Charged grain boundaries in germanium

Transcription

1 Charged grain boundaries in germanium A. Broniatowski To cite this version: A. Broniatowski. Charged grain boundaries in germanium. Journal de Physique, 1981, 42 (5), pp < /jphys: >. <jpa > HAL Id: jpa Submitted on 1 Jan 1981 HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Nous The It J. Physique 42 (1981) MAI 1981,. 741 Classification Physics Abstracts 73.40L I Charged grain boundaries in germanium (*) A. Broniatowski C.N.R.S., Laboratoire P.M.T.M., avenue J.B. Clément, Villetaneuse, France (Rep le 28 novembre 1980, accepté le 30 janvier 1981) Résumé étudions les propriétés d équilibre d un joint de flexion de faible désorientation dans le germanium, dans l hypothèse où les états liés au joint forment une bande partiellement remplie avec un niveau de neutralité situé dans la moitié inférieure de la bande interdite. L énergie électrostatique et l entropie des charges d écran jouent un rôle majeur pour déterminer l occupation des niveaux du joint. Les effets d écran dépendent fortement de la température et du taux de dopage : dans le germanium de type n, il apparait une couche de déplétion à basse température, avec une évolution graduelle vers une couche d inversion à haute température. Dans le germanium p la couche de déplétion se transforme en une couche d accumulation au passage de la température de neutralité. L effet de barrière électrostatique est beaucoup plus fort dans le germanium n que dans le germanium p, en accord avec les résultats expérimentaux actuellement connus Abstract. equilibrium properties of a low angle tilt boundary are discussed on the assumption that the boundary states form a partially filled band with a neutral level located in the lower half of the energy gap. The electrostatic energy and the entropy stored in the screening layer are major factors in determining the occupancy of the boundary levels. The screening effects depend strongly on temperature and the doping conditions : in n type germanium a depleted layer is found at low temperatures, with a gradual evolution towards an inversion layer in the high temperature range. In p type germanium the depleted layer transforms into an accumulation layer through a neutral temperature. The barrier effect is much stronger in n than in p type germanium, in agreement with available experimental results. 1. Introduction. is well known that there are internal potential barriers in polycrystalline germanium, associated with grain boundaries [1]. The latter result in a large increase in the rate of electronhole recombination [2, 3]. A discussion of these effects involves some description of the boundary in thermodynamic equilibrium and the purpose of this paper is to provide such a description in terms of a simple model. First consider the following experimental results. 1) Grain boundaries have an important effect on the conductivity of n type germanium but, in contrast, have little or no effect on the conductivity of p type germanium [1, 4]. 2) In n type germanium the barrier potential may be as much as a few tenths of a volt [1, 5] ; the barrier width is somewhat less than one micron [1]. 3) The boundary charge is apparently screened by a depleted layer [1]. An inversion layer might also form in some cases [59]. (*) Work supported in part by the C.N.R.S. under contract : ATP 3290 (1977). These results have suggested the existence of electron states localized at the boundary, which according to Shockley [10] might form a partially filled band. The properties of low angle tilt boundaries have been discussed along these lines by Mueller [11]. Other authors [9, 12] have suggested that the boundary states could form a discrete set of acceptor or donor levels. Impurity segregation could then be an important factor. Let us consider a bicrystal with a low angle (a few degrees) tilt boundary. The boundary is assumed to consist of a single set of parallel, equally spaced edge dislocations. Moreover, we suppose the bicrystal is uniformly doped, so that effects of impurity segrega tion can be neglected. Following Shockley [10] we suppose the dislocation states form a partially filled band. A characteristic level Eo is defined so that when the Fermi level coincides with Eo, the band is half filled and the dislocations are electrically neutral. More generally, the dislocations may act as donors or acceptors according to the position of the Fermi level with respect to Eo, thus becoming electrically charged. This model has been used by Schroter and Labusch [13] to interpret the electrical properties of deformed germanium, with the level Eo, located in Article published online by EDP Sciences and available at

3 The The 742 the energy gap, at 0.09 ev from the top of the valence band. (In this paper we shall also consider other values for the purposes of comparison.) Since Eo is located in the lower half of the energy gap, quite different effects are expected for n and p type germanium. In n type germanium, the boundary is always negatively charged, as the Fermi level lies above Eo. The boundary charge is screened by a depleted or an inversion layer, depending on the temperature and doping conditions (section 3) but in either case a nonohmic conductivity is expected, because of the repulsive nature of the potential barrier. In p type germanium the boundary can be made electrically neutral by adjusting the Fermi level to equal Eo. The condition EF Eo defines the neutral = temperature, To, as a function of the doping ratio and Eo (Fig. 1); for a given doping ratio, the higher Eo is, the higher the neutral temperature. For temperatures below To the boundary is positively charged, resulting in a depleted layer and a repulsive barrier stored in the screening layer. At low temperature, the equilibrium equation involves the electrostatic energy alone and becomes identical with Read s condition for the case of a single dislocation [14]. 2. Equilibrium occupancy of the boundary states. Equations are derived for an n type bicrystal and the corresponding conditions for a p type specimen are given at the end of the section. The occupancy of the boundary states is obtained with the following approximations : 1) the electrons are assumed to move independently, and therefore to form a Fermi distribution; and 2) the electron states are described within the rigid band approximation. We start with an electrically neutral boundary, and consider an energy level Ei within the bicrystal (1). When the boundary is charged, the level Ei is shifted by the amount q V(r), where q is the elementary positive charge and V(r) the electrostatic potential produced by the boundary and the screening charges (Fig. 2). At equilibrium, the average number of electrons in the level E; q V(r) is given by : Fig. 2. bending of the energy bands in an n type bicrystal. Equation (1) is complemented by Poisson s equation : Fig. 1. neutral temperature To as a function of the doping ratio, for various values of the level Eo. while for temperatures above To, the boundary charge is negative. In the latter case the majority charge carriers, holes, are attracted to the boundary and an accumulation layer forms. The boundary should then have little effect on the conductivity, as the barrier is of an attractive, rather than a repulsive type. As in the case of a dislocation, the electrostatic interactions play a major role in determining the occupation of the boundary states. The equilibrium occupation is derived in section 2 using a Thomas Fermi approximation. The dependence of the boundary charge and the screening effects on temperature is discussed in sections 3 and 4. A gradual evolution takes place through the extrinsic range, as an inversion layer forms in n type germanium, and an accumulation layer in p type germanium. Lastly (section 5) the occupation of the boundary states is discussed in terms of the electrostatic energy and the entropy where g is the dielectric constant and p(r) is the volume charge density, which, in turn is related to the occupation of the various energy levels at the point r. For the sake of simplicity the boundary plane is assumed to be uniformly charged, with a surface charge density a. Then Poisson s equation becomes : where p and V depend only the boundary. The boundary conditions for the potential follows : on the distance x from are as (1) Energies are measured from the top of the valence band in the unperturbed material; the electrostatic potential is also taken to be zero at this point.

4 743 and 2) there is a discontinuity in d V/dx of (JIB across the boundary plane. We now consider the relationship between a, p and the fractional occupation ratios of the various electron levels. a) The expression for a depends on the density of states in the dislocation band. We consider the case where the latter reduces to a single peak at the level Eo. Let 2 Nt be the number of boundary states per unit area, and /s their fractional occupation ratio. It then follows that : so that the boundary is electrically neutral for fb = V2. According to equation (1) fb is given by : where Vo is the electrostatic potential in the boundary plane, i.e. the barrier height. Nt is related to the misorientation angle 0 and the number of states per unit length of dislocation which we denote by 2/c. Note that in Read s model [14], c would be the distance between consecutive dangling bonds along the dislocation. For small values of 0, not exceeding a few degrees, Nt is given by : where b is the length of the Burgers vector and D is the distance between neighbouring dislocations. If we take b = c = 4 A and (J = 20, then D 115 A and 2.2 x 1013 CM 2. b) The screening charge density is made up of three contributions ; 1) the ionized impurities with density n (x); 2) the conduction electrons with density n(x) ; and 3) the holes in the valence band with density p(x). Let n (x) be the density of unionized impurities and Nd the doping ratio. Then In the extrinsic temperature range, n, p and nd are given by : where N c and N v are the effective number of states in respectively the conduction and the valence band, Eg is the energy gap and Ed is the donor level. The screening charge density is given by : Although the Fermi level depends, in principle, on the occupation of the boundary states, this dependence is only important when the size of the bicrystal is comparable with the thickness of the screening layer. We are only concerned with large specimens for which EF is fixed by the neutrality condition p = 0 in the unperturbed material. As an approximation we shall use the wellknown expression : As is shown in appendix A, the electrostatic energy density is given by : In particular consider the value of we in the boundary plane. According to Coulomb s theorem, the magnitude of the electric field is given by! cr 1/2 8 which means that we = a2 /8 8. Comparison with equation (16) gives : where no, po and n o are the densities of electrons, holes and unionized impurities in the boundary plane. The equilibrium properties of the boundary are determined by the set of equations (4), (5), (8) through (13), (15) and (17). An approximate solution is given in appendix B for the particular case in which /B its very close to 1/2. This solution is applicable for typical doping concentrations (between 1013 and 1018 cm 3) and temperatures. A similar set of equations is found for p type bicrystals as follows : where N, P and Nd refer to the densities of electrons, holes and unionized impurities in the bulk of the material which can be written as follows : where Na is the doping ratio, n is the density of unionized impurities and Ea is the acceptor level.

5 Let 3.1 (a) Since The properties of a tilt boundary in n type germanium. OCCUPANCY OF THE BOUNDARY STATES. us consider a moderately doped ( 1015 cm 3) n type bicrystal with a 20 tilt boundary. For numerical computation the neutral level is taken to be at 0.09 ev from the top of the valence band. Figure 3a shows the dependence of fb on temperature. The fractional occupation ratio is slightly higher than 1/2 with a minimum at T K, which means that the boundary charge remains negative, lying between 1011 and 1012 electrons. cm 2. of temperature. Such a decrease has been observed experimentally [1]. 3.3 PROPERTIES OF THE SCREENING LAYER. The on the den presence of an inversion layer, depends sity of holes in the boundary plane as compared with the densities of conduction electrons and ionized impurities. The doping ratio may be expressed as : Similar expressions are found for po, no and ndo as follows : no and ndo are easily shown to be very small as compared with Nd and therefore, only Nd and po need be considered. Figure 3c shows that po is an increasing function of temperature, equal to Nd (the doping ratio) for T T;. Therefore = a gradual evolution of an inversion layer takes place throughout the extrinsic temperature range : for T«Ti the screening layer consists of a depleted layer only (po Nd). At higher temperatures (T Z TJ the depleted layer splits and an inversion layer forms along the boundary plane. Figure 4 shows for two temperatures at T«Ti and T >> Ti, the variation of the potential and the screening charge density with distance from the boundary plane. The thickness of the screening layer is about one micron. It is not clear, whether the inversion layer can be properly described within the rigid band approximation. It might not be the case, as the holes are narrowly confined along the boundary plane (Fig. 4d) and a more rigorous quantum treatment is probably required. 3.4 THE FORMATION OF THE INVERSION LAYER. According to equations (19) and (20) the condition Po Nd gives; Fig. 3. The occupation ratio as a function of temperature in an n type bicrystal (doping ratio 1015 at. cm 3, tilt angle : 2 ). fb goes through a minimum at Ti 140 K. (b) The barrier height Vo as a function of temperature. (c) The hole density po in the boundary plane, as a function of temperature. At Ti, po Nd, the = doping ratio BARRIER HEIGHT. fb is very close to 1/2 the boundary level is practically coincident with the Fermi level. Consequently the energy bands are bent as shown in figure 2 giving rise to an electrostatic barrier of height : The dependence of Yo on temperature is shown in figure 3b. The numerical estimates compare well. with experimental data [1, 5]. According to equation (18) the barrier height is a decreasing function Equation (23) defines the temperature Ti formation of the inversion layer, for the as a function of the doping ratio and the level Eo. This equation parallels that for the neutrality condition in p type germanium, EF = Eo discussed in section 1. Inserting the appropriate expressions for the Fermi level given above (Eqs. (15) and (15 )) we find that for T; : and for To : Eo=kToLog(Nv/Na). When Na=Nd,T is approximately equal to To since Nc and N, are of the same magnitude in germanium, which means that figure 1 also represents the dependence of Ti on the doping ratio in n type germanium. Using the equilibrium equations given in section 2, it can be easily shown that fb has a minimum when

6 The Figure The 745 Fig. 4. electrostatic potential and the screening charge density as functions of distance from the boundary plane. (a) and (b) : 77 K. (c) and (d) : 300 K. We now consider the relationship of this minimum to the formation of the inversion layer. First suppose that T T ;. Then by equation (23), From figure 2, Eo is the minimum amount of energy required to promote a valence electron into a boundary state. Similarly, Eg EF is the minimum energy needed to promote a boundary electron into the conduction band. If the temperature is increased slightly, there will be changes in the filling of the boundary states. Since Eg EF Eo, more electrons are transferred from the boundary into the conduction band than from the valence band into the boundary. Therefore fb is a decreasing function of temperature. Conversely when T > Ti, Eg EF > Eo and fb becomes an increasing function of T. Thus fb must go through a minimum at T;, corresponding to the formation of the inversion layer. 3.5 THE SCREENING OF A SINGLE DISLOCATION. The previous discussion suggests a comparison with the properties of a single dislocation. Schroter and Labusch [13] assume a low screening charge density around the dislocations, and consequently use the Debye approximation. However, this assumption holds only when the dislocations are practically neutral, i.e. in p type germanium when T N To. The arguments in (3.3) and (3.4) also apply with figure 2 representing the bending of the energy bands near the dislocation. Depending on the temperature and the doping ratio the dislocation is surrounded with a depleted, an inversion or an accumulation layer. We understand that this fact has not been taken into account in discussing the results of Hall effect measurements in deformed germanium [13, 15]. 4. The properties of a tilt boundary in p type germanium. 5 shows the dependence of fb, Vo and po on temperature in a p type bicrystal with a 20 tilt boundary. Again a doping ratio of 1015 cm 3 is assumed. The neutral temperature is about 140 K. When T To the boundary becomes positively charged ( fb 1/2) and a depleted layer forms ( po Na) resulting in a repulsive barrier as in the case of n type germanium. However the barrier height is much lower in the present case (Fig. 5b). When T > To the boundary charge becomes negative, resulting in an accumulation layer (po > Na). 5. The electrostatic energy and the entropy stored in a bicrystal. occupancy of the boundary states is discussed in terms of the electrostatic energy and the entropy stored in the bicrystal. The electrostatic energy is of major importance at low temperatures, as Read has shown for the case of a single dislocation [14]. A different behaviour however prevails in the high temperature range, due to changes in the screening conditions, so that the dependence on the electrostatic energy is much reduced. On the other hand, the entropy stored in the screening layer is no longer negligible. Both effects favour a rapid build up of the boundary charge, as shown in figures 3 and 5. These properties will be studied for an n type bicrystal. Similar effects hold for p type specimens, but with an accumulation instead of an inversion layer.

7 (a) The 746 the change in W,,(a) for each additional electron is given by : where po is the screening charge density in the boundary plane. The quantity q(owe/oa)t will be evaluated in two limiting cases, (a) when the temperature is much lower and (b) much higher than T;. (a) T Ti (no inversion layer). According to equation (C. 8), The effect of varying the boundary charge is shown in figure 6a : an electron is transferred from A (the limit of the depleted layer) to B, the boundary plane, so that We is changed by q Vo, the work done against the repulsive electrostatic forces. An approximate form of the equilibrium condition is obtained by eliminating q Vo from equations (24) and (26), as follows : Fig. 5. The occupation ratio as a function of temperature in a p type bicrystal (doping ratio 1015 at. cm 3, tilt angle : 2 ). The boundary is electrically neutral at To 140 K. (b) The barrier height as a function of temperature. (c) The hole density po in the boundary plane, as a function of temperature. At To, po N 8 the = doping ratio. Solving equation (5) for the Fermi level gives : Equation (24) is a statement of the equality of the Fermi level with the chemical potential of the boundary electrons : Eo is the binding energy of an additional electron and k Log f B/( 1 f B) the corresponding change of entropy of the electrons in the boundary states. The term, q V o represents the part of the chemical potential associated with the electrostatic interactions. As a consequence of these interactions, changes in the boundary charge result in changes in the electrostatic energy of the bicrystal and the entropy of the electrons in the screening layer. Since fb is very close to 1/2, the quantity where fa is deduced from equation (27) in conjunction with equations (4) and (25). An equation similar to (27) has been given by Read [14] for a single dislocation («minimum energy approximation»). This equation holds only when there is no inversion layer : the entropy stored in the depleted layer is nearly independent of the boundary charge, and therefore the equilibrium condition involves the variation of the electrostatic energy alone. (b) T >> Ti (inversion layer). Varying the boundary charge now amounts to transferring electrons from C at the top of the inversion layer, to B (Fig. 6b), a local rearrangement as is discussed in appendix C. Therefore the electrostatic energy is not expected to depend very much on the boundary charge. However the entropy should have a strong variation as new holes are is very small as compared with the other terms in equation (24). Let We(6) be the electrostatic energy per unit boundary area. According to appendix C, Fig. 6. electron transfer corresponding to a variation of the boundary charge. (a) No inversion layer : A B. (b) Inversion layer : C + B.

8 The The 747 added to the inversion layer. We shall consider the case of a strong inversion layer, for which According to equation (C 10), Transferring internal energy of the bicrystal by an electron from C to B varies the an amount : Let 65 be the corresponding change in entropy : 6S consists of the entropy change of the boundary electrons, together with that of the holes in the screening layer. At equilibrium the free energy of the bicrystal is minimized with respect to the boundary charge and therefore it follows that : Neglecting the change in electrostatic energy and the change in entropy of the boundary electrons gives : A new form of the equilibrium condition is obtained by eliminating Eo from equations (24) and (31) so that : The quantity q Vo is now related to the variation of entropy of holes in the screening layer rather than the increase in electrostatic energy. The occupancy of the boundary states is found from equations (4), (20), (25) and (28) : the dependence on temperature is much stronger than in case (a). On the other hand, fb no longer depends on the doping ratio, which it to be expected as the electrons in the boundary states and the inversion layer form practically a closed system with respect to the rest of the bicrystal. 6. Conclusion. equilibrium properties of a tilt boundary have been discussed, assuming the boundary states to form a partially filled band with a neutral level located in the lower half of the energy gap. The density of boundary states is estimated to be about cm for a low angle boundary of a few degrees. In addition to the electrostatic energy, the entropy stored in the screening layer has been shown to play a major role in determining the occupancy of the boundary states. Both factors result in the boundary remaining practically neutral, with a net charge of typically electrons cm 2. The barrier potential is calculated to be a few tenths of a volt. The thickness of the screening layer is about one micron for a moderately doped (1015 cm 3) bicrystal. The properties of the screening layer depend on the type of semiconductor and show a strong dependence on the temperature and the doping ratio. In n type germanium a depleted layer is found at low temperatures, with a gradual transition towards an inversion layer at higher temperatures. A parallel evolution takes place in p type germanium, with a depleted layer changing into an accumulation layer, as the temperature is increased through the neutral temperature. In this paper the boundary states have been assumed to have the same energy for the sake of simplicity Similar results are expected for more complicated band structures, since the equilibrium properties of the boundary depend only on the density of states near the neutral level. The effects of impurity segregation have not been considered. Segregated impurities have apparently been observed near the core of boundary dislocations by high resolution electron microscopy [16]. However little is known at present concerning the effect of these impurities on electrical properties. Experiments are in progress to examine the most important features of this model, namely : 1) the existence of a neutral temperature in p type germanium ; 2) the formation of an inversion layer in n type germanium (eq. (23)); and 3) the dependence of the barrier potential on temperature (Eq. (18)). Acknowledgments. author is indebted to G. Faivre, P. Haasen, J. Labbe and G. Saada for many useful discussions. APPENDIX A Including the screening charge density (14) in Poisson s equation (3) gives : The electrostatic energy density is found by multiplying both sides in equation (A. I) by d V/dx and integrating, which gives : The constant C is found by setting x to a value much larger than the thickness of the screening layer, where both we and V are zero. Therefore and finally :

9 An 748 The value of w, in the boundary plane is a particular case (see text), as follows : P, Nd, n and nd are negligible in comparison to N, the latter being practically equal to Nd (see Eqs. (11), (12), (13), (21) and (22)). Therefore : Two limiting forms of equation (A. 5) are used in appendix C. When po Nd (no inversion layer), then : The second form applies when po >> ( qvolkt) Nd, so that : APPENDIX B Approximate solution of the equilibrium equations for an n type bicrystal. approximate solution is given for the case in which A is very close to 1/2. Solving equation (5) for qvo gives : According to (4), Assuming fb 1/2 means that : Equation (B. 1) gives to first order in a : According to (9) the hole density in the boundary plane is : Equation (B. 8) gives 6 as a function of temperature and the doping ratio. Lastly, fb can be deduced from a by means of (B. 2). The conditions of validity of this solution are discussed below. Equation (B. 8) shows that the boundary charge is independent of N,, i.e. the misorientation angle 0. Therefore, the condition (B. 3) is satisfied if 0 is large enough. Numerical computation shows that for 0 Z 10, fb remains very close to 1/2 throughout the extrinsic temperature range and over a wide range of doping ratios (between 1 O 13 and 1018 cm,3). For smaller 0, fb may deviate noticeably from 1/2 and the solution is no longer satisfactory. Moreover the boundary can no longer be considered as uniformly charged, because of the increased spacing between dislocations. APPENDIX C The electrostatic energy stored in an n type bicrystal. Let We( (J) be the electrostatic energy stored in the bicrystal per unit boundary area. We shall calculate the variation q(o We/0 (J)T when the boundary charge is varied by one electron at constant temperature. We shall also discuss the consequent modifications in the screening charge distribution. Similar considerations apply to a p type bicrystal. 1. THE VARIATION OF THE ELECTROSTATIC ENERGY. Let us suppose Poisson s equation is solved with the boundary charge 7 (Fig. 7a). According to Coulomb s theorem the electric field at the origin is u/2 8. Consider the point M (abscissa bx) where the electric field is ( 6 + bq)/2 s : clearly the part of the curve to the right of M describes the variation of the potential for the boundary charge Q + 6J, the plane of the boundary being shifted to the point of absissa bx. The relation between bx and 6J is found by considering the screening charge distribution (Fig. 7b) : according to Gauss s theorem the screening charge in the shaded area is one half of ba giving to first order in a : From (A. 5), (B. 4), (B. 6), (12) and (15) one may write : Figure 7c shows the density of electrostatic energy as a function of x. The variation of We is twice that of the electrostatic energy under the shaded area, as follows : According to (B. 3) the quantities involving a on the right hand side of (B. 7) are negligibly small, and therefore, First order expansions of (C. 1) and (C. 2) may be used for small ba, as follows : and :

10 (a) We 749 Therefore varying the boundary charge by one electron results in an increase of the electrostatic energy of the bicrystal by an amount : Homogeneous doping is a necessary condition for the validity of (C. 6), as translational invariance has been assumed for Poisson s equation. 2. THE RELATED CHANGES IN THE SCREENING LAYER. shall consider two cases, (a) when the temperature is much lower and (b) much higher than T;. a) T 1 (no inversion layer). In this case Po " qnd According to (C. 3) the width of the depleted layer is changed by : Therefore varying the boundary charge amounts to transferring some electrons between the conduction band and the boundary states, as is shown in figure 6a. According to (C. 6) the change in We per electron is a /8 ENd which when combined with (A. 6) gives : Fig. 7. The electrostatic potential as a function of distance from the boundary plane. (b) The screening charge distribution. (c) The electrostatic energy density. The figure is drawn for ba > 0. For a negative variation M is located on the dotted part of the curve (Fig. 7a) extending to x 0. where po and a 18 s are respectively the screening charge density and the electrostatic energy density in the boundary plane. Combining (C. 3) and (C. 4) gives : b) T > Ti (inversion layer). Now Po " qpo. Varying the boundary charge results in a change of the width of the inversion layer by an amount : The electron transfer takes place between the valence band and the boundary states as is shown in figure 6b. The variation of We will be found in the case where po» ( qvolkt) Nd. Then a /8 s kt po (A. 7) and therefore : References [1] TAYLOR, W. E., ODELL, N. H. and FAN, H. Y., Phys. Rev. 88 (1952) 867. [2] VOGEL, F. L., READ, W. T. and LOVELL, L. C., Phys. Rev. 94 (1954) [3] McKELVEY, J. P., J. Appl. Phys. 32 (1961) 442. [4] PEARSON, G. L., Phys. Rev. 76 (1949) 459. [5] MUELLER, R. K., J. Appl. Phys. 30 (1959) 546. [6] TWEET, A. G., Phys. Rev. 99 (1955) [7] LANDWEHR, G. and HANDLER, P., J. Phys. Chem. Solids 23 (1962) 891. [8] VUL, B. M. and ZAVARITSKAYA, E. I., Inst. Phys. Conf. Ser. 43 (1979) 421. [9] MATUKURA, Y., J. Phys. Soc. Japan. 17 (1962) [10] SHOCKLEY, W., Phys. Rev. 91 (1953) 228. [11] MUELLER, R. K., J. Appl. Phys. 31 (1960) [12] STRATTON, R., Proc. Phys. Soc. B 69 (1956) 513. [13] SCHRÖTER, W. and LABUSCH, R., Phys. Status Solidi 36 (1969) 539. [14] READ, W. T., Philos. Mag. 45 (1954) 775. [15] LABUSCH, R. and SCHETTLER, R., Phys. Status Solidi (a) 9 (1972) 455. [16] BOURRET, A., Electron Microscopy 1980 (P. Brederoo et G. Boom, The Hague), vol.1, p. 306.