 10 V/m as the electric conductivity increases. The ≅ × 5 δ × ≅ × 5 imum velocity is umin 1.5 10 m/s. For E = 5 streamer propagation velocity is u 2 10 m/s; for −6 ≅ × 5 ≅ × 4 10 m, we have umin 1.2 10 m/s. These calculated U = 50 kV, we have u 5 10 m/s. These values agree values coincide with the measured values of the mini- well with estimates obtained using expression (5). mum velocity [8]. Behind the streamer front, the temperature grows The characteristic features of the streamer discharge due to intense Joule heating, which causes ionization in cadmium sulfide deduced from formulas (5) and (6) and further increase in the electric conductivity. The are as follows. The streamer starts to form at t ≅ 3 ns at streamer transforms into a leader, whose velocity is less a maximum field of E ≅ 5.4 × 108 V/m. The strong ion- than the propagation velocity of the streamer front. The izing field is localized within a small region (δ = 3 × distribution of the temperature is highly nonuniform E both along and across the plasma channel; the temper- 10Ð6 m) and affects the crystal during very short time ature of the channel can attain T ≅ 104 K [9]. intervals (t ≅ 10Ð13 s). The streamer propagation velocity is u ≅ 4 × 106 m/s, the maximum electron density in ≅ × 24 Ð3 the streamer is n0 6 10 m , and the conductivity SLOW SPARK DISCHARGE in the streamer head is σ ≅ 2 × 102 S/m. IN A LOW-CONDUCTIVITY MEDIUM An electric breakdown in low-resistivity solid or liq- LEADER DISCHARGE IN A CONDENSED uid media may develop in a few microseconds after MEDIUM applying the voltage due to the melting and evaporation of the material, which leads to the formation of a gas The definition of a streamer allows one to define plasma. In this case, the solidÐliquidÐgasÐplasma more accurately the leader discharge. The leader is phase transitions occur consecutively due to Joule heat- known to form in a weakly ionized medium during the ing in the region where the electric field is nonuniform. onset of the ionizationÐheating instability; i.e., the pro- The initial scale length on which the electric field varies duction of current carriers is governed by Joule heating, is determined by the configuration of the discharge gap, whereas the rate at which they are produced is deter- which usually ranges from 10Ð6 to 10Ð2 m. The field gra-

TECHNICAL PHYSICS Vol. 46 No. 2 2001 SPARK DISCHARGES IN CONDENSED MEDIA 185 dient is limited only by the experimental facilities. The propagation velocities of slow and fast discharges are propagation velocity of the phase transition wave can estimated. be calculated using the rough formula A unified approach to describing slow and fast spark discharges in condensed media as phase transition j2δ u ≅ ------E- + v , (11) waves is proposed, which allows a quantitative expla- σ∆ε nation of the difference in the discharge propagation velocities. where ∆ε is the total change in the specific internal energy of a medium at the wave front. It is shown that the medium density changes signif- icantly only in slow discharges, which is related to the It follows from formula (11) that the propagation liquidÐgas phase transition. velocity of a discharge in a low-resistivity medium can vary over a wide range. However, the relation between the propagation velocity and phase transitions leads to REFERENCES the following restriction that is well known from exper- 1. N. G. Basov, A. G. Molchanov, A. S. Nasibov, et al., Zh. iments: the discharge propagation velocity in this case Éksp. Teor. Fiz. 70, 1751 (1976) [Sov. Phys. JETP 43, cannot exceed the speed of sound c, because phase tran- 912 (1976)]. sitions in condensed media are accompanied by a 2. N. I. Kuskova, Pis’ma Zh. Tekh. Fiz. 24 (14), 41 (1998) change in the volume. Thus, the formation time of a [Tech. Phys. Lett. 24, 559 (1998)]. τ δ slow discharge is limited from below: > /c. The dura- 3. V. V. Katin, Yu. V. Martynenko, and Yu. N. Yavlinskiœ, tion of the applied voltage pulse should be at least Pis’ma Zh. Tekh. Fiz. 13, 665 (1987) [Sov. Tech. Phys. longer than the discharge formation time. Lett. 13, 276 (1987)]. In slow discharges, the electric field pressure 4. W. F. Schmidt, IEEE Trans. Electr. Insul. EI-17, 478 becomes important after the transition of the material to (1982). the gaseous state. Since a gas is dielectric, both the 5. V. P. Gribkovskiœ, A. N. Prokopenya, and K. I. Rusakov, electric field strength and electric field pressure inside Zh. Prikl. Spektrosk. 60, 362 (1994). the produced gas bubble are higher than those in the 6. V. P. Gribkovskiœ, V. V. Parashchuk, and K. I. Rusakov, surrounding low-conductivity medium. The formation Zh. Tekh. Fiz. 64 (11), 169 (1994) [Tech. Phys. 39, 1178 of a slow spark discharge is completed by a gas dis- (1994)]. charge, which results in plasma production. The elec- 7. V. V. Vladimirov, V. N. Gorshkov, O. V. Konstantinov, tric field strength and electric field pressure in the et al., Dokl. Akad. Nauk SSSR 305, 586 (1989) [Sov. plasma channel decrease, whereas the temperature and Phys. Dokl. 34, 242 (1989)]. gas-kinetic pressure increase. 8. A. Z. Obidin, A. N. Pechenov, Yu. M. Popov, et al., Kvantovaya Élektron. (Moscow) 9, 1530 (1982). 9. V. M. Kosenkov and N. I. Kuskova, Zh. Tekh. Fiz. 57, CONCLUSION 2017 (1987) [Sov. Phys. Tech. Phys. 32, 1215 (1987)]. The definitions of a streamer and leader in condensed media are refined. The formation times and Translated by N. Ustinovskiœ

TECHNICAL PHYSICS Vol. 46 No. 2 2001

SOLIDS

Effect of the Saturation Conditions and Structure on the Retention of Helium in Structural Materials A. G. Zaluzhnyœ and A. L. Suvorov Institute of Theoretical and Experimental Physics, Bol’shaya Cheremushkinskaya ul. 25, Moscow, 117259 Russia Received April 25, 2000

Abstract—A comparison between the kinetics of helium desorption upon linear heating of samples saturated using various regimes is performed, and the effect of dislocations on the retention of helium in materials is estimated. In order to investigate the effect of the conditions of saturation of materials with helium on its retention, samples of austenitic stainless steel 0Kh16N15M3B saturated using various methods were studied, namely, helium irradiation in a cyclotron, in a magnetic mass-separation setup, inside IRT-2000 and BOR-60 reactors, and using the so-called “tritium trick” technique. The investigations show that when saturation of the samples with helium is accompanied by the introduction of radiation defects (in wide limits of helium concentrations and radiation damage), the kinetics of helium evolution from samples of this type is adequate to the kinetics of its evolution from samples irradiated in a reactor. The investigation of the kinetics of helium evolution from the samples of 0Kh16N15M3B steel both after a preliminary deformation and in the process of deformation showed that, in the process of heating, the helium atoms can migrate along dislocation pipes, resulting in a sig- niﬁcant effect on the release of helium and its redistribution in the volume of the material. The activation energy for helium pipe diffusion in austenitic steel 0Kh16N15M3B is about 0.7 eV. Mobile dislocations favor the ejection of helium onto the surface of the material, to grain boundaries, interphase interfaces, etc. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION ditions of material saturation with helium on its retention in the material, we investigated samples of an It is known that, under the effect of irradiation, austenitic stainless steel 0Kh16N15M3B saturated materials can change their physical and mechanical with helium using various methods. Upon simulation of properties. The main factors that restrict the reserve of the effect of reactor irradiation on the properties of a the material service life are radiation swelling, high- material, it is insufficient to specify definite values of temperature and low-temperature irradiation embrittle- the helium concentration and its distribution; one ment, radiation creep, etc. An important role in these should also ensure a specified scaling characteristic of detrimental phenomena belongs to gaseous products of the material damage. This characteristic is determined nuclear reactions, including helium. by the value of a coefficient K, which is expressed as Since the direct irradiation of materials in a reactor the ratio of the rate of helium formation (He, appm/s) requires much time and material cost, at present, in order to the rate of production of atomic displacements to imitate the accumulation of helium, various express (dpa/s). Thus, for a number of materials, we have methods are used, such as the irradiation of materials with K = 0.3 appm/dpa for irradiation in a fast reactor helium ions of various energies, irradiation with high- EBR-II and K = 70 appm/dpa for irradiation in an HFIR energy electrons and γ photons, saturation from a plasma, reactor [2]. etc. Naturally, the question arises to what extent this or that In order to study the effects of deformation on the technique is acceptable for adequately simulating reactor retention of helium in the 0Kh16N15M3B steel, we irradiation. In this work, we make an attempt to qualita- investigated the kinetics of helium evolution from tively compare the kinetics of helium desorption upon lin- undeformed materials, from materials subjected to a ear heating of samples saturated using various techniques preliminary cold working (to 15 and 50%), and in the and to estimate the role of dislocations in the retention of process of deformation. helium in materials.

EXPERIMENTAL RESULTS EXPERIMENTAL 1. Effect of the Saturation Conditions on the Kinetics The investigation of the kinetics of helium evolution of Helium Evolution from Materials from samples of structural materials in the process of uniform heating was performed in a high-vacuum mass In order to substantiate the methods of express imi- spectrometer setup [1]. To study the effects of the con- tation of the accumulation and retention of helium in

EFFECT OF THE SATURATION CONDITIONS AND STRUCTURE 187 structural materials upon reactor irradiation, we per- arb. units formed a comparative analysis of the spectra of gas C evolution from samples of 0Kh16N15M3B steel irradi- (a) B ated in a cyclotron [3], in an ILU-100 magnetic mass- A separation setup [4], in IRT-2000 and BOR-60 reactors [5], as well as saturated with helium using the “tritium trick” method [6]. Upon irradiation in a cyclotron, the C samples were uniformly saturated with helium to con- (b) B centrations of 1 × 103 at. % throughout the volume. The irradiation temperature did not exceed 400 K. The saturation of samples in an ILU-100 magnetic mass-separation setup was performed using bombard- α C ment of the material studied with particles with an (c) B energy of 70 keV to a fluence of 3 × 1020 ion mÐ2. The A range of α particles with this energy is about 200 nm. The temperature of irradiation did not exceed 400 K. The calculated concentration of helium in the near-sur- C face layer was about 1 at. %. (d ) C Irradiation of the samples in an IRT-2000 reactor was performed at temperatures below 420 K to fluences of 1.1 × 1023 neutron mÐ2 for fast neutrons (E > 2.6 MeV) and 2.6 × 1024 neutron mÐ2 for thermal neu- C trons. The experimentally determined helium concen- (e) tration in samples irradiated in the IRT-2000 reactor was 3.5 × 10Ð4 at. %. As the samples irradiated in the BOR-60 reactor, we used pieces cut out from the can of a fuel element that C had been irradiated to a fluence of 7.8 × 1026 neutron mÐ2 (f) (E > 0.1 MeV). The irradiation temperature was about A 850 K. The samples were studied both in the initial state and after the inner layer that was in contact with the nuclear fuel was etched away. The experimentally 400 600 800 1000 1200 T, K determined concentrations of helium in these samples Ð3 Ð3 proved to be 2.0 × 10 and 1.7 × 10 at. %, respec- Fig. 1. Spectra of the gas evolution rates upon uniform heat- tively. The characteristics of the samples are given in ing (7 K/min) of samples of 0Kh16N15M3B steel saturated the table. with helium using various techniques: (a) uniform saturation of the entire volume in a cyclotron; (b) irradiation with Figure 1 schematically displays the spectra of α particles using a magnetic mass-separation setup (energy the rate of helium evolution from the samples studied in of helium atoms 70 keV); (c) irradiation in an IRT-2000 the process of uniform heating at a rate of 7 K/min. Fig- reactor; (d) irradiation in a BOR-60 reactor (as-irradiated ure 1a displays the spectrum of helium evolution from state); (e) irradiation in a BOR-60 reactor after etching away α a surface layer from the side that was in contact with the a sample irradiated with particles in a cyclotron. The fuel; and (f) saturation using the “tritium trick” technique. identification of the peaks of the helium thermodesorption curve is given in [7], where, along with the study of the kinetics of helium evolution, electron micro- [8]. The activation energy of helium evolution at a tem- scopic investigations of the evolution of the dislocation perature corresponding to peak B was calculated from structure and the development of helium-induced ≈ the rate of growth of interstitial loops [9]. Helium evo- porosity were performed. Peak A (activation energy E lution at higher temperatures (at T > 1200 K) may be 0.7 eV) corresponds to helium release via pipe diffu- related to the migration of heliumÐvacancy complexes sion along dislocations that emerge onto the surface; He V . These conclusions are confirmed indirectly by peak B (E ≈ 0.8 eV) corresponds to growth of disloca- x y tion loops, their emergence onto the surface, and the the results of recent works concerning the investigation evolution of the related helium through dislocation of the evolution and distribution of helium in various pipes; peak C (E ≈ 2.4 eV) is ascribed to helium diffu- materials [10Ð12]. sion by the vacancy mechanism. The activation energy Note the complete identity of the spectra of the rate for the process of helium evolution at temperatures cor- of gas evolution from the samples uniformly saturated responding to peaks A and C, which are connected with in a cyclotron (Fig. 1a) and those irradiated in the helium diffusion, were calculated using the known IRT-2000 reactor (Fig. 1c). The spectrum of gas evolu- method based on the heating of samples at various rates tion from the sample irradiated with α particles with an

TECHNICAL PHYSICS Vol. 46 No. 2 2001

188 ZALUZHNYŒ, SUVOROV

T, K curve of helium thermodesorption from the initial sample correspond to the same mechanism of diffusion via 1400 vacancies: the ﬁrst peak is connected with the emer- 1300 gence of helium from the near-surface layer, while the 1200 second peak is related to the emergence of helium from 1100 the entire volume of the sample. Figure 2 shows the variation of the temperature of the maximum rate of gas 1000 evolution (calculated on the basis of [13]) correspond- 900 ing to helium diffusion by the vacancy mechanism upon linear heating of the sample at a rate of 7 K/min 10–1 100 101 102 l, µm as a function of the thickness l of the near-surface layer saturated with helium that determines the gas evolution Fig. 2. Variation of the temperature of the maximum rate of at this stage. It follows from a comparison of data pre- helium evolution through the vacancy mechanism upon lin- sented in Fig. 2 with the temperatures of the maxima of ear heating of the samples (at a rate of 7 K/min) as functions the gas-evolution rate (Figs. 1d, 1e) that the thickness of the depth of a near-surface layer saturated with helium µ (solid line, calculated) and the grain size (dashed line, of this layer in the initial sample is about 0.5 m, which experiment). agrees in order of magnitude with the range of α particles that are incorporated into the steel from the fuel element [14]. The gas evolution with a maximum at energy of 70 keV exhibits peaks B and C (Fig. 1b). The 1200 K, which is controlled by helium evolution from absence of peak A is possibly related to the fact that the the volume of the sample, is approximately 10 µm, related process is already realized during the irradiation which agrees with the average grain size in the samples due to the near-surface distribution of helium. The shift studied. It was shown in [15] that grain boundaries are of peak C toward lower temperatures can also be unsaturated traps for inert gases. Therefore, the evolu- related to the near-surface distribution of helium [13]. tion of inert gases from materials only occurs from near-surface grains of the material. Figure 2 displays The spectra of the rates of helium evolution from the the experimentally found temperatures of the maxi- 0Kh16N15M3B steel samples irradiated in the mum rate of helium evolution from samples uniformly BOR-60 reactor (Figs. 1d, 1e) show that helium is saturated with helium in a cyclotron as functions of the released only at high temperatures exceeding the irradi- depth of evolution and the grain size. This dependence ation temperature. This is likely to be due to the realiza- suggests that helium evolution from irradiated materi- tion of processes responsible for gas evolution at lower als occurs only from near-surface layers of the samples temperatures during irradiation. The gas evolution cor- (in this case, the thickness of this layer is about 10 µm). responding to the initial sample (Fig. 1d) exhibits two The rest of the helium is likely to be captured by inter- peaks with maxima of the evolution rate at 940 and nal traps such as grain boundaries, dislocations, radia- 1200 K, whereas in the spectrum of helium evolution tion defects, etc. from the sample with an etched-away inner layer (Fig. 1e), only one peak located at 1200 K is present. Figure 1f displays the curve of the helium evolution This difference in the kinetics of helium evolution from rate from a sample saturated using the “tritium trick” these samples is explained by the different distribution technique. This thermodesorption curve has two peaks, of helium in the material. Thus, in the initial sample the which correspond to energies of activation equal to 0.7 inner layer is enriched in helium by the introduction of and 2.4 eV. This indicates that when saturating the α particles from the fuel. In the sample with an etched- material using a “damage-free” method, such as the away inner surface, helium is distributed uniformly “tritium trick” technique, the gas atoms that are throughout the volume of the sample. The peaks in the retained in the material during the saturation time are

Characteristics of the samples studied Saturation Amount of helium, Energy Temperature, K Fluence, mÐ2 Parameter K [2] technique at. % (n, α) [5] Spectrum of the 420 2.0 × 1024 9.5 × 10Ð4 400 IRT-2000 reactor (n, α) [5] Spectrum of the 850 3.8 × 1026 2.0 × 10Ð3 0.6 BOR-60 reactor 1.7 × 10Ð3 He* [3] 0Ð29 MeV 400 1.0 × 1020 1.0 × 10Ð3 4000 He* [4] 70 keV 400 3.0 × 1020 1.0 T 3He [6] 300 5.0 × 10Ð7

TECHNICAL PHYSICS Vol. 46 No. 2 2001

EFFECT OF THE SATURATION CONDITIONS AND STRUCTURE 189 mainly linked with dislocations and vacancies. The arb. units absence of a noticeable helium evolution at tempera- (a) tures above 0.85Tmol indicates the absence of stable C heliumÐvacancy complexes. B Thus, we may conclude that, as to the behavior of A helium in structural materials, the simulation experiments that ensure the simultaneous introduction into the material lattice of both helium and radiation defects (b) (in wide limits of helium concentrations and radiation B C damage) sufficiently adequately reproduce reactor irra- A diation. Most probably, this is due to the stabilization of definite configurations of radiation defects by helium atoms. When helium is introduced by damage-free methods, in each case one should take into account the A specific features of its retention and its mobility. The (c) C neglect of these features can lead to an incorrect treatment of the results obtained. It has been found that B helium evolution upon linear heating of irradiated samples begins only at temperatures exceeding the irradiation temperature. This circumstance can be used to 400 600 800 1000 1200 T, K determine the temperature of sample irradiation. Fig. 3. Spectra of the rates of gas evolution upon a uniform heating of the samples (7 K/min) saturated with helium in a 2. Effect of Deformation on the Retention cyclotron: (a) initial state; (b) after a preliminary cold defor- of Helium in Materials mation to 15%; and (c) after a preliminary cold deformation to 50%. The investigation of the effect of deformation on the retention of helium in materials was performed using liminary deformation (with increasing density of dislo- samples of 0Kh16N15M3B steel uniformly saturated cations). with the gas as a result of bombardment with helium ions in a cyclotron [3] to a concentration of 1 × 10Ð3 at. %. Simultaneously, we determined the absolute The irradiation temperature did not exceed 400 K. The amounts of helium evolving in the process of isochro- samples were studied in an austenitized state and after nous annealings of the samples studied (annealing a preliminary cold deformation to 15 and 50%. Imme- duration was 0.5 h) using the method of evaporation of diately before the experiment, the samples (5.0 × 5.0 × the sample in a vacuum with subsequent mass-spectro- 0.1 mm in size) were subjected to electropolishing with scopic recording of the helium evolved [5]. These subsequent rinsing in alcohol to clean the surface. investigations showed that the amount of helium evolved at the low-temperature stage increases with Figure 3 displays the curves of helium desorption increasing deformation. The total amount of helium (in the process of uniform heating at a rate of 7 K/min) evolved in the process of isochronous annealings at from the austenitized and preliminarily deformed sam- temperatures up to 1373 K is approximately the same ples. It is seen from the figure that, in the case of for all the samples. The results obtained once more con- deformed samples (Figs. 3b, 3c), the low-temperature firm the assumption that helium evolution at tempera- peak A, which is ascribed to helium emergence onto the tures of 0.27 to 0.3 Tmol is related to the migration of surface by the mechanism of diffusion along disloca- helium atoms along dislocation pipes. The greater the tion pipes, becomes higher. The greater the preliminary deformation, the greater the number of dislocations that deformation, the greater the increase in the height of emerge onto the surface of a sample, and the greater the peak A. Peak B, which is related to the emergence onto amount of gas, which diffuses along the dislocation the surface of dislocation loops in the process of their pipes, that evolves from the sample. growth and to the subsequent evolution of helium by pipe diffusion, becomes somewhat lower. Note that a To further clarify the effect of dislocations on the significant evolution of nitrogen and of a small amount retention of helium in materials, we studied the release of other chemically reactive gases was also observed in of helium from samples of 0Kh16N15M3B steel in the this temperature range, which was likely due to the dis- process of deformation at a temperature of 543 K. This solution of Cottrell atmospheres in the process of the temperature corresponds to the low-temperature peak emergence of dislocation loops onto the surface upon of helium evolution, which was ascribed by us to the their growth. The contribution of this process to helium migration of helium atoms along dislocation pipes. To evolution from steel samples upon heating decreases, this end, we designed and constructed, at the cover as should be expected, with increasing amounts of pre- plate of the annealing chamber of the high-vacuum

TECHNICAL PHYSICS Vol. 46 No. 2 2001 190 ZALUZHNYŒ, SUVOROV

T, K locations, which in turn should manifest itself in the 573 (a) kinetics of its evolution from the sample. Such an 473 experiment was performed in the following way. An irradiated sample was fixed in grips in the annealing 373 chamber and deformed at a rate of 5 × 10Ð5 sÐ1. Then, it was heated at a constant rate of 1.5 K/min to a temper- ∆l, mm ature of 543 K, at which the stage of gas evolution was (b) observed (which was ascribed by us to the migration of 3 helium atoms along dislocations), held at this tempera- 2 ture until the gas evolution stopped, and deformed at 1 various rates up to failure. During the experiment, changes in the partial pressure of the evolved helium in the experimental volume of the setup were recorded. arb. units (c) The results of the investigation are given in Fig. 4. It is 8 seen that room-temperature deformation is not accom- 6 panied by helium evolution. An increase in the temper- 4 ature to 543 K leads to insignificant gas evolution. The activation energy of this process calculated using 2 the known technique of heating at various rates is E = 0 30 60 90 120 150 180 210 240 t, min 0.2 eV [8], which suggests that this peak of helium evolution is related to helium release from the material as Fig. 4. Deformation-induced helium evolution from sam- a result of its diffusion via the interstitial mechanism ples of 0Kh16N15M3B steel in the process of heating: [16]. This is related to the fact that after irradiation part (a) variation of the temperature; (b) sample elongation; and of the helium atoms are still located in interstices. After (c) rate of helium desorption. the heating is stopped, gas evolution is observed during 30 min of isothermal holding, which can be ascribed to diffusion of helium atoms along dislocation pipes. Fur- mass spectrometric setup for studying the kinetics of ther deformation at a rate of 5 × 10Ð5 sÐ1 for 20 min was gas evolution [1], a special attachment that permitted us accompanied by helium evolution at a virtually con- to deform the samples studied at various rates in the Ð5 Ð3 Ð1 stant rate. A twofold increase in the deformation rate range from 10 to 10 s . The samples for the investi- led to an increase in the gas evolution rate. No increase gation were stamped from a sheet 0.3 mm thick. The in the gas evolution was observed upon sample frac- samples were 100 mm in length and 4 mm in width. ture. The saturation with helium was performed using irradiation with α particles in a cyclotron [3]. The irradiation The results obtained suggest the possibility of both temperature did not exceed 400 K. The helium concen- diffusion of helium atoms along dislocation pipes and tration was 10Ð3 at. %. the entrainment of helium atoms onto the sample surface with mobile dislocations. The experimental procedure was as follows. First, the sample was heated to the test temperature (543 K) Since the deformation of the samples of and held at this temperature until helium stopped evolv- 0Kh16N15M3B steel leads to an increase in the evolu- ing. Then, the sample was deformed at a rate of 10Ð3 sÐ1. tion of helium from the material, we can suggest that The evolution of helium was observed only at the deformation at these temperatures may favor the trans- moment of sample failure. The amount of helium fer of helium to grain boundaries at temperatures that evolved was very small, at the level of the sensitivity of are lower than the characteristic temperatures of high- the technique, and was most likely related to its evolu- temperature radiation embrittlement. This may lead to tion from the fracture surface. The results of this exper- a degradation of the mechanical characteristics of the iment only permit us to make a conclusion on the low material in the case of thermomechanical loading under probability of interaction of mobile dislocations with conditions of generation and incorporation of helium helium atoms, which, after irradiation of the material atoms into the surface of materials in combination with with helium ions in the accelerator and preliminary iso- pulsed temperature actions, e.g., upon plasma disrup- thermal annealing at a temperature of 543 K, are most tion in fusion reactors. likely located in radiation vacancies, which under these The possibility of pipe diffusion of helium is con- conditions are the main sinks for interstitial helium firmed by the results of [17], where, in particular, the atoms. Assuming that the interstitial helium atoms effect of alternating loading on the kinetics of helium interact more efficiently with dislocations, we evolution from the A7 alloy (99.7% pure aluminum) attempted to “pin” them at dislocations by preliminar- was studied. The authors of [17] found that upon the ily (prior to heating) deforming the irradiated sample at action on the material of an alternating loading (trans- room temperature. In this case the interstitial helium verse bending of the sample at a frequency of 1 Hz) in atoms, upon subsequent heating, can be trapped by dis- the elastic region (the stresses calculated from the mag-

TECHNICAL PHYSICS Vol. 46 No. 2 2001 EFFECT OF THE SATURATION CONDITIONS AND STRUCTURE 191 nitude of flexure were ±0.6, ±1.2, ±1.8, and ±2.4 × locations can favor helium transfer to the surface of the 107 N/m2); the low-temperature peak of helium evolu- material, grain boundaries, or interphase interfaces. tion (at ~400 K) was shifted upon uniform heating toward lower temperatures in comparison with the state in the absence of a load, and the greater the load, the REFERENCES greater the shift. At the same time, as the stresses 1. A. G. Zaluzhnyi, V. P. Kopitin, and M. V. Chered- increase, the rate of helium evolution increases in this nichenko-Alchevskyi, Fusion Eng. Des., Part B 41, 129 temperature range. The observed changes in the kinet- (1998). ics of gas evolution with increasing mechanical stresses 2. V. F. Zelenskiœ, I. M. Neklyudov, and T. P. Chernyaeva, is ascribed to the fact that upon cyclic loads the trap- Radiation-Induced Defects and Swelling of Materials ping of helium with dislocations occurs not only during (Naukova Dumka, Kiev, 1988). the random walking of helium atoms, but also via their 3. V. F. Reutov and Sh. Sh. Ibragimov, USSR Inventor’s approach to the dislocation due to dislocation bowing. Certificate No. 531433, Byull. Izobret., No. 23 (1978). With increasing external load, the sag of the dislocation 4. A. G. Zaluzhnyœ, Yu. N. Sokurskiœ, N. P. Agapova, et al., bowing increases, thereby increasing the probability of At. Énerg. 40 (6), 490 (1976). trapping of helium atoms by a dislocation and their 5. A. G. Zaluzhnyœ, M. V. Cherednichenko-Alchevskiœ, and transfer to the surface. An increase in the load leads to O. M. Storozhuk, At. Énerg. 52 (6), 398 (1982). an increase in the absolute amount of helium whose 6. V. L. Arbuzov, S. N. Votinov, A. A. Grigor’yan, et al., At. reemission occurs through dislocations. Énerg. 55 (4), 214 (1983). 7. A. G. Zaluzhnyœ, O. M. Storozhuk, and M. V. Chered- nichenko-Alchevskiœ, Vopr. At. Nauki Tekh., Ser.: Fiz. CONCLUSION Radiats. Povrezhdeniœ Radiats. Materialoved. No. 2, 79 The investigation of the effect of helium introduc- (1988). tion into the lattice of a material on the kinetics of gas 8. V. S. Karasev, G. F. Shved, R. V. Grebennikov, et al., At. evolution shows that, as to the behavior of helium in Énerg. 34 (4), 251 (1973). structural materials, simulation experiments that ensure 9. D. M. Skorov, N. P. Agapova, A. G. Zaluzhnyœ, et al., At. the simultaneous introduction into the material lattice Énerg. 40 (5), 387 (1976). of both helium and radiation defects (in wide limits of 10. C. H. Zhang, K. Q. Chen, Y. S. Wang, and J. G. Sun, Pro- helium concentrations and radiation damage) satisfac- ceedings of the VIII International Conference on Fusion torily describe the effects of reactor irradiation. Most Reactor Materials, 1999, Part B, p. 1621. probably, this is caused by the stabilization of definite 11. A. Ryazanov, H. Matsui, and A. Kazaryan, Proceedings configurations of radiation defects by helium atoms. of the VIII International Conference on Fusion Reactor When helium is introduced using damage-free meth- Materials, 1999, Part C, p. 356. ods, in each case one should take into account the spe- 12. H. Matsui, M. Tanaka, M. Yamamoto, and M. Tada, cific features of its retention and its mobility in the J. Nucl. Mater. 191, 191 (1992). material. The neglect of these features can lead to an 13. A. G. Zholnin and A. G. Zaluzhnyœ, Poverkhnost, No. 10, incorrect treatment of the results obtained. The fact that 33 (1986). helium evolution upon heating of irradiated samples 14. I. Yu. Abdarashitov, K. V. Botvin, Sh. Sh. Ibragimov, begins only at temperatures that exceed the irradiation et al., Preprint No. 2-80, IYaF AN KazSSR (Institute of temperature can be employed to determine the irradia- Nuclear Physics, Kazakh Academy of Science, Alma- tion temperature. Ata, 1980). The investigation of the kinetics of helium evolution 15. A. G. Zaluzhnyœ, M. V. Cherednichenko-Alchevskiœ, O. M. Storozhuk, and A. G. Zholnin, At. Énerg. 56 (5), from the samples of 0Kh16N15M3B steel both after 314 (1984). their preliminary deformation and in the course of deformation showed that, in the process of heating, 16. F. Charsughi, Dissertation, Jül-2652 (Inst. für Fest- helium atoms can migrate along dislocation pipes, körperforschung, Jülich, 1992). resulting in a significant effect on helium evolution and 17. A. G. Zaluzhnyœ, A. P. Komissarov, N. A. Makhlin, et al., its redistribution in the volume of the material. The acti- Poverkhnost, No. 6, 51 (1983). vation energy for pipe diffusion of helium in the austenitic 0Kh16N15M3B steel is about 0.7 eV. Mobile dis- Translated by S. Gorin

TECHNICAL PHYSICS Vol. 46 No. 2 2001

Technical Physics, Vol. 46, No. 2, 2001, pp. 192Ð197. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 71, No. 2, 2001, pp. 61Ð66. Original Russian Text Copyright © 2001 by Bormontov, Levin, Vyalykh, Borisov.

SOLID-STATE ELECTRONICS

Theory of Laterally Nonuniform MOS Transistor under Weak Inversion: A Technique for Determination of Interface Parameters E. N. Bormontov, M. N. Levin, S. A. Vyalykh, and S. N. Borisov Voronezh State University, Universitetskaya pl. 1, Voronezh, 394693 Russia e-mail: [email protected] Received February 8, 2000

Abstract—The well-known model of currentÐvoltage (IÐV) characteristics of a MOS transistor (MOST) in weak inversion [1] was modiﬁed with regard to lateral nonuniformity of the semiconductor surface potential. A simple technique for determining the ﬂuctuation parameter and the spectral density of interface states from drain-current (output) and drainÐgate (transfer) single-threshold IÐV characteristics is developed. Combined with measurements of the MOST threshold voltage, it makes possible the calculation of the effective oxide charge. The technique is fairly accurate and is useful for IC process control. © 2001 MAIK “Nauka/Interperi- odica”.

INTRODUCTION It is the aim of this work to develop a basic model of It is known that interface states and lateral nonuni- a weak-inversion-channel MOS transistor [1] with formity (LN) of the semiconductor surface potential regard for lateral nonuniformity of the device. A solu- strongly affect the drain-current and drainÐgate charac- tion to this problem would allow designers to correctly teristics of a MOST [2]. In an effort to account for the treat postirradiation changes in the drainÐgate sub- drainÐgate curves of a low-voltage CMOS inverter, threshold characteristics and to elaborate a technique Swanson and Meindl [3] developed an approximate for determining the interface parameters that includes model of the subthreshold IÐV characteristics. Over- surface potential fluctuations. straeten, Declerck, and Muls proposed a useful method for determination of the interface state density from the MODEL OF A WEAK-INVERSION MOS MOST subthreshold IÐV characteristics [1]. They TRANSISTOR assumed that, under weak inversion conditions, the diffusion component of the current dominates and the To be definite, consider a p-channel MOST. φ semiconductor potential at the interface is uniform. The bulk potential B, surface potential ys, and drain However, most MOS structures are actually character- voltage VD are negative in this case. The channel poten- ized by a random distribution of the interface charge tial V varies from 0 at the source to VD at the drain. The density [4Ð7]. This results in a statistical scatter of the relationship between the gate voltage Vg, surface poten- potential ys. The scatter is usually described by the tial ys, and potential V at any point of the channel is Gaussian distribution [4] written as ()2 1 y s Ð y s Q kT qD kT Py()= ------exp Ð,------(1) sc ss s 2 V g =V FB Ð------++------y s ------------y s Ð V .(2) 2πσ 2σ Cox q Cox q φ φ σ Here, VFB = ms Ð Q0t/Cox is the flat-band voltage, ms is where ys is the mean value of the potential and is its standard deviation. the difference in the work functions between the metal and the semiconductor, Q0t is the fixed oxide charge, The nonuniformity of the surface potential substan- C is the geometric oxide capacitance, Q is the total tially increases when the transistor is subjected to ion- ox sc izing radiation [8Ð13]. However, a postirradiation charge of the space-charge region (SCR) of the semi- decrease in the slope of the drainÐgate characteristic is conductor, and Dss is the spectral density of interface usually associated only with an increase in the interface states. state density [14Ð16]. Despite the obvious need for tak- Our fluctuation model of the MOST IÐV character- ing into account the lateral nonuniformity of the radia- istics in the weak inversion range will be based on the tion-induced defect distribution, this has not been done model put forward in [1], where the formula for sub- before. threshold current was derived on assumption that,

THEORY OF LATERALLY NONUNIFORM MOS TRANSISTOR 193 under weak inversion, the mobile hole charge Qp in the total charge Qsc as a function of surface potential ys and SCR is much less than the charge of the depletion layer channel potential V [17]: ≈ (ionized donor impurity) QB; i.e., Qsc QB. The deple- ε tion layer charge QB was expanded in a series near the 2 skT ( ) midpoint of the weak inversion range y = 1.51lnλ, Qsc = ------exp ys Ð ys Ð 1-- s qLD where λ = n /N is the degree of doping of the semicon- (8) i D 1/2 ductor, ni is the intrinsic concentration of charge carri- λ2  qV qV + expÐ ys + ------ + ys Ð exp------ , ers, and ND is the donor concentration. Note that for- kT kT mula (2) incorporates the total space charge Qsc, which includes the charge of mobile minority carriers. It is where clear from the physical considerations that the effect of ε surface potential ﬂuctuations on the inversion layer skT L = ------charge is the strongest. Hence, a more rigorous D 2 q N D approach to the problem would be the expansion of Qsc rather than QB: is the Debye screening length. kT This expression can be rearranged to the form Q = Q* Ð ------C* (y Ð 1.51lnλ), (3) sc sc q sc s ( ) kT 2 exp Ð ys + qV/kT Q = 2εqN ------(Ð y Ð 1) 1 + λ ------where Qs*c and Cs*c are the SCR charge and capaci- sc D q s Ð y Ð 1 λ s tance, respectively, for the surface potential ys = 1.5ln . ( ) (9) Substituting (3) into (2) yields λ2 exp Ð ys + qV/kT = QB 1 + ------. Ð ys Ð 1 * kT λ Qsc qDss kT λ V g = V FB + 1.5------ln Ð ------+ ------1.5------ln Taking into account that, in the weak inversion q Cox Cox q (4) range, the charge of mobile holes in the channel is kT  C* qD  qD much less than that of ionized donors, + ------(y Ð 1.5lnλ) 1 + -----s--c + ------s-s Ð ------s-sV. q s  C C  C ox ox ox exp(Ð y + qV/kT) λ2------s------Ӷ 1, (10) The value in the brackets is the MOST gate voltage Ð ys Ð 1 λ V g* at which the surface potential equals 1.5ln . Let us introduce the notation [1] one can expand expression (9) into a series, leaving only zero- (QB) and ﬁrst-order (Qp) terms: Cox + Cs*c + qDss Cox + Cs*c n = ------, m = ------. (5) λ2 ( ) C C exp Ð ys + qV/kT ox ox Qsc = QB 1 + ------= QB + Q p, (11) 2 Ð ys Ð 1 In view of (5), expression (4) can be recast as where the mobile hole charge Qp is described by the kT( λ) qDss expression [1] Vg = V g* + ------ys Ð 1.5ln n Ð ------V. (6) q Cox q2ε N 1/2 Eventually, we obtain the relationship between the ( , )  s D  kT Q p ys V = ------(------)- ------surface potential ys and the gate voltage Vg: 2kT Ð ys Ð 1 q

2 q V g Ð V g* q Dss  q  y = 1.5lnλ + ------+ ------V × exp 2lnλ + ------V Ð y (12) s kT n kT nC  kT s ox (7) q V Ð V* qV qD = 1.5lnλ + ------g------g- + ------s-s------. kT  q  kT n kT C + C* + qD = C (y )------exp 2lnλ + ------V Ð y . ox sc ss d s q  kT s From Eq. (7), it follows that, at relatively low inter- Ӷ Here, C (y ) is the capacitance of the depletion layer face state densities (qDss Cox), the surface potential D s depends only slightly on V and depends largely on the between the inversion layer and the quasi-neutral inte- gate voltage and the semiconductor doping level. rior of the semiconductor. We now proceed to an analysis of the MOST IÐV If the surface potential is nonuniform, the SCR characteristics under weak inversion. In order to ﬁnd (depletion layer) capacitance CD must be averaged over the charge of mobile holes in the channel under weak the surface potential, which is distributed according to (1). inversion conditions, we write the expression for SCR As a result, we come to the expression for CD as a func-

TECHNICAL PHYSICS Vol. 46 No. 2 2001 194 BORMONTOV et al. tion of the mean surface potential ys [18]: * ( )kT ni q Vg Ð V g qV m = CD ys ------exp Ð------exp ------. y + 3σ q N kT n kT n s 2ε 1/2 D ( ) 1  q s N D  CD ys = ------∫ ------(------)- πσ 2kT Ð ys Ð 1 Although the drain current ID in the weak inversion 2 σ ys Ð 3 (13) range is of a diffusion character, it can be calculated from the general expression ( )2 × ys Ð ys exp Ð------dys. V 2σ2 D Zµ ( ) ID = Ð--- p ∫ Q p V dV, (19) Similarly, the total SCR capacitance, through which L the parameters m and n in [18] are deﬁned, is found by 0 averaging the total capacitance over the surface poten- as shown in [19]. Here, Z and L are the channel width tial: µ and length, respectively, and p is the hole mobility in 1.5 lnλ + 3σ the channel. ε s 1 Cs*c = ------∫ Substituting expression (18) into (19), we obtain the 2L 2πσ D 1.5 lnλ Ð 3σ ﬁnal expression for drain current in the subthreshold operating range of an LN MOS transistor: exp(y ) Ð 1 Ð λ2(exp(Ðy ) Ð 1) × ------s------s------(14) Z n kT 2 n exp(y ) Ð y Ð 1 + λ2(exp(Ðy ) + y Ð 1) I = ---µ ------i--C (y ) s s s s D L pm q  N D s D (20) ( λ)2 V Ð V* qV × ys Ð 1.5ln × q g g  D m exp Ð------dys . exp Ð------1 Ð exp------ , 2σ2 kT n kT n

The average surface potential and the gate voltage where the capacitance CD(ys ) is given by (13) and m are related by Eq. (7). This equation contains the volt- and n, by (5). age V* , which has the following form for an LN g Under weak inversion and for low interface state MOST: densities, the mean surface potential ys corresponding Q* qD kT λ sc ss kT λ to a given gate voltage Vg is determined by the expres- V g* = V FB + 1.5------ln Ð ------+ ------1.5------ln , (15) q Cox Cox q sion V Ð V* where Qs*c is calculated from the formula [18] λ q g g ys = 1.5ln + ------. (21) 1.5 lnλ + 3σ kT n 2qN D LD [( ) Qs*c = ------∫ expy Ð y Ð 1 Thus, formula (20) in combination with (13) and 2πσ (21) allows one to calculate the MOST IÐV characteris- 1.5 lnλ Ð 3σ (16) 2 tics under weak inversion with regard for the LN of the 2 1/2 (y Ð 1.5lnλ) + λ (exp(Ðy) + y Ð 1) ] exp Ð------dy. surface potential. The form of expression (20) is similar 2σ2 to that of the earlier relationship for subthreshold characteristics [1], but the parameters appearing in these Thus, with surface potential ﬂuctuations taken into expressions are different. For example, instead of the consideration, the charge of mobile holes in the channel depletion layer capacitance CD, the parameters m and n under weak inversion can be given in a form similar in expression (20) contain the total SCR capacitance to (12): Csc including the inversion layer capacitance. In addition, all of the capacitances in (20) are averaged over kT  q  Q (y , V) = C (y )------exp 2lnλ + ------V Ð y . (17) the surface potential. p s D s q  kT s The basic static parameter of a MOST, the threshold In view of (7), formula (17) can be recast as voltage VT, is calculated with consideration for surface potential ﬂuctuations by the formula * ( )kT ( λ) q Vg Ð V g Q p = Cd ys ------exp 0.5ln exp Ð------q kT n kT Q qD kT V = V + 2------lnλ Ð -----s--c--T- + ------s-s2------lnλ, (22) T FB q C C q qV  qD  ox ox × exp ------1 Ð ------s-s------(18) kT   Cox + Cs*c + qDss where the threshold SCR charge QscT corresponding to

TECHNICAL PHYSICS Vol. 46 No. 2 2001 THEORY OF LATERALLY NONUNIFORM MOS TRANSISTOR 195 the mean surface potential 2lnλ is given by [18] The parameter m can be found by dividing expres-

2 lnλ + 3σ sion (30) by (26). Using deﬁnition (5), one can obtain 2qN D LD [( ) an experimental value of the SCR capacitance in the QscT = ------∫ expy Ð y Ð 1 2πσ midpoint of the weak inversion range by the formula 2 lnλ Ð 3σ (23) Cs*c = (m Ð 1)Cox. For a given dopant concentration ND, 2 2 1/2 (y Ð 2lnλ) + λ ( ( ) ) ] the capacitance Cs*c depends only on the ﬂuctuation exp Ðy + y Ð 1 exp Ð------2------dy. 2σ σ parameter. Therefore, using the analytical Cs*c vs.

dependence [formula (14)] and the experimental Cs*c DETERMINATION OF INTERFACE value, one can calculate the fluctuation parameter σ. PARAMETERS σ Nomograms illustrating the Cs*c vs. dependences for Expression (20) may be recast in the more compact different dopant concentrations in the semiconductor form are presented in Fig. 1. They allow us to find the fluctuation parameter. q V g Ð V g* qV D m ID = I0 exp Ð------1 Ð exp------ , (24) kT n kT n If m and n are known, the spectral density Dss of where interface states can be calculated. In fact, from the definitions of m and n [formulas (5)], we have 2 n Zµ n kT i ( ) ( ) I0 = --- p--- ------ ------CD ys . (25) Cox + Cs*c  n  Cox + m Ð 1 Cox L m q N D D = ------Ð 1 = ------ss q m  q If the MOST drainÐgate characteristic is plotted in (31) tanα C the semilogarithmic coordinates lnID Ð qVg/kT, its slope ×  1  D ox  1  ------α----- Ð 1 = ------α------------α----- Ð 1 . is equal to tan D tan g q tan D ∂lnI kT D α 1 The density of interface states from formula (31) ------∂------= tan g = Ð--. (26) q V g n λ corresponds to the mean surface potential ys = 1.5ln , It should be noted that the parameter n is propor- i.e., to the midpoint of the weak inversion range. Since tional to the characteristic subthreshold voltage S = the energy dependence of the interface state density is ∂ ∂ Vg/ logID , which changes the drain current by one weak in this range [1, 4], the density can be set constant order of magnitude. Using formula (26), one obtains –8 –2 ∂ ∂ Csc, 10 F cm V g Vg kT S = ∂------= ∂------ln10 = ------nln10. (27) 10 logID lnID q As follows from expression (24), as the drain voltage grows, the drain current tends to the value 8 q V Ð V* I = I exp Ð------g------g- , (28) D max 0 kT n 6 which is called subthreshold saturation current. For- mula (24) can thus be recast as 16 –3 ND = 10 cm 4  ID  qV D m ln1 Ð ------ = ------. (29) I D max kT n 2 If the MOST subthreshold drain-current character- N = 1015 cm–3 istic is plotted in the coordinates D

14 –3  ID  qV D ND = 10 cm ln1 Ð ------ Ð ------, 0 1 2 3 4 5 I D max kT σ, kT/q its slope is equal to Fig. 1. SCR capacitance in the midpoint of the weak inver- m * σ tanα = --- . (30) sion range Csc vs. ﬂuctuation parameter nomograms. ND D n is the dopant concentration in the MOST substrate.

TECHNICAL PHYSICS Vol. 46 No. 2 2001 196 BORMONTOV et al.

log(1 – / ) mula (22) we obtain ID IDmax 0  kT  Q = Ð Q Ð C V Ð φ Ð 2------lnλ –1 0t scT ox T ms q  (32) –2 kT + 2qD ------lnλ, –3 ss q –4 where the space charge QscT is given by formula (23). –5 Note that, in our method, the MOST interface –6 parameters have been obtained without using the 1 assumption [14] that the interface state charge is equal –7 2 to zero in the “midgap.” 3 –8 0 0.1 0.2 0.3 0.4 0.5 | | VD , V RESULTS AND DISCUSSION Results of simulation of the drain-current and drainÐ Fig. 2. Output characteristics of the p-channel MOST under gate characteristics under weak inversion are presented weak inversion. Gate voltage V = Ð2 V, dopant concentra- σ 15 g Ð3 in Figs. 2 and 3. The fluctuation parameter was varied tion in the substrate ND = 10 cm , oxide thickness d0x = 14 200 nm, spectral density of interface states D = 5 × from 0 to 5; dopant concentration ND, from 10 to 10 −2 Ð1 ss 16 Ð3 10 cm eV , flat-band voltage VFB = Ð1 V, and channel 10 cm ; and interface state density Dss, from 0 to width-to-length ratio Z/L = 1. The fluctuation parameter σ = 1011 cmÐ2 eVÐ1. (1) 3, (2) 4, and (3) 5. It is seen that the slope of the ln(1 Ð ID/IDmax) vs. σ VD curve (i.e., the value of m/n) increases with , the log(ID) increase becoming significant at σ > 3 (Fig. 2). At the –2 3 same time, the slope of the lnID vs. Vg curve, i.e., the value of 1/n, remains almost constant at σ < 3 but sig- –4 nificantly decreases with growing σ for σ > 3 (Fig. 3). 2 σ ∆ The effect of on the threshold voltage shift VT = σ σ σ –6 1 VT( ) Ð VT ( = 0) is illustrated in Fig. 4. When > 2, the threshold voltage starts to grow markedly. Note also –8 that the higher the dopant concentration in the substrate σ ND, the faster the VT shift increases with (Fig. 4). –10 The effect of surface potential fluctuations on the char- σ –12 acteristic voltage S is shown in Fig. 5. At > 2, S also grows rapidly because of an increase in the SCR capac-

–14 itance Cs*c . 1 2 3 4 5 |V |, V g It should be noted that the slope of the drainÐgate characteristics decreases with an increase in both inter- Fig. 3. Transfer characteristics of the p-channel MOST σ face state density Dss and . In addition, interface states under weak inversion. The gate voltage VD = Ð1 V. The other parameters are the same as in Fig. 2. shift the MOST threshold voltage, increasing its absolute value. Let us apply our technique to determine the inter- with a high accuracy. Hence, the value of Dss found can face parameters from experimental drain-current and be extended to the entire subthreshold range. drainÐgate characteristics of a test p-channel MOST. The transistor was fabricated on a KÉF-4.5 silicon sub- Thus, from the slopes of the MOST drainÐgate and strate (resistivity 4.5 Ω cm, phosphorus concentration the drain-current characteristics plotted in semiloga- 1015 cmÐ3). The gate oxide (SiO ) thickness was d = σ 2 ox rithmic coordinates, the ﬂuctuation parameter and the 40 nm. A high LN of the surface potential was achieved spectral density D of interface states are found. γ ss by Co60 irradiation (energy of quanta Eγ ~ 1.2 MeV). With D and σ known and the threshold voltage V Plotted in semilogarithmic coordinates, the experimen- ss T tal drain-current and drainÐgate characteristics had 1/2 determined (for example, by extrapolating the ID (Vg) clearly deﬁned linear portions. Their slopes were α α curve in the strong inversion region to the Vg axis [17]), tan D = m/n = 0.89 and tan g = Ð1/n = Ð0.75. Using the effective oxide charge Q0t can be calculated. Indeed, these values, we calculated the SCR capacitance in the φ in view of the relationship VFB = ms Ð Q0t/Cox, from for- midpoint of the weak inversion range, Cs*c = (m Ð 1)C0x =

TECHNICAL PHYSICS Vol. 46 No. 2 2001 THEORY OF LATERALLY NONUNIFORM MOS TRANSISTOR 197

|∆ | VT , V Thus, from static subthreshold IÐV characteristics, 8 one can estimate the MOST charge parameters having 3 regard to the LN of the oxideÐsemiconductor interface. 7 The method is useful for IC process control. 6 5 REFERENCES 4 1. R. J. van Overstraeten, G. J. Declerck, and P. A. Muls, IEEE Trans. Electron Devices ED-22 (5), 282 (1975). 3 2 2. R. J. van Overstraeten, G. J. Declerck, and G. Broux, 2 IEEE Trans. Electron Devices ED-20 (12), 1154 (1973). 1 1 3. R. M. Swanson and J. D. Meindl, IEEE J. Solid-State Circuits SG-7 (4), 140 (1972). 0 1 2 3 4 5 4. E. H. Nicollian and A. Goetzberger, Bell Syst. Tech. J. 46 σ, kT/q (5), 1055 (1967). 5. J. R. Brews, J. Appl. Phys. 46 (5), 2181 (1975). Fig. 4. Threshold voltage shift ∆V due to surface potential σ T 14 15 6. C. Werner, H. Bernt, and A. Eder, J. Appl. Phys. 50 (11), fluctuations vs. . d0x = 200 nm; ND = (1) 10 , (2) 10 , and (3) 1016 cmÐ3. 7015 (1979). 7. K. Zeigler and E. Klausmann, Appl. Phys. Lett. 28 (11), 678 (1976). S, V/decade 0.45 8. Ionizing Radiation Effects in MOS Devices and Circuits, Ed. by T. P. Ma and P. V. Dressendorfer (Wiley, New 3 York, 1989). 0.40 9. H. Terletzki, A. Boden, F. Wulf, and W. R. Fahrner, Phys. 0.35 Status Solidi A 86 (8), 789 (1984). 10. R. K. Freitag, E. A. Burke, and D. B. Brown, IEEE 0.30 Trans. Nucl. Sci. 34 (6), 1172 (1987). 11. N. Saks and M. G. Ancona, IEEE Trans. Nucl. Sci. 34 0.25 2 (6), 1348 (1987). 12. M. A. Xapsos, R. K. Freitag, C. M. Dozier, and 0.20 1 D. B. Brown, IEEE Trans. Nucl. Sci. 37 (6), 1671 (1990). 0.15 13. R. K. Freitag, E. A. Byrke, C. M. Dozier, and D. B. Brown, IEEE Trans. Nucl. Sci. 35 (6), 1203 0.10 (1988). 0 1 2 3 4 5 σ, kT/q 14. P. J. McWhorter and P. S. Winokur, Appl. Phys. Lett. 48 (2), 133 (1986). Fig. 5. Characteristic subthreshold voltage S vs. σ. (1Ð3) The 15. D. M. Fleetwood, M. R. Shaneyfelt, J. R. Schwank, same as in Fig. 4. et al., IEEE Trans. Nucl. Sci. 36 (6), 1816 (1989). 16. Z. Shanfield and M. M. Moriwaki, IEEE Trans. Nucl. Sci. 34 (6), 1159 (1987). 1.57 × 10Ð8 F/cm2, and then (from the corresponding nomogram in Fig. 1) the fluctuation parameter σ = 2.8. 17. S. Sze, in Physics of Semiconductor Devices (Wiley, The spectral density of interface states calculated from New York, 1981; Mir, Moscow, 1984), Vol. 2. formula (31) was found to be D = 7.7 × 1010 cmÐ2 eVÐ1. 18. E. N. Bormontov and S. V. Lukin, in Proceedings of the ss 5th International Conference on Simulation of Devices 1/2 From the ID (Vg) curve in the strong inversion range, and Technologies, Obninsk, 1996, p. 35. we also estimated the threshold voltage: VT = Ð1.6 V. 19. R. J. van Overstraeten, G. J. Declerck, and G. Broux, σ IEEE Trans. Electron Devices ED-20 (12), 1150 (1973). Finally, substituting the values of VT, Dss, and into formulas (23) and (32), we calculated the effective × Ð8 2 oxide charge Q0t = 6.15 10 C/cm . Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 46 No. 2 2001

Technical Physics, Vol. 46, No. 2, 2001, pp. 198Ð201. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 71, No. 2, 2001, pp. 67Ð70. Original Russian Text Copyright © 2001 by Arutyunyan, Akhoyan, Adamyan, Barsegyan.

SOLID-STATE ELECTRONICS

Laser-Induced Implantation and Diffusion of Magnesium into Silicon V. M. Arutyunyan, A. P. Akhoyan, Z. N. Adamyan, and R. S. Barsegyan Yerevan State University, ul. A. Manukyana 1, Yerevan, 375049 Armenia e-mail: [email protected] (Aroutiounian) Received April 21, 2000

Abstract—Laser-induced diffusion (“implantation”) of magnesium atoms into silicon was studied experimentally. Neodymium-glass laser irradiation (λ = 1.06 µm, τ ~ 0.4 ms) was found to increase the diffusion coefﬁ- cient and solubility of magnesium in silicon. CurrentÐvoltage and capacityÐvoltage characteristics, as well as thermostimulated current spectra of 〈Si + Mg〉 crystals, were obtained. © 2001 MAIK “Nauka/Interperiodica”.

Infrared (IR) detectors operating in the atmosphere Reproducibility, uniformity, and degree of doping transmission ranges (3Ð5 and 8Ð14 µm) are still a focus are strongly dependent on diffusion source, diffusion of interest today. The characteristics and efficiency of annealing temperature, and cooling rate. However, the semiconductor IR detectors markedly depend on the small segregation coefficient and the high vapor pres- properties that an injected impurity imparts to the semi- sure of magnesium at the melting temperature of sili- conductor. The most important parameters in this con greatly hamper the introduction of Mg into silicon respect are impurity levels in the energy gap, cross sec- during growth. This has fostered a search for new tech- tion of carrier capture by these levels (hence, electron niques of doping silicon by magnesium. and hole lifetimes), photoionization cross section, and It is well known that laser irradiation (with an electroactive impurity concentration [1]. energy density W ~ 1 J/cm2) of semiconductor materi- Due to high manufacturability and high concentra- als gives rise to nonequilibrium effects that are charac- tion of electroactive impurity atoms, deep- and shal- terized by enhanced migration of impurity atoms dur- low-impurity silicon has recently started to play a sig- ing short laser pulses [2]. nificant role as a basic material for photodetectors oper- In this paper, we report on laser-induced “implanta- ating in the range indicated above. However, in spite of tion” of magnesium into silicon with a view to product- the advances in silicon technology, doping of silicon by ing Si〈Mg〉-based IR detectors. As a starting material, certain impurities with traditional methods is often a we took floating-zone p-Si(111) wafers of thickness challenge. Therefore, modern semiconductor electron- less than 1.0 mm and resistivity ρ ~ 20Ð40 kΩ cm. ics widely employs photonic technologies. Laser irradi- Chemically cleaned wafers were coated by a magne- ation is one of them [2Ð5]. sium film evaporated at a pressure of 4 × 10Ð5 mm Hg. It is well known that magnesium is a suitable dopant Properly selected deposition rates and substrate tem- µ perature (T = 250°C) provided for good adhesion of a for silicon infrared detectors operating in the 8Ð14 m µ range. Being an interstitial impurity, magnesium ~0.2- m-thick deposited layer. behaves as a helium-like double donor with ionization Then, the covered surface was irradiated by a energies E = 0.107 and 0.25 eV for the neutral (Mg0) focused beam from a Nd-glass free-running laser c (wavelength λ = 1.06 µm, pulse width τ ~ 0.4 ms). and singly ionized (Mg+) donor states, respectively [6Ð8]. Laser fluence was chosen according to the wafer sur- Hall measurements have revealed another four shal- face temperature evaluated from the dependence of the low donor levels with ionization energies of 0.04, temperature on the pulse energy and exposure time 0.055, 0.08, and 0.093 eV [8]. The 0.055 and 0.093 eV [2, 3, 11]. At the fluence W ~ 1.5 J/cm2, the surface is levels have also been discovered from photoconductiv- heated up to the MgSi eutectic temperature, ~950°C for ity spectra [9]. The presence of the latter level, as well 36.61 wt % Si [12]. Reflowing of the wafer surface was as of the deep levels indicated above, has been con- clearly observed under a microscope. Note that, at firmed by theoretical study [10]. Hall measurements smaller laser fluences, Mg2Si eutectic with 58 wt % Si performed in [7] on Mg-doped n-silicon have indicated (the eutectic temperature is about 637°C [12]) did not two plausible electrically different states of magnesium form. The beam scanned only half the wafer surface by ions. The first is the amphoteric state with the acceptor moving the objective table in two perpendicular direc- level Ec = 0.115 eV, and the second is the donor level tions. The other half of the wafer remained intact. After Ec = 0.227 eV. irradiation, which most likely resulted in the formation

LASER-INDUCED IMPLANTATION AND DIFFUSION 199 of MgSi eutectic, the wafers were placed into a diffu- n p sion chamber with a continuous flow of inert gas and standard thermal diffusion for 10 h at T = 1200°C was n p carried out. R , Ω cm R , kΩ cm After diffusion, the wafers were quickly cooled by n p immersion in water at room temperature. Then, the samples were mechanically and chemically treated. It 20 was found that the Mg-coated side of the wafer 1 changed the conductivity type from p to n over the entire surface. On the back side of the wafers, the con- 40 ductivity type remained the same as before diffusion (i.e., p-type). To measure the diffusion depth of Mg atoms, a skew metallographic section was made and the conductivity type across the wafer was determined. The 10 diffusion depth of Mg atoms estimated from the nÐp 2 20 junction position was found to be L* ~ 650 and L ≤ 500 µm in the irradiated and unirradiated parts of the wafer, respectively (here, L* and L are the depths of the nÐp junction in the respective parts of the surface). Along with these measurements, a four-point probe 100 300 500 700 900 method was used to determine the sample resistivity, d, µm which turned out to be smaller in the irradiated part (Fig. 1). Fig. 1. Resistivity profiles and nÐp junction depths for An extension of the doped layer and an increase in (1) unirradiated and (2) irradiated parts of the wafer. the conductivity of the irradiated part can be explained as follows. (1) Laser irradiation causes magnesium sili- I, µA cides, for example, MgSi, to form on the surface. They may serve as an infinite diffusion source [13]. (2) Laser 1 irradiation produces a liquid phase on the surface, 8 where the impurity diffusion coefficient is considerably 2 larger than in single-crystal (solid) silicon; assuming 6 3 that the extension of the magnesium-doped layer is 4 fully accounted for by the increased diffusion coeffi- 4 cient in the liquid phase and using the relationship 2 L* Ð L = Dτ –5 –4 –3 –2 –1 (where D is the diffusion coefficient and τ is the pulse 1 2 3 4 5 V, V duration), one obtains a diffusion coefficient in the melt –2 that exceeds the known values by several orders of magnitude. (3) Irradiation generates defects (for exam- 1 2 3 4 –4 ple, silicon vacancies), which not only accelerate diffusion [14, 15], but also increase the ultimate solubility of Fig. 2. IÐV characteristics of 〈Si + Mg〉 structures after irradiation with on energy density W = (1) 1.8, (2) 2.4, (3) 2.8, electroactive magnesium atoms [4]. In addition, ather- 2 mic, photoinduced diffusion is not ruled out either [16]. and (4) 3.3 J/cm . To display the advantages and characteristic fea- 2 tures of this laser technique, we studied currentÐvoltage sium silicides (Mg2Si or MgSi) at W = 1.8Ð2.5 J/cm or (IÐV) and capacitanceÐvoltage (CÐV) characteristics, as magnesium evaporation from the near-surface region at well as thermostimulated current (TSC) spectra, of high fluences (at W > 2.5 J/cm2, silicon melts). The laser-irradiated silicon wafers covered by a magnesium scatter in the voltages at which the forward current film (〈Si + Mg〉) for different laser fluences. To perform sharply rises is probably associated with the formation these measurements, we made alloyed Au + 1% Sb and of magnesium silicides with different potential barrier Al contacts to the n- and p-type sides of the samples, heights. The absence of reverse saturation current in respectively. Fig. 2 is probably related to current leak across the As seen from the IÐV characteristics shown in junction because of a spread in the beam intensity over Fig. 2, irradiation with a laser-fluence W = 1.8 J/cm2 the surface area. produces a rectifying structure. With an increase in the The CÐV characteristics for the irradiated samples laser fluence, the rectifying properties deteriorate, were taken at a frequency of 1 MHz (Fig. 3). From which may be due to either the formation of magne- Figs. 2 and 3, it follows that, at W = 1.8 J/cm2, when the

TECHNICAL PHYSICS Vol. 46 No. 2 2001

200 ARUTYUNYAN et al.

C, pF the expression 1 2  kT 16 1 ----- = ------U Ð U + ------, 2 εε 2 ( ) 0 e  14 C e 0 A N x

12 where A is the surface area of a test varactor, U0 is the barrier height at the metalÐsemiconductor junction, U 10 2 εε is the reverse bias, and x = ( 0A)/C, one obtains by dif- 8 3 ferentiation C3 dC Ð1 6 N(x) = Ð------. εε 2 dU e 0 A –6 –4 –2 0 2 4 6 V, V The calculated dependence N(x) is shown in Fig. 4. 〈 〉 It is seen that the impurity atom concentration in the Fig. 3. CÐV characteristics of Si + Mg structures after irra- 16 Ð3 diation with an energy density W = (1) 1.8, (2) 2.4, and surface layer, N ~ 10 cm , exceeds the known value (3) 2.8 J/cm2. of the ultimate solubility of magnesium in silicon [1]. To record TSC spectra at 80 K and above, the wafers were immersed in a liquid nitrogen cryostat with opti- N, cm–3 17 cal windows. The samples were in contact with a cop- 10 per unit having a heater to smoothly vary the wafer temperature. The temperature dependence of the current in 1016 the range indicated above was measured at a heating rate of ~0.4 K sÐ1. The TSC spectra were analyzed according to the fol- 1015 lowing procedure [18]. Placed into a vacuum cryostat, the sample was cooled in liquid nitrogen and then exposed to light from the fundamental band. Then, the 1014 sample was uniformly heated in darkness, and the tem- µ 0 5 10 15 20 d, m perature variation of the current was recorded. Compar- ing the resulting dependences with those obtained with- Fig. 4. Concentration proﬁle of magnesium impurity atoms. out preliminary low-temperature photoexcitation of the sample, one can discover TSC peaks due to trap depletion. These data clarify the energy of the traps in the ITSC, A energy gap. Sometimes, the trap concentrations and capture cross sections can also be determined.

–6 Under certain conditions, the initial stage of TSC 10 1 2 growth can be described as ∼ ( ) I exp ÐEi/kT , (1)

where Ei is the trap energy reckoned from the edge of the nearest allowed band [18]. 10–7 TSC spectra of the irradiated samples with the deposited magnesium ﬁlm (〈Si + Mg〉) are presented in Fig. 5. A peak indicating the appearance of a local center is seen at Tmax = 105 K. Note that laser irradiation of the reference silicon wafers (without the magnesium ﬁlms) does not give rise to the peak in the TSC spectra, 10–8 120 160 200 240 T, K thus pointing to the “magnesium” origin of this peak. The corresponding energy level determined by formula (1) from the initial stage of TSC growth (Fig. 5,

Fig. 5. TSC spectra of irradiated 〈Si + Mg〉 structures: curve 1) was equal to E = 0.13 eV, whereas that respon- (1) with and (2) without illumination. c sible for the equilibrium conductivity of the irradiated samples (Fig. 5, curve 2) was found to be Ec = 0.28 eV. rectifying properties are the best, the CÐV curve (Fig. 3, These values are in good agreement with the literature curve 1) closely agrees with the theoretical depen- data for magnesium levels in silicon [6Ð8]. dence. This curve was used to calculate the concentra- Thus, we can argue that laser irradiation of 〈Si + Mg〉 tion proﬁle N(x) of ionized impurity atoms [17]. From samples favors magnesium diffusion into silicon. At

TECHNICAL PHYSICS Vol. 46 No. 2 2001 LASER-INDUCED IMPLANTATION AND DIFFUSION 201 particular radiation intensities, the conductivity type 7. E. Ohta and M. Sakata, Solid-State Electron. 22, 677 changes and rectifying currentÐvoltage characteristics (1979). are observed. The magnesium concentration proﬁle 8. A. L. Lin, J. Appl. Phys. 53, 6989 (1982). constructed from the capacitanceÐvoltage characteris- 9. M. Kleverman, K. Bergman, and H. G. Grimmeis, Semi- tics shows that, in the near-surface region, the magne- cond. Sci. Technol. 1, 49 (1986). sium concentration exceeds the concentration of elec- 10. E. Janzen, R. Stedman, G. Grossmann, and H. G. Grim- troactive magnesium atoms reported elsewhere in the meiss, Phys. Rev. B 29, 1907 (1984). literature. As follows from the TSC spectra, laser irra- 〈 〉 11. J. Wang, R. Wood, and P. Pronko, Appl. Phys. Lett. 33, diation Si + Mg produces local centers with ioniza- 455 (1978). tion energies E = 0.13 and 0.28 eV. c 12. M. Hensen and K. Anderko, Constitutions of Binary Alloys (McGraw-Hill, New York, 1958), p. 1488. REFERENCES 13. B. I. Boltaks, Diffusion in Semiconductors (Fizmatgiz, 1. F. V. Gasparyan, Z. N. Adamyan, and V. M. Arutyunyan, Moscow, 1961; Academic, New York, 1963). Silicon Photodetectors (Yerevansk. Gos. Univ., Yerevan, 14. V. L. Vinetskiœ and G. E. Chaœka, Fiz. Tverd. Tela (Le- 1989). ningrad) 24, 2170 (1982) [Sov. Phys. Solid State 24, 2. V. I. Fistul’ and A. M. Pavlov, Fiz. Tekh. Poluprovodn. 1236 (1982)]. (Leningrad) 17, 854 (1983) [Sov. Phys. Semicond. 17, 15. V. N. Strekalov, Fiz. Tekh. Poluprovodn. (Leningrad) 20, 535 (1983)]. 361 (1986) [Sov. Phys. Semicond. 20, 225 (1986)]. 3. I. B. Khaœbulin and L. S. Smirnov, Fiz. Tekh. Polupro- 16. V. S. Vikhnin and M. K. Sheœnkman, Fiz. Tekh. Polupro- vodn. (Leningrad) 19, 569 (1985) [Sov. Phys. Semicond. vodn. (Leningrad) 19, 1577 (1985) [Sov. Phys. Semi- 19, 353 (1985)]. cond. 19, 970 (1985)]. 4. A. V. Dvurechenskiœ, G. A. Kachurin, E. V. Nidaev, and 17. L. V. Pavlov, Methods for Determination of Parameters L. S. Smirnov, Pulsed Annealing of Semiconducting of Semiconducting Materials (Vysshaya Shkola, Mos- Materials (Moscow, 1982). cow, 1987). 5. S. Yu. Karpov and Yu. V. Koval’chuk, Fiz. Tekh. Polupro- 18. A. G. Milnes, Deep Impurities in Semiconductors vodn. (Leningrad) 20, 1945 (1986) [Sov. Phys. Semi- (Wiley, New York, 1973; Mir, Moscow, 1977). cond. 20, 1221 (1986)]. 6. J. W. Chen and A. G. Milnes, Annu. Rev. Mater. Sci. 10, 157 (1980). Translated by A. Sidorova-Biryukova

TECHNICAL PHYSICS Vol. 46 No. 2 2001

Technical Physics, Vol. 46, No. 2, 2001, pp. 202Ð206. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 71, No. 2, 2001, pp. 71Ð76. Original Russian Text Copyright © 2001 by Sopinskiœ, Romanenko, Indutnyœ.

SOLID-STATE ELECTRONICS

Profiling of Holographic Diffraction Gratings by Using SilverÐArsenic Triselenide Interaction N. V. Sopinskiœ, P. F. Romanenko, and I. Z. Indutnyœ Institute of Semiconductor Physics, National Academy of Sciences of Ukraine, Kiev, 03028 Ukraine e-mail: [email protected] Received May 10, 2000

Abstract—Rulings of holographic gratings were proﬁled by interaction of silver with chalcogenide glassy semiconductors. The shape of the rulings was determined with an atomic force microscope. Angular and spectral dependences of the diffraction efﬁciency of the gratings were found, and a relation between these dependences and the grating surface pattern was analyzed. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION phase. This phase differs in properties from both the Profiled (blazed) gratings make it possible to con- metal and the semiconductor [10, 11]. Profiling was centrate energy in a given spectrum range. Of special accomplished by transforming starting symmetric interest are profiled holographic gratings (PHGs), since (unprofiled) HGs written on CGS films. Here, we took they combine the advantages of ruled gratings with advantage of the fact that the etching selectivity for Ag- those of conventional (unprofiled) holographic gratings doped CGS films is much higher than the photoinduced (CHGs), namely, high diffraction efficiency in a given selectivity. spectrum range and low level of light scattering. One In deciding on a CGS material, preference was method of fabricating PHGs is making them from given to As2Se3. First, the use of arsenic triselenide CHGs. To do this, an initial symmetric grating is usu- makes feasible writing of a starting symmetric HG by ally etched by an inclined ion beam [1]. means of a HeÐNe laser [12], which was used in this The recent trend in CHG technology is the use of work. Second, the rate of interaction between As2Se3 chalcogenide glassy semiconductor (CGS) films [2–5]. and Ag is one of the highest among CGSÐMe systems: These films are high-resolution inorganic photoresists. intense interaction proceeds both during and after dep- Due to light-induced structure transformations, their osition of the metal even at room temperature [9]. solubility, particularly, in organic alkaline solvents, changes. Based on this effect, symmetric CHGs are PROFILING TECHNIQUE AND STUDY obtained when an interference pattern is recorded (writ- OF SYMMETRIC AND ASYMMETRIC ten) on a CGS film with subsequent selective etching in HOLOGRAPHIC GRATINGS organic alkaline etchants [2Ð5]. One of today’s problem in this field is the transfor- Starting gratings were written on relatively thick mation of symmetric CGS-based CHGs into asymmet- (800Ð1000 nm) As2Se3 films deposited onto mirrorlike ric ones. Publication [5] reports PHGs prepared by ion glass substrates by thermal evaporation of glassy etching of symmetric CHGs formed on CGS films. The As2Se3. The gratings were exposed on a holographic unique properties of CGS films allow for other, setup based on the wave amplitude division method. unusual, methods for transforming their surface pat- The spatial frequency of the gratings was 600 mmÐ1. tern. In [6, 7], we developed a method of fabricating Writing was carried out with a HeÐNe laser (λ = blazed HGs. In this method, symmetric rulings of an 632.8 nm). The radiant exposure was ~10Ð1 J/cm2. initial grating are made asymmetric by using additional After exposure, the samples were etched in an amine- oblique monochromatic or polychromatic irradiation based alkaline solution. During etching, the CGS film and chemical etching of the irradiated grating. was selectively dissolved with the formation of rulings In this work, we made an attempt to fabricate PHGs of regular shape. through interactions that take place when silver layers The next procedure was profiling. A thin (1Ð10 nm) are vacuum-deposited onto CGS films. AgÐCGS inter- Ag layer was applied on the gratings at a certain angle action begins during deposition [8] and continues at a ϕ varying from 10° to 80° with respect to the normal to different rate (depending on a specific AgÐCGS sys- the grating. The grating was mounted in such a way that tem) after the deposition process is terminated [9]. The the flux of evaporated silver was directed normally to metal penetrates into the semiconductor to form a the rulings. Ag penetrates into the As2Se3 layer at the metal-enriched (to several tens of atomic percent) instant of deposition and also as a result of subsequent

PROFILING OF HOLOGRAPHIC DIFFRACTION GRATINGS 203 chemically and thermally stimulated diffusion to form h, nm a metal-enriched semiconductor layer (reaction prod- 150 uct). The etch rate of Ag-doped CGS films in alkaline etchants is much lower than for undoped ones; there- 100 fore, in subsequent etching of the grating, the former served as a protective mask. Unprotected regions of the 50 As2Se3 layer were etched off, and the grating (rulings) became asymmetric. In this way, we obtained profiled 0 (blazed) gratings. Profiling was performed in the same amine-based etchant, which was used to pattern the starting gratings. –50 To estimate the profiling effect, we recorded angular –100 and spectral dependences of the absolute diffraction 0 1 2 3 4 5 6 efficiency η for the starting and transformed gratings in x, µm the first diffraction order (η is defined as the ratio of the first-order diffraction intensity to the incident inten- Fig. 1. Ruling profile in the initial symmetric holographic sity). Prior to optical measurements, the as-prepared grating. and profiled gratings were covered by a 100-nm-thick reflection aluminum film. η vs. β curves (β is the angle of incidence of light) were taken with an LGN 208 A formed [2, 3]. These experiments and also the simula- laser (λ = 632.8 nm, β = 0°Ð80°). Spectral measure- tion of surface patterning suggest that not only the ments of η were made using Littrow’s autocollimation depth but also the shape of the pattern depends on the scheme. The angle between the incident and diffracted initial thickness of the layers, their properties, exposure beams was about 8°; and the spectral range, 400Ð conditions, etch times, and etchant selectivity. Holo- 800 nm. Both the spectral and angular dependences of graphic diffraction gratings with rulings of sinusoidal η were measured for s- and p-polarized light (E is per- and cycloidal shapes have been obtained. pendicular and parallel to rulings, respectively), as well We performed experiments with gratings that had as for unpolarized light. sinusoidal rulings, which are the most studied. Figure 1 The surface pattern of the gratings was examined shows a typical AFM image of the starting symmetric with a Dimension 3000 scanning probe microscope grating with a spatial frequency of 600 mmÐ1. As seen (Digital Instruments) in the AFM tapping mode. from the image, the shape of the rulings is truly close to ≈ sinusoidal. The depth of the grooves is h0 150 nm; ≈ hence, the modulation depth h0/d 0.09 (the spacing of RESULTS AND DISCUSSION the grating is 1667 nm). The distance between identical The geometric and diffraction properties of the as- points on the tops of the neighboring rulings (surface prepared and profiled gratings show that the effect of distance) is 1700 nm, or 2% greater than the horizontal profiling depends, to some extent or another, on many distance between the rulings (which equals the spacing process parameters. Among them are the amount of Ag of the grating). The mean angle of inclination of the deposited on the as-prepared grating and the angle of facets (facet angle) is ~10°, the steepest being 15°. For deposition, time of AgÐAs2Se3 interaction, thickness of the wavelength and angle-of-incidence ranges used in the experiments, the diffraction efficiency of such grat- the As2Se3 film on which the initial symmetric grating is written, exposure and etching conditions for prepara- ings was the same (within the accuracy of measure- tion of the starting grating, and etching conditions for ment: 2% for angular and 5% for spectral measure- profiling. The fact that the desired effect is related to a ments) when they were irradiated from opposite direc- wide variety of physical and chemical factors implies tions perpendicular to the rulings. its significant stability. In addition, this allows us to Figure 2 shows the profile of a typical asymmetric assume that the properties of PHGs thus obtained can grating made from a symmetric one by additional evap- 〈 〉 be varied in wide limits. On the other hand, it becomes oration of a silver film of thickness hAg = 3.6 nm (this difficult to estimate the relative contribution of the value is averaged over the grating area) at 50° to the above factors to the PHG performance. In general, they normal to the substrate surface, and subsequent etch- can be subdivided into three groups: those associated ing. The ruling depth in the new, asymmetric, grating is with the CGS film (primarily its thickness and deposi- more than twice that in the initial one, 355 nm, and the tion rate), those that govern the fabrication of the sym- ≅ modulation depth is h0/d 0.21. The minimum is metric grating, and those closely related to AgÐCGS 469 nm away from the left peak and 1198 nm from the interaction. right one; that is, the projection of the larger facet onto The first two have been much studied. Empiric the substrate surface is nearly three-fourths of the grat- experimental studies and numerical simulation of HG ing spacing. The smaller facet has a gradually increas- writing in CGS-based inorganic resists have been per- ing steepness with an average angle of 37°; the steepest

TECHNICAL PHYSICS Vol. 46 No. 2 2001

204 SOPINSKIŒ et al.

h, nm of the left-hand slope of initial rulings. As we move 150 upward along the left-hand slope, the etch depth of the 100 initial grating increases, the PHG bottom lying almost in the middle of the back slope. 50 0 The degree of asymmetry of the profiled grating was studied by taking the angular and spectral dependences –50 of the diffraction efficiency η. For p polarization, the –100 angular dependences are plotted in Fig. 3. Here, –150 curves 1 and 2 were obtained when light was incident on the larger and the smaller facet, respectively. For –200 normal incidence (ϕ = 0, symmetric arrangement of –250 diffraction orders), the diffraction efficiencies mea- 0 1 2 3 4 5 6 , µm sured on the two facets differ by a factor of 8.5. This x points to the considerable asymmetry of the ruling shape. For the larger facet, the maximum efficiency was Fig. 2. Ruling profile in the grating transformed from the initial grating shown in Fig. 1. observed when the angle of incidence was close to the mean facet angle. For the smaller one, two peaks appear: the angular position of one of them (Ð50°) is η p, % close to the mean slope of this facet, while that of the other (20°) coincides with the position of the peak from 50 the larger facet and is likely to be associated with 1 rereflections of the light incident on the larger facet. 40 The share of total energy accounted for by conjugate diffraction orders is maximal at 20°; in other words, the maximum amount of light is reflected and rereflected at 30 this angle. Figure 4 shows the angular dependences for s polarization. At ϕ = 0, the diffraction efficiency for the 20 larger and the smaller facet is, respectively, 44 and 8.5%; i.e., the difference is fivefold. Unlike p polariza- 2 10 tion, here distinct anomalies are observed. The anom- aly at β = 15° is associated with the disappearance (appearance) of the second order; while that at 38°, 0 β with the appearance (disappearance) of the conjugate –80 –40 0 40 80 , deg order. The anomalies are more pronounced for the reflection from the smaller facet possibly because of the Fig. 3. Angular dependences of the diffraction efficiency of the PHG for light polarized parallel to the ruling direction. greater effect of the larger facet on the smaller than vice λ = 632.8 nm. versa. Thus, the angular dependences in the case of p polarization more adequately depict the ruling shape. portion is inclined 80° with respect to the substrate. Spectral dependences of the diffraction efficiency With the larger facet, the situation differs. It can be for unpolarized light that were taken in the autocollima- divided into three parts: top (gently sloping), middle tion regime on the larger facet of the rulings (curve 1), (nearly a plateau), and bottom (the steepest portion). on their smaller facet (curve 2), and for the symmetric The height of the gentle (top) portion somewhat grating (curve 3) are shown in Fig. 5. Throughout the exceeds the ruling half-height; here, the average incli- spectral range (400Ð800 nm), the diffraction efficiency nation is about 10°. The inclination of the bottom por- for the larger facet exceeds that for the smaller one. For tion is close to the maximum steepness of the smaller wavelengths of 620, 660, and 700 nm, these values dif- facet. On average, the larger facet angle is about 16°. fer by a factor of 5.7, 4.5, and 4.5, respectively. Also, From a comparison between the ruling shapes of the the efficiency for the larger facet exceeds that for the initial and transformed gratings, it follows that the front symmetric grating, except for the short-wave part of the facets of rulings of the initial grating remain practically interval. This means the profiling has a significant intact. This means that the reaction product nearly com- effect between 500 and 800 nm, and at 620 nm, the pletely protected the CGS layer from dissolution during effect is the greatest. If the wavelength 620 nm is taken as the blaze wavelength, one can determine the blaze profiling etching. The first (gentle) portion of the larger ϕ slope virtually copies the profile of the initial sinusoidal angle b (the larger-facet angle). This angle is found grating. The back (left-hand in Fig. 2) facets were from the formula [13] affected by etching much more noticeably. Here, small ϕ λ plateaus correspond to the partially etched bottom part 2d sin b = m b, (1)

TECHNICAL PHYSICS Vol. 46 No. 2 2001 PROFILING OF HOLOGRAPHIC DIFFRACTION GRATINGS 205

λ η where b is the blaze wavelength, d is the lattice spac- s, % ing, and m is a diffraction order. 50 The blaze angle equals 11°. Comparing the value of 1 the blaze angle obtained from spectral measurements λ 40 with the facet angles, we see that b nearly coincides with the slope of the gentle portion of the larger facet. The peak at the blaze angle is not very pronounced 30 because the three portions on the larger facet greatly differ in slope. 20 Simple geometric simulation of Ag deposition will allow us to describe AgÐAs2Se3 interaction in quantita- 2 tive terms. Let the x axis be directed normally to the rul- 10 ings and the y axis, normally to the substrate (Fig. 6). The density of Ag deposited on the ruling surface (and, 0 hence, the thickness of the Ag film) depends on the –80 –40 0 40 80 β, deg amount of the metal evaporated and the angle between the Ag flux and each specific deposition area on the rul- Fig. 4. The same as in Fig. 3 for light polarized perpendicu- ing surface. Then, the thickness of the metal at a point lar to the ruling direction. x in the cross section perpendicular to the grating relief is given by η, % ( ) Θ( ) hAg x = K cos x , (2) 60 where K is a proportionality coefficient, which depends on the amount of Ag deposited, and Θ(x) is the angle of 50 1 incidence of the Ag flux on the ruling surface at the point x. 40 We also have Θ(x) = ϕ Ð α(x), (3) 30 ϕ 3 where is the angle of incidence of the Ag flux on the 20 substrate that is reckoned from the normal to the sub- α strate and (x) is the ruling inclination to the flat sur- 10 2 face of the substrate at the point x. Equation (2) is valid if the sticking coefficient of the 0 metal is independent of the angle of incidence on the 400 500 600 700 800 λ, nm CGS film surface and on the amount of the metal deposited. The profile of the initial sinusoidal grating is Fig. 5. Spectral dependences of the diffraction efficiency for described by the expression unpolarized light. ( ) ( )( ( π )) h x = h0/2 1 + cos 2 x/d , (4) hAg, nm Y, nm where d is the grating spacing and h0 is the height of the 5.0 profile. 160 The inclination of this grating to the substrate sur- 4.5 140 face is then expressed as 120 4.0 100 α(x) = arctan{[πh /d][Ðsin(2πx/d)]}, (5) 0 3.5 ϕ 80 where h0/d is the modulation depth for the initial 60 grating. 3.0 40 20 Figure 6 illustrates the distribution of the Ag film 2.5 thickness along the ruling of the initial grating. The α 0 thickness was calculated by formulas (2), (3), and (5). 2.0 –20 Comparing this distribution with the distortions of the 0 0.25 0.50 0.75 1.00 initial grating due to the metal deposition and subse- x/d quent etching, one can conclude that etching is virtually Fig. 6. Profile of a ruling of the sinusoidal grating with h0 = absent for hAg(x) > 3 nm. When hAg(x) is less than 3 nm, 150 nm and d = 1667 nm (Y, dashed line) and the distribution the etch rate sharply grows. Since the Ag films (of of the Ag deposit along this ruling (h , solid line). The 〈 〉Ag thickness less than 10 nm) deposited on the noninter- mean thickness of the Ag layer is hAg = 3.6 nm.

TECHNICAL PHYSICS Vol. 46 No. 2 2001 206 SOPINSKIŒ et al. acting substrate are discontinuous (form islands), they etch rate. The maximum ratio of the facet projections is could not play the role of a mask during profiling etch- ~0.72d/0.28d in this case. ing. It appears that protection is provided by the top metal-doped CGS layer. The Ag thickness critical for the formation of the mask is seen to be about 3 nm. This CONCLUSION value is close to that obtained earlier upon calculating Thus, our results indicate that the profile of a holo- the Ag amount incorporated into the As2Se3 film when graphic grating can be changed by using AgÐAs2Se3 the metal is thermally evaporated on this film [8]. The interaction. The blaze angle in the PHGs depends on etch rate is maximal at the point one-fourth of the spac- the modulation depth of the initial sinusoidal gratings ing away from the top of the ruling back facet, where and also on the angle of deposition and the Ag layer the Ag thickness estimated is minimal, 2.3 nm. thickness. By controlling the Ag distribution along the From the measurements and simulation of the ruling cross section through a change in the deposition shapes of the initial and profiled gratings, an empiric angle and the Ag layer thickness, one can substantially relationship ∆h(h ) can be derived, where ∆h is the change the grating profile and, hence, the blaze angle, Ag which specifies the reflection properties of the grating. thickness of that part of the As2Se3 film dissolved during etching and hAg is the amount of the Ag deposit. As follows from correlation analysis data, this empiric REFERENCES relationship is fairly accurately approximated by the 1. J. Flammand, F. Bonnemason, A. Thevenon, and sum of two exponentials: J. X. Lerner, Proc. SPIE 1055, 288 (1989). ∆ ( ) { ( ) } 2. I. Z. Indutnyi, I. I. Robur, P. F. Romanenko, and h hAg = 204.93exp Ð hAg Ð 2.412 /0.125 (6) A. V. Stronski, Proc. SPIE 1555, 248 (1991). { ( ) } + 61.51exp Ð hAg Ð 2.412 /1.553 . 3. I. Z. Indutnyi, A. V. Stronski, S. A. Kostioukevich, et al., Opt. Eng. 34, 1030 (1995). With (6), we simulated the transformation of the ini- 4. R. R. Gerke, T. G. Dubrovina, P. A. Dmitrikov, and tial symmetric gratings into asymmetric ones. The ini- M. D. Mikhaœlov, Opt. Zh. 64 (11), 26 (1997) [J. Opt. tial grating was assumed to be sinusoidal with a modu- Technol. 64, 1008 (1997)]. lation depth of h0/d = 0.09. Variable parameters were 5. A. V. Lukin, A. S. Makarov, F. A. Sattarov, et al., Opt. angle of deposition of Ag particles and the thickness of Zh. 66 (12), 73 (1999) [J. Opt. Technol. 66, 1071 the metal deposited. (1999)]. The simulation confirms experimental data suggest- 6. P. F. Romanenko, M. V. Sopinski, and I. Z. Indytnyi, ing that the profiling effect takes place in a wide range Proc. SPIE 3573, 457 (1998). of these parameters. For example, it was shown that, as 7. P. F. Romanenko, N. V. Sopinskiœ, I. Z. Indutnyœ, et al., the Ag film gets thinner (at a given angle of Ag deposi- Zh. Prikl. Spektrosk. 66 (4), 587 (1999). tion) and etching of the Ag-doped grating becomes 8. M. T. Kostyshin, V. L. Gromashevskiœ, N. V. Sopinskiœ, selective, the initially symmetric profile transforms into et al., Zh. Tekh. Fiz. 54 (6), 1231 (1984) [Sov. Phys. an asymmetric one because of a slight (~0.01d) shift of Tech. Phys. 29, 709 (1984)]. its maxima and a more considerable (~0.05d) shift of its 9. I. Z. Indutnyœ, M. T. Kostyshin, O. P. Kasyarum, et al., minima to the left of the initial grating rulings. The Photostimulated Interactions in Metal-Semiconductor modulation depth remains practically unchanged in this Structures (Naukova Dumka, Kiev, 1992). case, since the minima and maxima lower roughly 10. A. G. Fitzgerald and C. P. McHardy, Surf. Sci. 152/153, 〈 〉 1255 (1985). equally. With further decreasing hAg , the ruling shape can be approximated by an asymmetric trapezoid. The 11. V. Honig, V. Fedorov, G. Liebmann, and P. Suptitz, Phys. modulation depth in such a grating is somewhat larger Status Solidi A 96 (2), 611 (1986). 〈 〉 12. P. F. Romanenko, I. I. Robur, and A. V. Stronskiœ, than in the initial symmetric one. As hAg continues to decrease, the shape becomes nearly triangular, the pro- Optoélektron. Poluprovodn. Tekh. 27, 47 (1994). jections of the ruling facets on the x axis being about 13. M. C. Hutley, Diffraction Gratings (Academic, London, 0.7d for the larger and 0.3d for the smaller. For an 1982). extremely thin Ag film, the larger facet has two drastically differing slopes because of a sharp increase in the Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 46 No. 2 2001

Technical Physics, Vol. 46, No. 2, 2001, pp. 207Ð211. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 71, No. 2, 2001, pp. 77Ð81. Original Russian Text Copyright © 2001 by Shuaibov, Shimon, Dashchenko, Shevera.

OPTICS, QUANTUM ELECTRONICS

An Electric Discharge Emitter Operating Simultaneously in the 308 [XeCl(BÐX)], 258 [Cl2(D'ÐA')], 236 [XeCl(DÐX)], 222 [KrCl(BÐX)], 175 [ArCl(BÐX)], and 160 [H2(BÐX)] nm Bands A. K. Shuaibov, L. L. Shimon, A. I. Dashchenko, and I. V. Shevera Uzhgorod State University, Uzhgorod, 88000 Ukraine e-mail: [email protected] Received March 31, 2000

Abstract—An experiment to optimize an ultraviolet (UV)Ðvacuum ultraviolet (VUV) multiwave emitter using chlorine molecules and chlorides of heavy inert gases is reported. The emitting medium was an ArÐKrÐCl2 or an ArÐKrÐXeÐCl2 (HCl) mixture kept at a pressure ranging from 1 to 30 kPa. Excitation was effected by means of a transverse volume discharge with spark preionization. Emission spectra were examined. The dependences of the emission intensity on the total pressure of the medium, partial pressures of its components, charging voltage, and number of discharge pulses were studied. It is demonstrated that such a discharge emits simultaneously in the 308, 258, 236, 222, 175, and 160 nm bands due to the transitions XeCl(BÐX), Cl2(D'ÐA'), XeCl(DÐX), KrCl(BÐX), ArCl(BÐX), and H2(BÐX), respectively. It was established that the respective intensities are close to each other if the partial pressures are as follows: P = 10Ð20 kPa; P = 0.4Ð0.6 kPa; P = 0.2Ð0.4 kPa, Ar Kr, Xe Cl2 P = 0.08 kPa, and P = 0.5Ð1.0 kPa. It was found that the addition of H to the medium decreases the inten- HCl H2 2 sities of the excimer bands, increases the emission resource (to 104 pulses or higher), and expands the operating wavelength range. The last-named effect is due to Lyman H2 bands (at 158Ð161 nm). © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION energies, we started looking for ways to expand the operating range into the VUV region (≤190 nm). Early Today, electric discharge excimer lamps offer the results were obtained with a source emitting at highest power and the best selectivity among sources of * spontaneous emission in the wavelength range 200Ð 258 (Cl2 ) and 175 (ArCl*) nm, the medium consisting 350 nm [1Ð3]. For this reason, they are widely of He (Ne), Ar, and Cl2 (HCl) [11]. employed in microelectronics, high-energy chemistry, This paper reports on an experiment to optimize a biology, and quantum electronics [4Ð6]. On the other UVÐVUV emitter operating simultaneously at 308, hand, the majority of them operate at a single wave- 258, 236, 222, 175, and 160 nm. Operating mixtures length determined by the emitting medium. It seems were composed of Ar, Kr, Xe, Cl (HCl), and H . worthwhile to explore the feasibility of multiwave 2 2 lamps, namely, those using particular BÐX transitions of RX* molecules, where R is Ar, Kr, or Xe and X is F EXPERIMENTAL SETUP or Cl. These sources could ﬁnd application in pulse photometry, biochemistry, biophysics, and medicine. The medium was excited by transverse volume dis- For example, they could serve to calibrate detectors of charge (TVD). Spark preionization was effected auto- short-wave light pulses or to simultaneously treat matically in the interelectrode space. A sketch of the selected molecular bonds in compounds having high emitter is shown in [8]. A TVD plasma occupied a vol- chemical or biological activity. We designed such emit- ume of 18 × 2.2 × 1.0 cm3, and the electrodes were ters for the range 353Ð222 nm. Some of them operate at spaced 2.2 cm apart. TVD was initiated by a two-loop 222 (KrCl), 249 (KrF), 308 (XeCl), and 353 (XeF) nm, LC circuit including a 30-nF storage ceramic capacitor with F and Cl being provided by CF2Cl2 molecules and 20 pulse-sharpening ceramic capacitors (KVI-3, [7, 8]. Other emitters use multicomponent media (con- 470 pF, 20 kV), the latter providing a total capacitance sisting of He, Kr, Xe, HCl, and SF6), which include two of 9.4 nF. The capacitors were combined into two banks heavy inert gases and differing halogen-containing sealed with an insulating compound. The banks were molecules [9, 10]. To generate photons with higher placed near the TVD electrodes and preionization spark

208 SHUAIBOV et al. gaps inside a discharge chamber. This excitation circuit EMISSION AND RESOURCE closely resembles the system used for generating fast CHARACTERISTICS ionization waves in pulsed longitudinal discharge [12Ð Figure 1 shows an emission spectrum of the TVD 14], since the main part of the circuit has a very small plasma in an ArÐKrÐCl mixture. Note that the UVÐ inductance (less than 10 nH), the discharge plasma is 2 separated from the grounded shields by a high-permit- VUV power is emitted largely at wavelengths of 175, tivity insulator (the compound), and the pulses of TVD 222, and 258 nm, which, respectively, correspond to the current are narrower than 30 ns. transitions ArCl(BÐX), KrCl(BÐX), and Cl2(D'ÐA'). If small amounts of Xe and H2 are added to the medium, Spectra were examined with a 0.5-m vacuum mono- the spectrum also exhibits bands at 308, 236, and 158Ð chromator including a diffraction grating with 161 nm, representing the transitions XeCl(BÐX), 1200 lines per millimeter. The reciprocal linear disper- XeCl(DÐX), and H2(BÐX), respectively. (The third tran- sion of the spectrophotometer was 1.4 nm/mm. The sition produces Lyman H bands.) With the ArÐCl emission emerged from the hermetically sealed dis- 2 2 charge chamber via a CaF window. The photodetector (HCl) mixture, only two bands are pronounced, at 258 2 and 175 nm. If Cl is replaced with HCl, the emission was built around an FÉU-142 photomultiplier with a 2 LiF window. The chambers containing the photomulti- due to Cl2(D'ÐA') and ArCl(BÐX) is less intense for all plier and the grating, respectively, were evacuated to a of the mixtures. It was found that the intensity at residual pressure P ≤ 10Ð3 Pa. The monochromatorÐ 175 nm is maximal if the partial pressure of Cl2 is 0.2Ð photomultiplier system operates in the spectral band 0.4 kPa and that of HCl, 0.08 kPa. This is because HCl 130Ð350 nm. The spectrophotometer was calibrated and Cl2 have strong and zero VUV absorption, respectively. With the ArÐCl mixture, the optimal value of against the continuous emission of H2 in the region 2 165Ð350 nm. P(Ar) is within 8Ð15 kPa at a reasonable level of the charging voltage U across the storage capacitor (U = 4Ð 15 kV). If the total pressure P in the mixture is raised 222 [KrCl( )] nm above 30 kPa, the TVD becomes inhomogeneous. As U B–X is increased from 5 to 15 kV, the intensity linearly rises by a factor of 3Ð5 for all of the bands observed. Figures 2 and 3 show the intensities emitted in the excimer bands and the Cl2(D'Ð A') band against PKr for the ArÐKrÐCl2 mixture and the ArÐKrÐXeÐCl2 mixture, ≥ respectively. In the former case, a rise in PKr to 0.05 kPa leads to an increase in the intensity for KrCl(BÐX) and a decrease in that for ArCl(BÐX). For Cl2(D'ÐA'), the radiation yield is maximal at PKr = 0.3 kPa. If PKr = 0.6 kPa, the intensities of the three bands are close to each other. At PKr < 0.05 kPa, a rise in the Kr concentration leads to an increase in the intensity for ArCl(BÐX) 175 [ArCl(B–X)] nm (probably, due to changes in disharge parameters such ≥ as ne and Te). At PKr 0.05 kPa, the decrease for ArCl(BÐX) and the attending increase for KrCl(BÐX), * 258 [Cl2] nm which proceed at equal rates, stem from the fact that Ar atoms are replaced by Kr atoms in the formation of excimer molecules. It has been demonstrated [15] that the largest rate constant k in this process is that of 3 energy transfer from Ar( P2) to Kr with the formation × −12 3 of Kr[5p(3/2)2]: k = 5.6 10 cm /s. The process runs vigorously in an ArÐKr plasma at PKr = 0.1Ð1.0 kPa, which agrees with Fig. 2.

With the ArÐKrÐXeÐCl2 mixture (Fig. 3), the intensity curves run in a more complicated fashion. As PKr is increased, the intensity for KrCl(BÐX) rises again, whereas the intensities for XeCl(BÐX) and ArCl(BÐX) exhibit a peak if P(Xe) = 0.4 kPa or a decrease if P is 160 200 240 λ, nm Xe larger. For Cl2(D'ÐA'), the curve has a valley at PKr = 0.4 kPa. Such behavior is mainly due to energy transfer 3 Fig. 1. TVD plasma emission spectrum of the ArÐKrÐCl2 from Ar( P2) to Kr and Xe atoms [15Ð17] and to spe- mixture with P = 13.3, P = 0.6, and P = 0.24 kPa. Ar Kr Cl2 ciﬁc features of interaction between excited Kr and Xe

TECHNICAL PHYSICS Vol. 46 No. 2 2001 AN ELECTRIC DISCHARGE EMITTER OPERATING 209 atoms in a KrÐXe plasma [18, 19]. Compared with the J, arb. units emitting media of electric discharge excimer lasers and high-pressure (P ≥ 100 kPa) excimer lamps, the recom- 2 bination reaction Ar+ + ClÐ + (Ar) ArCl* + (Ar) plays a less important part in TVD at P = 1Ð30 kPa, 120 since its rate constant may be one order of magnitude smaller at such low buffer gas pressures (e.g., 10 kPa rather than 100 kPa) [20]. Instead, the initial TVD stage is governed by the “harpoon” reaction Ar(m) + Cl2 80 ArCl* + Cl [21, 22], which requires that the plasma contain excited (metastable) atoms of heavy inert gases. This reaction to produce ArCl(B) may proceed at 3 a rate as high as 7 × 10Ð10 cm3/s [23]. For P = 13.3, Ar 40 P = 0.4, and P = 0.24 kPa, the respective intensities Xe Cl2 1 emitted in the excimer bands and the Cl2(D'ÐA') band are comparable to each other if PKr = 0.4Ð1.2 kPa. This result allows using such an emitter in UVÐVUV pulse photometry. 0 0.4 0.8 1.2 1.6 2.0 From Fig. 4, where the emission intensity is plotted PKr, kPa against the number of discharge pulses N for the KrCl(BÐX) band of the ArÐKrÐCl2 mixture, one can Fig. 2. TVD emission intensity vs. Kr partial pressure evaluate the emission resource (the number of pulses at a wavelength of (1) 175 (ArCl), (2) 222 (KrCl), and after which the intensity is halved). In the experiment, (3) 258 (Cl2* ) nm for the ArÐKrÐCl2 mixture with PAr = 13.3 kPa and P = 0.24 kPa (U = 12.5 kV). the emission resource was found to be as high as Cl2 (1−2) × 104 pulses. The resource characteristic was obtained under stationary conditions (in the absence of J, arb. units gas flow), the discharge chamber having a passive volume of 10 l. The emission resource is determined 2 mainly by the purity of the buffer gas (we used com- 100 mercial-grade Ar) and the materials of the electrodes and the discharge chamber (stainless steel, aluminum, acrylic plastic, fluoroplastic, and quartz). Intensity fluctuations could be suppressed by pumping fresh portions 4 5 of the gas through the emitter at a small rate (≤0.01 m/s). Alternatively, one could employ a solid- 50 3 state source of Cl2 and a regenerator of the waste emitting mixture; the former would be switched on if the × intensity falls below a given level. Also, note that the 1 resource of an electric discharge laser operating at 308 (XeCl) and 222 (KrCl) nm and using a HeÐRÐHCl mixture at P ≥ 100 kPa can be increased by two orders 0 0.4 0.8 1.2 1.6 2.0 of magnitude by introducing a small amount of H2 PKr, kPa (P ≤ 130 Pa) into the lasing medium [24]. H2 To expand the operating band of the emitter by Fig. 3. TVD emission intensity vs. Kr partial pressure at a wavelength of (1) 175, (2) 222, (3) 236, (4) 258, and means of Lyman H2 bands, we investigated the effect of (5) 308 nm for the ArÐKrÐXeÐCl2 mixture with PAr = 13.3, H2 on TVD emission. Figure 5 shows the respective P = 0.4, and P = 0.24 kPa (U = 12.5 kV). Xe Cl2 intensities in the ArCl, H2*, and Cl2* bands as functions of P for the ArÐHClÐH mixture. It is seen that the H2 2 ArCl* and H2 intensities are reduced considerably if characteristic develops a plateau (curve 2 in Fig. 4). P is raised to a sufficiently high level (≥0.1 kPa). The Consequently, the emission resource increases consid- H2 spectrum exhibits Lyman bands, at 158Ð161 nm, if erably (N > 104 pulses). The intensities of the RX* P ≥ 0.3 kPa. At P = 0.57 kPa, the ArCl* and H bands fall mainly because of intense energy transfer H2 H2 2 intensities are the same. As P is increased further, the from Ar(m) to H2 [25] and the reduction of HCl when H2 Lyman bands become dominant. If a small amount of the decay products of the former process interact with H2 is added to the ArÐKrÐHCl mixture, the intensity in H2. With the mixtures containing H2, strong continuous the ArCl(BÐX) band slightly decreases and the resource emission was also observed in the region 200Ð400 nm

TECHNICAL PHYSICS Vol. 46 No. 2 2001 210 SHUAIBOV et al.

J, arb. units data concerning the efﬁciency with which VUV emission is excited are available. With ArÐKrÐHClÐH2 mix- 1.00 tures, these processes lead to a decrease in the intensities of the excimer bands when the H2 concentration is raised.

0.75 2 CONCLUSION We have established that a TVD plasma in ArÐKrÐ Cl2 and ArÐKrÐXeÐCl2 mixtures is an efﬁcient multiwave source of pulsed emission at 308, 258, 236, 222, and 175 nm due to the transitions XeCl(BÐX), Cl (D'Ð A'), 1 2 0.50 XeCl(DÐX), KrCl(BÐX), and ArCl(BÐX), respectively. For a reasonable charging voltage (4Ð15 kV), the respective intensities are close to each other if PAr = 10Ð 20 kPa, P = 0.4Ð0.6 kPa, P = 0.2Ð0.4 kPa, and 0 2.5 5.0 7.5 10.0 Kr, Xe Cl2 3 PHCl = 0.08 kPa. With a small amount of H2 added to the N, 10 puls ≤ mixture, PH2 0.1Ð0.2 kPa, the resource characteristic Fig. 4. TVD emission intensity vs. the number of discharge acquires a plateau and the emission resource increases. pulses at a wavelength of (1) 222 (KrCl) and (2) 175 (ArCl) nm If PH2 is raised above 0.5Ð1.0 kPa, the intensities of the for the (1) ArÐKrÐCl2 and (2) ArÐKrÐHClÐH2 mixtures with bands corresponding to the excimer molecules and P = 13.3, P = 0.4, P = 0.24, P = 0.12, and P = Ar Kr Cl2 HCl H2 0.13 kPa. Cl2* fall by one order of magnitude and Lyman H2 bands appear at wavelengths of 158Ð161 nm. The opti- J, arb. units mal composition of the emitting medium is determined by the energy transfer processes Ar(m)ÐKr, Xe, H2, and Kr(m)ÐXe and by the formation of the ArH* and 3 Σ+ 40 H2(a g ) molecules.

2 1 REFERENCES 1. T. Gerber, W. Luthy, and P. Burkhard, Opt. Commun. 35, 20 242 (1980). ᭝ 2. B. A. Koval’, V. S. Skakun, V. F. Tarasenko, and 3 E. A. Fomin, Prib. Tekh. Éksp., No. 4, 244 (1992). ᭝ × 3. V. M. Borisov, V. A. Vodchits, et al., Kvantovaya Élek- tron. (Moscow) 25, 308 (1998).

0 0.5 1.0 PH2, kPa 4. V. Yu. Baranov, V. M. Borisov, and Yu. Yu. Stepanov, Electric Discharge Excimer Lasers on Inert Gas Halo- genides (Énergoatomizdat, Moscow, 1988). Fig. 5. TVD emission intensity vs. H2 partial pressure at a wavelength of (1) 160 [H2(BÐX)], (2) 175 (ArCl), and 5. U. Kogelshanz and H. Esrom, Laser Optoelectron. 22, 55 * (1990). (3) 258 ( Cl2 ) nm for the ArÐHClÐH2 mixture with PAr = 13.3 and PHCl = 0.12 kPa (U = 12 kV). 6. Yu. G. Basov, Pumping Sources for Microsecond Lasers (Énergoatomizdat, Moscow, 1990). 7. A. K. Shuaibov, L. L. Shimon, and I. V. Shevera, Prib. ≥ at PH2 0.5 kPa. This may be attributed to the formation Tekh. Éksp., No. 3, 142 (1998). of ArH* molecules [26] or to continuous emission from 8. A. K. Shuaibov, L. L. Shimon, I. V. Shevera, and 3Σ+ 3 Σ+ O. J. Minya, J. Phys. Stud. 3 (2), 157 (1999). H2 (a g Ð b u ). In this experiment, we were unable to distinguish between these types of continuous spec- 9. A. K. Shuaibov, Teploﬁz. Vys. Temp. 36, 508 (1998). trum. For longitudinal pulse discharge, which develops 10. A. K. Shuaibov, Zh. Tekh. Fiz. 68 (12), 64 (1998) [Tech. in the form of a fast ionization wave, it has been dem- Phys. 43, 1459 (1998)]. onstrated that this system offers a strong continuous 11. A. K. Shuaibov, L. L. Shimon, A. I. Dashchenko, et al., ≥ hydrogen spectrum in the UV range if PH 1.33 kPa Pis’ma Zh. Tekh. Fiz. 25 (11), 29 (1999) [Tech. Phys. 2 Lett. 25, 433 (1999)]. and E/N ≥ 100 Td, the excitation rate of the a3 Σ+ state g 12. É. I. Asinovskiœ, L. M. Vasilyak, and V. V. Markovets, × 20 3 of H2 being 1.3 10 cm /s [27]. However, in [27], no Teploﬁz. Vys. Temp. 21, 371 (1983).

TECHNICAL PHYSICS Vol. 46 No. 2 2001 AN ELECTRIC DISCHARGE EMITTER OPERATING 211

13. A. G. Abramov, É. I. Asinovskiœ, and L. M. Vasilyak, Fiz. 21. E. B. Gordon, V. G. Egorov, V. T. Mikhkel’lson, et al., Plazmy 14, 979 (1988) [Sov. J. Plasma Phys. 14, 575 Kvantovaya Élektron. (Moscow) 15, 285 (1988). (1988)]. 22. V. É. Peét, E. V. Slivinskiœ, and A. B. Treshchalov, Kvan- 14. B. B. Slavin and P. I. Sopin, Teploﬁz. Vys. Temp. 30, 1 tovaya Élektron. (Moscow) 17, 438 (1990). (1992). 23. É. M. Vrublevskiœ, A. V. Gusev, A. G. Zhidkov, et al., 15. L. G. Piper and D. W. Setser, J. Chem. Phys. 63, 5018 Khim. Vys. Énerg. 24, 356 (1990). (1975). 24. T. J. McKee, D. J. James, W. S. Nip, et al., Appl. Phys. 16. C. H. Chen, J. P. Judisch, and M. G. Payne, J. Phys. B 11, Lett. 36, 943 (1980). 2189 (1978). 25. B. P. Lavrov and A. S. Mel’nikov, Opt. Spektrosk. 85, 17. J. Galy, K. Aoume, A. Birot, et al., J. Phys. B 26, 477 729 (1998) [Opt. Spectrosc. 85, 666 (1998)]. (1993). 26. F. V. Bunkin, V. I. Derzhiev, V. A. Yurovskiœ, and 18. J. D. Cook and P. K. Leichner, Phys. Rev. A 31, 90 S. I. Yakovlenko, Kvantovaya Élektron. (Moscow) 13, (1985). 1828 (1986). 19. J. D. Cook and P. K. Leichner, Phys. Rev. A 43, 1614 27. S. V. Pancheshnyœ, S. M. Starikovskaya, and A. Yu. Sta- (1991). rikovskiœ, Fiz. Plazmy 25, 435 (1999) [Plasma Phys. Rep. 25, 393 (1999)]. 20. M. R. Flannery, in Applied Atomic Collision Physics, Ed. by E. W. McDaniel and W. Nigan (Academic, New York, 1982; Mir, Moscow, 1986), Vol. 3. Translated by A. Sharshakov

TECHNICAL PHYSICS Vol. 46 No. 2 2001

RADIOPHYSICS

Synthesis of Current on a Strip from a Given Realizable Radiation Pattern S. I. Éminov Mudryœ State University, Sankt-Peterburgskaya ul. 41, Novgorod, 173003 Russia Received March 14, 2000

Abstract—A theory of synthesis of current on a strip from a given realizable radiation pattern is developed. The theory chooses the space of currents from the condition that the near-zone ﬁeld is limited. The space of patterns is deﬁned as the image of the space of currents due to current-to-pattern mapping. For these spaces, the Hilbert structure is introduced and the basis is constructed. As a result, the problem of synthesizing current from a given pattern is reduced to expansion over the basis. A numerical example is considered. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION AND STATEMENT have been implemented for closed surfaces, but none OF THE PROBLEM for unclosed surfaces. A relationship between the density of surface cur- Whereas the problem of synthesizing current on a rents induced on both sides of a strip of electrical width closed surface from a given radiation pattern has been 2a, j(t), and a radiation pattern F(χ) is given by the inte- covered adequately [6], many problems concerning gral equation [1Ð3] unclosed surfaces, in particular, the problem of choosing the space of currents, remain unsolved. 1 ()χ ()χ() The purpose of this paper is to develop a criterion F ==Kaja∫ exp iatjtdt. (1) for choosing the space of currents, to construct the Ð1 space of currents for a strip and the space of radiation Real angles correspond to |χ| ≤ 1. However, it is patterns, and to study radiation described by formula (1) common practice to consider the function F(χ) at any in appropriate spaces. real χ. Equation (1) has been studied in many investiga- ENERGY INTEGRAL: H-POLARIZATION tions. Analytical methods of synthesizing currents from PROBLEM a given realizable radiation pattern have been developed in [2, 3]. Variational methods that calculate cur- In this section, we consider the H-polarization problem: the strip is located in the plane y = 0, the generatrix rents producing a radiation pattern close to a given one τ (not necessarily realizable) are addressed in [4]. In of the strip is parallel to the z axis, and the currents jx( ) these publications, it is assumed that the current j(τ) pass parallel to the x axis (perpendicularly to the strip belongs to the space L ; i.e., edge) and vanish at the edge. In our opinion, the space 2 of currents should be chosen from physical consider- 1 ations. Namely, we must assume that not only the far- jt()2dt<+∞ (2) field power but also the near-field power is limited. It is ∫ the condition that ensures the uniqueness of solutions Ð1 to the Maxwell equations [7]. or, by virtue of Parseval’s identity, Let us find a condition that provides the finiteness of ∞ the field power in the near zone. To this end, we inte- ()χ 2 χ< ∞ grate the Poynting vector along a closed line l encir- ∫F d + . (3) cling the cross section of the strip and contract this line Ð∞ into a segment:

However, the selection of the space L2 has not been 1 1 a˜ substantiated in the publications cited above. P ==--- []EH, *ndlÐ--- E ()τ j*()ττd, (4) 2∫ 2∫x x Synthesis of current on an unclosed surface has also l Ð1 been considered in [5], where the current is found with regard for the Meixner conditions imposed on its edge where E and H are the ﬁelds produced by the currents τ behavior. However, to date, methods developed in [5] jx( ), 2a˜ is the width of the strip, the asterisk means

SYNTHESIS OF CURRENT ON A STRIP 213 complex conjugation, and the normal n is parallel to the and substitute this expression into (9) to obtain y axis. +∞ µ ˜ π Next, we express the electric field Ex in terms of the a 1 (χ) 2 χ τ P = i -ε------∫ ------F d . (11) current density jx( ) [1] as 8a χ2 Ð∞ Ð 1 1 µ 1 ∂ ∂ (2) For energy integral (12) to be finite, it is sufficient E (τ) = ------j (t)----H (a τ Ð t )dt x ε 4a∂τ ∫ x ∂t 0 that Ð1 (5) +∞ 1 1 2 a µ (2) (χ) χ < ∞ Ð -- --- j (t)H (a τ Ð t )dt ∫ ------F d + . (12) ε ∫ x 0 χ2 4 Ð∞ Ð 1 Ð1 and represent the Hankel function in formula (5) as the A comparison of (12) and (3) shows that the space Fourier integral L2[Ð1, 1] has become narrower: there exist currents that satisfy (12) but do not belong to L2[Ð1, 1]. +∞ ( χ(τ )) (2)( τ ) i exp Ðia Ð t χ H0 a Ð t = π-- ∫ ------d , χ2 Ð 1 (6) SPACE OF CURRENTS Ð∞ IN THE H-POLARIZATION PROBLEM 2 2 χ Ð 1 = i 1 Ð χ , χ ≤ 1. We introduce the space of currents by means of the operator Substituting (5) into (4) in view of (1) and (6), we obtain +∞ 1 1 +∞ (Au)(τ) = -- ∫ χ ∫ cos[χ(τ Ð t)]u(t)dtdχ µ π π a˜ χ2 (χ) 2 χ P = Ði -ε------∫ Ð 1 F d . (7) 0 Ð1 (13) 8a +∞ 1 Ð∞ 1 ≡ ------χ exp(Ðiχ(τ Ð t))dtdχ. Energy integral (7) converges if 2π ∫ ∫ Ð∞ Ð1 +∞ χ2 (χ) 2 χ < ∞ It is known [8] that the operator A is symmetric and ∫ Ð 1 F d + . (8) positive definite and its domain of definition D(A) is Ð∞ everywhere dense in L2[Ð1, 1]:

As follows from (8) and (3), the fact that the currents 2 2 (Au, u) ≥ γ (u, u), ∀u ∈ D(A), γ > 0. (14) belong to the space L2[Ð1, 1] does not suffice for the field power to be finite. To meet this condition, it is nec- Hereafter, (. , .) means a scalar product in L2[Ð1, 1]. essary to narrow the space L2[Ð1, 1]. The positive definiteness of the operator A allows one to introduce the energy space HA [9], which is defined as a completion of D(A) on the norm ENERGY INTEGRAL: E-POLARIZATION PROBLEM 2 [u] = (Au, u), (15) Consider the E-polarization problem: the currents τ and the scalar product in this space is defined as jz( ) pass parallel to the z axis (parallel to the edge of the strip) and tend to infinity at the edges. In this case, [u, v ] = (Au, v ). integration of the Poynting vector yields

1 Next, let us establish a relationship between HA and ˜ 1 a the Sobolev space H1 ([Ð1, 1]), which can be regarded P = -- [E, H*]ndl = Ð-- E (τ) j*(τ)dτ. (9) -- 2∫ 2 ∫ z z 2 t Ð1 ∞ [10] as a completion of C0 ([–1, 1]) (the set of infinitely We write the electric field Ez in terms of the current differentiable finite functions with the support [Ð1, 1]) τ density jz( ), on the norm

1 +∞ µ (τ) a ( ) (2)( τ ) 2 1 ( χ ) (χ) 2 χ Ez = Ð-- --- jz t H0 a Ð t dt, (10) u 1 = ------1 + u˜ d , (16) ε ∫ -- π ∫ 4 2 2 Ð1 Ð∞

TECHNICAL PHYSICS Vol. 46 No. 2 2001 214 ÉMINOV ∞ where Designate the Fourier transform for the space Hs(Ð , ˜ 1 +∞) as Hs (Ð∞, +∞). In the latter, the norm is also (χ) ( ) ( χ ) ∞ ∞ u˜ = ∫ u t exp i t dt. defined by relationship (20). The spaces Hs(Ð , + ) ˜ Ð1 and Hs (Ð∞, +∞) are Hilbert spaces with the scalar The positive definiteness of the operator A immedi- product ately yields the equivalence of norms (15) and (16). +∞ Consequently, the energy space HA coincides with the 1 2s 〈 , v〉 ------( χ ) ˜ (χ)v˜ (χ) χ Sobolev space H1 ([Ð1, 1]). A significant advantage of u s = ∫ 1 + u d . (21) -- 2π 2 Ð∞ the norm and the scalar product introduced above is that they allow one to analytically introduce an orthonormal The operator K1, or the Fourier transform as an basis of HA [8] in the form operator, which maps from the space H1 ([Ð1, 1]) into -- 2 2 ϕ (τ) = ------sin[narccos(τ)]; n = 1, 2, …, ˜ n the space H1 (Ð∞, +∞), is an isomorphism; this map- πn -- 2 (17) 0, m ≠ n ping does not change the norm. Therefore, the image of (Aϕ , ϕ ) =  m n H1 ([Ð1, 1]) is a closed set on which the inverse opera- 1, m = n. --  2 Ð1 To conclude this section, write the operator A in the tor K1 is defined and limited. At the same time, the coordinate form: Sobolev space H1 ([Ð1, 1]) coincides with the energy -- 1 2 1 ∂ ∂ 1 (Au)(τ) = ------∫ u(t)---- ln------dt. (18) space HA, in which the basis is analytically specified. π∂τ ∂t τ Ð t Therefore, we will assume that the operators act in the Ð1 energy space HA. The equivalence of representations (13) and (18) for the operator A on a dense set is validated by the well- We consider again Eq. (1), known expression 1 1 1 cos[χ(τ Ð t)] Ð 1 F(χ) = K j = a exp(iaχt) j(t)dt , (22) ln------= C + ∫------dχ a ∫ τ Ð t χ Ð1 0 (19) +∞ and map it from the energy space j ∈ H into the Sobo- cos[χ(τ Ð t)] A + ------dχ, ˜ ∫ lev space F ∈ H1 (Ð∞, +∞). Define the space of patterns χ -- 1 2 where C is the Euler constant. as the image of HA due to the mapping Ka : Im(Ka). On the set Im(Ka), the scalar product is given by

SPACE OF PATTERNS IN THE H-POLARIZATION +∞ PROBLEM: BASIS ( , ) 1 χ χ Kau Kav 1 = ------KauKav d , (23) -- π ∫ 2 2 We introduce Sobolev spaces [11] on the straight Ð∞ ∞ ∞ ∞ ∞ ∞ line Hs(Ð , + ) as a completion of the set C0 (Ð , + ) of infinitely differentiable finite functions on the norm which is equivalent to scalar product (21). This statement follows from the positive definiteness of the oper- +∞ ator A. The set Im(Ka), being a closed set, is a Hilbert 2 1 2s 2 u˜ = ------(1 + χ ) u˜ (χ) dχ < +∞, (20) space with scalar product (23). s 2π ∫ Ð∞ We define the basis functions f a of this space where n ϕ through the basis functions n of the space of currents: +∞ u˜ (χ) = u(t)exp(iχt)dt. a ϕ ∫ f n = Ka n. (24) Ð∞ ϕ The constant factor preceding the integral in the With definition (17) of the basis functions n (17) right-hand side of (20) is introduced for convenience. and the definition of the operator Ka, the basis functions

TECHNICAL PHYSICS Vol. 46 No. 2 2001 SYNTHESIS OF CURRENT ON A STRIP 215

a f n are found as product deﬁned as a(χ) 1 f n (u, v ) , = u(t)v (t)q(t)dt. (31) ( χ) 2 q ∫ ( )k Ð 1 π( )J2k Ð 1 a Ð1 2 2k Ð 1 ------χ------, n = 2k Ð 1 (25) Ð1 = ( χ) Consider an operator L that maps the weight space k Ð 1 J2k a i(Ð1) 4πk------, n = 2k, 2 L ρ into the weight space L Ð1 , where ρ(t) = 1 Ð t . χ 2, 2, ρ where J are Bessel functions. Chebyshev polynomials of the ﬁrst kind Tn(t) = n cos[(n Ð 1) arccos(t) ] (n = 1, 2, …) form a basis of the Functions (25) were derived using the integrals of space L Ð1 , while weighted Chebyshev polynomials the products of trigonometric functions and Chebyshev 2, ρ a ρ polynomials of the second kind. The basis functions f Tn(t)/ form a basis of the space L2, ρ. The following n relationship is valid for the operator L [12]: are orthonormal by construction:

 ≠ ln2, n = 1 a a 0, m n  ( f , f )1 =  (26) n m -- L(T /ρ)(τ) =  (32) 2 1, m = n. n 1 (τ) ≠ ------Tn , n 1. n Ð 1 Relationship (26) can also be verified using the tab- ulated integral Let I denote an identity operator that maps from the +∞ (χ) (χ) space L , ρÐ1 into the space L2, ρ and relates a function Jm Jn 2 m Ð n 2 ∫ ------dχ = ------sin------π. (27) u(τ) to a function u(τ)/ρ(τ). We consider the operator χ π(m2 Ð n2) 2 0 IL: L , ρ L , ρ. (33) An arbitrary function F belonging to the space of 2 2 patterns Im(Ka) can be expanded over the orthonormal As follows from relationship (32), this operator is basis f a : positive. The positiveness (this property alone) of the n operator IL also allows the introduction of the energy ∞ + space HL [9], which is defined as a completion of L2, ρ a a F(χ) = (F, f )1 f (χ). (28) on the norm ∑ n -- n 2 n = 1 2 [u] = (ILu, u) , ρ. (34) From an expansion for the pattern, we immediately 2 obtain an expansion for the current: The space HL is a Hilbert space with the scalar +∞ product (τ) ( , a) ϕ (τ) j = ∑ F f n 1 n . (29) [ , ] ( , ) -- u v = ILu v 2, ρ. (35) 2 n = 1 As a result, the problem of finding the current from According to (32), the orthonormal basis of this a given realizable radiation pattern is reduced to the space has the form problem of expanding over a given orthonormal basis. Thus, the above-stated problem of finding the current 1 1 π------, n = 1 from a given realizable radiation pattern in the case of ln2 1 Ð τ2 H polarization has been completely solved. ψ (τ) = n [( ) (τ)] 2(n Ð 1) cos n Ð 1 arccos > ------π------, n 1, SPACE OF CURRENTS 1 Ð τ2 IN THE E-POLARIZATION PROBLEM 0, m ≠ n ( ψ , ψ ) (36) We introduce the space of currents by means of the IL m n =  operator 1, m = n.

1 1 1 The power space HL can be shown to coincide with (Lu)(τ) = -- ∫ u(t)ln------dt. (30) π τ the Sobolev space H1 ([–1, 1]), which is deﬁned as a Ð t -- Ð1 2 ∞ We also need the weight spaces L2, q with the scalar completion of C0 ([–1, 1]) (the set of inﬁnitely differ-

TECHNICAL PHYSICS Vol. 46 No. 2 2001 216 ÉMINOV entiable ﬁnite functions with the support [Ð1, 1]) on the We deﬁne the space of radiation patterns as the norm image of HL due to the mapping Ka : Im(Ka). In this +∞ space, we introduce a scalar product equivalent to (21) (χ) 2 by the formula 2 1 u˜ χ u 1 = ------∫ ------d , (37) -- 2π 1 + χ 2 ∞ χ χ Ð ( , ) C 1 f ( )g( ) Ð f (0)g(0) χ f g 1 = --- f (0)g(0) + ------d -- π π ∫ χ where 2 2 χ < 1 1 (40) 1 f (χ)g(χ) u˜ (χ) = ∫ u(t)exp(iχt)dt. + ------dχ, 2π ∫ χ Ð1 χ > 1

Calculate the Fourier transform of the basis func- where f = Kau, g = Kav, and C is the Euler constant. tions: χ ψ If f or g is zero at = 0 (all functions K1 n possess 1 this property at n > 1), scalar product (40) simplifies to K ψ = exp(iχt)ψ (t)dt +∞ 1 n ∫ n (χ) (χ) ( , ) 1 f g χ Ð1 f g 1 = ------d . -- π ∫ χ 2 2 π Ð∞ (χ) ------J0 , n = 1 (38) ln2 The set Im(Ka), being a closed set, is a Hilbert space = ( )k Ð 1 π( ) (χ) with scalar product (40). As follows from (32), the Ð1 2 2k Ð 1 J2k Ð 2 , n = 2k Ð 1 a basis functions gn of this space have the form i(Ð1)k Ð 1 4πkJ (χ), n = 2k. 2k 1 a ( χ )ψ ( ) Using formula (38) and methods presented in [8], gn = ∫ exp ia t n t dt one can easily prove that norms (34) and (37) are equivalent. Note in conclusion that the energy space H cor- Ð1 L π responds to the E-polarization problem: if the function ( χ) ------J0 a , n = 1 (41) describing the current belongs to HL, then energy inte- ln2 gral (12) is finite. = ( )k Ð 1 π( ) ( χ) Ð1 2 2k Ð 1 J2k Ð 2 a , n = 2k Ð 1 ( )k Ð 1 π ( χ) SPACE OF PATTERNS IN THE E-POLARIZATION i Ð1 4 kJ2k a , n = 2k. PROBLEM: BASIS An arbitrary function F belonging to the class of The operator K1, or the Fourier transform as an realizable patterns Im(Ka) can be expanded over the a operator, mapping from the space H1 ([Ð1, 1]) into the -- orthogonal basis gn : 2 ˜ +∞ space H1 (Ð∞, +∞), also is an isomorphism; this map- -- (χ) a(χ) 2 F = ∑ cngn , (42) ping does not change the norm. Therefore, the image of n = 1 H1 ([Ð1, 1]) is a closed set on which the inverse opera- -- where 2 Ð1 a (F, g )1 tor K1 is defined and limited. The Sobolev space n ------2- H1 ([Ð1, 1]) coincides with the energy space HL, in a a , n = 1 -- ( , ) gn gn 1 2 cn = -- which the basis is given analytically. Therefore, we 2 a consider Eq. (1) once more, (F, g )1, n > 1. n -- 1 2 (χ) ( χ ) ( ) a F = Ka j = ∫ exp ia t j t dt , (39) It should be noted that the basis gn is orthonormal Ð1 at n > 1. From an expansion for the pattern, we imme- ∈ diately obtain an expansion for the current: and map it from the energy space j HL into the Sobo- ˜ +∞ lev space F ∈ H1 (Ð∞, +∞). Equation (39) differs from -- (τ) ψ (τ) 2 j = ∑ cn n . (43) Eq. (1) by a constant factor. n = 1

TECHNICAL PHYSICS Vol. 46 No. 2 2001 SYNTHESIS OF CURRENT ON A STRIP 217

M χ M(a FN ( ) QN ) 1.0 1.0 M = 0

0.8 0.8

0.6 0.6 1 M = 0 1 2 3 0.4 0.4 2 0.2 3 0.2

0 0.2 0.4 0.6 0.8 1.0 0 χ 0.2 0.4 0.6 0.8 1.0 a/10

Fig. 1. Fig. 2.

jM(τ)/maxjM(τ) M(τ)/max M(τ) N N j N jN 1.0 1.0

a = 9 0.8 0.5 0.6 a = 12 0 0.4

a = 6 0.2 –0.5 0 –1.0 –0.2 a = 1 M = 2 M = 1 –0.4 –1.5 –0.6 –2.0 –0.8 0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0 τ τ

Fig. 3. Fig. 4.

The problem of ﬁnding the current from a given ever, in this case, the inequality M < N must be satisﬁed realizable pattern in the case of E polarization has also M (in Figs. 1Ð4, N = 10). The radiation pattern F (χ) has been solved completely. N only a weak dependence on the length of a radiator a. However, the radiator length strongly affects other NUMERICAL RESULTS parameters. Figure 2 plots the radiated-to-total power We consider an example of calculating the current ratio against a. At small a, the radiated power is much from a given radiation pattern, e.g., in the H-polariza- smaller than the total power. As a increases, the ratio tion problem. Let the pattern have the form grows. Figures 3 and 4 illustrate curves for the current.

Ð1 At small a, the current oscillates and, as calculations N 2χ2 M(χ) ( χ2)M ( χ) χ  a  show, is large. Though narrow-beam secondary-lobe- FN = 1 Ð sin a a ∏ 1 Ð ------ . π2n free patterns can be produced at small a’s, such an n = 1 approach is extremely inefﬁcient. As a increases, the This radiation pattern is realizable [2]. By selecting current distribution becomes smoother. The narrower appropriate M and N, one can obtain various narrow- the pattern, the larger a must be to increase the power beam patterns without secondary lobes (Fig. 1). How- ratio and, accordingly, to smooth out the current.

TECHNICAL PHYSICS Vol. 46 No. 2 2001 218 ÉMINOV

REFERENCES S. Flügge (Springer-Verlag, Berlin, 1961; Mir, Moscow, 1964), Part 1, p. 218. 1. E. V. Zakharov and Yu. V. Pimenov, Numerical Analysis of Radio-Wave Diffraction (Radio i Svyaz’, Moscow, 8. V. N. Plotnikov, Yu. Yu. Radtsig, and S. I. Éminov, Zh. 1982). Vychisl. Mat. Mat. Fiz. 34 (1), 68 (1994). 2. E. G. Zelkin, Construction of Radiating Systems from 9. S. G. Mikhlin, Variational Methods in Mathematical Given Radiation Pattern (Énergoizdat, Moscow, 1963). Physics (Nauka, Moscow, 1970; Pergamon, Oxford, 3. B. M. Minkovich and V. P. Yakovlev, Theory of Synthesis 1964). of Antennas (Sov. Radio, Moscow, 1969). 10. J.-L. Lions and E. Magenes, Problemes aux limites non 4. L. D. Bakhrakh and S. D. Kremenetskiœ, Synthesis of homogénes et applications (Dunod, Paris, 1968; Mir, Radiating Systems: Theory and Computing Methods Moscow, 1971). (Sov. Radio, Moscow, 1974). 11. G. I. Éskin, Boundary-Value Problems for Elliptic 5. Ya. N. Fel’d, Radiotekh. Élektron. (Moscow) 32, 1137 Pseudodifferential Equations (Nauka, Moscow, 1973). (1987). 12. I. I. Vorovich, V. M. Aleksandrov, and V. A. Babeshko, 6. B. Z. Katsenelenbaum, Approximability of Electromag- Nonclassical Mixed Problems in Theory of Elasticity netic Field (Nauka, Moscow, 1996). (Nauka, Moscow, 1974). 7. H. Hönl, A. W. Maue, and K. Westpfahl, in Handbuch der Physik, Vol. 25: Kristalloptik. Beugung, Ed. by Translated by A. Khzmalyan

TECHNICAL PHYSICS Vol. 46 No. 2 2001

RADIOPHYSICS

Spectra of Exchange Dipole ElectromagneticÐSpin Waves in Asymmetric MetalÐInsulatorÐFerromagneticÐInsulatorÐ Metal Systems V. E. Demidov and B. A. Kalinikos St. Petersburg State University of Electrical Engineering, St. Petersburg, 197376 Russia e-mail: [email protected] Received May 29, 2000

Abstract—Spectra of exchange dipole electromagneticÐspin waves in tangentially magnetized asymmetric planar-layer metalÐinsulatorÐferromagneticÐinsulatorÐmetal (MIFIM) systems were studied theoretically. It is shown that symmetry breakdown due to a difference in the permittivities of the insulating layers may substantially enhance interaction between the spin and electromagnetic waves. This improves the electrical controllability of the dispersion properties of the spin waves, for example, by varying the permittivity of one of the layers that are in contact with the ferromagnetic ﬁlm. Optimal geometries of the layer structures that provide the best electrical control are suggested. © 2001 MAIK “Nauka/Interperiodica”.

Spin waves in ferromagnetic films and layer struc- asymmetric tangentially magnetized MIFIM systems tures are used to advantage in various microwave and at finding the optimum geometry that provides the devices. The basic advantage of spin-wave devices is best controllability and is easy to implement in prac- the possibility of their electrical tuning by varying the tice. permanent magnetic field applied to the ferromagnetic film. However, this way of tuning suffers from draw- Consider an unbounded plane-parallel layer struc- backs, such as large dimensions of the magnetic sys- ture in the y0z plane (Fig. 1). It includes an isotropic tems, low speed, and high power consumption. Tuning ferromagnetic film of thickness L that has a saturated ⑀ of spin-wave devices can be improved by using layer magnetization Ms and a permittivity L. The film is sep- structures containing both ferromagnetic and ferroelec- arated from two perfectly conducting metal screens by tric layers. In such structures, a new mechanism of con- insulating layers of thicknesses a and b and permittivi- ⑀ ⑀ trolling the dispersion characteristics of spin waves ties a and b, respectively. The origin is placed at the propagating in layer waveguide systems appears, since center of the ferromagnetic film. It is assumed that the ferroelectrics can change their permittivity under the film is magnetized to saturation by a permanent mag- action of a permanent magnetic field. This mechanism netic field of strength H applied along the z axis. The offers high speed and is not power-hungry. 0 spins on the surfaces of the film are assumed to be non- It seems therefore logical to study the wave spectra interacting. in layer structures in which ferroelectric and ferromagnetic layers are in contact. Since ferroelectrics are of For convenience, we introduce the second coordi- high permittivity, a theory of spin waves that includes nate system ξηζ rotated through an angle ϕ about the x electromagnetic delay (which is usually neglected) axis. The ζ axis coincides with the direction of wave should be elaborated. propagation. In [1], we developed a theory of exchange dipole electromagnetic and spin waves propagating in sym- x, ξ metric layer MIFIM systems. It was shown that such a systems allow the control of the spin wave spectrum by L varying the permittivity of the ferroelectric layers. H0 b Based on this effect, a variety of spin-wave devices for z ϕ processing microwave signals tuned through a change η 0 kζ in the constant electric field can be designed. However, the geometry of the symmetric layer structure is hardly ζ feasible. y This work is aimed at investigating the spectrum of exchange dipole electromagnetic and spin waves in Fig. 1. Layer structure geometry.

220 DEMIDOV, KALINIKOS

In the structure considered, a dispersion relation for C2 = cosh(γ b)(Ð1)n Ð cosh(γ (b + L)), exchange dipole spin and electromagnetic waves is n L L derived by jointly integrating the complete set of the C3 = sinh(γ (a + L))(Ð1)n Ð sinh(γ a), Maxwell equations and equations for magnetization n L L motion using the Green tensor function formalism for n n C = sinh(γ b)(Ð1) Ð sinh(γ (b + L)), planar-layer structures [2, 3]. The result is a transcen- 4 L L ω dental equation that relates the eigenfrequency of the D = sinh(γ (a + L)) nth mode of spin waves to the longitudinal wave num- a L γ ber kζ: L (γ ) (γ ( )) + -γ--- tanh aa cosh L a + L , (Ω ω xx)[Ω ω ( yy 2 ϕ zz 2 ϕ)] a nk Ð M An nk Ð M An cos + An sin (1) D = sinh(γ (b + L)) (ω ω xz ϕ)(ω ω zx ϕ) b L Ð + M An sin Ð M An sin = 0, γ L (γ ) (γ ( )) where + -γ--- tanh bb cosh L b + L , b 2 2γ xx kζ 1 kζ L 2 γ A = Ð 1 + ------Ð ------T = sinh(γ a) Ð ----L tanh(γ b)cosh(γ a), n γ 2 κ2 sinh(γ d)N(γ 2 κ2) L(1 + δ ) a L γ b L L + n L L + n 0n b × [ 1( ( )n ) 2( ( )n)] γ Cn Db Ð1 + T b Ð Cn Da Ð T a Ð1 , (γ ) L (γ ) (γ ) T b = Ð sinh Lb + -γ--- tanh aa cosh Lb , a γ 2 xz 1 kζ 2 A = ------L------γ 2 n (γ ) 2 2 ( δ )  L  sinh Ld (γ + κ ) L 1 + 0n (γ ) (γ ) (γ ) L n N = sinh L L 1 + γ------γ----tanh aa tanh bb  a b × [C3(D (Ð1)n + T ) + C4(D Ð T (Ð1)n)], n b b n a a γ γ  + cosh(γ L) ----L tanh(γ b) + ----L tanh(γ a) , 2 2 γ L γ b γ a  yy k 1 k 2 b a A = ------0---L----- Ð ------0--L------L------n 2 2 y 2 2 2 ( δ ) γ κ (γ ) (γ κ ) L 1 + 0n L + n sinh Ld N L + n d = a + b + L. ⑀ γ ⑀ γ The elements Ny, T y , and Dy are obtained from N, × 3 L a (γ ) y( )n L b (γ ) y i i Cn⑀------γ----tanh aa Db Ð1 Ð ⑀------γ----tanh bb T b T , and D by the replacement a L b L i i γ ⑀ γ γ ⑀ γ ⑀ γ ⑀ γ ----L ----L------a ----L ----L------b 4 L b (γ ) y L a (γ ) y( )n γ ⑀ γ , γ ⑀ γ . Ð Cn⑀------γ----tanh bb Da + ⑀------γ----tanh aa Ta Ð1  , a a L b b L b L a L The other designations are the same as in [1]. γ 2 zx 1 kζ 2 In parallel with dispersion relation (1) for the A = ------L------n (γ ) 2 2 2 ( δ ) exchange dipole spin waves, one can deduce a disper- sinh Ld N(γ κ ) L 1 + 0n L + n sion relation for the natural waves of the structure in the absence of magnetization in the layer L. For the TE γ γ × 1 L (γ ) ( )n L (γ )  mode, Cn-γ--- tanh aa Db Ð1 Ð -γ--- tanh bb Tb a b N γ---- = 0, (2) γ γ L 2 L (γ ) L (γ ) ( )n + Cn-γ--- tanh bb Da + -γ--- tanh aa Ta Ð1  , b a and for the TM mode, γ y 2 γ 3 L N = 0. (3) zz γ 1 2 A = Ð ------L------+ ------L------n 2 2 (γ ) 2 2 2 ( δ ) γ κ sinh Ld N(γ κ ) L 1 + 0n Note that each of the factors in Eqs. (2) and (3) L + n L Ð n depends both on wave number kζ and on frequency ω. γ γ Practical spin-wave devices usually employ two × 3 L (γ ) ( )n L (γ )  Cn-γ--- tanh aa Db Ð1 Ð -γ--- tanh bb Tb types of spin waves: longitudinal waves, propagating in a b the direction of the permanent magnetic ﬁeld (ϕ = 0), γ γ and transverse waves, traveling at a right angle to the 4 L (γ ) L (γ ) ( )n ﬁeld (ϕ = π/2). Ð Cn-γ--- tanh bb Da + -γ--- tanh aa Ta Ð1  , b a Longitudinal spin waves interact with lower order TM electromagnetic waves, which are free of cutoff 1 (γ ( ))( )n (γ ) 0 Cn = cosh L a + L Ð1 Ð cosh La , [1]. For longitudinal spin waves, a dispersion relation

TECHNICAL PHYSICS Vol. 46 No. 2 2001 SPECTRA OF EXCHANGE DIPOLE ELECTROMAGNETICÐSPIN WAVES 221

ω/2π, GHz (a) (b) (c) 12.0

11.8

11.6

11.4

11.2

11.0 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120 –1 kζ, cm

Fig. 2. Change in the spectra of electromagneticÐspin waves when symmetry breakdown is due to different thicknesses of the a and b layers.

ζ may relate the slope of the dispersion curve for the TM0 tive and negative directions along the axis. Dashed ⑀ ⑀ waves with a and b. In symmetric MIFIM structures, lines are the dispersion curves for TE1 electromagnetic such a control mechanism was shown to be inefficient waves and transverse spin waves. In the former case, [1]. As follows from Eq. (1), if the symmetry breaks the curves were calculated for the unmagnetized L layer down, this mechanism becomes still less efficient, (fast waves), and in the latter, the magnetostatic because any asymmetry weakens interaction between approximation (slow waves) was used. Solid lines longitudinal spin waves and electromagnetic TM0 depict the spectrum of the electromagnetic and spin waves. waves that follows from Eq. (1). It is seen that, in the asymmetric layer structure, the transverse spin waves Transverse spin waves extensively interact with TE1 ζ electromagnetic waves. They have a cutoff frequency propagating in the opposite directions of the axis var- that varies with the permittivity of the ferroelectric iously interact with the electromagnetic waves. Com- layer. If the cutoff frequency lies close to the frequency pared with the case of symmetric MIFIM structures, for range of long (low-frequency) spin waves, the disper- the waves traveling in the positive and negative direc- sion curve for the spin waves considerably changes. tions, the interaction efficiency grows and diminishes, respectively. This can easily be explained by consider- Thus, transverse spin waves appear to be the most ing the cross-sectional distribution of the magnetic promising for control of the dispersion by varying the fields hξ and hζ, which provide interaction between permittivity of ferroelectrics. Therefore, in subsequent transverse spin and TE electromagnetic waves. It turns analysis of asymmetric layer structures, we will con- 1 sider transverse spin waves alone. out that the polarizations of these fields for both waves mainly coincide in one case and differ in the other. Magnetostatic studies of the dispersion characteristics of spin waves in asymmetric layer MIFIM struc- Figure 3 shows the variation of the spectrum of the tures [2, 3] show that transverse spin waves have the transverse spin waves when symmetry breakdown is property of nonreciprocity; that is, their dispersion associated with different permittivities of the layers. µ ⑀ µ characteristics depend on the sign of the longitudinal Here, L = 20 m, L = 14, a = b = 300 m, and H0 = ⑀ ⑀ wave number kζ. One more nonreciprocity mechanism 3400 Oe. In Fig. 3a, a = b = 1000; in Figs. 3b and 3c, ⑀ ⑀ emerges when electromagnetic delay is taken into con- a = 1630 and b = 14. As in the case when the layers a sideration. In this case, transverse spin waves propagat- and b are heterogeneous in thickness, a difference in ing in the opposite directions along the ζ axis interact their permittivities causes the nonreciprocity of the with electromagnetic waves with various efficiencies. waves traveling in the opposite directions of the ζ axis. Figure 2 shows the spectrum of transverse electro- For those traveling in the negative direction, interaction magneticÐspin waves when symmetry breakdown is is stronger; while for those propagating in the negative due to various thicknesses of the insulating layers. In direction, weaker, as compared with the symmetric the calculations, the parameter values were as follows: structure. µ ⑀ ⑀ ⑀ L = 20 m, L = 14, a = b = 1000, and H0 = 3400 Oe. The above analysis implies that the asymmetric In Fig. 2a, a = b = 300 µm; in Figs. 2b and 2c, a = structure can provide stronger interaction between 100 µm and b = 500 µm. The thicknesses of the ferro- transverse spin and electromagnetic waves. Note also electric layers were taken so as not to change the cutoff that the layer structure shown in Fig. 3c is quite feasi- frequency of the TE1 electromagnetic wave. In Figs. 2b ble: the properties of the layers L and b are close to and 2c, the waves propagate, respectively, in the posi- those of epitaxial YIG films grown on GGG substrates.

TECHNICAL PHYSICS Vol. 46 No. 2 2001 222 DEMIDOV, KALINIKOS

ω/2π, GHz (a) (b) (c) 12.0

11.8

11.6

11.4

11.2

11.0 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120 –1 kζ, cm

Fig. 3. Change in the spectra of electromagneticÐspin waves when symmetry breakdown is due to different permittivities of the a and b layers.

Enhanced interaction between electromagnetic and approximation, and solid lines show dispersion curves ⑀ spin waves in asymmetric layer structures makes it pos- obtained from Eq. (1) at various a (the associated val- sible to improve the efficiency of controlling the spec- ues are indicated by figures). trum of exchange dipole spin waves by varying the per- A comparison of our Fig. 4 with Fig. 4 in [1], show- mittivities of the layers. Thus, the asymmetric layer ing the control of the dispersion characteristics for structures offer greater promise for devices with electri- transverse spin waves in the symmetric structure, cally tunable properties. clearly points to a better controllability of the spin wave ⑀ Figure 4 exemplifies the control of the dispersion spectrum in the asymmetric structure. As a varies, the characteristics of transverse spin waves traveling in the spin wave dispersion curve changes in a much wider negative ζ-axis direction. Here, the permittivity of the range of longitudinal wave numbers. Moreover, in the layer a in the asymmetric layer structure (L = 20 µm, asymmetric structure, the frequency range of the upper ⑀ µ ⑀ L = 14, a = b = 300 m, and b = 14) varies from 1400 branch of hybridized dispersion curves is higher than in to 800. The calculation is made for the permanent mag- the symmetric one. This provides single-mode propa- netic field H0 = 3000 Oe. The dashed line is the disper- gation of transverse spin waves throughout the range ⑀ sion curve of the spin waves in the magnetostatic of a. Thus, the asymmetric structures are candidates for ω/2π, GHz spin-wave devices intended for processing microwave 11.0 signals. They are easier to fabricate and offer better tuning capabilities than the symmetric ones.

ACKNOWLEDGMENTS 10.5 This work was ﬁnancially supported by the Russian Foundation for Basic Research (grant no. 99-02-16370) 800 and by the Ministry of Education of the Russian Feder- ation (grant no. 97-8.3-13). 1000 10.0 REFERENCES 1. V. E. Demidov and B. A. Kalinikos, Pis’ma Zh. Tekh. 1200 Fiz. 26 (7), 8 (2000) [Tech. Phys. Lett. 26, 272 (2000)]. 1400 2. V. F. Dmitriev and B. A. Kalinikos, Izv. Vyssh. Uchebn. 9.5 Zaved., Fiz. 31 (11), 24 (1988). 0 50 100 150 200 –1 3. B. A. Kalinikos, Izv. Vyssh. Uchebn. Zaved., Fiz. 24 (8), kζ, cm 42 (1981).

Fig. 4. Control of spin wave dispersion characteristics in the asymmetric layer structure. Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 46 No. 2 2001

ELECTRON AND ION BEAMS, ACCELERATORS

A Charged Ellipsoidal Bunch in a Magnetic Field A. S. Chikhachev Khrunichev State Space Research and Production Center, Moscow, 121087 Russia Received November 9, 1999; in ﬁnal form, March 13, 2000

Abstract—The states of a long rotating charged ellipsoidal bunch in a longitudinal uniform magnetic ﬁeld are studied. The states are described using two integrals of motion that couple the transverse velocities xú and yú ω with the x and y coordinates; the frequency H = eH/mc (where H is the total magnetic ﬁeld); and the quantities ω1 and ω2, which characterize the Coulomb repulsion in the x and y directions. It is shown that equilibrium states with a high charge density per unit length (ν տ 1) can exist. © 2001 MAIK “Nauka/Interperiodica”.

ω ω Rapid progress in accelerator technology during the parameters 1 and 2 describe the Coulomb repulsion last few decades has required the development of meth- of the particles in the ellipsoidal bunch. ods for solving problems related to the dynamics of In the frame of reference related to the principal charged particles under different conditions. Of special axes, we have interest are situations in which charged particles inter- 2 2 act with the self-fields of dense bunches. In this paper, 2 νc 2 ν c ω ==------, ω ------, a charged bunch rotating in a magnetic field is studied 1 R ()R + R 2 R ()R + R with allowance for the interaction of the particles with x x y y x y the self-field of the bunch. where Note that the behavior of long charged-particle 3e2N bunches (or beams) with elliptic cross sections in a lon- ν = ------2- gitudinal magnetic field is as yet poorly investigated. In mc Rz [1], a beam with an elliptic cross section in a magnetic field in the presence of an external quadrupole structure is the charge density per unit length and N is the total was studied. A nonsteady ellipsoidal bunch in the number of particles in the bunch. It is assumed that longitudinal size of the bunch is much larger than its trans- absence of a magnetic field was investigated in [2]. ӷ verse dimensions (Rz Rx, Ry). Diamagnetic properties of beams with circular cross Equations (1) have periodic solutions x ~ eiΩt and y ~ sections were studied within the so-called rigid rotator iΩt model [3]. In this paper, the ellipsoidal bunch is e with described using another model that differs from the ω2 Θú2 Θúω ω2 ω2 rigid rotator model in that the average angular velocity Ω2 H++2 2 HÐ 1Ð 2 12, = ------is assumed to depend on the angle. The effect of the 2 interaction delay, which may be important for dense (2) ω2 ω2ω22 bunches, is also studied. ±Θú H()ω2()ω2ω21Ð2 + ------HÐ21+2------. 1. We consider the motion of charged particles in a 2 2 magnetic field aligned with the z-axis with allowance From Eq. (8), which will be derived below, it fol- for repulsion caused by the particle space charge. Let lows that physically allowable (real) values of Rx /Ry the x- and y-axes be directed along the principal axes of can be obtained when the elliptic cross section of a rotating bunch. In this case, the equations of motion have the form ω2ω2ω 2 ω 2 1+2H ω 2 H 2 > ------Ð ------==0 Ð ------K 0 0. 2 4 24 4 ω2ω xúú Ðyú2()Θú+ω ÐxΘú+ ------H- =ωÐ ------H- x, H 214 This inequality substantially limits the domain of (1) parameters in which nonsteady equilibrium states can 2 4 exist. Taking into account the relationship ω2ω yúú +xú2()Θú+ω ÐyΘú+ ------H- =ωÐ ------H- y, H 2 22 2 4 4 2ω∆ δ= Ð4K------H- + Θú + ----- 0 0 024 where Θú is the angular rotational velocity of the bunch, ω ()∆ω2 ω2 H = eH/mc, H is the total magnetic field, and the =1Ð2,

Θú ω Ӎ ω2 ω (Θú 2 Θú ω ω2)2 it also follows from Eq. (8) that + H/2 0 / H, E = + H + 2 which results in real values of Ω. If Θú and ω are con- 2 H  α (ω + 2Θú ) α  stant, then Eqs. (1) have the following invariants: × ------1------H------+ -----2- . (Θú 2 Θú ω ω2 Ω2)2 Ω2  2 2 + H + 2 + 1 2  ω2 + Θú + Θú ω  I = xú + (ω + 2Θú )------2------H------y For the distribution function speciﬁed above, the 1  H ω2 Θú 2 Θú ω Ω2  2 + + H + 1 semiaxes of the elliptic cross section of the bunch are determined by the formulas 2 2  (ω + 2Θú )Ω Θú + Θú ω + ω2  + ------H------1------yú + Ð------H------1-x , 2 2Ω2(Θú 2 Θú ω ω2 Ω2)2 2 2 2 Ω Rx v 0 1 + H + 1 + 2 ω + Θú + Θú ω + Ω 1  ----- = ------, 2 H 1 B α α (Θú 2 Θú ω ω2)2(Ω 2 Ω2)2 1 2 + H + 1 1 Ð 2 2 2 2 (5)  ω Θú Θú ω  2 2 1 + + H 2 2Ω2(Θú Θú ω ω2 Ω2) I = yú Ð (ω + 2Θú )------x Ry v 0 2 + H + 2 + 1 2 H 2 2 2  ω Θú Θú ω Ω  ----- = ------2------. 1 + + H + 2 A α α (Θú 2 Θú ω ω2) (Ω 2 Ω2)2 1 2 + H + 2 1 Ð 2  (ω Θú )Ω Θú 2 Θú ω ω2  2 H + 2 2 + H + 2 ω2 ω2 + ------xú + Ð------y . The expressions for 1 and 2 and Eq. (5) yield 2 2 2 Ω ω Θú Θú ω Ω 2  1 + + H + 3 2 2 v Ω (Θú + Θú ω + ω2 + Ω2) ------0------1------H------1------2------Setting Rx = 2 α α (Θú + Θú ω + ω2)(Ω 2 Ð Ω2) α α 1 2 H 1 1 2 I = 1 I1 + 2 I2, 2 α (ω + 2Θú ) Ω2 where 1, 2 are positive constants representing the dis- × α α H 1 1 + 2------(6) tribution function in the form (Θú 2 Θú ω ω2 Ω2)2 + H + 2 + 1 κδ( 2) f = I Ð v 0 , ω = νc------2------, and integrating this function with respect to velocities, ω ω2 ω2 we obtain that the density is nonzero in an elliptic 1 1 + 2 region in the x and y coordinates. The complete expres- Ω (Θú 2 Θú ω ω2 Ω2)2 sion for the integral of motion I is v 0 2 + H + 2 + 1 R = ------y 2 2 2 2 2 2 2 2 α α (Θú Θú ω ω )(Ω Ω ) 1 2 + H + 2 1 Ð 2 I = Axú + Byú + 2C1 xúy + 2C2 xyú + Dx + Ey , (3) where (ω Θú )2Ω2 × α H + 2 2 α 2 1------+ 2 (7) (ω + 2Θú ) Ω 2 (Θú 2 Θú ω ω2 Ω2)2 α α H 2 + H + 1 + 2 A = 1 + 2------, (Θú 2 Θú ω ω 2 Ω2)2 + H + 1 + 2 ω = νc------1------. 2 (ω + 2Θú ) Ω 2 ω ω2 ω2 α H 1 α 2 1 + 2 B = 1------+ 2, (Θú 2 Θú ω ω 2 Ω2)2 δ + H + 2 + 1 Below, we consider the case 0 0; i.e., Ω Ӎ Ω Ӎ Ω δ Ω Ӷ (ω Θú )(Θú 2 Θú ω ω 2) 1 2 0, 0/ 0 1. C1 = H + 2 + H + 2 Then, from Eqs. (6) and (7), it follows that  α α  1 2 ν Ӷ × ------+ ------ , v0/ c 1 in a rotating frame of reference. Dividing Θú 2 Θú ω ω2 Ω2 Θú 2 Θú ω ω2 Ω2 + H + 2 + 1 + H + 1 + 2 Eq. (6) by Eq. (7) results in the equation ω Θú 2 Θú ω ω2 Θú H + H + 1 ω2 + ------Ð K0 C = Ð------C , (4) Rx 2 2 2 2 2 1 = = ------. Θú Θú ω ω ------2 (8) + H + 2 R ω ω y 1 Θú H + ------+ K0 (Θú 2 Θú ω ω2)2 2 D = + H + 1 2. Let us calculate the longitudinal magnetic self- 2  α (ω + 2Θú )  ﬁeld of the bunch. The coordinates x and y in the rest × -----1- + α ------H------, 1 1 2 2 2 frame are related to the coordinates x and y in the rotat- Ω (Θú 2 Θú ω ω2 Ω2)  1 + H + 1 + 2 ing frame of reference with the axes aligned with the

TECHNICAL PHYSICS Vol. 46 No. 2 2001 A CHARGED ELLIPSOIDAL BUNCH IN A MAGNETIC FIELD 225 principal axes of the elliptic cross section of the bunch Using expressions (9), Eq. (11) can be transformed as follows: into x = xcosΘ Ð ysinΘ, y = xsinΘ + ycosΘ. ω Ω νΘú ν Ω 1 1 H Ð H = Ð 0 . (12) Integrating the expressions Taking into account the interaction delay in the lower order approximation leads to the correction to the j (r')(r ⊥r⊥' )dr' 1 x1 1 bunch potential Ax = --∫------, 1 c r'3 Θú 2ρ (1) dr' ( )( ) Φ (x , y ) = ------0 ------(x'2 Ð y'2) 1 jy r' r1⊥r⊥' dr' 1 1 2 ∫ 1 r 3 Ay = --∫------2c ' 1 c r'3 × [(x2 Ð y2)cos2Θ + 2x y sin2Θ] and taking into account the equalities 1 1 1 1 Θú 2ρ (13) C C 0 Rx Ð Ry 1 2 ρ eN π ------xú = Ð-----y, yú = Ð----- x, 0 = ------, = ------2-- Rx Ry A B 4π 2c Rx + Ry ------R R R 3 x y z × [(x2 Ð y2)cos2Θ + 2x y sin2Θ]. we obtain 1 1 1 1 3. Let us analyze Eqs. (8) and (12) taking into 2πR R ρ  C  account the relationship A = ------x------y------0-x R -----2 Ð Θú x1 c(R + R ) 1 x B  x y   ω  ∆ 2K Θú + -----H- Ð --- Ӎ 0, (14) 0 2  2 C  sin2Θ C  1 Ð cos2Θ + R -----1 + Θú ------+ y R -----2 Ð Θú ------(9) y A  2 1 x B  2 which follows from the condition  ω  Θ  δ Ӎ 0 Θú + -----H- > 0 . C1 Θú  1 + cos2 0   Ð Ry----- +  ------, 2 A 2  From Eq. (8), it follows that 2πR R ρ  Θ C ω ω2 x y 0 1 + cos2  2 Θú  H 0 Ay = ------x1 Ð------Ð Rx Θú + ------= ------1 ( )   ω c Rx + Ry  2 B 2 H and 1 Ð cos2ΘC  + ------1 + Θú R (10) 2  A  y ω2 ω Ω = -----0- Ð -----H- . 0 ω 2 Θ  H C2 Θú  C1 Θú  sin2 Ð y1 ----- Ð  Rx + ----- +  Ry ------. ω2 ω2 B A 2  When H > 2 0 , Eq. (12) yields ω2 From Eqs. (9) and (10), we find the magnetic self- Ω ω ( ν) ν 0 field of the bunch H = H 1 + Ð 2 ω------. (15) H ∂ ∂ ( ) A Ax in y1 1 The external field is attenuated by rotating particles Hz = ------Ð ------. ∂ ∂ 2 2 x1 y1 Ω ω ω ω so that H > H. For H < 2 0 , the sets of Eqs. (8) and Ω The external field is characterized by H = (12) have no real solutions. When the delay of the elec- eH(ext)/mc, and the self-field is characterized by the dif- tric field is taken into account, Eqs. (9), (10), and (12) ω Ω and the equality ference H Ð H, which can be expressed as follows: ∂ C R C R (1) 1 A ∇Φ(1) ω Ω νΘú ν 1 y 2 x  E = Ð ----∂----- Ð H Ð H = + ------Ð ------ . (11) c t A Rx + Ry B Rx + Ry δ yield the following expressions for ω2 and ω 2 : According to Eqs. (4), for 0 0, we have 1 2 ω ω ν 2 C C C1 Θú H  C2 Θú H ω2  c Θú  2 1 ----- = Ð + ------Ð K0 , ----- = + ------+ K0. 1 = ----------- Ð Rx----- + Ry----- , (16) A 2 B 2 Rx + Ry Rx B A

TECHNICAL PHYSICS Vol. 46 No. 2 2001 226 CHIKHACHEV

2 2 ν  c  C2 C1 xú = (xú Ð yΘú )cosΘ Ð (yú + xΘú )sinΘ, ω = ------+ Θú R ----- + R ----- . (17) 1 2 R + R R  x B y A  x y y ( Θú ) Θ ( Θú ) Θ yú1 = xú Ð y sin + yú + x cos . Accordingly, instead of Eq. (8), we obtain Since the average quantities satisfy the relationships ω ω 2 ú  ú H  H ω + νΘ Θ + ------Ð K Θú + ------Ð K C1 C2 R 2  2 0 2 0 xú = Ð -----y, yú = Ð----- x, -----x = ------= ------. (18) A B Ry 2  ω  ω ω νΘú Θú H Θú H 1 +  + ------+ K0 + ------+ K0 the maximum average particle velocities in the rest 2 2 frame are It follows from here that ω ω ( )  H  ( )  H  ω ω xú1 max = ------Ð K0 Ry, yú1 max = ------+ K0 Rx, Θú H Θú H Ω  ω2 νΘú Ω 2 2 2 + ------  + ------+ 0  = 0 Ð 0 . (19) 2 2 where The external field can be found from the following νcω νcω set: R = ------1-- , R = ------2-- . y ω ω x ω ω 2 0 2 1 0 2 2 2Θú (4 + ν)Θú + ------[ω (ν Ð 1) + Ω (ν + 2)] ν H H When ω ӷ ω and ω2 Ӎ ω2 /2, the inequalities (20) 1 2 0 H ω ν ( ) Ӷ ( ) Ӷ ν տ H ω Ð 2 Ω  ω2 xú1 max c and yú1 max c are valid for 1. + ---ν--- H ------+ H Ð 0 = 0, 2 The mean-square velocities are on the order of v0 Ӷ ν (note that v0 c at any ). This means that dense ω2 ω Ω 2 2 H  H Ð H bunches can be confined by the external field. The par- ω Ð ------+ ------0  ν  ticles in a long bunch expand rather slowly along the Θú 2 = ------ω------Ω------. (21) field lines. ω H Ð H H + 2------Thus, in this paper, the states of a long charged high- 2 density bunch rotating in a magnetic field have been studied and the effect of the interaction delay on these Assuming that the term νΘú |Ω | in Eqs. (18) and 0 states has been estimated. (19) is small, for the external field we obtain ω ω2 3 ν2 H 0 REFERENCES 2 4 ------Ð ------  ω  2 ω Ω Ӎ ω + ν ω Ð 2-----0- + ------H------, (22) 1. A. S. Chikhachev, Zh. Tekh. Fiz. 55, 1179 (1985) [Sov. H H  H ω  2 Phys. Tech. Phys. 30, 673 (1985)]. H ω H 2. A. S. Chikhachev, Radiotekh. Élektron. (Moscow) 39, ν where the term proportional to 2 is always positive; 453 (1994). i.e., when the delay in the potential is taken into 3. R. C. Davidson, Theory of Nonneutral Plasmas (Ben- account, the attenuation of the external field is stronger jamin, New York, 1974; Mir, Moscow, 1978). than in the case when the delay is ignored. 4. The particle velocities in the rest frame are Translated by I. Efimova

TECHNICAL PHYSICS Vol. 46 No. 2 2001

Technical Physics, Vol. 46, No. 2, 2001, pp. 227Ð233. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 71, No. 2, 2001, pp. 98Ð104. Original Russian Text Copyright © 2001 by Volkolupov, Dovbnya, Zakutin, Krasnogolovets, Reshetnyak, Mitrochenko, Romas’ko, Churyumov.

ELECTRON AND ION BEAMS, ACCELERATORS

Electron Beam Generation in a Magnetron Diode with Metal Secondary-Emission Cathode Yu. Ya. Volkolupov, A. N. Dovbnya, V. V. Zakutin, M. A. Krasnogolovets, N. G. Reshetnyak, V. V. Mitrochenko, V. P. Romas’ko, and G. I. Churyumov National Research Center, Kharkov Institute of Physics and Technology, Kharkov, 61108 Ukraine Received August 24, 1999; in ﬁnal form, February 15, 2000

Abstract—The effects of electric and magnetic ﬁeld intensities, the triggering-pulse droop rate, and the electrode diameter on the processes of electron beam formation and generation were studied experimentally. The results of mathematical simulation of the secondary-emission multiplication of the electron ﬂow are presented. Tubular electron beams with a wall thickness of 1.5Ð2 mm, a current density of 1Ð70 A/cm2, and a particle energy of 5Ð100 keV were obtained. It was shown that several electron bunches could be obtained during a single voltage pulse. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION 10 ns); a focusing solenoid (4) (magnetic field intensity, up to 3500 Oe; longitudinal nonuniformity, ~8%); a The pulse and mean power of microwave sources, as vacuum chamber (3) containing a secondary-emission well as their service life, depend on the type of the cath- diode with a central copper cathode (5) and a tubular ode [1Ð4]. It is well known that magnetron diodes stainless-steel anode (6) (electrode length, 100Ð based on cold secondary-emission metal cathodes have 140 mm; pressure in the chamber is maintained at a a prolonged service life (~100 000 h [2]) and provide ≤ Ð6 2 level of 10 torr using an ion pump); an indication high density electron emissions (~50 A/cm [4]). These system consisting of current and voltage detectors and diodes have a rather simple design and can be effec- a Faraday cup (7) with a calorimetric power meter; and tively used as microwave sources. However, certain a synchronization system. The electron energy was effects relevant to beam generation in magnetron determined by measuring the electron absorption by diodes are insufficiently understood, both theoretically aluminum foil. The beam spot size was measured using and experimentally. In particular, the process of space the beam spot image on X-ray film and molybdenum charge formation in the anodeÐcathode gap in crossed foil. The electron beam produced by the magnetron electric and magnetic fields is virtually obscure [5]. To diode powered by a pulse modulator at a pulse repeti- solve this problem, physical processes in the magnetron tion rate of 10Ð50 Hz was studied. diode in the magnetic field exceeding the Hull cutoff 2 2 Secondary emission was induced both by an exter- field (H = 6.72(U)1/2[r (1 Ð r /r )]Ð1, where H is the Hull a k a nal voltage pulse applied to the anode from separate longitudinal magnetic field (Oe); U is the voltage pulse sources (U = 1Ð15 kV, tdroop ~ 2Ð100 ns) and by a across the diode (V); and rc and ra are the cathode and voltage surge droop in the initial part of the cathode anode radii (cm), respectively) should be studied both voltage pulse (self-triggering). The amplitude of this theoretically and experimentally. The goal of this work was to study the processes of electron layer formation near the secondary-emission 34 567 metal cathode and electron beam generation in a magnetron diode.

EXPERIMENTAL SETUP AND METHODS r Parameters of the electron beams generated by magnetron diodes were studied using the experimental R1R2 setup shown in Fig. 1. The setup consisted of a high- R voltage pulse modulator (1) (voltage amplitude, 5Ð 3 µ 200 kV; pulse duration, 2Ð10 s; triggering pulse 1 2 amplitude, up to 15 kV; triggering pulse duration, ~70 ns); a high-voltage generator (2) for inducing secondary emission (amplitude, up to 3.5 kV; duration, 1Ð Fig. 1. Diagram of the experimental setup.

228 VOLKOLUPOV et al.

U, kV ing electric field and a static magnetic field along cyc- 90 loidal trajectrories increases. This causes electron bombardment of the cathode. As the electric field near the 70 cathode decreases, at a certain moment the electron energy becomes sufficient for removing secondary electrons with a secondary-emission coefficient σ 50 1 greater than one. Secondary electrons can be removed both by electrons with the energy of 0.3Ð1 keV incident 30 2 at a right angle and by electrons with lower energies incident at smaller angles. Once the electrons have been 10 emitted, the stage of fast voltage droop begins. This stage is characterized by an avalanche-like increase in the number of bombarding and knocked-out electrons. I, A 25 A sharp increase in the number of electrons causes a decrease in the voltage level, thereby enhancing the 20 voltage droop rate. As a consequence, the energy of 15 3 electrons bombarding the cathode increases and reaches a point where σ > 1. This causes a further 10 increase in the number of electrons near the cathode 5 until a dynamic equilibrium is attained, and the electron layer near the cathode is formed. On attaining equilibrium, the stage of steady-state secondary-electron mul- 0 1 2 3 4 5 6 7 8 9 , µs tiplication begins and the beam is generated. It was t revealed experimentally that the beam was generated when the electron-drift velocity v = E/H was equal to Fig. 2. Oscillograms of the cathode voltage pulse: (1) modulator idling; (2) beam generation mode; and (3) Faraday 0.1Ð0.2 s (depending on experimental conditions). cup beam current mode. It was found experimentally that the fast droop stage duration and, therefore, the beam current buildup time depended on the anode and cathode diameters and the surge varied from 10 to 100 kV; the droop duration was voltage droop rate and duration. For example, if the about 1 µs (Fig. 2). cathode diameter exceeded 0.2 cm, the anode diameter exceeded 2.2 cm, the droop duration was 0.1Ð0.5 µs, µ EXPERIMENTAL RESULTS and the droop rate exceeded 20 kV/ s, the beam current AND DISCUSSION pulse buildup time exceeded 10 ns. Experiments with magnetron diodes with cathode and anode diameters of 1. Electron beam formation. The processes of 0.2 and 1.0 cm, respectively, showed that the beam cur- electron beam formation were studied using magnetron rent was generated within a shorter time interval (1Ð diodes with different values of the ratio ra/rc (1.2Ð15). 10 ns). The duration of the time interval was equal to Typical oscillograms of the cathode voltage pulse and the droop duration. The droop rate was 1200Ð Faraday cup beam current obtained using a magnetron 300 kV/µs. At such a high droop rate, the number of gun (ra = 1.3 cm; rc = 0.25 cm) are shown in Fig. 2. Sec- primary autoelectrons is rather small. However, ondary emission was induced by the cathode voltage because of a high pulse droop rate, the energy of these pulse surge droop. As seen from the oscillograms, the autoelectrons acquired in 10Ð20 gyroperiods reaches a electron layer is formed and the beam is generated upon value at which σ > 1. the surge droop. The process of beam current genera- Collective motion of electrons during their multipli- tion during the voltage droop occurs in two stages: a cation can cause space charge oscillations. This relatively slow voltage droop (hundreds of nanosec- becomes possible at the steady-state stage, when the onds) and a fast voltage droop (nanoseconds or tens of space charge reaches a certain minimum density and nanoseconds). During the first stage (curve 1), auto- the electron layer begins to shield the cathode, which electronic emission and accumulation of autoelectrons results in potential sagging in the anodeÐcathode gap in the anodeÐcathode gap occur (idle run of modulator). and a decrease in the electric field strength near the During the second stage (curve 2), the electron layer is cathode. The energy of electrons is changed as a result formed and the beam is generated. It should be noted of their interaction with the fields induced by space- that duration of the slow droop stage depends on the charge oscillations. The noise accompanying this inter- electric intensity E. A decrease in the electric intensity action can be observed in the beam current pulse. It was from 120 to 40 kV/cm caused an increase in the slow found experimentally that the noise was stronger when droop stage duration from 250 to 600 ns. a large-diameter cathode (80 mm) was used; i.e., the Autoelectrons are emitted during the slow droop noise increased with decreasing electric intensity. The stage. The energy of the electrons moving in a decreas- noise amplitude reached 20% of the beam current

TECHNICAL PHYSICS Vol. 46 No. 2 2001

ELECTRON BEAM GENERATION IN A MAGNETRON DIODE 229 amplitude. The noise wave period was equal to several I, A nanoseconds. 40 2. The dependence of the beam generation cur- 1 rent on the diode dimensions. The dependence of 30 electron beam parameters on the diode dimensions was studied. The cathode diameter d ranged from 2 to 20 80 mm; and the anode diameter D, from 10 to 140 mm. The values of the Faraday cup beam current I, cathode 2 voltage U, and the magnetic field intensity H for vari- 10 ous anode and cathode diameters are given in the table for eight modifications of the magnetron diode. The electric field strength near the cathode varied from 20 0 10 20 30 40 to 125 kV/cm (at a constant voltage pulse amplitude). dc, mm It was found that the beam current obeyed the Lang- muirÐChild law. At a given voltage, the beam current Fig. 3. Dependence of the beam current on the cathode amplitude could be maximized by tuning the magnetic diameter. field strength. Electron beams with a current ranging from 1 to 50 A and a particle energy ranging from 5 to 100 keV were obtained. 10 A by changing the magnetic field strength from 1100 to 2000 Oe. At the initial stages of beam generation, short (~1 µs) surges were observed on the beam current pulse The diode working range dependence on the cath- plateau. These surges were caused by gas desorption ode voltage amplitude at a given magnetic intensity was from the cathode surface and gas ionization [4]. After studied. It was found that an approximately 20% varia- the diode was aged, the beam current pulse amplitude tion in the voltage amplitude had virtually no effect on and shape remained invariant. the process of electron beam formation in a static magnetic field. As the cathode voltage approached the ∆U The dependence of the beam current on the cathode limit from above or below, the conditions for beam gen- diameter is shown in Fig. 3. These curves were eration were violated, and the voltage pulse instability obtained at a voltage of 24 kV and anodeÐcathode dis- caused secondary emission breakdown (Fig. 4). tance of 5 (curve 1) and 20 mm (curve 2). It can be con- The use of this effect in experiments with a magne- cluded from these curves that the beam current is tron diode with anode and cathode diameters of 78 and inversely proportional to the logarithm of the ratio 40 mm, respectively, allowed the beam current to be between the anode and cathode diameters. This result is completely modulated in amplitude at a carrier fre- in good agreement with the dependence obtained for quency of 1 MHz (Fig. 5). For this purpose, sinusoidal classical magnetrons [6]. As seen from Fig. 3, the beam modulation of the voltage pulse peak was performed, current decreases with increasing anodeÐcathode dis- and electron bunches were obtained at the diode output tance. However, one of the advantages of the diode is on the dips of the sinusoid. Complete modulation of that it operates at rather low magnetic intensities current at a carrier frequency of ≥1 MHz was also (≤1000 Oe). In this case, the magnetic coil heating is attained in experiments with a magnetron diode of the insignificant, so that it is not necessary to adjust the same dimensions by tuning the magnetic field distribu- magnetic field. In addition, this allows a magnetron tion over the diode axis. Multispiking generation of the diode with specified parameters (current, magnetic electron beam could be attained by varying the ampli- field intensity, and dimensions) to be designed. tude and longitudinal distribution of the magnetic field. 3. Effect of electric and magnetic fields on electron beam formation. The effect of the magnetic field strength on beam generation at a constant cathode volt- Table age amplitude was studied. There was a sharp rise, a d, mm D, mm U, kV I, A H, Oe plateau, and a sharp decrease in the Faraday cup beam current amplitude with increasing magnetic field. Such 2 10 7 1.6 2100 behavior was caused by changes in the electron trajec- 5 26 32 14 1900 tories and processes of energy accumulation by elec- 5 50 60 1 1400 trons in the anodeÐcathode gap in an increasing magnetic field. If the anodeÐcathode spacing was signifi- 5 50 60 10 2000 cantly larger, the dependence became smoother. This 16 50 17 5 600 allowed beam current tuning over a wide range. The 40 50 30 50 2200 experimental data for the magnetron diode (anode 40 78 100 50 1800 diameter, 50 mm; voltage, 60 kV) are given in the table. It follows that the current can be varied from 0.5 to 80 100 19 8 1100

TECHNICAL PHYSICS Vol. 46 No. 2 2001

230 VOLKOLUPOV et al. The beam spot size on the collector was measured. 10 kV It was determined that the beam cross section consisted of concentric rings with a uniform azimuthal intensity U distribution. The beam diameter was approximately equal to the cathode diameter, and the beam wall thickness was 1Ð2 mm. Therefore, the energy of electrons in the electron layer was about 0.5Ð1 keV, which approx- I imately corresponded to the maximum secondary emission coefficient for copper. Experiments with a magne- 5 A tron diode with a cathode 80 mm in diameter showed µ that, if the cathode was exposed to a nonuniform mag- 0.5 s netic field with a transverse component (about 5%), the beam spot image was partly sharp and partly fuzzy. The Fig. 4. Oscillograms of the cathode voltage U and beam current I; the cathode diameter is 5 mm, the anode diameter is thicknesses of the sharp and fuzzy parts were about 2 26 mm, and H = 1800 Oe. and 3Ð4 mm, respectively. This fuzziness was caused by the nonuniformity of the magnetic field. U, kV; I, A 100 THEORETICAL STUDY OF THE ELECTRON FLOW MULTIPLICATION 80 Theoretical simulation of the secondary emission 60 development and the steady-state stage of the secondary-emission cathode operation was performed to study 40 U specific features of the processes occurring in a magnetron diode. The steady-state stage was analyzed using a 20 I three-dimensional mathematical model of a magnetron diode. The model was based on a self-consistent set of 0 2 4 6 8 10 simultaneous differential equations of motion (for elec- t, µs tron flow) and the Poisson equation (for calculating the space-charge forces). Fig. 5. Oscillograms of the beam current I and cathode voltage U; H = 700 Oe. The magnetron diode simulated by this model was assumed to have the following parameters: cathode diameter, 5 mm; anode diameter, 26 mm; peak anode Ua, kV voltage (0 < t < t1), 75 kV; anode voltage plateau (t2 < t < t3), 35 kV; voltage droop duration (t1 < t < t2), 60 1.25 µs; voltage plateau duration, 60 µs; magnetic intensity, 2000 Oe; and cathode length, 90 mm. A linear approximation of the shape of the voltage pulse applied 30 to the anode is shown in Fig. 6. The value of the anode voltage within the interval t2 < t < t3 was taken as the working value. The large-particle method described in [7] was used to study the dynamics of the electron flow 0 t1 t2 5 t3 10 formation processes. t, µs Consider specific features of electron processes in Fig. 6. Shape of the anode voltage pulse. the magnetron diode. These processes are nonsteady (secondary-emission multiplication occurs at the anode ≤ ≤ voltage pulse surge droop (t1 t t2)). On the other For example, an electron beam pulse with an amplitude hand, the spatial distribution of the electron flow should of ~15 A and duration of 8 µs was obtained at a cathode be studied using a three-dimensional cylindrical coor- voltage of 55 kV and a magnetic intensity of ~1150 Oe dinate system (r, ϕ, z). The mathematical model of the (cathode diameter, 40 mm; anode diameter, 78 mm). As magnetron diode is based on the self-consistent set of the magnetic intensity decreased to 700 Oe, the beam simultaneous equations of motion for the electron flow current pulse assumed a spiky shape with a generation ∂ µ v r r 2 period of ~1 s and a current amplitude of 30 A (dura------= ηE (t) + rv ϕ Ð ω rv ϕ, tion of each spike was 10Ð30 ns). ∂t 0 c

TECHNICAL PHYSICS Vol. 46 No. 2 2001 ELECTRON BEAM GENERATION IN A MAGNETRON DIODE 231

∂( Θ) v v v N σ r η 1 ϕ( ) r ϕ ω r c ------= ----E0 t Ð 2------+ c-----, ∂t r2 r r 750 1 1.50 (1) ∂v z ∂r ------z = ηE (t), ---- = v , ∂t 0 ∂t r 500 80 0.75 ∂ϕ ∂z Θ v -∂----- = , -∂--- = z, 40 t t 2 ω where vr, ϕ, z are the electron velocity components, c = η η 0 B0 is the cyclotron frequency, and = e/m is the 0 0.5 1.0 1.5 reduced electron charge; and on the Poisson equation W/Wmax (for calculating the space-charge ﬁeld)

2 2 Fig. 7. (1) Dependence of the secondary-emission coeffi- 1 ∂  ∂U 1 ∂ U ∂ U ρ σ ------r------+ ------+ ------= Ð----, (2) cient on the particle energy and (2) dependence of the ∂  ∂  2 2 2 ε number of primary electrons N on the particle energy r r r r ∂ϕ ∂z 0 c (Ua/Ucr = 0.7). where ρ = ρ(r, ϕ, z) is the space charge (SC) density. The set of simultaneous equations (1) and (2) was solved in the quasi-stationary approximation for a con- r stant anode voltage within the limits of the motion equation integration step (beginning at t = t1). The step ∆ π ω 1 was taken to be T = (1/10)Tc, where Tc = 2 / c is the 3 2 cyclotron oscillation period. The equation of motion was solved numerically using the RungeÐKutta method ρ ρ' of the fourth order. The quiet start model [8] was used as the model for determining the initial coordinates and velocities of the particles in the magnetron diode interaction space. The Poisson equation (2) was solved by the Hock- ney method of finite differences using fast Fourier transforms [9, 10] at given initial and boundary conditions. Numerical differentiation of the SC potential at the nodes of the finite-difference mesh was used for Fig. 8. Ua/Ucr = (1) 0.7; (2) 0.5; and (3) 0.3. determining the electrostatic field intensity E0 = −gradU. Discrete values of the SC potential (least- squares method) were locally smoothed to reduce fluc- for voltage normalization was determined from the tuations of the calculated SC field. Hull cutoff field. For the magnetron diode under study, The results of the simulation are presented in Figs. 7 the critical potential was found to be 139 kV. and 8. The energy distribution of the primary electrons The theoretical radial distribution of the SC density (large particles (macroparticles) with a charge of q = in the diode interaction space and the experimental 0.8 × 10Ð13 C) and the theoretical approximation of the radial distribution of the beam intensity at the collector experimental dependence of the secondary-emission surface are shown in Fig. 8. The maximum SC densities coefficient for copper on the primary electron energy in the three cases under consideration were calculated: [11] are shown in Fig. 7. As seen from Fig. 7, the num- ρ ρ ρ ρ in the first case, / br = 0.19; in the second case, / br = ber of low-energy macroparticles (i.e., particles with ρ ρ ρ σ ≤ 0.16; in the third case, / br = 0.1, where br is the Bril- energy corresponding to 1) is more than 60% of the louin density of the space charge. In the third case total number of macroparticles bombarding the cath- ≈ (Ua/Ucr 0.3), the theoretical and experimental distri- ode. The presence of low-energy particles affects the butions of SC density are in satisfactory agreement. SC field distribution near the cathode surface (within a ∆ The presence of electrons in the paraxial region is distance equal to the selected mesh size r = (ra Ð caused by cycloidal motion of electrons. rc)/32). Changes in the number of low-energy particles affect the cathode current (limitation of emission by the Numerical simulation of secondary-electron emis- SC field). The number of macroparticles with energies sion from the cathode surface at the steady-state volt- ≤ ≤ higher than the first critical potential (i.e., the energy age stage (t2 t t3) was performed. The secondary- for which σ > 1) depends on the voltage applied. The electron emission was caused by bombardment of the macroparticle energy increases with increasing voltage. cathode with primary electrons with energies of 300, It should be noted that the critical potential value used 560, and 700 eV. The plateau voltage was equal to U =

TECHNICAL PHYSICS Vol. 46 No. 2 2001 232 VOLKOLUPOV et al.

N, arb. units 4

0 10 20 30 40 50 t, ns

Fig. 9. Time dependence of the number of electrons in the electron layer (E = 300 eV).

N, arb. units 4

0 10 20 30 40 50 t, ns

Fig. 10. Time dependence of the number of electrons in the electron layer (E = 700 eV).

Φ 4 , 10 V 40 kV; the magnetic field intensity, H = 2000 Oe. A hot 1.8 cathode was the source of primary electrons. When the 1.7 hot cathode had worked for several nanoseconds, it was 1.6 switched off. The initiated process of electron multiplication was studied. When the energy of the primary 1.5 electrons reaches 700 eV, irregular disturbances of the 1.4 electron density of the space charge occur with a delay 1.3 of >30 ns (Fig. 10). If the energy of the primary electrons is about 300 eV, these disturbances are insignifi- 1.2 cant (Fig. 9). These results are in qualitative agreement 1.1 with the results presented in [5]. As seen from the 0 10 20 30 40 50 t, ns obtained results, the secondary-emission coefficient of the cathode material should be maximal at an incident Fig. 11. Time dependence of the potential in the middle of electron energy of ~500 eV. The time dependence of the the anodeÐcathode gap. potential in the middle of the anodeÐcathode gap is

TECHNICAL PHYSICS Vol. 46 No. 2 2001 ELECTRON BEAM GENERATION IN A MAGNETRON DIODE 233 given in Fig. 11. As seen in Fig. 11, there is a decrease switches with characteristic commutation times of sev- in the potential caused by thermoelectrons in the begin- eral nanoseconds. ning of the curve. When thermionic emission ceases, the accumulated electrons disperse, causing a slight increase in the potential. Then, there is a decrease in the REFERENCES potential (the duration of the decrease is 10Ð15 ns) 1. V. M. Lomakin and L. V. Panchenko, Élektron. Tekh., caused by the completion of transient processes and Ser. 1, No. 2, 33 (1970). formation of the space charge maintained by the sec- 2. J. F. Skowron, Proc. IEEE 61 (3), 69 (1973). ondary emission. 3. S. A. Cherenshchikov, Élektron. Tekh., Ser. 1, No. 6, 20 (1973). The results of calculations are in satisfactory agree- 4. A. N. Dovbnya, V. V. Zakutin, et al., in Proceedings of ment with the results of experiments on electron beam the 5th European Particle Accelerators Conference, Sit- generation using magnetron diodes. ges, 1996, Ed. by S. Myers, A. Pacheco, R. Pascual, et al. (Inst. of Physics Publ., Bristol, 1996), Vol. 2, p. 1508. 5. A. V. Agafonov, V. P. Tarakanov, and V. M. Fedorov, CONCLUSION Vopr. At. Nauki Tekh., Ser. Yad.-Fiz. Issled., Nos. 2Ð3, 137 (1997). It was shown that magnetron diodes with metal sec- 6. G. G. Sominskiœ, D. K. Terekhin, and S. A. Fridrikhov, ondary-emission cathodes can be used to generate Zh. Tekh. Fiz. 34 (9), 1666 (1964) [Sov. Phys. Tech. straight electron beams with high current densities. Phys. 9, 1286 (1965)]. Tubular electron beams with a current density of up to 7. R. Hockney and J. Eastwood, Computer Simulation 50Ð70 A/cm2, the outer diameter of 3.5Ð84 mm, a wall Using Particles (McGraw-Hill, New York, 1984; Mir, thickness of 1.5Ð2 mm, and a particle energy of 5Ð Moscow, 1987). 60 keV were obtained (pulse duration was 10 µs). It 8. A. S. Roshal’, Modeling of Charged Beams (Atomizdat, was revealed experimentally that a beam current pulse Moscow, 1979). train could be generated using a single voltage pulse. 9. R. W. Hockney, J. Assoc. Comput. Mach. 12 (1), 95 Magnetic ﬁeld variation was shown to allow 10- to (1965). 20-fold tuning of the current. The processes of electron 10. J. W. Cooly and J. W. Takey, Math. Comput., No. 19, 161 multiplication and the steady-state stage of secondary (1965). emission were studied theoretically. It was shown that 11. I. M. Bronshteœn and B. S. Fraœman, Secondary Electron magnetron diodes could be used as electron sources for Emission (Nauka, Moscow, 1969). high-power microwave devices and charged-particle accelerators. They also can be used as fast high-voltage Translated by K. Chamorovskiœ

TECHNICAL PHYSICS Vol. 46 No. 2 2001

SURFACES, ELECTRON AND ION EMISSION

Processes in Open Systems on Crystal Surfaces with Low Miller Indices V. A. Voœtenko St. Petersburg State Technical University, ul. Politekhnicheskaya 29, St. Petersburg, 195251 Russia Received March 31, 2000

Abstract—Exposure characteristics that were obtained when growing various ﬁlms on natural low Miller index surfaces of several crystals were collected and analyzed. An evolution theory that explains their special form was constructed. The type of dose characteristics obtained suggests that the surface underwent a reconstruction, i.e., a nonequilibrium phase transition that occurs on the surface. A quantitative analysis of the experiments available has been performed. In particular, a quantitative estimation was obtained of to what extent coverage with lead hinders the oxidation of the surface of a nickel crystal. Upon intense light or electron irradiation of silicon, divacancies are the predominant centers of the formation of point and extended radiation defects, as well as of local regions of melting. For some two-dimensional systems (divacancies, sulfur atoms on the surfaces of passivated semiconductors, oxide ﬁlms), delay times and evolution times for the self-organized structure formed were determined. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION film vanishes and the superlattice becomes destroyed. The processes of film nucleation in an open growth On the other hand, with decreasing evaporation temper- system on a crystal surface are characterized by peri- ature, the film grown (e.g., a photocathode) proves to be odic fluctuations of the adatom concentration. Various very coarse and highly defective because of the ways of the development of two-dimensional nuclei of decreased diffusion along the surface of contact. The a new phase, e.g., by the mechanism of VolmerÐWeber magnetic structure of a film or the crystal structure of or StranskiÐKrastanov, are possible, but the predomi- individual layers of a photocathode without deteriorat- nant factor always is the appearance of large-scale peri- ing the sharpness and flatness of the layer boundaries odic fluctuations [1]. can be preserved by using buffer layers. In recent works [2, 3], magnetic films were applied The maintenance of stable growth of the energeti- (using a shrouded sublimation source cooled by liquid cally unfavorable free surface of iron upon gas-phase nitrogen) on the surface of semiconductors such as Ge deposition of fcc iron films on a sulfided (100) surface and GaAs subjected to sulfide passivation. Analogous of semiconductors [2, 3] is ascribed to the interaction of evaporation of oxygen onto the surface of pure and sulfur atoms with the growing surface. The decrease in lead-covered nickel was performed in [4]. Passivation the growth entropy appears to be due to the floating out of the silver surface with chlorine upon the evaporation of a sulfur layer on top of the surface. The processes in an ultrahigh vacuum was studied in [5]. At present, that occur on the surface are controlled by Auger elec- automated methods of controlling growth surfaces are tron spectroscopy (AES) [2Ð4] and low-energy electron widely developed. One of such methods is based on the diffraction (LEED) [2, 3]. In spite of the great impor- measurement of the temporal dependence of the inten- tance of the problem from the viewpoint of practical sity of electron-diffraction patterns [6] (which is stud- application and the large number of theoretical models ied in our work as well). These investigations con- [9, 10], the evolution processes that determine the firmed the result known from biology [7] and econom- adhesion of thin films with low-index surfaces (sur- ics [8] according to which the properties of open faces with low Miller indices) have yet been clarified at systems are connected with the character of their present insufficiently. microscopic interactions more closely than in the equi- On a microscopic level, the guarantee of the forma- librium case. Nevertheless, one can usually reveal the tion of an ideal film on a semiconductor surface appears general empirical regularities. This work is devoted to to consist in that the applied atoms have no electron their explanation for the case of thin films. In particular, orbitals that could produce energy levels in the forbid- the role of evaporation temperature upon film prepara- den zone of the main crystal. In this case, no charge tion is discussed. On the one hand, with increasing transfer occurs onto the bonding orbitals, and the ada- evaporation temperature (e.g., upon deposition of mag- toms can freely move along the surface or a boundary. netic films or upon passivation or oxidation), a signifi- It is exactly upon such movement that the above-men- cant interdiffusion of the main contacting components tioned coarse-scale fluctuations arise that serve as a occurs. In this case, the magnetic moment of the iron basis for the formation of clusters of a new phase, just

PROCESSES IN OPEN SYSTEMS ON CRYSTAL SURFACES 235 as upon the diffusional decomposition of solid solution of adatoms on the surface; F is the elastic force tions [1]. On the whole, the main factor that determines [12] due to the lattice-parameter dependence on the the structure, morphology, and sharpness of the grow- film composition; and b is the mobility. The quantities ing film is the diffusion of the lighter atoms (sulfur, G(n) and F in Eq. (2) are considered to be functions of oxygen) along the boundary between the heavier sub- the concentration n. According to reaction (1), the for- stances (semiconductor and iron, nickel, and lead). For mation of one sulfide bond that favors the adhesion of example, the probability of the formation of sulfide the iron film (coating) to the sulfided semiconductor bonds on a semiconductor surface upon sulfiding is substrate requires the simultaneous localization of two determined by the diffusion of oxygen vacancies in the sulfur atoms at one immobile iron atom. The nickel oxide film [11]. The three-dimensional diffusion theory oxide has a variable composition NiOx with x = 0.98Ð of growth that was applied in [4] to the case of oxida- 1.7 and therefore [13] can be regarded as a solid solution of pure nickel failed in explaining the first plateau tion of two compounds, namely, NiO and NiO2. Thus, in the exposure dependence of the intensity of the oxy- the probability of the independent localization of a pair gen Auger peak. The exposure curve of oxidation of of oxygen atoms near one nickel atom is important for nickel with a lead coating [4] cannot be interpolated at the formation of the oxide film as well. all; no model can be associated with the experimental The homogeneous lattice chemical reactions, such points. The authors of [2, 3] also give no approximation as the formation of complex secondary radiation for the exposure dependence that they obtained for the defects [14, 15], or oxidation or synthesis of molecules intensity of the LEED pattern. All this was done in the (e.g., by Eq. (1)), satisfy the balance equation present paper. We show that a satisfactory agreement between the theory and experiment can be obtained ( ) n assuming that the evolution processes on the surface G n = -τ-. (3) begin from the coalescence of a pair of diffusing adatoms. The grid of nanoclusters that arises in the process In the two-dimensional case [11, 15Ð17], Eq. (3) can of deposition on the surface is determined by the num- lead to a bistability of the atomic coating, i.e., to its ten- ber of initial protopairs. Formally, the situation is as if, dency to be in one of two states with different steady- during the entire process of condensation of adatoms, state concentrations n. For this to occur, it is necessary the critical nuclei contain only one pair of adatoms. that the rate of generation G(n) of primary quasiparticles be a nonlinear function of the concentration n. Among the variety of open systems, objects of this type STRANSKIÐKRASTANOV THEORY occur quite frequently. Thus, the transition to bistable OF EVOLUTION PROCESSES solutions of the homogeneous balance Eq. (3) occurs, The adhesion of iron films to sulfided (100) surfaces e.g., in the case of heat-conducting [16] or optical [17] of semiconductors is stabilized owing to the continuous systems, as well as upon radiation defect formation segregation of sulfur through an ordered c (2 × 2) iron [14, 15]. Such a structural phase transition can be due layer. The stabilization of the process, as follows from to the fact that the formation of chemical bonds at the the similarity of dose characteristics, occurs also upon initial and final stages of a reaction occurs at different oxidation of the (100) nickel surface; here, the move- rates. For example, the formation of iron sulfide FeS ment normal to the surface occurs inward. In both molecules is a process with a higher activation energy, cases, the key event is the formation of molecules such characterized by a smaller probability G1 than the prob- as FeS2 and NiO2, e.g., ability of attachment (G2) of a sulfur atom to a layer that already contains such molecules. This means that Fe + 2S = FeS2. (1) G2 > G1 and that the curve of the G(n) function has the Suppose that only the lightest adatoms (sulfur, oxy- form shown in Fig. 1 by the solid line. The existence of gen) participate not only in the transverse motion but several stages of oxidation was reliably established, also in the movement along the interface, whereas the e.g., for the oxidation of nickel with oxygen [4] or sil- migration of heavy atoms (iron, lead) is absent. Upon ver with chlorine [5]; they were shown to include the deposition of an iron film, diffusional motion in the (1) chemisorption of an oxidizer on the free surface, preliminarily adsorbed layer of sulfur atoms is initiated (2) complete oxidation of the surface up to the forma- and is accompanied by the floating of sulfur on top of tion of one or several monolayers of an oxide, and (3) a the free surface in the perpendicular direction. The dif- slow thickening of the oxide film. fusion occurs according to the equation The multistage character of the reaction causes a ∂ nonmonotonic reaction rate as a function of n, which n ( ∇ ) ( ) n -∂---- + div nbF Ð D n = G n Ð -τ-. (2) corresponds to a nonlinear G(n) dependence. As was t indicated above, Eq. (3) can have three solutions if G(n) Here, n is the concentration of adatoms per unit surface is nonlinear. Among the solutions shown in Fig. 1, cor- area; D is the coefficient of atomic diffusion parallel to responding to “sparse” (nl), intermediate (nc), and the surface; τ is the lifetime of atoms on the surface “dense” (nh) states of the system of adatoms or other with respect to desorption; G(n) is the rate of genera- quasiparticles, two extreme solutions represent stable

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236 VOŒTENKO

G(n) [15, 18], we obtain the following expression for the linear addition to the steady-state concentration:

G2 δn = A(t)exp{i(ky Ð ωt)} f (x), (4) where k and ω are the wave vector and the frequency of the wavelike disturbance. In the same approximation, we represent the functions G(n) and F(n) in the form of expansions in a small G1 addition δn: 1  F(n) = Cδn, G(n) = -- + ν δn. (5) nl nc nh n τ 

Fig. 1. Schematic dependences of the incoming (G(n), solid Here, C is the coefficient of the linear expansion that is line) and outgoing (n/τ, dashed line) terms of the kinetic determined by the elastic force [12] that is due to the Eq. (2) on the concentration of particles n. Along the hori- effect of the irradiating beam of particles, and the fre- zontal axis, stable (nl and nh) and unstable (nc) solutions of the stationary homogeneous Eq. (3) are indicated; along the quency ν is defined as vertical axis, the limiting values of G and G for the func- 1 2 ∂G 1 tion G(n) are given. ν = ------Ð --. (6) ∂n τ states. In the intermediate range of concentrations nl < Substituting Eqs. (4)Ð(6) into Eq. (2), we obtain (for n < n , the balance Eq. (3) leads to bistability of the details, see [15]) that a longitudinal wave is established h ∂ ∂ homogeneous state of the system. The arising instabil- along the surface; for a constant gradient n/ x, it is characterized by the frequency ω = knCb and damping ity of solutions of the diffusion Eq. (2) can be consid- γ 2 ν ered by analogy with bifurcations known from hydro- = k D Ð . Therefore, the condition sufficient for the dynamics [18]. solutions to Eq. (2) to be stable in the Lyapunov criterion sense is a significant sink of particles, i.e., Two regimes of the loss of stability of solutions of ∂ 1 ≥ G differential equations are known from hydrodynamics, -τ- -∂-----. (7) which differ in their symmetry [18]. Their experimental n effects are quite various; for surfaces, they consist in Condition (7) is fulfilled in the region of the extreme the formation of geometrical patterns of various shapes. roots nl and nh, where the chemical lattice reaction They are observed, e.g., on metallic electrodes upon occurs steadily and uniformly. If the sink of particles is electropolishing [19] or in the form of local melting insufficient, so that ν > 0, then a negative damping is regions on illuminated silicon [20]. The regime of iso- possible, γ = k2D Ð ν < 0. This occurs in the central por- tropic turbulence of the Qouette flow (e.g., in the space tion of Fig. 1, where, in a certain region near the inter- between coaxial cylinders) corresponds in the case of mediate root of Eq. (3), the straight line n/τ has a slope Eq. (2) to zero elastic force F = 0. This regime leads to that is smaller than that of the G(n) curve. If the stabil- a transformation of closed isolines. Another regime that ity criterion (7) is violated, the amplitude A(t) increases admits a linear analysis of stability using the Lyapunov in accordance with the Lyapunov exponent [22] criterion (see, e.g., [21]) is characterized by plane sym- because of the negative damping γ. However, this metry and corresponds to Poiseuille flow [18]. The growth is limited by the nonlinear nature of Eq. (2). The Lyapunov exponents transform a laminar flow into a equation for the squared modulus of the amplitude A(t) turbulent regime; a generalization of this law onto the averaged over the wave period, which describes its sat- phase space determines a new characteristic time of the uration, can be obtained by the Landau method [18]. Certainly, there are restrictions inherent in this method, system [22]. It is exactly the plane regime that is real- but the comparison with the experiment [2Ð4] (see ized upon growing iron films on the surface of semicon- below) indicates that apparently all of these restrictions ductors such as Ge and GaAs. Upon the floating out of are met. It is the real boundary of stability found from the layer of sulfur atoms that passivates the boundary of the Lyapunov criterion [21, 22] that should be deter- the semiconductor through the layer of iron atoms, the mined by that unique type of disturbances and by the concentration n is maximum at the boundary and falls same frequency ω(k) that yield zero damping (γ = 0). In off to vanish at the free surface because of evaporation. this case, we can expand the averaged temporal deriva- This kinetic state is described approximately by the set- tive of |A|2, i.e., 〈∂|A|2/∂t〉, in A, in which, for the reasons tling of a linear gradient ∂n/∂x = const. As the x axis, the that were indicated in [18], it is sufficient to retain only perpendicular to the surface plane was taken. By per- a few lowest terms that do not vanish upon averaging. forming a Fourier transform with respect to the longitu- The terms of odd orders in amplitude A necessarily dinal coordinate y, which is oriented parallel to F contain a periodic factor and vanish upon averaging.

TECHNICAL PHYSICS Vol. 46 No. 2 2001 PROCESSES IN OPEN SYSTEMS ON CRYSTAL SURFACES 237

Among the terms of even, e.g., fourth, orders in A, there ln(N0/N – 1) 2 2 | |4 are terms such as A A* = A , which do not vanish 1 upon averaging. Note here, returning to the restrictions of the 0 method, that the above temporal derivatives of the type 〈∂|A|2/∂t〉 are direct analogs of the “probabilities of tran- –1 sition per unit time” that are calculated in perturbation –2 theory on the basis of the nonstationary Schrödinger equation of quantum mechanics [23]. In this sense, the –3 Landau method is applicable to nondegenerate systems. At a temperature close to absolute zero, any sys- –4 tem is in a nondegenerate state. For open systems, the 2 4 6 8 10 12 14 16 analog of temperature appears to be the intensity of an t, min external action, so that the theory that is developed here is applicable for the case of negligibly small external Fig. 2. Time variation of the intensity of the [1/2, 1/2] flows. In economics [8], the conclusions that will be S/Fe(100) diffraction reflection in the process of iron depo- made below may be of interest in the area of insufficient sition at 150°C. The experimental points are given accord- | |2 ing to [2]; the curve was obtained by the least-squares financing. Our interest in the squared modulus A in method using Eq. (10). The adjustable parameters are a = this case is related to the fact that the probability of coa- τ τ Ð1 t0/ l = 1.66 and b = 1/ l = Ð0.37 min . lescence of a pair of atoms per unit area or of the formation of other similar microstructures, e.g., divacancies, is proportional to the statistical weight, i.e., the in a single constant, which is designated in Eq. (10) as τ number of combinations C2 = n(n Ð 1)/2. By averaging exp(t0/ l). There are grounds to believe that the con- n stant t that occurs in this equation is precisely that time the expansion of this number into a series in the ampli- 0 | |2 that is required for the kinetic energy that is necessary tude A, we obtain precisely A , and the rate of genera- to “swing” a corresponding fluctuation to be accumu- tion of sulfide bonds, molecules of FeS2 or NiO2, or lated. For example, a delay t0 can take place between divacancies in silicon wafers is equal to the beginning of the bombardment of the surface with ∂N 1 ∂ A 2 iron atoms and the floating out of sulfur atoms from the ------W-- = ------. (8) surface because of the inertia of the latter ones. The ∂t 2 ∂t delay time t0 may be considered independent of the This rate is given by the average temporal derivative intensity of the external action I, in particular, of the from the quantity of this type. By writing the Landau density of the flux of atoms from the outside. Note the τ constant α, which was introduced in [18] in a similar dependence of the evolution time I on the intensity of expansion, through the limiting concentration of dou- action I. Taking into account what we said above about ble bonds N0W, we obtain the smallness of the external action I, we perform an τÐ1 ∂N N  N  expansion of the reciprocal time into a series in I: I = ------W-- = -----W-- 1 Ð ------W--- , (9) ξ η ξ η ∂t τ  N  + I, where and are positive phenomenological l 0W constants. To complete the consideration of the Stran- τ where l is the characteristic time determined by the skiÐKrastanov theory of evolution processes, we note Lyapunov exponent [22]. that it is exactly that mechanism that is realized upon It was shown previously [11, 15] that coarse-scale growth of lateral structures such as quantum filaments fluctuations δn are settled in the system after this time, and quantum dots. They are formed via the diffusion which lead to the evolution of the structure of semicon- mechanism of the development of a single monolayer, ductor wafers and, in some cases, can be observed in which is accompanied by decomposition through the the form of characteristic patterns on the surface. The StranskiÐKrastanov mechanism. On the contrary, for solution to Eq. (9) is the exposure dose characteristic of growing superlattices, a vicinal surface should be used the evolution type [24], and, in this case, thermal diffusion leads to the destruction of the growth steps. N0W NW = ------[---(------)----τ-----]. (10) 1 + exp t0 Ð t / l ANALYSIS OF EXPERIMENTAL RESULTS Scaling considerations permit us to unambiguously correlate the parameters of deposition of the films in The structure of iron films deposited on passivated [2Ð4] or burning out of molten regions in [20] with the surfaces of semiconductors was controlled in [2, 3] by theoretical constants that describe them; actually, this the method of low-energy electron diffraction (LEED). has already been done in Eq. (10). Since Eq. (9) is a The above theory permits one to describe the character- first-order differential equation, its integration results istic evolution type of the exposure characteristic con-

TECHNICAL PHYSICS Vol. 46 No. 2 2001 238 VOŒTENKO

ln(N0/N – 1) on the surface of iron (S/Fe(100) [1/2, 1/2] LEED 2 spots) in the process of iron deposition at a temperature of 150°C according to [2]. In the process of deposition of iron and floating of sulfur atoms, a chain is formed 0 around pairs of sulfur atoms, and, in this way, a cluster of the square lattice of sulfur atoms is formed. We assume that the brightness of the S/Fe(100) [1/2, 1/2] –2 LEED spots is proportional to the number of nanoclusters that are formed in this way, i.e., to the number of the initial pairs (FeS2 molecules). The straight line in –4 Fig. 2 represents the fitting of the dose dependence of the above diffraction reflections using Eq. (10). In the caption to Fig. 2, we give the numerical values of the –6 coefficients that describe the straight line y = a + bt (fit- τ 0 5 10 15 20 25 ting parameters); the corresponding evolution time l = t, min τ Ð1/b for Fig. 2 turned out to be l = 2.70 min. It was the fulfillment of the conditions of the theory of partially Fig. 3. The height of the oxygen Auger peak as a function of the time of oxygen deposition onto a pure nickel surface inhomogeneous reconstruction of the surface (see (100). The experimental points are given according to [4]; Eq. (10)) that, in our opinion, predetermined the reten- the curve was obtained by the least-squares method using tion of magnetic properties of the films [2]. It is of inter- τ Eq. (10). The adjustable parameters are a = t0/ l = 2.77 and est that this high quality was obtained at a moderate τ Ð1 b = 1/ l = Ð0.33 min . temperature of deposition T = 150°C, which turned out to be possible due to the advantages of the technique of ln(N0/N – 1) sulfiding [11]. The adjustable delay time proved to be t02 = 4.48 min. The magnetic structure of the film is controlled by the formation of sulfide pairs inside the 2 film. The division of the entire iron layer into domains with a low (nl) and a high (nh) concentration of sulfur 0 explains the retention of the magnetic properties of the film upon its saturation with sulfur. In this case, the magnetic domains become closed through the regions –2 with a low concentration of sulfur ni. Figures 3 and 4 display dose characteristics of the –4 processes of oxidation of the pure and lead-coated surfaces of nickel. The straight lines represent the fitting of the experimental data points (squares) obtained by 0 25 50 75 100 125 150 170 200 Auger spectroscopy using Eq. (10) by the linear least- t, min τ squares method. The evolution time l = Ð1/b was Fig. 4. Same as in Fig. 3 for the case of oxygen deposition obtained to be 3.00 min for the pure surface (Fig. 3) and onto a (100) surface coated with lead. The adjustable param- 26.3 min for the surface coated with lead (Fig. 4). The τ τ Ð1 delay times t also differ by about an order of magni- eters are a = t0/ l = 1.84 and b = 1/ l = Ð0.038 min . 0 tude, namely, t01 = 8.3 min for the pure surface and t02 = 48.4 min for the surface coated with lead. The good structed using LEED investigations. Upon oxidation of agreement between the experiment and the theory pure and lead-coated surfaces of nickel [4], the intensi- developed, just as in the case of iron films, indicates the ties of the oxygen Auger signal were measured, which high quality of the films; oxygen films were deposited were proved to depend on the exposure time too. In the at room temperature. The difference in the times for the latter case, the lead film also was subjected to a partial two different regimes of oxidation characterizes the oxidation; the process of oxidation consisted in the dif- protecting effect of the lead coating and also confirms fusion of oxygen into the space between the nickel and the adequacy of the theory. the lead films. On the whole, the situation was the same as upon the deposition of iron films on the sulfided surface. A satisfactory agreement between the theory and REFERENCES the experiment indicates the nonuniform reconstruction of semiconductor and metallic surfaces in the process 1. V. G. Karpov and M. Grimsditch, Phys. Rev. B 51 (13), of deposition. 8152 (1995). Figure 2 displays the intensity of the signal of dif- 2. G. W. Anderson, P. Ma, and P. R. Norton, J. Appl. Phys. fraction reflections from a square lattice of sulfur atoms 79, 5641 (1996).

TECHNICAL PHYSICS Vol. 46 No. 2 2001 PROCESSES IN OPEN SYSTEMS ON CRYSTAL SURFACES 239

3. G. W. Anderson, M. C. Hanf, and P. R. Norton, Phys. 15. V. A. Voœtenko and S. E. Mal’khanov, Zh. Éksp. Teor. Rev. Lett. 74 (14), 2764 (1995). Fiz. 114 (3), 1067 (1998) [JETP 87, 581 (1998)]. 4. C. Argile, Surf. Sci. 409, 265 (1998). 16. A. V. Subashiev and I. M. Fishman, Zh. Éksp. Teor. Fiz. 5. B. V. Andryushechkin, K. N. El’tsov, and V. V. Martynov, 93 (6), 2264 (1987) [Sov. Phys. JETP 66, 1293 (1987)]. Poverkhnost, No. 8, 72 (1999). 17. N. N. Rozanov, Optical Bistability and Hysteresis in 6. V. N. Petrov, V. N. Demidov, N. P. Korneeva, et al., Zh. Distributed Nonlinear Systems (Nauka, Moscow, 1997). Tekh. Fiz. 70 (5), 97 (2000) [Tech. Phys. 45, 618 18. L. D. Landau and E. M. Lifshitz, Course of Theoretical (2000)]. Physics, Vol. 6: Fluid Mechanics (Nauka, Moscow, 7. E. G. Rapis, Zh. Tekh. Fiz. 70 (1), 122 (2000) [Tech. 1986; Pergamon, New York, 1987). Phys. 45, 121 (2000)]. 19. V. V. Yuzhakov, Hsueh-Chia Chang, and A. E. Miller, 8. A. Cavagna, J. P. Garrahan, I. Giardina, and D. Sher- Phys. Rev. B 56, 12608 (1997). rington, Phys. Rev. Lett. 83 (21), 4429 (1999). 20. Ya. V. Fattakhov, I. B. Khaœbullin, R. M. Bayazitov, 9. J. A. Venables, G. D. T. Spiller, and M. Hanbucken, Rep. et al., Poverkhnost, No. 11, 61 (1989). Prog. Phys. 47 (4), 399 (1984). 21. G. A. Korn and T. M. Korn, Mathematical Handbook for 10. S. A. Kukushkin and V. V. Slezov, Dispersive Systems on Scientists and Engineers (McGraw-Hill, New York, Solid Surfaces: Mechanism of Formation of Thin Films 1961; Nauka, Moscow, 1973). (Nauka, Leningrad, 1996). 22. A. K. Pattanayak, Phys. Rev. Lett. 83 (22), 4526 (1999). 11. V. A. Voœtenko and S. E. Mal’khanov, Poverkhnost, 23. L. D. Landau and E. M. Lifshitz, Quantum Mechanics: No. 10, 72 (1999). Non-Relativistic Theory (Nauka, Moscow, 1956; Perga- 12. I. P. Ipatova, V. G. Malyshkin, V. A. Shchukin, et al., mon, New York, 1977. Phys. Low-Dimens. Struct. 3Ð4, 23 (1997). 24. I. L. Aleœner and R. A. Suris, Fiz. Tverd. Tela (St. Peters- 13. A. Ya. Kipnis, Popular Library of Chemical Elements burg) 34 (5), 1522 (1992) [Sov. Phys. Solid State 34, 809 (Nauka, Moscow, 1977), Vol. 1. (1992)]. 14. V. A. Voœtenko and S. E. Mal’khanov, Zh. Éksp. Teor. Fiz. 112 (2), 707 (1997) [JETP 85, 386 (1997)]. Translated by S. Gorin

TECHNICAL PHYSICS Vol. 46 No. 2 2001

SURFACES, ELECTRON AND ION EMISSION

Interaction of Graphite with Atomic Hydrogen E. A. Denisov and T. N. Kompaniets St. Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, 198904 Russia Received January 31, 2000; in ﬁnal form, March 29, 2000

Abstract—The kinetics of sorption of atomic hydrogen by pyrolytic, quasi-single-crystal, and RGT commercial-grade graphite was studied. The processes of sorption and subsequent thermal outgassing are shown to proceed in a similar manner for all the three types of graphite. Thermal desorption spectra obtained during linear heating of hydrogen-saturated samples have two peaks. A mathematical model including features of the thermal desorption kinetics that are observed when heating is terminated is suggested. According to this model, two types of traps with binding energies of 2.4 and 4.1 eV are present in graphite. The physical justiﬁcation of the model is given. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION have been irradiated by atomic hydrogen. Based on the kinetic information, one can develop models of pro- Over recent years, most of studies on hydrogenÐ cesses occurring in this system and, in particular, sepa- graphite interaction have been concerned with applica- rate surface and volume effects contributing to atomic tions, namely, with the use of graphite for protecting hydrogen sorption. the first wall of fusion reactors. Therefore, interactions of several-keV hydrogen ion beams with commercial- grade graphites have largely been considered. A large MATERIALS body of both experimental and theoretical data is avail- ρ able in this field [1–12]. In these works, emphasis has Commercial-grade RGT graphite ( = 2.20Ð been given to chemical erosion of the graphite surface 2.26 g/cm3) contains about 7.5 at. % of titanium and is [8, 13Ð18]. As to desorption, only that of hydrocarbons produced by unidirectional hot pressing of carbonÐtita- has been studied extensively in spite of the fact that the nium powder [19]. Most of the RGT grains are disk- amount of hydrogen incorporated into hydrocarbons shaped and lie parallel to the basal plane of the graphite leaving the surface during thermal desorption is much lattice. The mean grain size is about 10 µm. smaller than the amount of hydrogen desorbed as H2 Pyrolytic graphite is of density ρ = 2.186 g/cm3. On molecules. In addition, in most of the studies it is a microscale, it has a layered structure with a spacing of assumed that atomic hydrogenÐgraphite interaction is a 0.5Ð1.0 µm in the growth (c-axis) direction because of purely surface process and the possibility of hydrogen the step growth kinetics. We studied two types of pyro- dissolution in graphite is usually ignored. We believe lytic graphite that differed in temperature of final that, when graphite is exposed to an atomic hydrogen annealing: true pyrolytic graphite (PG) and quasi-sin- flux, the possibility of hydrogen dissolution may mark- gle-crystal graphite (QSCG). The former looked like edly increase in comparison with the situation when commercial-grade graphite with a rough surface. equilibrium molecular hydrogen is involved. At least QSCG had a much smoother and shiny surface, which some hydrogen atoms may occur beneath the adsorbent indicates its higher structural order. surface even at room temperature. Unfortunately, the simulation of atomic hydrogen sorption by graphite with volume processes taken into account has been EXPERIMENTAL reported only in two of the works cited above [17, 18]. The sorption/desorption kinetics was studied with As was noted, most of sorbed hydrogen molecules thermal desorption spectrometry. Graphite samples × × leave graphite as H2 molecules as the temperature rises. were made in the form of ribbons measuring 1 40 Available literature data on the kinetics of this process 0.5 mm. The surface of the ribbon was parallel to the are, however, scarce and refer mostly to the estimation basal graphite plane. The sample was attached to of the number of hydrogen molecules from thermal current leads and placed into a vacuum chamber. desorption spectra (TDS). The only attempt to deter- The residual (mainly hydrogen) pressure was kept at mine the energy of activation of desorption was based 10Ð8 torr. The temperature, which can be varied accord- on the a priori assumption that this process is of a dis- ing to a specified law, was measured with a W/WRe sociative nature [8]. Therefore, is was the aim of this thermocouple. The desorbed hydrogen was detected by work to determine the kinetic and energy parameters of a magnetic sectorial mass spectrometer. Prior to sorp- desorption processes that take place after the graphites tion experiments, the samples were annealed for a long

INTERACTION OF GRAPHITE WITH ATOMIC HYDROGEN 241 time at 1200°C. At the end of annealing, the tempera- T, °C Jdes. H2, arb. units ture was momentarily raised to 1400°C. 1400 Purified hydrogen was supplied to the chamber through a diffusion filter. Atomization was carried out 8 1200 when the gas passed near a 100-µm-diam. tungsten fil- 1000 ament heated to 2500°C. The filament was arranged 6 parallel to the sample at a distance of 5Ð8 mm so that 800 the atomic hydrogen flux struck the surface parallel to the basal plane of the sample. The irradiation dose was 4 600 calculated with regard for the inlet hydrogen pressure, atomization yield, and mutual arrangement of the sam- 400 ple and the atomizer. 2 200 The probability of hydrogen atomization on tungsten heated to 2100°C was taken equal to 0.3 [20]. Dur- 0 Ð2 0 10 20 30 40 50 60 ing exposure, the hydrogen pressure was 10 torr. The t, s flux of hydrogen atoms toward the front side of the sample was estimated at 5 × 1013 H0/(cm2 s) in view of Fig. 1. Typical TDS of PG irradiated by hydrogen atoms. the experiment geometry. Linear heating with a rate of 25 K/s.

° Jdes. H2, arb. units T, C EXPERIMENTAL RESULTS 6 1200 Unlike the RGT samples [21], the PG and QSCG 1 samples did not sorb molecular hydrogen at pressures 5 1000 below 5 torr and temperatures below 600°C. During 2 4 linear heating, noticeable desorption of H2 was 800 observed only after these graphites had been irradiated 4 by atomic hydrogen. A typical postirradiation TDS for 3 600 the PG is presented in Fig. 1. For the other types of 3 2 graphites, the spectra are similar. No marked differ- 3 2 1 400 ences were also observed in the atomic hydrogen sorp- 1 tion kinetics. Dependences of the sorbed hydrogen 4 200 quantity on the sorption temperature and irradiation dose for all the three graphites are much the same. The 0 structure of the graphites and their purity seem to insig- 0 20 40 60 80 100 , s nificantly affect the sorption/desorption kinetics with t atomic hydrogen involved. Fig. 2. TDS obtained after sorption of H0 when linear heat- Two TDS peaks at 850 and 1250°C in Fig. 1 are ing was terminated at different temperatures. PG, heating related to hydrogen release from two states with differ- rate 25 K/s. ent binding energies. The temperatures of the desorption peaks were independent of the initial hydrogen obtained under the assumption that hydrogen is present concentration, indicating the first order of the desorp- only on the surface. tion kinetics. If heating is linear and the kinetics is of α 2 In the next series of experiments, the sorption con- the first order, ln( /Tm ) should vary linearly with ditions were identical, but the final heating tempera- α Ed/kTm ( is the rate of heating, Tm is the temperature of tures were different (Fig. 2). When heating was termi- the desorption maxima, Ed is the energy of activation of nated (Fig. 2, curves 1Ð4), the desorption rate began to desorption, and k is the Boltzmann constant) [22]. fall sharply, the time of fall being almost independent From the slope of this dependence, one easily obtains of the final temperature of heating. Within 10Ð20 s, the the energy of activation of desorption. For the first desorption rate decreased to several percent of its max- state, the energy of activation estimated for three rates imum; i.e., hydrogen release was nearly completely of heating was found to be 2.4 eV. For the second state, stopped. The desorption rate sharply decreases in spite the energy of activation can be estimated only roughly, of the fact that the sample still contains a large amount since the temperature of heating was close to that of the of hydrogen, which is released at a subsequent rise in second desorption peak (at repeated heating to higher the temperature (Fig. 3, curves 1Ð3). In the experiments temperatures, the sample rapidly broke down). The the results of which are illustrated in Fig. 3, the sample value of Ed for the second state was evaluated at about was first heated to some intermediate temperature, 4 eV. Note that the above energies of activation were which was kept for 45 s, and then heating was contin-

TECHNICAL PHYSICS Vol. 46 No. 2 2001

242 DENISOV, KOMPANIETS ° It should be noted that the unusual hydrogen desorp- Jdes. H2, arb. units T, C 6 1200 tion kinetics was also observed after the graphites had 1 been irradiated by ~100-eV hydrogen ions. The hydro- 5 gen was desorbed in the same temperature interval; the 2 1000 desorption rate sharply dropped after heating had been 4 terminated; and after the second rise in the temperature, 3 800 the remaining hydrogen was entirely desorbed. The only difference was the absence of two distinct peaks 3 600 (Fig. 4).

2 400 1 3 2 DISCUSSION 3 2 1 200 An adequate kinetic model is the first step toward understanding basic processes underlying gasÐsolid 0 0 20 40 60 80 100 interaction. The model of choice must take into consid- t, s eration the following. Upon irradiation, hydrogen is implanted into graphite part way down the surface (tens Fig. 3. TDS obtained after sorption of H0 when linear heat- of nanometers for ion energies of several hundred elec- ing was first terminated at different temperatures and then tron volts [6]), and mass transfer conclusively plays a continued to 1200°C. PG, heating rate 25 K/s. part in the outgassing process. Since the process kinetics are similar after irradiation by atoms and by ions, ° one can suggest that a considerable fraction of hydro- Jdes. H2, arb. units T, C 9 1400 gen atoms penetrates into the volume of the sample; hence, the outgassing kinetics also includes mass trans- 8 fer in the bulk of the graphite. Note, however, that all 1200 7 the features of the hydrogen evolution kinetics (for example, the presence of two peaks) cannot be covered 1000 6 by the simple diffusion model. 5 800 Thus, graphite outgassing should be described in 4 terms of at least three processes two of which are 600 strongly temperature-dependent (i.e., have a large 3 400 energy of activation) and show up as the TDS peaks. 2 The third one, which is responsible for a decrease in the 200 desorption rate when heating is stopped, must depend 1 on temperature only slightly. This may be hydrogen 0 diffusion between adjacent graphite layers [23]. This 0 20 40 60 80 100 , s process requires a very small activation energy largely t because these layers are loosely bonded and spaced at more than 3 Å apart. From our estimates [24], the Fig. 4. TDS of PG irradiated by ~200-eV H+ ions. Linear energy of activation of diffusion in RGT graphite is no heating was terminated at 1000 and 1200°C, heating rate 25 K/s. more than 0.5 eV. Therefore, the following model seems to be the most plausible. During sorption of the atoms or ions, a hydrogen-saturated layer of some ued at the same rate to 1200°C. After the second heat- thickness forms beneath the surface and two sorts of ing, the remaining hydrogen was totally released, its traps arise in the graphite. These traps, having large amount being practically the same as in the case when energies of activation, capture the hydrogen. Upon it was desorbed at similar temperatures but under con- heating, the hydrogen escapes from the traps into the mobile phase, with its migration over the sample being tinuous heating. The kinetic characteristics of hydrogen accompanied with reverse trapping. It is also assumed release also remained unchanged. These features were that the rate of desorption from the surface far exceeds typical of all three graphites. the rate of the other processes and is not the limiting Such an unusual desorption kinetics cannot be stage of outgassing. explained in terms of classical desorption from surface. To mathematically represent this model (diffusion For the first-order desorption kinetics, when heating is with reverse capture by traps of two sorts), we used a stopped, the desorption rate must decrease exponen- set of differential equations from our previous work tially with a time constant proportional to exp(ÐEd/kT); [25]. ° hence, at Ed = 2.4 eV, the rate of fall at 600 C and Calculations which follow substantiate the determi- 800°C would differ by several orders of magnitude. nation of the activation energy of hydrogen detrapping

TECHNICAL PHYSICS Vol. 46 No. 2 2001

INTERACTION OF GRAPHITE WITH ATOMIC HYDROGEN 243

α 2 ln(α/T2 ) using the slope of the ln( /Tm ) vs. 1/Tm curve. Within m the model suggested, we evaluated TDS for linear heat- + ing with various rates. The energy of hydrogen evolution from traps of the first sort was assumed to be –10 2.4 eV, and the diffusion coefficient was varied Ð7 Ð4 2 between 10 and 10 cm /s. It turned out that the slope +2.25 eV α 2 –11 of the ln( /T m ) vs. 1/Tm curve obtained in the model was practically coincident with that of the curve con- + structed from experimental data (Fig. 5). Therefore, the –12 2.36 eV value of 2.4 eV was taken as the energy of hydrogen 2.38 eV 2.25 eV evolution from traps of the first sort. Also noteworthy is that the energy of activation of detrapping can be deter- –13 mined without knowing the exact value of the diffusion 9.0 9.5 10.0 10.5 11.0 11.5 coefficient. As the diffusion coefficient value, we took 1/kT the one obtained in experiments on sorption of molec- m Ð6 2 α 2 ular hydrogen by RGT graphite: D = 10 cm /s [24]. Fig. 5. ln( /Tm ) vs. 1/Tm for the first desorption peak As follows from calculations, with such a diffusion within the diffusionÐrecapture model. D = (᭹) 10Ð7, (ᮀ) − coefficient and the saturated layer thickness of the order 10 6, (᭡) 10Ð5, and (+) 10Ð4 cm2/s. of the RGT grain size, the falls in the desorption curves when heating is switched off are adequately described , arb. units if unactivated capture by traps of the first sort has a time Jdes. H2 constant of 25 sÐ1. 5 Capture by traps of the second sort must be of an activation nature, since transitions from one state to the 4 other were observed at none of the rates and final temperatures of heating used. The model adequately 7 describes both thermal desorption under linear heating 3 to 1200°C and thermal desorption falls when heating is 3 6 terminated (Fig. 6). The parameter values used were as 2 follows: preexponential in the expression for diffusion 5 Ð6 2 1 2 4 coefficient D0 = 10 cm /s, energy of activation of dif- 1 fusion ED = 0, preexponentials in trapping rate constants for traps of the first and second sort r1 = 25 and r = 7.0 × 107 sÐ1, energies of trapping activation for 0 10 20 30 40 50 60 70 2 t, s traps of the first and second sort E = 0 and E = r1 r2 2.0 eV, preexponentials in detrapping rate constants for Fig. 6. Model TDS calculated for final linear heating tem- × 11 peratures (1) 730, (2) 770, (3) 820, (4) 1050, (5) 1100, traps of the first and second sorts b1 = 2.0 10 and (6) 1140, and (7) 1190°C. Heating rate 25 K/s. × 13 Ð1 b2 = 5.0 10 s , and energies of detrapping activation for traps of the first and second sort E = 2.4 and E = b1 b2 higher than on the (1120 ) plane by 0.8 eV [32]. Thus, 4.1 eV. the difference in bond energy per molecule for the dif- Note that the vast majority of studies where hydro- ferent planes at the boundary of a graphite layer is gen ionÐgraphite interaction was simulated points to 1.6 eV. This value is in good agreement with the above the necessity of considering hydrogen capture by traps results. localized in the bulk of graphite. The binding energies of trapped hydrogen vary in wide limits: from 0.5 to We believe that hydrogen capture by the traps of the 4.5 eV [1Ð3, 5, 7Ð11, 26Ð31]. first or the second sort can be related to the production The nature of the traps and specific sorption/desorp- of CÐH bonds with carbon atoms belonging to edge dis- tion processes still remains unclear. Dangling bonds of locations on the (1120 ) or (1010 ) planes, respectively, carbon atoms in the (1010 ) and (1120 ) planes seem to in the bulk of graphite. During sorption, transport of be the most probable sites of hydrogen sorption. Linear hydrogen in the form of atoms saturates dangling bonds defects like dislocations (dangling boundaries of of the edge dislocations to the maximum possible graphite layers) may readily form in graphite because extent. During thermal desorption, the hydrogen is of creeping of one layer on another during pyrolysis. detrapped in the form of molecules and migrates along The CÐH bond energy on the (1010 ) graphite plane is graphite layers until it reaches the crystallite boundary

TECHNICAL PHYSICS Vol. 46 No. 2 2001 244 DENISOV, KOMPANIETS and leaves the material through the pore network, grain 9. S. Fukuda, T. Hino, and T. Yamashina, J. Nucl. Mater. boundaries, or other macrodefects. 162Ð164, 997 (1989). It should be noted that this is only a possible physi- 10. D. K. Brice, Nucl. Instrum. Methods Phys. Res. B 44, cal interpretation of the mathematical model. Basically, 302 (1990). thermal desorption spectrometry cannot elucidate the 11. R. A. Causey, M. I. Blaskes, and K. L. Wilson, J. Vac. physical nature and location of the traps. In layered Sci. Technol. A 4, 1189 (1986). structures like graphite, sorption/desorption processes 12. D. Federici and C. H. Wu, J. Nucl. Mater. 186, 131 proceed in a rather unusual manner. For example, for- (1992). eign atoms and molecules may be accumulated in large 13. P. C. Stangeby, O. Ausiello, A. A. Haasz, and B. L. Doyle, amounts between graphite layers [33]. Unfortunately, J. Nucl. Mater. 122Ð123, 1592 (1984). intercalation in the hydrogenÐgraphite system has yet 14. P. Hucks, K. Flaskamp, and E. Vietzke, J. Nucl. Mater. received little attention. One can assume, however, that 93Ð94, 558 (1980). hydrogen atoms falling between graphite layers recom- 15. J. W. Davis, A. A. Haasz, and P. C. Stangeby, J. Nucl. bine to form molecules and become “blocked” in inter- Mater. 155Ð157, 234 (1988). planar gaps. In this case, hydrogen release would 16. I. S. Youle and A. A. Haasz, J. Nucl. Mater. 182, 107 require either molecule dissociation or severe deforma- (1991). tion (up to local destruction) of the graphite plane; 17. T. Tanabe and Y. Watanabe, J. Nucl. Mater. 179Ð181, 231 hence, energy must be spent to activate desorption. To (1991). conﬁrm the validity of the above assumptions, atomic- 18. M. Balooch and D. R. Olander, J. Chem. Phys. 63, 4772 hydrogen-induced modiﬁcations of the graphite surface (1975). must be examined. 19. A. P. Zakharov, Report on the Contract N 7/4 between Sintez NTO, St. Petersburg, Russia and Fusion Centre (Moscow, 1995). CONCLUSION 20. J. Smith and W. Fait, in Interaction of Gases with Sur- Thus, when atomic hydrogen interacts with PG, faces (Mir, Moscow, 1965), p. 362. QSCG, and commercial-grade RGT graphites, sorp- 21. E. Denisov, T. Kompaniets, A. Kurdyumov, and tion/desorption processes are virtually independent of S. Mazaev, J. Nucl. Mater. 233Ð237, 1218 (1996). the material type, being governed by processes in the 22. D. Woodruff and T. Delihar, Modern Techniques of bulk. Hydrogen release during heating is well described Space Science (Cambridge Univ. Press, Cambridge, by the mathematical model including diffusion with 1986; Mir, Moscow, 1989). reverse capture by traps of two sorts. Physically, the 23. B. M. U. Scherzer, J. Wang, and W. Moller, Nucl. traps are treated as dangling bonds of edge dislocations Instrum. Methods Phys. Res. B 45, 54 (1990). 24. E. A. Denisov, T. N. Kompaniets, A. A. Kurdyumov, and on the (1120 ) and (1010 ) planes. Diffusion of atoms S. N. Mazaev, J. Nucl. Mater. 212Ð215, 1448 (1994). goes both along and across graphite layers, while 25. E. Denisov, T. Kompaniets, A. Kurdyumov, and molecular diffusion takes place in interplanar gaps S. Mazaev, Plasma Devices Op. 6, 265 (1998). only. 26. E. Hoinkis, J. Nucl. Mater. 182, 93 (1991). 27. W. R. Wampler, B. L. Doyle, R. A. Causey, and K. Wil- REFERENCES son, J. Nucl. Mater. 176Ð177, 983 (1990). 1. W. Moller, J. Nucl. Mater. 162Ð164, 138 (1989). 28. J. Redmond and P. L. Walker, J. Chem. Phys. 64, 1093 (1960). 2. K. L. Wilson and W. L. Hsu, J. Nucl. Mater. 145Ð147, 121 (1987). 29. A. P. Zakharov, Processes of Accumulation and Reemis- sion of Hydrogen Isotopes in Carbonic Materials on 3. R. A. Causey, J. Nucl. Mater. 145Ð147, 151 (1987). Interaction with Ion and Plasma Streams (Report) (Inst. 4. P. G. Fischer, R. Hecker, H. D. Rohrig, and D. Stover, Fiz. Khim. Ross. Akad. Nauk, Moscow, 1991). J. Nucl. Mater. 64, 281 (1977). 30. S. L. Kanashenko, A. E. Gorodetsky, V. N. Chernikov, 5. K. Nakayama, S. Fukuda, T. Hino, and T. Yamashina, et al., J. Nucl. Mater. 233Ð237, 1207 (1996). J. Nucl. Mater. 145Ð147, 301 (1987). 31. V. N. Chernikov, A. E. Gorodetsky, S. N. Kanashenko, 6. W. R. Wampler and C. W. Magee, J. Nucl. Mater. 103, et al., J. Nucl. Mater. 217, 250 (1994). 509 (1982). 32. J. P. Chen and R. T. Yang, Surf. Sci. 216, 481 (1989). 7. G. Hansali, J. P. Biberian, and M. Bienfait, J. Nucl. Mater. 171, 395 (1990). 33. S. A. Solin and H. Zabel, Adv. Phys. 37, 87 (1988). 8. V. Phyllipps, E. Vietzke, M. Erdweg, and K. Flaskamp, J. Nucl. Mater. 145Ð147, 292 (1987). Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 46 No. 2 2001

EXPERIMENTAL INSTRUMENTS AND TECHNIQUES

On the Possibility of Determining Spalling Strength of Metals in a Submillisecond Range from Experiments with a Penetrating Radiation Effect on a Target T. A. Andreeva and S. N. Kolgatin St. Petersburg State Technical University, ul. Politekhnicheskaya 29, St. Petersburg, 195251 Russia e-mail: [email protected] Received December 1, 1999

Abstract—The strength of a metal under the action of short pulses is of great interest. Meanwhile, the problem concerning the magnitude of a tensile stress that leads to cleavage has not yet been solved theoretically. In this paper, an attempt is made to clarify, using the mathematical simulation method, the parameters of pulse radiation that can yield the clearest answer to the question of which of the existing theories of strength works best in metallic materials subjected to short pulses of micro- or nanosecond range. Recommendation are given on the optimal experimental design. © 2001 MAIK “Nauka/Interperiodica”.

τ INTRODUCTION pulse, and the power of the heat flux on the surface q0. The strength of a metal under the action of short By varying one or two of these parameters, we can pulses of micro- or nanosecond range is of great practi- change (in certain limits) the criterion of momentari- cal interest. Meanwhile, the question of the magnitude ness [7] and, consequently, the amplitude of the shock of a tensile stress that leads to cleavage has not yet been pulse in metal. Of practical interest is the dependence solved theoretically. There are facts [1] that favor the of the time for which the metal exists in the state of ten- kinetic theory of strength [2]. Some authors [3] believe sion on the pulse parameters, as well as the intervals of χ τ that the strength attainable in the submillimeter range is , pulse, and q0 in which one can distinguish the exact independent of the pulse duration and is a result of instant at which spalling occurs. motion of elementary carriers of plastic deformation. We describe the formation and propagation of a The final choice of this or that theory must be made on compression wave in the target by a set of equations of the basis of an analysis of additional experimental data. gas dynamics, which, in the Lagrangian coordinates s = It would be convenient to extract such data from exper- x ρ (ξ)dξ (where ρ is the density, x is the Euler coordi- iments on the effects of pulse penetrating radiation on a ∫0 flat plate using the interferometric method of measur- nate, and ξ is the integration variable), has the following the velocity of motion of the back surface of a tar- ing form: get. In this case, the understanding of gas-dynamics ∂()1/ρ ∂ v ∂ x ∂ v ∂ p processes that occur upon the absorption of energy in ------===------, ------v , ------Ð ------, (1)Ð(3) the target would favor the correct interpretation of the ∂t ∂ s ∂ t ∂ t ∂ s results obtained. In this work, we made at attempt, ∂ε ∂ ∂ ∂ v W (), T using the mathematical simulation method, to clarify -----∂ ==Ð p ------∂ - Ð ------∂ +Qst,W Ð k------∂, (4), (5) the variation of which parameters of pulse radiation can t s s s yield the clearest (from the viewpoint of the problem p = f ()ρ, T ,ε = f ε()ρ, T ,k = ρ f λ()ρ, T .(6)Ð(8) stated) results. Recommendations are given on the opti- p mal experiment design. Here, t is the time; v is the velocity; p, the pressure, ε, the internal energy; W, the heat flow due to thermal conductivity; Q, the specific energy contribution; k and λ, RESULTS AND DISCUSSION the thermal diffusivity and thermal conductivity coeffi- There exist numerous types of devices based on cients, respectively; fp and fε, the thermal and caloric pulsed sources of penetrating radiation that can be equations of state, respectively, taken from [8]; and fλ, employed for generating compression waves in metal- the dependence of the thermal conductivity on the den- lic targets (relativistic electron beams [4], lasers [5], sity and temperature, which is defined as in [7], using X-ray sources based on Z pinches [6], etc.). The radia- the dependences from [9, 10]. The energy contribution tions differ in penetrating power, which can arbitrarily Q was calculated by the Bouguer law be characterized by parameters such as the average χχ() mass coefficient of absorption χ, the pulse duration Qq= 0exp Ðsft, (9)

246 ANDREEVA, KOLGATIN σ p, kbar where St is the StefanÐBoltzmann constant and T0 is the initial temperature. 30 On the second surface (s = s ), we have 1 M ( , ) ( , ) 20 2 p t sM = p0, T t sM = T 0. (13) The set of gas-dynamics equations (1)Ð(13) was 10 3 4 solved by the finite-difference method [11]. To provide 5 the stability of calculations, a linear artificial viscosity 0 was used. As a sample for the investigation, a copper plate 7 mm thick was taken. In order to answer the –10 question posed at the beginning of the paper, the amplitude and the duration of the energy pulse, as well as the magnitude of the mass absorption coefficient, were –8 –6 –4 –2 0 2 4 6 8 varied. x, mm Figure 1 displays the evolution of a compression Fig. 1. Distribution of pressure over the thickness of the wave in the plate under the effect of a radiation pulse plate at various time moments. µ χ 2 with the parameters tpulse = 1 s, = 2.8 m /kg, and q0 = 6 × 1013 W/m2. Curve 1 refers to the time moment t = v, m/s 0.5 µs (radiation maximum); curve 2 refers to the µ instant tpulse = 1.0 s (the end of the pulse); curve 3, to 100 4 µ µ the instant tpulse = 2.0 s; curve 4, to tpulse = 2.6 s; and 75 5 µ curve 5, to tpulse = 3.1 s. At the initial stage, there is 50 3 formed a pressure peak at the wall (curve 1), which then 25 propagates to the target depth (curves 2, 3). After the 2 compression pulse emerges onto the free back surface 0 1 of the target, a rarefaction region is formed near it –1000 (curve 5). The size of the rarefaction region is virtually independent of the parameters of the radiation pulse –2000 χ such as tpulse and , whereas the amplitude of the tensile –3000 stress is mainly dependent on q0. The maximum tensile –4000 stress and the rate of deformation are observed at a –8 –6 –4 –2 0 2 4 6 8 point that is located about 2.0 mm from the back sur- x, mm face (this corresponds to the distance passed by the sound in a period equal to half the pulse duration). Fig. 2. Velocity fields inside the target at various time moments. The pressure in the plate can be estimated from the velocity of motion of the back surface, which in turn can be measured by the interferometric methods [12]. where q0 is the density of the energy flux on the surface; The velocity fields at various time moments for the χ is the mass-averaged absorption coefficient (a constant variant under consideration are shown in Fig. 2. At the quantity); and ft is a function of the temporal shape of the initial stage, an evaporated gas layer is stripped off pulse, which approximately was assumed to be from the front surface of the target; the velocity of this layer in the coordinate system chosen is negative, t 1 Ð 1 Ð ------, t ≤ 2t whereas in the condensed medium a flow is formed that t pulse is directed toward the radiation flow (curves 1, 2). As f t = pulse (10) > the shock wave propagates into the metal, the velocity 0, t 2tpulse. of the rear surface first increases gradually (curves 3, 4) The initial conditions for the above set of equations and then decreases (curve 5). had the form The allowance for the fracture of the target material ( , ) ( , ) was performed according to both above-mentioned the- v 0 s = 0; p 0 s = p0; (11) ories. According to the first of them (the temporal the- ρ( , ) ρ ( , ) 0 s = 0; T 0 s = T 0. ory of strength [2]), the material service life depends on the applied stress and temperature, which activate the On the surface (s = 0), the following boundary con- process of breaking interatomic bonds. According to ditions were used: the second theory [3], the strength properties of the material in a submillimeter range are the result of ( , ) ( , ) σ ( 4 4) p t 0 = p0, W t 0 = St T Ð T 0 , (12) motion of elementary carriers of plastic deformation

TECHNICAL PHYSICS Vol. 46 No. 2 2001 ON THE POSSIBILITY OF DETERMINING SPALLING STRENGTH OF METALS 247 such as dislocations, disclinations, and grain bound- vw, m/s aries. 200 1 The simulation of fracture in terms of the temporal theory of strength was performed by the following algorithm. After the temperature, velocity, density, 150 Euler coordinate, internal energy, and pressure have 2 been calculated, the elements of the array of deformation rates were calculated at a new time step at each kth 100 3 point of the difference scheme: 50 ( p (t) Ð p (t Ð δt)) b(k) = (Ð1)------k------k------; (14) δt 0 then, for points where the pressure proved to be nega- 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 tive (pk(t) < 0), the data obtained were compared with t, µs the limiting value of b∗. In this variant of the calcula- Fig. 3. Time dependence of the velocity of the rear surface tion, the scabbing occurred after b∗ reached a limiting of the plate. value equal to [1]

kbar b ≈ 100------. (15) * µs logq In the second variant of the calculation, the fracture 17 condition was assumed to occur when the negative pressure at one of the nodes of the difference mesh σ 15 (k) = (Ð1)pk(t) reached the limiting value equal to [1] 13 σ ≈ 50 kbar. (16) 10 10 8 8 If the conditions of the target fracture were fulfilled 6 6 at the kth node, then the following boundary conditions x, m2/kg 4 4 t, µs were set at this node, p(t, sk) = p0 and T(t, sk) = T0, and 2 2 the subsequent calculation was performed only for the nodes from the (k + 1)th to the last node. Fig. 4. Surfaces of the rear scabbing. Figure 3 displays the variation of the velocity of the back surface Vw accessible for the experimental deter- the sample can serve as arguments in favor of this or mination as a function of time. Curve 1 corresponds to that physical model of fracture. calculations using Eq. (15); curve 2 was calculated by Eq. (16). It is seen that in the second case fracture REFERENCES occurs later. If the material does not fail, the Vw(t) dependence falls off smoothly (curve 3 in Fig. 3). Note 1. N. A. Zlatin and B. S. Ioffe, Zh. Tekh. Fiz. 42 (8), 1740 that the presence of scabbing simulated by any of the (1972) [Sov. Phys. Tech. Phys. 17, 1390 (1972)]. two models under consideration leads to a discernible 2. V. R. Regel’, A. I. Slutsker, and É. E. Tomashevskiœ, difference in the calculated dependence for Vw . Kinetic Nature of Solid Strength (Nauka, Moscow, 1974). Conditions necessary to check which of the theories 3. Yu. I. Meshcheryakov, S. A. Atroshenko, T. V. Balicheva, is valid can be judged from Fig. 4, which gives the et al., Preprint No. 24, Leningradskiœ Filial Inst. Mashi- results of numerical simulation of scabbing on the rear novedeniya im. A.A. Blagonravova, Akad. Nauk SSSR χ surface in a wide range of the pulse parameters tpulse, , (Leningrad Branch of Blagonravov Institute of Engi- and q0. The lower surface corresponds to condition (15); neering Science, Russian Academy of Sciences, Lenin- the upper one, to condition (16). It is seen that, at the grad, 1989). given values of the tensile strength, there is a significant 4. V. G. Volkov, K. G. Gaœnulin, S. L. Nedoseev, et al., Vopr. gap between the surfaces in the whole interval of the At. Nauki Tekh., Ser. Termoyad. Sintez 1 (7), 36 (1981). pulse parameters. When the pulse parameters fall into 5. N. E. Andreev, M. E. Veœsman, V. V. Kostin, and this gap, the scabbing or the retention of the integrity of V. E. Fortov, Teplofiz. Vys. Temp. 34 (3), 379 (1996).

TECHNICAL PHYSICS Vol. 46 No. 2 2001 248 ANDREEVA, KOLGATIN

6. K. S. Dyabilin, M. E. Ledebev, and V. P. Smirnov, ena (Nauka, Moscow, 1966; Academic, New York, Teploﬁz. Vys. Temp. 34 (3), 479 (1996). 1966, 1967), Vols. 1, 2. 10. I. M. Bespalov and A. Ya. Polishchuk, Preprint No. 1- 7. T. A. Andreeva, S. N. Kolgatin, and K. A. Khishchenko, 257, IVTAN (Institute of High Temperatures, USSR Zh. Tekh. Fiz. 68 (5), 44 (1998) [Tech. Phys. 43, 518 Academy of Sciences, 1988). (1998)]. 11. A. A. Samarskiœ and Yu. P. Popov, Difference Methods 8. A. V. Bushman, G. I. Kanel’, A. L. Ni, and V. E. Fortov, for Solution of Gas Dynamics Problems (Nauka, Mos- Thermal Physics and Dynamics of Severe Pulse Actions cow, 1992). (Inst. Khim. Fiz. Akad. Nauk SSSR, Chernogolovka, 12. N. A. Zlatin and G. S. Pugachev, Zh. Tekh. Fiz. 43 (9), 1988). 1961 (1973) [Sov. Phys. Tech. Phys. 18, 1235 (1973)]. 9. Ya. B. Zel’dovich and Yu. P. Raizer, in Physics of Shock Waves and High-Temperature Hydrodynamic Phenom- Translated by S. Gorin

TECHNICAL PHYSICS Vol. 46 No. 2 2001

EXPERIMENTAL INSTRUMENTS AND TECHNIQUES

Effect of Fullerene-Containing Additives on the Bearing Capacity of Fluoroplastics under Friction B. M. Ginzburg and D. G. Tochil’nikov Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, 199178 Russia E-mail: [email protected] Received June 15, 2000

Abstract—A modified procedure for determining the bearing capacity or maximum admissible pressure on a friction unit has been developed. The new feature of the procedure is a test of the wear capacity of a material as a function of pressure. The procedure has been tested on fluoroplastics doped with fullerene-containing soot lubricated with water. F-4 fluoroplastic with 1% fullerene-containing soot added to it shows a 30% higher bearing capacity. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION stant speed (Fig. 1). In the specimen, a crater forms of length U that increases with time. The crater depth is The subject of this work, which is the bearing capac- always much less than the disk diameter. Initially, the ity of modified fluoroplastics under friction, happens to specimen is loaded with a small load F = q. In the be at the junction of two branches of science—the beginning, the counterfaces are nonconformal, their physics of strength and polymer materials science. The contact area is extremely small, and the unit pressure at bearing capacity of a material under friction is, in the contact is very high. As the crater deepens, the pres- essence, a test of its strength under more complicated sure drops rapidly and the crater length asymptotically test conditions than simple stretching, compression, approaches a constant value (Fig. 1a). As soon as the bending, and so on. It is, in fact, a shear deformation measured crater length stops increasing (the so-called accompanied by contact interactions, often in the pres- stable crater), a load F = 2q is instantly applied to the ence of a lubricant (producing a wedging action around friction unit. The crater length first increases rapidly, microcracks or the Rebinder effect), which does not then the process slows down, and the crater length affect the main features of the phenomenon. asymptotically approaches a new stable value. At the In the pioneering work of Khrushchov [1] a proce- moment when the higher load is applied, the pressure P dure has been developed for determining the maximum jumps up and then decreases (as the contact area grows admissible pressure Pmax on a material, which is run-in due to attrition), finally settling at a higher level. as one of the counterfaces in a friction unit. But the actual quantity P defined in this procedure is the In a similar way, loads F = 3q, 4q, and so on are max applied to the friction unit, and the above process is strength characteristic of the material, which, by anal- repeated until, starting with a load q = nq, the ulti- ogy with the bearing capacity of a friction unit (defined max as the maximum admissible pressure on a friction unit, mate pressure on the stabilized crater no longer exceeding which may cause scouring), we call the bear- increases at this and still higher loads. Furthermore, the ing capacity of a material. Note that there is one more pressure seldom stays at the level reached; more often, closely related performance characteristic: the maxi- it drops with every new load despite further increases in mum admissible operating pressure of a friction unit the rate of attrition and the crater length. Pw . This parameter is usually chosen to be lower than In Fig. 1b, a diagram is given of the crater lengths max and pressures in U, P coordinates. To a first approxima- the bearing capacity of a friction unit by some arbitrary tion, the crater area is equal to the area of its projection quantity so that the wear of a friction unit is small or on the specimen plane; therefore, F ≈ PUb (where b is w close to zero at pressures of P < Pmax . the crater width). Lines F = const in U, P coordinates are hyperbolas (point 1 lies on hyperbola F = q, The purpose of this work was to modify the proce- points 1', 2 on hyperbola F = 2q, and so on). The line dure for determining the bearing capacity of materials linking points 1, 2, 3, etc., is the locus of points corre- proposed in [1] to adapt it to new promising materials sponding to the lengths of stable craters. Area A can be such as fullerene-containing fluoroplastics. defined as corresponding to friction with low wear; The procedure for determining Pmax proposed in [1] area B, to friction with imperfect lubrication (boundary is based on the stepwise loading of a planar specimen friction); and area C, to pressures unattainable for the pressed against the periphery of a disk rotating at a con- given friction unit.

250 GINZBURG, TOCHIL’NIKOV

ω (a) (b) U U 6 6' U P 5 5' 5 6 A 4 4 4' 3 6' 3 3' 2 4' 5' 2 1 3' 2' 2' 1 1' 1' 4 5 6 3 2 B C 1 Pmax Pmax

t P

Fig. 1. Schemes of displaying the data used in [1]: (a) presentation of the dependence of the crater length U and pressure P on test duration (or the number of disk rotations); (b) same data in UÐP coordinates; in the inset: schematic of the friction unit.

U, mm 100 Pb Cd 80 Sn Al

60 Zn 40

20 Cu

0 100 200 300 400 500 600 700 P, MPa Fig. 2. Test results for six commercial purity metals in [1].

Figure 2 shows the diagrams in U, P coordinates for surface. According to this hypothesis [3], any structural six commercial purity metals (lead, tin, cadmium, zinc, changes in a polycrystalline specimen which hinder the and copper) lubricated with kerosene [1]. The UÐP dia- migration of dislocations and the process of shear gram (magnitude of Pmax, slope of the initial part of the deformation of crystallites should cause an increase in U(P) curve, the curve behavior at F > qmax, and so on) Pmax. This approach makes the lack of correlation proved to be typical for a given friction unit, especially between Pmax and the Brinell hardness in metals com- of a metalÐlubricant combination. It has been found prehensible [1]. that the Pmax values do not correlate with the Brinell This experimental technique has a number of draw- hardness. backs. The wear particles sinking to the crater bottom The existence of Pmax has as yet no physical expla- can be abrasive; as the crater becomes deeper, the fric- nation. An assumption made in [1] is that the surface tion of the lateral surfaces against the crater walls temperature of the specimen can rise up to the metal becomes more signiﬁcant, and building the UÐP dia- melting point (or the temperature which destroys some gram is quite laborious. In this study, an attempt has protective layer at the surface) for a short time after the been made to eliminate the above drawbacks of the application of a load in excess of qmax. The local tem- technique and apply the modiﬁed technique to some perature rise was discussed in a number of works and new and promising materials. The basic approach is the has been observed experimentally [2]. same as in [1]. However, our opinion, as well as that of the author of [1], is that such a large temperature rise under abun- EXPERIMENTAL TECHNIQUE dant lubrication is unlikely to occur. In our studies, a hypothesis has been propounded that shear deforma- A friction shaft-plane unit (Fig. 1) was used in the tion can lower the melting point of crystal grains at the tests. This arrangement provides an easy escape for the

TECHNICAL PHYSICS Vol. 46 No. 2 2001

EFFECT OF FULLERENE-CONTAINING ADDITIVES 251 wear particles from the tribological contact zone; no FN lateral friction problems arise as the wear groove width a (which is an analog of the crater length in [1]) increases in size. The tests were carried out in a standard 2070 SMT-1 roller friction machine. The specimen tested was a rectangular plate 10 mm thick clamped in a specially designed holder. The plate was approached from below by a roller of diameter D = 46 mm and thick- 1 ness 16 mm made of wear-resistant chromium-nickel- 2 molybdenum steel of grade 18Kh2NChMA (GOST a 3 4543-71) rotating at ω = 400 minÐ1, which corresponds to a linear sliding velocity of 1 m/s. The roller width ω was always larger than the specimen width b. The roller was submerged 2 mm into a 200 ml water bath. Lubri- 4 cation with water prevented the surface temperature from rising; even at the heaviest loads, the water temperature in the bath did not exceed 50°C, which is too low to cause significant changes of the tribological characteristics of the specimens studied. The tests were carried out for two experimental set- Fig. 3. Schematic of the friction unit test: (1) specimen; (2) specimen holder; (3) rotating steel disk; (4) water bath; ups. In setup I (basic), the specimen and the roller were a is the width of the wear groove. in contact along a line. The initial load F was 100 N. The friction torque M was continually monitored. Every 300 s, the test was interrupted, the groove width wear intensities. Details of the processing of measure- a was measured under an optical microscope, and then ment data and calculations of the tribological character- the specimen was loaded with a higher load. It was istics have been given in earlier works [3Ð5]. determined beforehand that 300 s is enough time for the groove width and the pressure at the contact to reach a quasi-stable value. The load was increased to a maxi- MATERIALS AND PREPARATION mum of 1600 N in increments of 200 N. From the mea- OF SPECIMENS sured wear, the groove width nominal pressures in the Materials for the tribological tests were grade F-4 beginning (Ps) and end (Pe) of the loading step were polytetrafluoroethylene (GOST 10007-80) and an determined. As distinct from work [1], a Pe(F) depen- F-4K20 composite (TU6-05-1412-76), which is a poly- dence was built using the obtained data. The load value tetrafluoroethylene containing 20 wt % of coal coke. beyond which the quantity Pe stopped to grow was All tested materials also contained 1 wt % fullerene assumed to be the maximum admissible load and the soot introduced under the same conditions as the coal corresponding Pe the maximum admissible pressure. coke. In tests with setup II, relatively small area grooves The fullerene soot was produced in the plasma of an of various widths were first abraded to determine the electric arc [6]. About 8% of a fullerene mixture linear wear rate I at different initial loads. Note that I h h (mainly C60 and C70 in the ratio ~75/25) was extracted depends not only on pressure, but also on the initial from the soot using toluene as a solvent. The soot intro- load. Therefore, for the data comparisons to be correct, duced into the fluoroplastics was that which remained we carried out tests for the same load F = 1600 N, after extraction, and it still contained a considerable which is the highest of the loads in the range investi- amount of higher fullerenes (~12%) [7]. About 1% of gated. As will be seen below, in tests with setup II, an fullerene soot was introduced in the material under additional criterion, the closeness of Ih to zero, could be study under the same technical conditions as when applied for estimating the maximum admissible pres- introducing coal coke. sures, which is especially important in cases where the quantity Pe could not be determined unambiguously RESULTS AND DISCUSSION from Pe(F) data. In every version of the tests (kind of setup, fluoro- In tests using setup I, it was found that following plastic specimen, load and pressure magnitudes), three every increase in the load F, the rate of wear processes to eight experiments were carried out. The resulting taking place in the tribological contact rapidly declines characteristics represent arithmetic means of the data and stabilizes. As a result, the wear groove width a for the particular version of the test. Relative values of (analog of the crater length in [1]) and depth h initially the root-mean-square error for the arithmetic means of increase drastically and afterwards, as the contact test tribological characteristics are ~5% for nominal pres- goes on, change only slightly. The wear rate of the tri- sures P, ~6% for friction coefficients, and ~10% for the bological contact is determined by the pressure, which

TECHNICAL PHYSICS Vol. 46 No. 2 2001 252 GINZBURG, TOCHIL’NIKOV

–8 Pe, MPa Ih, 10 16 250 1 4 14 2 200 3 12 150 3 1 2 4 10 100

8 50

6 0 0 400 800 1200 1600 F, N 0 20 40 60 80 100 P, MPa Fig. 4. Dependence of the pressure Pe at the end of test step on the load F applied to the specimen. (1) F-4 fluoroplastic; (2) F-4 + 1% fullerene soot (F-4SZh1); (3) fluoroplastic Fig. 5. Dependence of the linear wear rate Ih on pressure composite F-4K20; (4) F-4K20 + 1% fullerene soot in the contact: (1) F-4; (2) F-4SZh1; (3) F-4K20; (F-4K20SZh1). (4) F-4K20SZh1. depends on the contact area. Alongside the increase and located. According to the other, the fullerenes penetrate subsequent stabilization of the wear groove depth and the crystallites, which induces high concentrations of the tribological contact area, a rapid decrease and lev- dislocations in them, thus impeding shear deformations eling off of the pressure in the tribological contact of the crystallites and of the material as a whole. Which occurs. Thus, the growth of the tribological contact area of the two mechanisms is applicable can be determined and lowering of pressure are interrelated; their rates and using different structural methods, and such an attempt ultimate values depend on the load on the contact and will be undertaken in our next study. In a recent report the wear groove width, the latter being, in turn, the [8], a simple technique for identifying the easiest slip function of the wear resistance of the material. planes in polymer crystallites was proposed. This technique can be used for estimating the effect of fullerenes In Fig. 4, Pe(F) curves for the fluoroplastic specimens are shown. It is seen that P initially rises as F is on the susceptibility of polymer crystallites to shear e deformation. increased and that, at F values of 800Ð1000 N, Pe levels off at different levels. Introduction into F-4K20 of the fullerene soot strengthens this material still further: the admissible One exception is the starting F-4 fluoroplastic, for pressure rises to 16 MPa. which two stable pressure levels have been observed: In Fig. 5 the results of the tests for setup II are Pe = 8 MPa at F = 400 N and Pe = 11 MPa at F = 1400 N. But, as shown below, low wear is observed at shown. The introduction of 1% fullerene soot resulted in a significant reduction of the linear wear rate for the pressures of Pe < 11 MPa, so this value should be taken as the maximum admissible value. same pressures: by a factor of 2Ð3 in F-4 (specimen F-4SZh1) and a factor of 1.3Ð1.7 in F-4K20 (specimen For the F-4K20 fluoroplastic containing 20% car- F-4K20SZh1). In this figure, it is seen that at P < Pmax bon coke, the admissible pressure is higher (~14 MPa), the quantity I approaches zero. Using the data in possibly due to significant transformation of the defor- h Fig. 5, an unambiguous choice of Pe value for the F-4 mation mechanism of the polytetrafluoroethylene crys- fluoroplastic could be made. Note that, in contrast to tallites. The presence of numerous defects due to the metals, pressures in excess of P are quite feasible, coke impurity makes the dislocation mechanism of max shear less effective. but the wear rates will be higher. Therefore, operating pressures for a material can be only those below Pmax. It is remarkable that the introduction of just 1% soot produces an even slightly greater rise of the admissible pressure to Pe = 15 MPa. The cause, in our opinion, is CONCLUSIONS the high chemical activity of fullerenes, which is partic- To conclude, a modified procedure for determining ularly seen in the formation of a fullerene-polymer net- the bearing capacity of a material in a friction unit has work on the friction surfaces [5]. been proposed. In addition to the dependence of pres- At least two mechanisms of the growth of Pe involv- sure on the applied load, this procedure yields data on ing fullerenes can be suggested. According to one, the linear wear rate as a function of pressure. The range fullerenes penetrate only amorphous regions of the of admissible pressures for a material or its bearing amorphous-crystalline network where its points are capacity is defined as a range in which the pressure is

TECHNICAL PHYSICS Vol. 46 No. 2 2001 EFFECT OF FULLERENE-CONTAINING ADDITIVES 253 not rising (with increasing load) and the wear is not 4. Yu. P. Kozyrev, B. M. Ginzburg, N. D. Priemskii, et al., high. Wear 171, 71 (1994). 5. B. M. Ginzburg, M. V. Baœdakova, O. F. Kireenko, et al., Zh. Tekh. Fiz. 70 (12), 87 (2000) [Tech. Phys. 45, 1595 REFERENCES (2000)]. 1. M. M. Khrushchev, Run-In of Bearing Alloys and Trun- 6. W. Krätschmer and D. R. Huffman, Philos. Trans. R. nions (Akad. Nauk SSSR, Moscow, 1946). Soc. London, Ser. A 343 (1667), 33 (1993). 7. F. Beer, A. Gügel, K. Martin, et al., J. Mater. Chem. 7 2. Contact Interaction of Solid States and Calculation of (8), 1327 (1997). Friction and Deterioration Forces: Collection of Arti- cles, Ed. by A. Yu. Ishlinskiœ and N. B. Demkin (Nauka, 8. B. M. Ginzburg and N. Sultanov, Zh. Tekh. Fiz. 71 (2), Moscow, 1971). 129 (2001) [Tech. Phys. 46, 258 (2001)]. 3. B. M. Ginzburg, Pis’ma Zh. Tekh. Fiz. 26 (8), 65 (2000) [Tech. Phys. Lett. 26, 345 (2000)]. Translated by B. Kalinin

TECHNICAL PHYSICS Vol. 46 No. 2 2001

Technical Physics, Vol. 46, No. 2, 2001, pp. 254Ð257. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 71, No. 2, 2001, pp. 125Ð128. Original Russian Text Copyright © 2001 by Babichev, Kozlovskiœ, Romanov, Sharan.

EXPERIMENTAL INSTRUMENTS AND TECHNIQUES

Parameter Optimization of a Silicon Pressure-Sensitive Element Based on Dual-Drain MIS Transistors G. G. Babichev, S. I. Kozlovskiœ, V. A. Romanov, and N. N. Sharan Institute of Semiconductor Physics, National Academy of Sciences of Ukraine, Kiev, 252650 Ukraine Received April 18, 2000

Abstract—The object of investigation was an integrated pressure transducer with a differential pressure-sensitive element based on two parallel-connected p-channel dual-drain MIS transistors. The transistors were made on the planar side of a silicon membrane with the rigid central region near the edges of membrane’s thin part (valley). The optimum design of the transducer and its basic characteristics were determined. © 2001 MAIK “Nauka/Interperiodica”.

The progressively growing demand for semiconduc- same, since their layout and geometries are identical; tor pressure transducers [1] stimulates a search for new hence, the output signal Uout (Fig. 2) is zero provided types of high-sensitivity elements that combine low that load resistances are equal: RD1 = RD2 = RD. power consumption and low intrinsic noise. In the previous work [2], we studied the sensitive elements based When pressure is applied, the membrane converts on silicon ﬁeld-effect dual-drain pressure-sensitive the load distributed over the surface to uniaxial elastic transistors subjected to uniform uniaxial elastic tensile (compressive) strain in the regions where the µ mechanical strain. transistors are located. Because of this, the mobility p In this work, we consider a more general case when the strain varies linearly with the MIS transistor (MIST) channel length. Such a distribution of elastic 4 (100) strain occurs in the valley of a specially shaped rigid- center membrane upon membrane loading. It has been 3 shown [3, 4] that this membrane, combined with a dif- 1 ferential sensitive element, decreases the effect of “par- 2 asitic” mechanical stresses due to transducer packag- ing. Moreover, it also provides a linear output characteristic of high-sensitivity thin-membrane transducers, in which the bending of the membrane upon loading is comparable to its thickness. In what follows, we will optimize the design of the [011] n-Si pressure transducer with a differential sensitive ele- - ment built around two parallel-connected p-channel [001] Ly/2 dual-drain MISTs. The schematics of the silicon mem- y 0 brane and the crystallographic orientation of the tran- –Ly/2 sistor channels are given in Fig. 1. The pressure-sensi- Σ S tive transistors are placed on the planar side of the (x) p-Si membrane near the edges of its thin part (valley). When [011] the membrane is loaded uniformly and the transistors D2 are placed symmetrically about the plane passing W through the center of the valley so that it is parallel to D1 the edges of the valley and perpendicular to its plane, elastic mechanical stresses in the vicinity of the devices x [010] are equal in magnitude and opposite in sign [3, 4]. Figure 2 shows the layout and the connection dia- Fig. 1. Arrangement of the sensitive element on the mem- gram of the sensitive element. With appropriate bias brane and the crystallographic orientation of the transistor channels: 1, dual-drain MIS strain-sensitive transistors (S, voltages (U0, UG) applied and in the absence of pres- source; D1 and D2, drains); 2, rigid central island; 3, valley; sure, the drain currents of transistors T1 and T2 are the and 4, membrane support.

PARAMETER OPTIMIZATION OF A SILICON PRESSURE-SENSITIVE ELEMENT 255 of majority carriers (holes) becomes anisotropic and with the boundary conditions additional strain-induced transverse and longitudinal L ∂ϕ ∂ϕ electric fields arise in the FET channel [2]. They cause y = ±-----y, a(x)------+ ------= 0, (7) current disbalance in the drain circuits, as a result of 2 ∂x ∂y which the output current appears. x = 0, L ϕ(0, y) = 0, ϕ(L , y) = U . (8) Below, we calculate a strain-induced potential x x DS change in the MIST channel and the relative, SR, and Equations (6)Ð(8) are solved by the Fourier method: absolute, SA, sensitivities of the transducer (SR is also ϕ( , ) called the conversion efficiency). x y = UC Ð UG Let the MIST channel be bounded by the regions (9) 2  U  U x Ð (U Ð U ) Ð 2 U Ð U Ð -----D----S -----D---S---- + Φ(x, y), ≤ ≤ ≤ ≤ ≤ ≤ C G  C G  0 x Lx, ÐLy/2 y Ly/2, 0 z Lz. (1) 2 Lx The potential will be calculated under the following where assumptions and simplifications. 2a 2 (i) The conductivity in the xy plane is anisotropic Φ( , ) 0  UDS  x y = ---π-----UDS Ð ----(------)- due to uniaxial elastic mechanical stress Σ aligned with 2 UC Ð UG the [110] direction (Fig. 1). The Σ(x) dependence has ∞ (10) , , , π π the form [3, 4] × Fn(k UC UG UDS)  ny  nx ∑ ------(--π------)--sinh------ sin------ , Σ( ) Σ ( ) ncosh nLy/2Lx Lx Lx x = 0 k0 + k1 x/Lx , (2) n = 1 | | | | ≤ Σ ( , , , ) where k0 , k1 1 and k0 0 is the mechanical stress at Fn k UC UG UDS the point x = 0. 1 (k + k z)sin(πnz) (11) (ii) Elastic mechanical stresses and the anisotropy = ∫------0------1------dz. parameter [2] [ ( ) ] ( )2 0 1 Ð 2 UC Ð UG Ð UDS UDSz/ UC Ð UG ( ) Π Σ ( ) ( ) a x = 44 0 k0 + k1 x/Lx /2 = a0 k0 + k1 x/Lx (3) Figure 3 depicts the distribution of the transverse Π potential difference ∆ϕ(x) along the channel when the ( 44 is the piezoresistive coefficient of shear for p-Si) are sufficiently low. Hence, the inequality max|a(x)| Ӷ 1 is anisotropy parameter depends in various ways on the coordinate x: valid (in calculations, a0 will be set equal to 0.06) and the conductivity tensor components can be written as ∆ϕ( ) ϕ( , ) ϕ( , ) x = x Ly/2 Ð z ÐLy/2 (12) σ σ ≅ σ( ) σ σ σ( ) Φ( , ) Φ( , ) xx = yy x , xy = yx = x ax. (4) = x Ly/2 Ð x ÐLy/2 .

(iii) The transistor channel is relatively thin (Lx, Ly The form of the anisotropy parameter is seen to con- ӷ Lz), and its conductivity in the smooth approximation siderably affect the longitudinal distribution of ∆ϕ(x). [5, 6] is given by Note that a problem similar to Eqs. (6)Ð(8) was numerically solved for the case when conductivity anisotropy σ(x) µ C 2  U  U x (5) T = ----p----- (U Ð U ) Ð 2 U Ð U Ð -----D----S -----D---S----, 2 C G  C G  T1 Lz 2 Lx where C is the capacity per unit area of the gateÐsemi- conductor structure, UDS is the potential difference between the source and the drain, U is the cutoff volt- C Pressure age (gate voltage at which the channel conductivity + vanishes), and UG is the gate voltage. We assume that U0 UDS < UC Ð UG (UDS, UC, UG > 0) and the transistor is in the on state. RD1 RD2 In view of the above simpliﬁcations, the potential distribution ϕ(x, y) in the channel can be considered as Uout quasi-planar; then, in the ﬁrst approximation with T1 T respect to the anisotropy parameter, the spatial distribu- 2 tion of the potential is expressed as U G – ∂2ϕ ∂2ϕ ∂ln(σ(x))∂ϕ ------+ ------+ ------= 0 (6) 2 2 ∂x ∂x ∂x ∂y Fig. 2. Layout and connection diagram of the transistors.

TECHNICAL PHYSICS Vol. 46 No. 2 2001 256 BABICHEV et al.

∆ϕ, V was induced by an external magnetic field. The ∆ϕ(x) ≠ 0.10 curve for uniform strain (k0 0, k1 = 0) agrees well with the numerically calculated distribution of Hall voltage 0.08 in the channel of an MIS magnetotransistor [7]. 1 Now let us calculate the conversion efficiency SR and the absolute sensitivity S of the pressure trans- 0.06 A ducer. Let the drains D1 and D2 be separated by a gap 2 W (Fig. 1) and have the same width LD = (Ly Ð W)/2. The 0.04 values of SA and SR are found by integration [2]: 3 ÐW/2 0.02 ∂Φ S = 2Cµ R (U Ð U Ð U ) ------(L , x)dy,(13) A p D C G DS ∫ ∂x x ÐLy/2 0 0.2 0.4 0.6 0.8 1.0 ÐW/2 x/Lx (U Ð U Ð U ) ∂Φ S = Cµ ------C------G------D---S------(L , y)dy. (14) Fig. 3. Distribution of the transverse potential difference R p 0 ∫ ∂x x 2ID along the channel. UDS = 3 V and UC Ð UG = 5 V. Coeffi- ÐLy/2 cients k0 and k1 in the expression for the anisotropy param- 0 0 0 eter a(x) = a0(k0 + k1x/Lx) are (1) k0 = 1, k1 = 0; (2) k0 = 0, Here, ID1 = ID2 = ID are the drain currents of transis- k1 = 1; and (3) k0 = k1 = 0.5. tors T1 and T2 in the absence of strain: –11 SA, 10 V/Pa 0 µ LD UDS ID = pC------UC Ð UG Ð ------ UDS. (15) Lx 2 9 1 Integrating Eq. (13) and (14) in view of Eq. (10) and 2 (11) yields Π ( ) 6 3 44 UC Ð UG Ð UDS Lx Ψ SR = ------(------)------, (16) 2 UC Ð UG LD ( ) 3 Π 0 UC Ð UG Ð UDS Lx Ψ SA = 2 44 ID RD------(------)------, (17) UC Ð UG LD where

0 0.5 1.0 1.5 2.0 2.5 3.0 ∞ n 2 (Ð1) F (k, U , U , U ) Ly/Lx Ψ = -- ∑ ------n------C------G------D---S--- π n n = 1 Fig. 4. SA as a function of the channel geometry. (1)Ð(3) the (18) same as in Fig. 3. (π ) × cosh nW/2Lx ------(---π------)- Ð 1 . –10 cosh nLy/2Lx SR, 10 The absolute and relative sensitivities of the strain- 5 sensitive transistor as functions of the channel geome- 1 try are shown in Figs. 4 and 5. The run of the SA vs. Ly /Lx curve is almost the same for the uniform and non- 4 uniform distributions of mechanical strain (Fig. 4): SA ≥ 2 monotonically increases and saturates at Ly /Lx 2. For k1 = 0, this is consistent with the similar analytical 3 dependence for an MIS magnetotransistor [7]. The 3 dependence of SR on Ly /Lx (Fig. 5) features a peak at ≠ Ly /Lx = 0.1Ð0.2 for the nonuniform case (k1 0). From 2 the curves represented in Figs. 4 and 5, one can ﬁnd the optimal value of Ly /Lx at which SR and SA are maximal.

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Figures 6 and 7 depict analytical dependences of SA L /L y x on the drainÐsource, UDS, and gateÐsource, UG, poten- µ Ð2 tial differences for CRD p = 10 . As UDS grows (Fig. 6), Fig. 5. SR as a function of the channel geometry. (1)Ð(3) The same as in Fig. 3. the absolute sensitivity linearly increases and peaks at

TECHNICAL PHYSICS Vol. 46 No. 2 2001 PARAMETER OPTIMIZATION OF A SILICON PRESSURE-SENSITIVE ELEMENT 257

–11 –11 SA, 10 V/Pa SA, 10 V/Pa 8 6 1 2 3 6 4 1 2 3 2 4

2 0 1 2 3 4 0 1 2 3 4 5 , V UDS UG, V

Fig. 6. SA vs. UDS for UC Ð UG = 5 V. Fig. 7. SA vs. UG for UC = 10 V and UDS = 3 V.

≈ UDS (UC Ð UG)/2. The run of curves 1Ð3 to a great twofold and provides a better linearity of the load char- 0 acteristic compared to the single-transistor design [2]. extent is deﬁned by the dependence ID (UDS) [expressions (15) and (17)]. With an increase in UG, SA drops practically linearly (Fig. 7, curves 1Ð3). As follows REFERENCES from formula (16), SR is virtually independent of UDS 1. J. Bryzek, Sens. Actuators A 56, 1 (1996). and UG. 2. G. G. Babichev, S. I. Kozlovskiœ, and N. N. Sharan, Zh. Tekh. Fiz. 70 (10), 45 (2000) [Tech. Phys. 45, 1276 To conclude, consider brieﬂy one more basic prop- (2000)]. erty of a pressure transducer: linearity of the load char- 3. A. Yasukawa, M. Shimazoe, and Y. Matsuoka, IEEE acteristic. It is known [3, 4, 8] that possible reasons for Trans. Electron Devices ED-36 (7), 1295 (1989). the nonlinearity of piezoresistive membrane transduc- 4. M. Nishihara, K. Yamada, and Y. Matsuoka, Hitachi Rev. ers are nonlinear conversion of pressure to elastic strain 30 (6), 285 (1981). by that part of the membrane where the sensitive ele- 5. I. M. Vikulin and V. I. Stafeev, Physics of Semiconductor ment is located and also nonlinearity of the piezoresis- Devices (Radio i Svyaz’, Moscow, 1990). tive effect. Experimental and theoretical studies [3, 4, 6. R. S. Muller and T. Kamins, Device Electronics for Inte- 8] suggest that the nonlinearity of the load characteris- grated Circuits (Wiley, New York, 1986; Mir, Moscow, tic due to the above reasons is greatly reduced by the 1989). differential design of the sensitive element, where one 7. P. W. Fry and S. F. Hoey, IEEE Trans. Electron Devices ED-16 (1), 35 (1969). of the transistors is under compression and the other, 8. K. Suzuki, T. Ishihara, M. Hirata, and H. Tanigawa, under stretching. IEEE Trans. Electron Devices ED-34 (6), 1360 (1987). Thus, the design of the pressure transducer proposed in this work increases the absolute sensitivity Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 46 No. 2 2001

EXPERIMENTAL INSTRUMENTS AND TECHNIQUES

X-ray Diffraction Determination of Easiest Slip Planes in AmorphousÐCrystalline Polymers B. M. Ginzburg* and N. Sultanov** *Institute of Problems in Machine Science, Russian Academy of Sciences, Vasil’evskiœ Ostrov, Bol’shoœ pr. 61, St. Petersburg, 199178 Russia **Tajik State University, ul. Rudaki 17, Dushanbe, 734000 Tajikistan e-mail: [email protected] Received May 15, 2000

Abstract—A simple X-ray diffraction technique for determining the easiest slip planes in amorphousÐcrystal- line polymer grains is suggested. The efﬁciency of the technique was demonstrated with polyethylene and polyamide-6. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION tallization processes to occur [4]. Macromolecules in Determination of the easiest slip planes in solids is the crystallites are then oriented largely along the draw- a complex, while well-established, procedure. Usually, ing direction. If the film is amorphized or does not crys- bulk single crystals are deformed and then examined tallize during the orientation process (for example, if with electron microscopy, as well as electron and X-ray orientation drawing is carried out at temperatures diffraction methods [1Ð4]. In the case of polymers, below the glass-transition temperature), it can be many small single crystals grown in dilute solutions are annealed. The annealing conditions should be selected settled out on plastic substrates that are subsequently in such a way as to preserve the preferential orientation deformed [4]. of the molecules along the drawing direction (e.g., annealing can be performed at a fixed length of the It would be logical to assume that the easiest slip specimen). The need for initially thick films may show planes observed in polymer single crystals are also up at this stage as well, since the film geometry must present in mosaic polycrystals; however, this statement prevent plane texturing of the crystallites. It is desirable is in doubt. Moreover, in polymer crystals, the easiest that the film texture be axial relative to the texture axis, slip planes usually coincide with planes of chain fold- which coincides with the direction of macromolecules ing, and the direction of the latter depends on growth in the crystallites. When an X-ray beam is incident nor- conditions for both bulk single crystals and single-crys- mal to the drawing direction, a large-angle diffraction tal grains. At least, folding planes in the crystals and pattern for such films is similar to a rotating-crystal pat- grains may differ. Therefore, determination of slip tern and exhibits a set of curved reflections on the layer planes (and the easiest slip planes, in particular) in lines (primarily on the zero layer line, or equator). The polycrystalline polymer materials seems to be a topical equatorial reflections are those from crystallographic problem, since this is necessary for a better understand- planes parallel to the axes of macromolecules. It is just ing of deformation mechanisms in amorphousÐcrystal- along these planes that share strain in polymer crystals line polymers. takes place [4]. In this work, we present a simple technique for finding the easiest slip planes in amorphousÐcrystalline Once the highly oriented amorphousÐcrystalline polymers. film has been prepared, the next orientation drawing (reorientation) is carried out at a large angle (from 20° to 70°; 45° seems to be the best) to the direction of the EXPERIMENTAL first drawing. To make this procedure easier, a piece is First, a sufficiently thick highly oriented amor- cut out of the initial specimen in such a way that its long phous–crystalline film of a crystallizable polymer is sides make the given angle with the first drawing direc- formed by any one of the preparation procedures. Its tion (Fig. 1a). thickness, dictated by transmission X-ray diffraction During the second drawing, diffraction patterns are analysis, usually ranges from 0.1 to 2 mm. Thick films taken from strained specimens from time to time. The are structurally more uniform; in addition, substrate X-ray beam remains perpendicular to the specimen all effects are avoided in this case. Highly oriented crystal- the time. A typical change in the large-angle diffraction lites are prepared, for example, by orientation drawing pattern is demonstrated in Figs. 1b and 1c. The diffrac- at temperatures sufficient for crystallization or recrys- tion pattern turns toward the direction of the second

X-RAY DIFFRACTION DETERMINATION 259

(a) (b) (c) W1 W1 2 2 1 α α 1

W2 W2

Fig. 1. (a) Specimen preparation: W , direction of the first orientation drawing of the polymer film at a temperature T ; W , direction 1 α 1 2 of the second orientation drawing at T2; and , angle between these directions. (b) Large-angle X-ray diffraction pattern for the initial specimen (after the first drawing); 1 and 2 are equatorial reflections. (c) Large-angle X-ray diffraction pattern for the specimen stretched in the direction W2; the reflections are turned toward this direction, reflection 2 from the easiest slip planes being ahead of reflection 1. drawing, but the equatorial reflections from the easiest Figure 2 shows a typical large-angle diffraction pat- slip planes are ahead of those from other planes: the tern obtained under recoverable strain (~50%) in the α ° former are rotated through a larger angle toward the experiments with T1/T2 = 85/20 for = 75 . The second drawing axis. Note that such a criterion for 200 reflection is an angle Ψ ahead of the 110 reflection determination of the easiest slip planes is valid not only when they rotate toward the vertical direction of a new for reversible but also for low plastic strains. equator. Like 110, other equatorial reflections, hk0 and In addition, slip, or shear strain, inside the crystal- 0k0, are also left behind. The same is true for the lites may lead to radial broadening of reflections from 100 reflection of the monoclinic modification of the the slip planes; this effect is, however, noticeable only polyethylene (the closest to the center of the pattern). for intense reflections. Moreover, it shows up only The last-named reflection is likely to appear because of when reorientation occurs in a system of highly ori- martensitic transformation. The azimuth angle of ented crystallites where the macromolecule axis is aligned with the texture axis. If these axes do not coincide or the angle of reorientation is close to right angle, shear strain of the crystallites may cause the reverse effect: radial narrowing of the reflections [5].

RESULTS AND DISCUSSION

Polyethylene. First, we applied our technique to W2 low-density polyethylene. Grains of commercial polyethylene of molecular weight M = 25 × 103 were used to mold 1.4-mm-thick films. The operating temperature and pressure were 200°C and 5 MPa, respectively. The films were quenched in room-temperature water. Then, 110 they underwent uniaxial stretching to an elongation by ° 200 a factor of 5 (first orientation) at T1 = 85 C and to an ° elongation by a factor of 9 at T1 = 20 C. Next, specimens cut out of the films at various angles α to the first orientation direction (Fig. 1a) were subjected to step ψ strain (second orientation) at temperatures T2 = 20 and 85°C in a special chamber inside the X-ray chamber. After each step, the stressed specimens were held for 12 h and large-angle diffraction patterns were recorded Fig. 2. Large-angle X-ray diffraction pattern for the polyeth- ° on a flat X-ray film. The X-ray beam was directed nor- ylene specimen after the first drawing at T1 = 85 C and after uniaxial elastic stretching (~50%) at an angle α = 75° to the mally to the X-ray film plane. Ni-filtered CuKα radia- ° first drawing direction for T2 = 20 C. At the center, the asso- tion was used. ciated low-angle pattern is shown.

TECHNICAL PHYSICS Vol. 46 No. 2 2001

260 GINZBURG, SULTANOV elastic strain, this broadening is known to reversibly change [6]. It is associated with shear strain of the crystallites due to intracrystallite slipping. The sign of the shear strain is the distorted low-angle diffraction pattern at the center of Fig. 2. Similar results were obtained for the other temperature conditions of the first and second drawings: T1/T2 = 20/20 and 85/85. In Fig. 3, a large-angle pattern for the 20/20 series is depicted; the strain of the specimen, 50%, is not fully recoverable: after load removal, the W2 residual strain was ~15%. However, in this case, too, the 200 reflection leads and the angle of advance Ψ is small. The low-angle diffraction pattern at the center of Fig. 3 is an indication of the shear strain in the crystallites. In polyethylene, the (110) plane is the most closely packed. Usually, it is also the easiest slip plane. In our 110 case, however, the easiest slip plane is (200), second in 200 packing density [7]. It appears that here the (200) plane is the folding plane, which in many ways explains the results observed. Ψ Thus, in highly oriented mosaic amorphousÐcrystal- line polyethylene, at least two slip systems, (200)[001] and (110)[001], are present, the former being the system of easiest slip. This is consistent with the literature Fig. 3. The same as in Fig. 2 for the specimen drawn at T = data [7]. ° ° α ° 1 20 C and stretched to 50% at T2 = 20 C for = 45 . The residual strain after load removal is 15%. For right-angle reorientation (α = 90°), one might expect the (200)[020] slip system; however, this system was not revealed. The crystallites first were at short- advance Ψ is small, no more than several degrees. This term unstable equilibrium and then were separated into means that slipping along other paratropic planes pro- two sets with the (200)[001] and (110)[001] slip sys- ceeds with almost the same easiness; specifically, it tems. obviously proceeds along the (110) planes. Polyamide-6. In polyamide-6, the macromolecule The (110) reflection is the most intense; therefore, axes are aligned with the b axis. Intermolecular interac- its radial broadening is seen immediately from the dif- tion in polyamide-6 is known to strongly depend on the fraction patterns without any measurements. Under crystallographic direction because of hydrogen bonds

(a) (b) (c) 002 002 W1 002 200 200 W 2 W2

200

Fig. 4. Large-angle X-ray diffraction pattern for the oriented polyamide-6 ﬁlm annealed at 200°C and then uniaxially stretched (sec- α ° ° ond drawing) at an angle = 45 to the initial orientation at T2 = 20 C; the second drawing was performed in the horizontal direction. The strain ε = (a) 0, (b) 22% (fully recoverable strain), and (c) 300% (irreversible strain to prerupture with necking).

TECHNICAL PHYSICS Vol. 46 No. 2 2001

X-RAY DIFFRACTION DETERMINATION 261 formed in the (002) planes. Slipping transverse to these between by intermolecular interaction force. It is there- planes is therefore greatly hindered in polyamide-6. fore hoped that our technique for determination of eas- Let us see how this well-known situation is reflected iest slip planes in mosaic amorphousÐcrystalline poly- in experiments using our approach. Polyamide-6 spec- mers will be applied to studying other crystallizable imens were prepared by annealing (2 h, 200°C) ori- polymers. ented commercial PK-4 films of fixed length. Further procedures were the same as with the polyethylene. REFERENCES Figure 4 demonstrates large-angle diffraction patterns obtained after reorientation under room condi- 1. E. Schmid and W. Boas, Kristallplastizät, mit beson- α ° derer berücksichtigung der Metalle (J. Springer, Berlin, tions for = 45 . When the strain was recoverable 1935; Gos. Ob”edinennoe Nauchno-Tekhnicheskoe Izd. (~22%), the angles of rotation for the equatorial reflec- NKTP SSSR, Moscow, 1938). tions drastically differed: the 002 reflections turned ° 2. R. Berner and H. Kronmüller, Plastische Verformiung through nearly 30 , while the positions of the von Einkristallen, in Moderne Probleme der Metal- 200 reflections remained almost unchanged. Hence, the physik, Ed. by A. Seeger (Springer-Verlag, Berlin, 1965; easiest-slip plane is (002). Mir, Moscow, 1969). Two points seem to be noteworthy in our opinion. 3. B. I. Smirnov, Dislocation Structure and Hardening of First, note that the 200 reflections turn insignificantly Crystals (Nauka, Leningrad, 1981). even at a subsequent plastic strain of 300%, when the 4. B. Wunderlich, Macromolecular Physics, Vol. 1: Crystal 002 reflections are set on a new equator. Second, the Structure, Morphology, Defects (Academic, New York, 002 reflections markedly broaden in the radial direc- 1973; Mir, Moscow, 1976). tion. In the case of plastic strain, this may be associated 5. B. M. Ginzburg, V. N. Baranov, V. I. Gromov, and with melting of initial crystallites and the nucleation of K. B. Kurbanov, Mekh. Polim., No. 1, 3 (1974). new, smaller, ones; for elastic strain, this indicates the 6. B. M. Ginzburg, N. Sultanov, and D. Rashidov, J. Mac- presence of shear strain in the crystallites. romol. Sci., Phys. 9 (4), 609 (1974). Thus, our approach was tested on two polymers 7. V. A. Marikhin and L. P. Myasnikova, Supramolecular with extremely different intermolecular interactions. It Structure of Polymers (Khimiya, Leningrad, 1977). can be assumed that the vast majority of other crystallizable polymers are close to those studied or lie in Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 46 No. 2 2001

Technical Physics, Vol. 46, No. 2, 2001, pp. 262Ð265. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 71, No. 2, 2001, pp. 133Ð136. Original Russian Text Copyright © 2001 by Kravchuk, S. Novosad, I. Novosad.

BRIEF COMMUNICATIONS

Luminescence from Mn-Doped PbI2 Crystals I. M. Kravchuk*, S. S. Novosad**, and I. S. Novosad** * Lviv Technologies State University, Lviv, 79013 Ukraine ** Franko National University, Lviv, 79000 Ukraine Received January 21, 2000

Abstract—Luminescent properties of PbI2 and PbI2 : 0.5 mol % MnCl2 crystals under X-ray or N2-laser excitation are studied experimentally. The measurements are performed at temperatures ranging from 85 to 295 K. For PbI2 crystals under laser excitation, spectral bands with peaks near 495 and 512 nm, respectively, are observed at 85 K. With X-ray excitation at the same temperature, luminescence is observed in the 515- and 715-nm bands. The doping decreases the intensity in the 515-nm band, increases it for longer wavelengths, and shifts the highest peak to 700 nm. At 85 K, the doping has an insigniﬁcant effect on the excitation energy accumulated by trapped electrons. Certain PbI2 crystals also exhibit a peak in a region of 580Ð595 nm. This peak becomes much higher if the crystal is treated with an N2 laser at room temperature or if it is heated to 450Ð 485 K. As the measurement temperature rises from 85 to 295 K, luminescence intensity decreases considerably. With X-ray excitation at room temperature, the yield of PbI2 : Mn luminescence peaked at 660 nm for doped crystals is about three times larger than the yield peaked at 555 nm for nondoped crystals. The spectral curves and underlying radiative processes are discussed. © 2001 MAIK “Nauka/Interperiodica”.

The luminescence from PbI2 crystals excited by consists of bands at 515Ð520 and 715 nm. Some of the X-ray pulses was addressed in [1]. With temperature specimens also exhibited a peak in the region of 580Ð raised from 77 to 293 K, it was revealed that the emis- 595 nm under both types of excitation (see curve 1 in sion intensity decreases by a factor of ~25 and the spec- Fig. 1). This peak becomes much higher after the crys- tral peak shifts from 520 to 550 nm. A remarkable fea- tal (in the cryostat) is heated to 450Ð485 K or treated ture of the crystals is that the temperature rise has an with N2 laser radiation at room temperature. These insignificant effect on the pulse luminescence decay crystals retain their luminescent properties for about six time, which is 0.4 ns at most. The high density and months in storage at the laboratory. short luminescence decay time of PbI2 crystals suggest If the measurement temperature is raised from 85 to that they may be used as fast scintillators. Regarding 295 K, luminescence decreases considerably. With X- the activated PbI2 crystals, little attention has been ray excitation at room temperature, the spectral curve given to their luminescent properties under pulsed or has a low-intensity band centered at 555 nm and a wide stationary X-ray excitation. plateau between 640 and 780 nm (see curve 2 in Fig. 1). This experimental study examines the luminescence from PbI2 : Mn crystals under stationary excitation by I, arb. units X-rays or laser radiation. We used an LGI-21 nitrogen 1 laser operating at λ = 337.1 nm. Our aim was to explore the effect of Mn doping on the accumulation of excitation energy by trapped electrons, the specific light yield, and the luminescence spectrum for temperatures ranging from 85 to 295 K. The crystals were grown by the StockbargerÐBridg- man technique [2]. During their growth, the melt was doped with ~0.5 mol % MnCl2. The starting material (PbI2) was of analytical-grade purity and was further purified by oriented crystallization [2]. Luminescent properties of the crystals were studied according to the 2 technique described in [3]. ×0.2 First, let us consider the luminescence spectra of nondoped crystals. With laser excitation at 85 K, we 500 600 700 800 λ, nm detected an intense narrow band with a peak near 512 nm and a faint band centered at 495 nm. With X- Fig. 1. Luminescence spectra of a PbI2 crystal under X-ray ray excitation at the same temperature, the spectrum excitation for (1) 85 and (2) 295 K.

LUMINESCENCE FROM Mn-DOPED PbI2 CRYSTALS 263

Similar shapes have spectra obtained with laser excita- I, arb. units tion near the higher temperature limit. At 85 K, trapped electrons accumulate a small 1 amount of energy during laser or X-ray excitation. 2 When the temperature is raised to 295 K, nonelementary peaks were observed in the thermoluminescence curves at about 117 K. ×0.05 Now, let us proceed to Mn-doped PbI2 crystals. Consider the case of X-ray excitation. At 85 K, luminescence intensity was found to decrease by three times in the 515-nm band and to increase for longer wavelengths, with the highest peak shifted to 700 nm (curve 1 in Fig. 2). The plateau between 590 and 610 nm stems from a peak at 595 nm, which character- 500 600 700 800 λ, nm izes the thermally treated PbI2 crystals. A rise in temperature crucially reduces the peak at Fig. 2. Luminescence spectra of a PbI2 : Mn crystal under 700 nm. Specifically, for 150Ð295 K, its height is about X-ray excitation for (1) 85 and (2) 295 K. 5% of that for 85 K. At 295 K, the spectrum is characterized by a low-intensity wide nonelementary band I, arb. units centered at 660 nm (curve 2 in Fig. 2). At this tempera- 1 ture, the specific light yield is about three times larger than that of nondoped crystals. The energy accumulated under X-ray excitation is small again. Let us consider laser excitation. At 85 K, the spec- 2 trum has a low-intensity band centered at 700 nm as well as an intense narrow band at 515 nm (curve 1 in Fig. 3). If the temperature is raised to 295 K, the luminescence intensity is strongly quenched and the crystal emits at wavelengths longer than 550 nm (curve 2 in × Fig. 3). The falling segment of the spectral curve indi- 0.05 cates emission peaks at 660 and 700 nm. Figure 4 shows the luminescence intensity in the bands at 515 and 700 nm against temperature for the case of laser excitation. It is seen that the two curves differ markedly. As temperature increases from 85 to 180 K, the intensity in the 515-nm band decreases rapidly and monotonically (curve 1), whereas that in the 700-nm band increases slightly until 95 K is reached and then decreases monotonically (curve 2). Eventu- 500 600 700 800 λ, nm ally, intensity falls to a very low level in both cases. Thus, the above results demonstrate that doping Fig. 3. Luminescence spectra of a PbI2 : Mn crystal under laser excitation for (1) 85 and (2) 295 K. with Mn changes the PbI2 luminescence spectrum. For laser excitation, a new band arises. For X-ray excitation, the intensity is redistributed in favor of longer photons corresponding to band-to-band absorption, an wavelengths. At low temperatures, a similar effect was induced signal can be detected at 77 K. The signal rep- observed in crystals doped with CdI2. Based on [4Ð7], resents electron paramagnetic resonance (EPR) and we reason that interstitial Mn ions have a valency of 2. exhibits a line with a g-factor of 2.009. This EPR signal 2+ + No luminescence from Mn centers in the crystals was is ascribed to Pb centers in PbI2. They result from elec- observed near the lower temperature limit. This may tron trapping by irregular Pb2 ions located near stem from the fact that the excited states of doping ions extended defects or at sites where leadÐiodine bonds are in the conduction band or that the dopant forms no are broken. Another EPR signal, which has a compli- efficient recombination centers that could compete with cated nature and reflects hyperfine interaction, is asso- those of the matrix [8]. ciated with a hole trapped by two IÐ ions with the for- Some authors attribute the edge emission at 512 nm Ð Ð mation of an I2 molecular ion [10, 11]. to I2 centers in PbI2 [9, 10]. The low-intensity peak located near 495Ð500 nm is associated with the radia- In single crystals and polycrystals of PbI2, the tive relaxation of free excitons. If PbI2 is irradiated with 715-nm band has a high quantum yield mainly under

TECHNICAL PHYSICS Vol. 46 No. 2 2001

264 KRAVCHUK et al.

I/Im rise in the concentration of the centers of long-wave 1.0 luminescence both in the bulk and near the surface. An abrupt decrease in the intensityÐtemperature characteristic measured at 515 nm may result from the temperature-stimulated delocalization of holes from Vk centers [8]. Some of the holes are then trapped by deeper centers; hence, a slight increase in the curve for 700 nm is observed. In this band, luminiscence quench- ing near the higher temperature limit may be induced 0.5 2 by both the temperature-stimulated delocalization of 1 holes from the centers, followed by their radiationless recombination at deep electron-trapping centers, and ionic and photochemical processes [2]. A peak in the region 580Ð595 nm, which is characteristic of a near-surface region in nondoped crystals, is probably related to oxygen [17]. As noted above, this peak can be made higher by heat or laser treatment of the crystal. The luminescence intensity markedly 100 200 T, K decreases as the temperature rises from 85 to 200 K. Fig. 4. Normalized luminescence intensity vs. temperature We also studied the kinetic properties of lumines- for (1) the 515- and (2) the 700-nm band in the case of a cence from PbI crystals under β-, γ-, or X-ray excita- PbI : Mn crystal under laser excitation. 2 2 tion according to the techniques described in [1, 17]. The results agree with [1]. This allows us to infer that, optical excitation near the absorption band edge. It is for the 515- to 520-nm and the 555-nm band, doped crystals under X-ray excitation also have a short lumi- conceivable that the EPR signal with g = 2.009 and the τ ≈ red band of photoluminescence have a common mech- nescence decay time ( 1 0.5 ns). The longer lumines- τ ≈ × Ð5 anism, since they both obey a boat-shaped temperature cence decay time ( 2 1.5 10 s) of nondoped crys- characteristic in a range of 80Ð150 K [10]. In all likeli- tals measured between 500 and 800 nm at room tem- hood, the red band is caused by defects in the crystal perature seems to stem from the radiative relaxation of bulk, and the photoluminescence probably stems from excitons conﬁned to defects. recombination. This explains why a nondoped perfect PbI crystal does not luminesce in the 715-nm band 2 REFERENCES when it is irradiated with an N2 laser operating at a wavelength for which the crystal offers strong intrinsic 1. A. S. Voloshinovskiœ, P. A. Rodnyœ, and A. Kh. Kkhudro, absorption in a near-surface layer, but does luminesce Opt. Spektrosk. 76 (3), 428 (1994) [Opt. Spectrosc. 76, when it is irradiated with X-rays, which have a higher 382 (1994)]. penetrating power. 2. Wide-Gap Layered Crystals and Their Physical Proper- ties, Ed. by A. B. Lyskovich (Vishcha Shkola, Lviv, Based on [10], one could associate this optical emis- 1982). 3 1 2+ sion with the P1 S0 radiative transitions in Pb 3. O. M. Bordun, I. M. Bordun, and S. S. Novosad, Zh. ions induced by the recombination of holes with Pb+ Prikl. Spektrosk. 62 (6), 91 (1995). electron-trapping centers. Investigating the photovol- 4. V. G. Abramishvili and A. V. Komarov, Fiz. Tverd. Tela taic properties of PbI2 crystals has revealed that this (Leningrad) 31 (4), 68 (1989) [Sov. Phys. Solid State 31, material is an n-type semiconductor if its temperature is 583 (1989)]. below 180 K [12]. Otherwise, it is a p-type semicon- 5. M. S. Brodin, A. O. Gushcha, B. E. Derkach, et al., Fiz. ductor. We therefore suggest that the 715-nm emission Tverd. Tela (Leningrad) 30 (11), 3481 (1988) [Sov. is produced by the recombination of electrons and the Phys. Solid State 30, 1999 (1988)]. holes conﬁned to halogen ions in the neighborhood of + 6. V. G. Abramishvili, A. V. Komarov, S. M. Ryabchenko, Pb ions. The latter ions are associated with anionic et al., Fiz. Tverd. Tela (Leningrad) 29 (4), 1129 (1987) vacancies [13] that are always present in PbI2 as a result [Sov. Phys. Solid State 29, 644 (1987)]. of the volatilization of iodine during crystal growth 7. M. S. Brodin, I. V. Blonskiœ, and V. N. Karataev, Ukr. Fiz. [14]. In short, we connect the 715-nm emission with α- Zh. (Russ. Ed.) 34 (4), 526 (1989). centers (see also [15]). Since Mn2+ has a smaller ionic 2+ 2+ 8. E. D. Aluker, D. Yu. Lusis, and S. A. Chernov, Electron radius compared to Pb [16], the presence of Mn ions Excitations and Radioluminescence in Alkali Metals in a PbI2 crystal causes lattice disordering and hence a (Zinatne, Riga, 1979).

TECHNICAL PHYSICS Vol. 46 No. 2 2001 LUMINESCENCE FROM Mn-DOPED PbI2 CRYSTALS 265 9. J. Arends and J. Verwey, Phys. Status Solidi B 23 (1), 14. M. S. Brodin, V. A. Bobik, I. V. Blonskiœ, and N. A. Davi- 137 (1967). dova, Fiz. Tverd. Tela (Leningrad) 32 (2), 403 (1990) 10. I. Z. Irdutnyœ, M. T. Kostyshin, O. P. Kasyarum, et al., [Sov. Phys. Solid State 32, 232 (1990)]. Photostimulated Interactions in MetalÐSemiconductor 15. I. A. Parﬁanovich and É. E. Penzina, Electron Color Structures (Naukova Dumka, Kiev, 1992). Centers in Ionic Crystals (Vostochno-Sibirskoe Knizh- noe Izd., Irkutsk, 1977). 11. A. V. Patankor and E. E. Schneider, Phys. Chem. Solids 27 (3), 575 (1966). 16. G. B. Bokiœ, Crystal Chemistry (Nauka, Moscow, 1971). 12. Takeshi Hagihara, Shunji Nigata, and Nobuvoso Ayai, 17. V. V. Averkiev, I. M. Bolesta, I. M. Kravchuk, et al., Ukr. Jpn. J. Appl. Phys. 20 (5), 1003 (1981). Fiz. Zh. (Russ. Ed.) 25 (8), 1392 (1980). 13. I. Baltag, S. Lefrant, L. Miyut, and R. P. Mondescu, J. Lumin. 63, 309 (1995). Translated by A. Sharshakov

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Glow Discharge with an Annular Hollow Cathode E. I. Gyrylov Section of Physical Problems at the Presidium of Buryat Science Center, Siberian Division, Russian Academy of Sciences, Ulan-Ude, 670047 Russia e-mail: [email protected] Received April 10, 2000; in ﬁnal form, August 4, 2000

Abstract—A discharge system with peripheral discharge chambers (Penning cells) and common hollow cathode with only one cell supplied with power is studied. It is shown that a jumpwise transition from a dark discharge to a glow discharge is accompanied by the penetration of plasma into the hollow cathode. © 2001 MAIK “Nauka/Interperiodica”.

Due to a high plasma density in a glow discharge pressure of 11.7 Pa is shown in Fig. 2. When the glow with a hollow cathode, such discharges are widely used discharge was ignited at the minimum value of the dis- as plasma sources of charged particles [1, 2]. In this charge current required for this type of a discharge to paper, we study a discharge system with an annular hol- exist (1Ð1.4 mA), there was plasma in the hollow cath- low cathode and two peripheral discharge chambers ode and the probe current attained 8 µA. As the current (Penning cells) [3], one of which is supplied with power, whereas the anode of the other chamber is connected to the cathode (Fig. 1). The penetration of the 342 51 plasma into the hollow cathode was established by the ion current from a single probe, which was set at the end of the hollow cathode near the aperture of the working cell. The discharge was ignited at a residual pressure of air in the vacuum chamber. Experiments show that the transition from a dark discharge to a glow discharge occurs in a jump. At a pressure of 13.3 Pa and voltage of 70 V, the discharge current I changes rapidly from 1Ð50 µA to 1Ð10 mA and the discharge voltage V falls by 5Ð10 V. This transition is accompanied by penetration of the plasma into the hollow cathode, the probe current being 1 µA. The Fig. 1. Diagram of the discharge system: (1) hollow cathode, (2) anode, (3) ﬂat cathode, (4) permanent magnet, and absence of the effect of a hollow cathode at low current (5) probe. is related to the fact that, in this case, the charged particles do not enter the hollow cathode from the Penning cell. The plasma and electrons do not penetrate into the I, A hollow cathode because of the drop in the cathode 100 potential in front of the cathode aperture, whereas the penetration of ions is hindered by their scattering in 10–1 front of the aperture due to the speciﬁc shape of the boundary of the cathode dark space. 10–2

As the current increases, the length L of the region –3 of the cathode voltage drop decreases. At certain criti- 10 cal parameters of the reﬂex discharge, the cathode dark 10–4 space shrinks so that the ion shell in front of the hollow cathode aperture breaks and the plasma penetrates into 10–5 the cathode. The condition for penetrating is L Ӷ R, where R is the radius of the hollow cathode [1]. 10–6 0 100 200 300 400 The maximum current of the dark discharge attained V, V 200 µA at a discharge voltage of 950 V and a pressure of 8 Pa; in this case, the probe current was zero. The Fig. 2. The discharge current I vs. voltage V at a pressure of dependence of the discharge current on the voltage at a 11.7 Pa.

GLOW DISCHARGE WITH AN ANNULAR HOLLOW CATHODE 267

∆V, V V, V 250 400

200 300 6 150 5 200 4 3 * * * 100 * 2 100 1 50 0 10 20 30 0 9 10 11 12 P, Pa P, Pa Fig. 4. Dependence of the discharge voltage V on the argon Fig. 3. The decrease in the voltage after the ignition of a pressure P for different values of the discharge current: I = glow discharge vs. pressure. (1) 10, (2) 50, (3) 100, (4) 150, (5) 200, and (6) 250 mA. decreased further, the glow discharge transformed into hollow cathode when the discharge switches from the the dark discharge; in this case, there was no plasma in dark mode to the glow one. the hollow cathode and the probe current was zero. The decrease in the voltage after the ignition of a glow discharge versus pressure is shown in Fig. 3. The REFERENCES higher the ignition voltage, the larger the decrease in 1. Yu. E. Kreœndel’, Plasma Sources of Electrons (Atomiz- the voltage; the ratio ∆V/V ranges within 0.5Ð1. dat, Moscow, 1977). Figure 4 presents the dependence of the discharge 2. A. S. Metel’, in Proceedings of the 1st All-Union Confer- voltage on the argon pressure for different values of the ence on Plasma Emission Electronics, Ulan-Ude, 1991, discharge current in the case when only one cell is sup- p. 77. plied with power, whereas the other cell is insulated. At 3. E. I. Gyrylov and A. P. Semenov, Zh. Tekh. Fiz. 65 (1), a constant discharge current, the discharge voltage 189 (1995) [Tech. Phys. 40, 106 (1995)]. decreases with increasing pressure. Thus, it is experimentally shown that, in the discharge system under study, plasma penetrates into the Translated by N. Ustinovskiœ

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Energy Filter in the Form of a Three-Electrode Plane Capacitor with End Diaphragms L. P. Ovsyannikova and T. Ya. Fishkova Ioffe Physicotechnical Institute, Russian Academy of Sciences, St. Petersburg, 194021 Russia e-mail: [email protected] Received April 27, 2000

Abstract—An electrostatic system based on a plane capacitor is suggested to provide monochromatization of the beams of charged particles. In addition to monochromatization, this system also straightens the beam trajectory. The parameters of the system are calculated numerically within a broad range of variation of its strength. © 2001 MAIK “Nauka/Interperiodica”.

Two-electrode plane capacitors are widely used for In the preceding works [2, 3], we studied the work- the energy filtration of charged particles. For this pur- ing modes of an energy filter based on a cylindrical pose, the capacitor should be operated in the mirror capacitor as charged particles entered through the end mode with the beam inlet and outlet through its plate. electrode. This system allows a more convenient A number of electrodes with linear potential variation arrangement of the source, detector, and other ele- are placed along the edges to provide field homogene- ments. If the beam intersects the capacitor axis at two ity. To simplity the system design, these electrodes can points (double intersection), this system provides a straight beam-propagation trajectory. In this case, the be replaced by end diaphragms, with a potential equal system is of considerable interest for developing cer- to the potential of any of the capacitor plates. Although tain physical devices based on the monochromatic both the linear dispersion and energy resolution of such beams of charged particles. a device are on par with similar characteristics of a clas- The goal of this work was to solve similar problems sical plane capacitor, the former has a much simpler using plane electrodes. The cross section of the sug- design than the latter [1]. To ensure angular beam gested device is composed of three plane-parallel plates focusing in the mirror mode, both the source and the and two plane end diaphragms and is shown in Fig. 1. detector of the charged particles should be near the The diaphragms are placed close to the outer plane capacitor. However, in some cases, this requirement electrodes, and their holes are coated with grids to cannot be met. allow the beam to enter and exit. Fieldmaking poten-

x, mm

O 9 O = –75 s V s a

3 O O O 0 6 1218 24 30 36 42 48 54 z, mm –3 O a O

–9 V = –75

L –15

Fig. 1. A three-electrode plane capacitor with end diaphragms providing a straight trajectory of beam propagation.

ENERGY FILTER IN THE FORM OF A THREE-ELECTRODE PLANE 269 x/a L/a D/a respectively. The trajectories of propagation of an elec- ε 1.4 tron beam with an initial energy of 0 = 100 eV in one 0.8 8 of the working modes of the device are shown in Fig. 1. 1.2 The parameters of the system were chosen to make the inlet and outlet coordinates of the axial beam trajec- 0.7 7 3 tory equal to one another: x = x0 = x1 (Fig. 2, curve 1). 1.0 In this case, dependence of the total length of the sys- 2 ε 0.6 6 tem L on its strength eV/ 0 (where V is the potential dif- 0.8 ference between the outer and middle plates of the 1 capacitor) is characterized by curve 2 (Fig. 2), whereas 0.5 5 the dependence of the coefficient of linear dispersion ∆ ε ∆ε 0.6 D = z 0/ is characterized by curve 3 (Fig. 2). It fol- 0.4 4 lows from Fig. 2 that the stronger the system, the smaller its length and dispersion and the closer the 0 0.5 0.6 0.7 0.8 0.9 1.0 beam is to its axis. If the inlet size of a parallel beam is ε ∆ eV/ 0 x0 = 0.1a, the size of the beam at the first intersection site does not exceed 0.02a, whereas the outlet size of ∆ ∆ Fig. 2. Parameters of the system shown in Fig. 1 as functions the beam is xj < 1.2 x0. The exit angle of the axial tra- of its strength: (1) distance from the axis to the beam jectory in any working mode does not exceed 0.5°, and entrance and exit in case of an axial beam trajectory; (2) sys- the maximum deviation of peripheral trajectories is less tem length; and (3) linear energy dispersion. than ±3°. The energy filter with straight beam trajectory tials are applied to the outer electrodes, whereas both described in this work is based on a three-electrode the inner plate of the capacitor and the end diaphragms plane capacitor with end diaphragms. The parameters are grounded. The length of the system is several times of the energy filter are close to those of a similar system longer than its width. based on a cylindrical capacitor. In terms of design The geometrical electric parameters of the system characteristics, this energy filter is compatible with var- were calculated numerically using the TEO computer ious optoelectronic systems composed of plane ele- program for a plane problem [3]. If the beam inlet ments. It should be noted that such purely electrostatic direction is parallel to the longitudinal axis of the sys- systems are similar to a Wien filter with crossed elec- tem and the inlet site is at a certain distance from the trostatic and magnetic fields. axis, the strength of the system and its total length should be chosen to meet the following condition: the REFERENCES focused beam spot size at the site of the first intersection should be as large as possible, whereas the output 1. L. P. Ovsyannikova and T. Ya. Fishkova, Zh. Tekh. Fiz. beam should be as parallel as possible. Therefore, this 65 (3), 113 (1995) [Tech. Phys. 40, 288 (1995)]. system provides a straight trajectory of beam propaga- 2. L. P. Ovsyannikova and T. Ya. Fishkova, Zh. Tekh. Fiz. tion and its energy filtration according to the size of the 62 (5), 179 (1992) [Sov. Phys. Tech. Phys. 37, 582 first slit in the middle plate. (1992)]. The results of numerical calculations of the param- 3. L. P. Ovsyannikova, S. V. Pasovets, and T. Ya. Fishkova, Zh. Tekh. Fiz. 62 (12), 171 (1992) [Sov. Phys. Tech. eters of the energy filter described above are given in Phys. 37, 1215 (1992)]. Fig. 2. In these calculations, the distance between the plane electrodes and the gap between the outer and end electrodes were taken to be a = 10 mm and s = 0.2a, Translated by K. Chamorovskiœ

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The Effect of Boundary Conditions on the Superconducting Transition Critical Temperature in Superlattices I. N. Askerzade Institute of Physics, Academy of Sciences of Azerbaijan, pr. Dzhavida 33, Baku, 370143 Azerbaijan Received May 15, 2000

Abstract—For a superconducting superlattice consisting of alternating layers of two materials with thicknesses a and b, an analogue of the Cooper formula is obtained for boundary conditions in the general form. The effect of the boundary conditions on the critical temperature is studied, and the possibility of order parameter localization in the layers with a higher critical temperature is demonstrated. © 2001 MAIK “Nauka/Interperiodica”.

The phenomenon of high-temperature superconduc- It is interesting to trace a relationship between the tivity has recently generated interest in experimental transition matrix and the superlattice critical tempera- [1, 2] and theoretical [3Ð5] investigation of supercon- ture. For this purpose, we will study the effect of the ducting superlattices. The advanced deposition tech- boundary conditions upon the density of states on the niques enable one to produce artificial superlattices Fermi surface and, hence, upon T (a, b) in terms of the with smoothly varying parameters. Therefore, superlat- c tices have a great potential for simulating properties of BCS theory in the effective mass approximation. Such layered superconductors. a method was applied within the Eliashberg approach [6] to derive the critical temperature for an ellipsoidal Let a superlattice consist of two alternating super- Fermi surface. conducting layers A and B of thicknesses a and b and have a total period L = a + b. In the classical statement We consider the Debye frequencies and the con- of the Cooper problem, the boundary conditions stants of four-fermion interaction in the BCS theory to imposed on the wave function (WF) of a Cooper pair be the same; this assumption is justified because of a are the continuity of the WF and its first derivative at weak dependence of T on ω and on the order of the the interface: c d electronÐphonon interaction constants (the weak cou- 1 ∂Ψ 1∂Ψ pling model). The layers are considered to differ in Ψ ==Ψ , ------, (1) A B m ∂z m∂z mass only. Due to the periodic structure of superlat- A A B B tices, the carrier masses can be renormalized and the where mA, B is the effective mass of charge carriers and concept of longitudinal and transverse effective masses subscripts A and B relate to the materials on each side can be introduced [7]. Then, employing the transition of the interface. matrix method, the electron spectrum is readily obtained from the solution of the Schrödinger equation Suppose that the critical temperature in layer A, TcA, for the wave function envelope satisfying the general- is higher than in the other, TcB. In the general case, the WF and its derivative on one side of the interface ized boundary conditions. The effective masses at dif- should be joined with a linear combination of Ψ and ferent points of a Brillouin zone can be found by ∂Ψ/∂z at its other side: expanding the energy in terms of the wave vector. Anal- ysis shows that, at the top of a mini-Brillouin zone, lon- ΨΨ Ψ Att B B gitudinal and transverse masses are expressed as =11 12 =Tˆ , (2) ϕϕ ABϕ At21 t22 B B κ mA sinh b m|| = h------, (4) ()2 where ka+ bCa ∂Ψ ∂Ψ ϕ  ϕ mA where A==l------∂ - , B l------------∂ - , (3) z A mB z B ζ()ζ2 α2 2 Ð1()2 2 αÐ2 h =t22 + t12 +t11 + t21 l is an arbitrary constant length, and Tˆ stands for the AB ()α αÐ1 κ transition matrix at the interface. Ð2 t11t12 + t21t22 coth b,

THE EFFECT OF BOUNDARY CONDITIONS 271

Ð1 Ð1 Ð1 where the order parameter is localized. A similar situa- m⊥ = m θ + m (1 Ð θ), A B tion is observed in bulk superconductors [8], where it is 2 (5) reasoned by the presence of Tamm levels. With ordi- θ Ca  sinka = ----(------)-a1 + ------ , nary boundary conditions (1), the order parameter 2 a + b ka localization does not occur and we come to the Cooper limit; i.e., T smoothly varies between T and T .  κ c cA cB 2 ( )  sinka  sinh b  These results are of interest for the interpretation of Ca = 2 a + b a1 + ------ + b------+ 1  ka κb experimental data on order parameter localization on a dislocation network [1]. Ð1 1 + coska κb  In conclusion, by choosing a transition matrix, one × β  ------κ------t11 Ð t12 tanh------. can control the minizone structure of an electron in a cosh b + 1 2  superlattice and, consequently, the density of states on the Fermi surface. The latter parameter, in turn, is The other designations follow [7]. The effective responsible for the localization effect experimentally constant of interaction in the BCS expression for criti- observed in superconducting superlattices. cal temperature γ λ 2 ω Ð1/ eff REFERENCES T c = --π--- De (6) 1. O. A. Mironov, S. V. Chistyakov, I. Yu. Skrylev, et al., can be evaluated from Pis’ma Zh. Éksp. Teor. Fiz. 50, 300 (1989) [JETP Lett. 50, 334 (1989)]. λ = N (0)g. (7) eff eff 2. H. C. Yang, L. M. Wang, and H. E. Horng, Phys. Rev. B Introducing the longitudinal and transversal effec- 59, 8956 (1999). tive mass means that the Fermi surface is an ellipsoid. 3. S. Takahashi, T. Hirai, M. Machida, and M. Tachiki, For this spectrum, the density of states is given by Physica C (Amsterdam) 235Ð240, 2585 (1994). 4. M. V. Baranov, A. I. Buzdin, and L. N. Bulaevskiœ, Zh. P0 Éksp. Teor. Fiz. 91, 1063 (1986) [Sov. Phys. JETP 64, N (0) = ----- m||m⊥. (8) eff π2 628 (1986)]. 5. V. M. Gvozdikov, Fiz. Nizk. Temp. 15, 636 (1989) [Sov. A feature of our model is that the effective mass and, J. Low Temp. Phys. 15, 358 (1989)]. hence, the critical temperature depend on the general- 6. M. Prohammer, Physica C (Amsterdam) 157, 4 (1989). ized boundary conditions. Let us trace the behavior of 7. N. F. Gashimzade and E. L. Ivchenko, Fiz. Tekh. Polu- the Tc (a, b) function derived in view of the expressions provodn. (Leningrad) 25, 323 (1991) [Sov. Phys. Semi- for m|| and m⊥. The generalized boundary conditions cond. 25, 195 (1991)]. ⇒ lead to the localization of the order parameter: as a 0, 8. I. M. Suslov, Zh. Éksp. Teor. Fiz. 95, 949 (1989) [Sov. Tc tends not to TcB but to T* > TcB, where T * is deﬁned Phys. JETP 68, 546 (1989)]. by elements of the transition matrix Tˆ AB . As this takes place, the high-Tc layers behave as planar defects, Translated by A. Sidorova-Biryukova

TECHNICAL PHYSICS Vol. 46 No. 2 2001