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Hall coefficient factor in polar semiconductors
J. Phys. C/lem. Solids. 1973, Vol. 34, pp. 9-13.
HALL
Pergamon Press.
Printed in Great Britain
COEFFICIENT
FACTOR
IN
POLAR
SEMICONDUCTORS* D. KRANZER
Lehrkanzel fiir Physikalische Elektronik, T H - W i e n , Austria
(Received 8 September 1971 ; in revised form 13 October 1971) A b s t r a c t - T h e Hall coefficient factor in polar semiconductors is calculated avoiding any serious approximation in the calculation. Lattice scattering (polar optical, acoustic deformation potential and piezoelectric scattering), combined lattice and ionized impurity scattering, and the effect of a nonparabolic conduction band are considered. Results are given both in the limit of vanishing magnetic field and for arbitrary values of the magnetic field. Furthermore our results are compared with experimental values reported previously. 1. INTRODUCTION
TO DETERMINE the carrier concentration in a semiconductor Hall measurements are usually performed. The carrier concentration n is obtained from the expression n = - - r n / e R n (Rn being the Hall factor, rH the scattering or Hall coefficient factor, and e the electronic charge) which is valid as long as one type of carrier is present. The scattering factor r, depends on the scattering mechanisms, the applied magnetic field, and the degeneracy of the carriers. This scattering factor can be evaluated directly from the Boltzmann equation. In polar semiconductors, however, this calculation is complicated by the fact that no momentum relaxation time exists for temperatures below the optical phonon temperature. Originally Lewis and Sondheimer[1] calculated the temperature and magnetic field dependence of the scattering factor in polar semiconductors using a variational method. Delves[2] performed a numerical integration of the Boltzmann equation and calculated rH at several lattice temperatures. His results agree within 25 per cent with those of Lewis and Sondheimer[1]. Further evaluations have been performed by Delvin *This work was sponsored by the Ludwig Boltzmann Gesellschaft.
[3] and Fortini et a/.[4]. However, the results given in the literature[l, 3,4] in the limit of vanishing magnetic field agree only qualitatively (see Fig. 2). In addition, scattering by lattice imperfections other than polar optical phonon and the effect of nonparabolicity have received little attention [5] even though it is well known that both appreciably affect the transport properties of polar semiconductors. Recently, Stillman et al.[6] reported experimental results concerning the scattering factor obtained from epitaxial n-GaAs samples. The values obtained on extremely pure samples (total ionized impurity concentration of N~ = 6.9 × 10j3 c m -3) show a qualitative agreement with calculations performed previously[l, 3, 4]. Samples, however, with larger N~ did not show the characteristic behavior of rn as compared with the extremely pure sample. Stillman et al.[6] suggested that the contribution of ionized impurity scattering masks the temperature variation of the scattering factor due to polar optical mode scattering. In the present paper the scattering factor of polar semiconductors is reconsidered and the calculations are extended to arbitrary values of the magnetic field, including all relevant scattering mechanisms and the effect of nonparabolicity. A brief description of the theory is given in Section 2. Our results are
10
D. KRANZER
then compared with both experimental[6] and previous theoretical values[l, 3,4]. In addition, the effect of the nonparabolicity of the conduction band is discussed (Section 3). 2, THEORY
Assuming a homogenous material, an isotropic conduction band, and low electric fields the distribution function may be written as [1] f(p)=fo(e)+p"
c(e).
2m*e (1 + ~%) '
(2)
where m* is the effective mass at the edge of the conduction band and % is the energy gap which is assumed to be independent of the lattice temperature T. If we consider an electric field (E~, Eu, 0) and a magnetic field (0,0, B), two components of the unknown vector c do not vanish; that is, % and %. It is easily shown that the vector components c~ and cu are solutions of the simultaneous equations [1]
(3)
(Opcu~
B. p
dE
Eo=
jx = -- e f ~
dE up ~ 3 ( axE.r + auE u) -~p
= 0.zxEx + 0.xuEy j~ = -- e f ~
(flxEx + fluEs) ~d e ,op 3
= 0.u~Ex + 0.uuEv.
(6)
Inserting equation (5) into equation (3) we obtain az = flu,
au = - - f l ~ .
(7)
Considering equations (6) and (7) the conductivities become 0.x~ = - 0.~x.
(8)
a~ and o~ are finally determined from
(9)
The scattering factor is given by (4) (4)
is the Planck distribution function of the phonons, to1 is the angular frequency of the Nq
The current density is given by
--aax+(OP°~'~ = 0 . \ at /~
P df0 b=-EoNq dp'
m* eo31 (__~ -~o) 4.rrh -•
(5)
Opotz ---ff~jc+ ao~= b
= E~,. b,
where a----Eo" Nq dp'
% = flxEz + fiuEu.
0.xx = 0 . ~ ,
(Opc~] at /c + a ' c u= E x . b
- - a . c~ + \ - - ~ - / c
Cx = axEz + au Eu
(1)
where f0 is the thermal equilibrium function. The energy-momentum relation is given by Kane [7] as /9 2 =
longitudinal optical phonons, K= and K0 are the dielectric constants for high and low frequencies, respectively, and (Opc/at)c is the collision term. We obtain the collision term as presented in the literature[8] by inserting the matrix elements for lattice scattering [9, 10, 11] and the Brooks-Herring formula [12] for ionized impurity scattering. For low electric fields we may write
rH =
RHne -- ne
0.xu
B 0.2 + 0 . 2 "
(10)
In the limit of vanishing magnetic fields X Y is small compared with 0.2xx and we obtain a
HALL
COEFFICIENT
II
FACTOR
simplified expression for r,,
0.. B-O.05 . ... 8=0.5 13-
(11)
----
In this case aZ and OL, are solutions equations
of the
i g
!wlr (
at c >
=aa
2’
ref (6)
0=0.05
Vhf
N,= N,;O 64
,5 -3 B=DSVs/m xl0 cm 1
N,=O
12.A’
@ z z 0
VS/d vs/m2 I
.I
lb, -
(12)
In this limit mZZ is independent of the magnetic field and cZU is proportional to the magnetic field. Consequently rH is independent of the magnetic field. In the limit of high magnetic fields, on the other hand, o-!,~ is small compared with o-&,. It is easily shown that
r=+ Fig. 1. Dependence of the scattering factor on y for several magnetic fields and impurity concentrations. The material parameters used in the calculation are those given by Wolfe et al. [ 171. Polar optical, piezoelectric, and deformation scattering and nonparabolicity are taken into account.
deviation exists in the lattice temperature range where the maximum of the scattering factor occurs. Concerning this point we want en 1 to emphasize that Hall measurements using ‘“=x.--&=1, van der Pauw methods are extremely difficult at low magnetic fields and are usually less accurate than those at larger magnetic fields. where % is given by At magnetic fields B 2 0.05Vs/m’ p X B is large compared with 1 in the lattice tempera(14) ture range where piezoelectric and deformation potential scattering predominate [ 141. and % is independent of the scattering Consequently, piezoelectric and deformation mechanisms under consideration. potential scattering have no noticable effect on the scattering factor as shown in Fig. 1. 3. RESULTS The general behavior of r, can be underWe applied the theory as outlined in Section stood as follows: The deviation of r, from 1 2 to re-analyse the experiments made by at vanishing magnetic fields is a consequence Stillman et a1.[6]. Equations (9) are solved of the energy dependence of the scattering numerically (the method used is described in mechanisms. For simplicity we consider Ref. [ 131) and the results are given in Fig. 1. scattering by polar optical modes only. The For lattice (polar optical, acoustic deformamobility Al. is expected to be nearly energy tion potential, and piezoelectric) scattering independent at lattice temperatures T- 8 and with a magnetic field of 0.5 Vs/m2 our [ 151 and T -G 8, [ 163 (0 is the optical Debye theoretical results agree quantitatively (in temperature defined by 8 = fi~~/k~). Accordthe whole lattice temperature range con- ing to Ehrenreich [ 151 for T < 8 p is at first sidered) with the experimental values[6] of inversely and then directly proportional to sample a (Ni = 6.9 X 1013cm-3). At lower some small positive power of the electron magnetic fields (B = 0.05 Vs/m’), however, a energy as the temperature is decreased. For
12
D. KRANZER
T > 0 p~ is proportional to E "s. The scattering factor at low magnetic fields reflects this behavior; that is, rn becomes approximately 1 when /z is nearly energy independent and deviates appreciably from 1 w h e n / z depends on the c a r d e r energy (Fig. 2). If, in addition, scattering by ionized impurities is taken into account, the combined mobility changes noticeably below a certain lattice temperature To only. According to the dominant scattering mechanism/x is proportional to eats far below To and approaches the energy dependence due to polar optical scattering above To. Consequently, a minimum of rn exists at a lattice temperature where the effect of the ionized impurity scattering on the energy dependence of the combined mobility is partly compensated by polar optical scattering. It has been shown previously[l] that a maximum of the /x vs. T curve exists. T h e corresponding temperature depends on the total ionized impurity concentration. F o r Nt = 6.3 × 10]Scm-a (sample e of Ref. [6]) the maximum of /~ appears at T = 85°K. This temperature is close to the temperature where
1.3
3 I
I ..ref./I /
2..xef./3/ 3...ref,/4/and
~
work
present
.--GoAs
l/I" \~ 1"2
I
""'~
---"
lnAs
\
o
the minimum of the rz vs. y curve (y = O/T) occurs in Fig. 1. Below this temperature rn increases as T decreases and approaches the value due to scattering by ionized impurities. A larger amount of impurity scattering (as compared with Ni = 6.9 × l0 ]3 cm -a) tends to smooth out the rH vs. y curve as has been suggested [6]. The effect of a nonparabolic conduction band on the rn vs. y curve is shown in the limit of vanishing magnetic fields in Fig. 2 for scattering by polar optical modes only (InAs, Eo = 0.43 eV). Results obtained previously [1,3,4] using a parabolic energy-momentum relation are included in Fig. 2. In this case our results for a parabolic band are in complete agreement with those of Fortini et a/.[4] in the whole lattice temperature range considered. In addition results for perfectly pure n - G a A s which include scattering by polar optical modes, piezoelectric and acoustic deformation potential scattering are also shown in Fig. 2. In the calculation we used the material parameters given by Wolfe et al. [17]. A noticeable calculated difference is found between parabolic and nonparabolic energy bands even in the case of n - G a A s where the effect of the nonparabolic conduction band on the low field mobility is negligible [14] for T < 0. Considering Fig. 1 the agreement between theory and experiment is improved substantially when the nonparabolic conduction band is taken into account. 4. CONCLUSION
.E o
I
2
3
4
5
6
7
8
Fig. 2. Dependence of the scattering factor on 7 in the limit of vanishing magnetic fields. 1, 2, 3 parabolic band, polar optical scattering only; . . . . . . nonparabolic band, polar optical, piezoelectric, and deformation potential scattering.
We want to emphasize that in our analysis any serious assumption is avoided. T h e measured and the theoretical scattering factor evaluated from the Boltzmann equation agree very well. T h e nonparabolic conduction band affects the value of rz much more than it affects the low field mobility. T h e contribution of ionized impurity scattering to the scattering factor is large at low temperatures and can be completely explained by considering the c a r d e r energy dependence of the mobility. F o r the special case of a parabolic conduction
HALL C O E F F I C I E N T F A C T O R
band and very small magnetic fields our results are in complete agreement with those of Fortini et al.[ 14]. Acknowledgements--We wish to thank Prof. Dr. H. W. PiStzl for his interest in our work and many valuable discussions. We are very thankful to the Institute fiJr Numerische Mathematik, Technische Hochschule Wien for the use of their computer.
REFERENCES
1. LEWIS B. F. and S O N D H E I M E R E. H. Proc. R. Soc. A227, 241 (1954). 2. DELVES R. T. Proc. Phys. Soc. 73, 572 (1959). 3. DELV1N S. S. Physics and Chemistry of I I - V I Compounds (Edited by M. Aven and J. S. Prener), p. 551, North-Holland, Amsterdam (1967). 4. FORTINI A., D I G U E T D. and L U G A N D J., J. appl. Phys. 41, 3121 (1970).
13
5. E H R E N R E I C H H., J. Phys. Chem. Solids 9, 129 (1959). 6. STILLMAN G. E., W O L F E C. M. and D I M M O C K J. O.J. Phys. Chem. Solids 31, 1199 (1970). 7. K A N E E. O.J. Phys. Chem. Solids 1,249 (1959). 8. G R I G O R E V N. N., D Y K M A N I. M. and TOMC H U C K P. M., Soviet Phys.-solidSt. 10, 837 (1968). 9. F R O H L I C H H. and P A R A N J A P E B. V. Proc. Phys. Soc. B69, 21 (1956). 10. MEIJER H. J. G. and POLDER D., Physica 19, 255 (1953). 11. BARDEEN J. and SHOCKLEY W., Phys. Rev. 80, 72 (1950). 12. BROOKS H., Adv. Electron. Electron Phys. 7, 85(1955). 13. K R A N Z E R D., Phys. Status Solidi 46, Nr. 2 (1971). 14. RODE D. L., Phys. Rev. B 2, 1012 (1970). 15. E H R E N R E I C H H.,J. appl. Phys. 32, 2155 (1961). 16. HOWARTH D. J. and S O N D H E I M E R E. H., Proc. R. Soc. A219, 53 (1953). 17. W O L F E C. M., STILLMAN G. E. and L I N D L E Y W. T.J. appl. Phys. 41, 3088 (1970).
HALL
Pergamon Press.
Printed in Great Britain
COEFFICIENT
FACTOR
IN
POLAR
SEMICONDUCTORS* D. KRANZER
Lehrkanzel fiir Physikalische Elektronik, T H - W i e n , Austria
(Received 8 September 1971 ; in revised form 13 October 1971) A b s t r a c t - T h e Hall coefficient factor in polar semiconductors is calculated avoiding any serious approximation in the calculation. Lattice scattering (polar optical, acoustic deformation potential and piezoelectric scattering), combined lattice and ionized impurity scattering, and the effect of a nonparabolic conduction band are considered. Results are given both in the limit of vanishing magnetic field and for arbitrary values of the magnetic field. Furthermore our results are compared with experimental values reported previously. 1. INTRODUCTION
TO DETERMINE the carrier concentration in a semiconductor Hall measurements are usually performed. The carrier concentration n is obtained from the expression n = - - r n / e R n (Rn being the Hall factor, rH the scattering or Hall coefficient factor, and e the electronic charge) which is valid as long as one type of carrier is present. The scattering factor r, depends on the scattering mechanisms, the applied magnetic field, and the degeneracy of the carriers. This scattering factor can be evaluated directly from the Boltzmann equation. In polar semiconductors, however, this calculation is complicated by the fact that no momentum relaxation time exists for temperatures below the optical phonon temperature. Originally Lewis and Sondheimer[1] calculated the temperature and magnetic field dependence of the scattering factor in polar semiconductors using a variational method. Delves[2] performed a numerical integration of the Boltzmann equation and calculated rH at several lattice temperatures. His results agree within 25 per cent with those of Lewis and Sondheimer[1]. Further evaluations have been performed by Delvin *This work was sponsored by the Ludwig Boltzmann Gesellschaft.
[3] and Fortini et a/.[4]. However, the results given in the literature[l, 3,4] in the limit of vanishing magnetic field agree only qualitatively (see Fig. 2). In addition, scattering by lattice imperfections other than polar optical phonon and the effect of nonparabolicity have received little attention [5] even though it is well known that both appreciably affect the transport properties of polar semiconductors. Recently, Stillman et al.[6] reported experimental results concerning the scattering factor obtained from epitaxial n-GaAs samples. The values obtained on extremely pure samples (total ionized impurity concentration of N~ = 6.9 × 10j3 c m -3) show a qualitative agreement with calculations performed previously[l, 3, 4]. Samples, however, with larger N~ did not show the characteristic behavior of rn as compared with the extremely pure sample. Stillman et al.[6] suggested that the contribution of ionized impurity scattering masks the temperature variation of the scattering factor due to polar optical mode scattering. In the present paper the scattering factor of polar semiconductors is reconsidered and the calculations are extended to arbitrary values of the magnetic field, including all relevant scattering mechanisms and the effect of nonparabolicity. A brief description of the theory is given in Section 2. Our results are
10
D. KRANZER
then compared with both experimental[6] and previous theoretical values[l, 3,4]. In addition, the effect of the nonparabolicity of the conduction band is discussed (Section 3). 2, THEORY
Assuming a homogenous material, an isotropic conduction band, and low electric fields the distribution function may be written as [1] f(p)=fo(e)+p"
c(e).
2m*e (1 + ~%) '
(2)
where m* is the effective mass at the edge of the conduction band and % is the energy gap which is assumed to be independent of the lattice temperature T. If we consider an electric field (E~, Eu, 0) and a magnetic field (0,0, B), two components of the unknown vector c do not vanish; that is, % and %. It is easily shown that the vector components c~ and cu are solutions of the simultaneous equations [1]
(3)
(Opcu~
B. p
dE
Eo=
jx = -- e f ~
dE up ~ 3 ( axE.r + auE u) -~p
= 0.zxEx + 0.xuEy j~ = -- e f ~
(flxEx + fluEs) ~d e ,op 3
= 0.u~Ex + 0.uuEv.
(6)
Inserting equation (5) into equation (3) we obtain az = flu,
au = - - f l ~ .
(7)
Considering equations (6) and (7) the conductivities become 0.x~ = - 0.~x.
(8)
a~ and o~ are finally determined from
(9)
The scattering factor is given by (4) (4)
is the Planck distribution function of the phonons, to1 is the angular frequency of the Nq
The current density is given by
--aax+(OP°~'~ = 0 . \ at /~
P df0 b=-EoNq dp'
m* eo31 (__~ -~o) 4.rrh -•
(5)
Opotz ---ff~jc+ ao~= b
= E~,. b,
where a----Eo" Nq dp'
% = flxEz + fiuEu.
0.xx = 0 . ~ ,
(Opc~] at /c + a ' c u= E x . b
- - a . c~ + \ - - ~ - / c
Cx = axEz + au Eu
(1)
where f0 is the thermal equilibrium function. The energy-momentum relation is given by Kane [7] as /9 2 =
longitudinal optical phonons, K= and K0 are the dielectric constants for high and low frequencies, respectively, and (Opc/at)c is the collision term. We obtain the collision term as presented in the literature[8] by inserting the matrix elements for lattice scattering [9, 10, 11] and the Brooks-Herring formula [12] for ionized impurity scattering. For low electric fields we may write
rH =
RHne -- ne
0.xu
B 0.2 + 0 . 2 "
(10)
In the limit of vanishing magnetic fields X Y is small compared with 0.2xx and we obtain a
HALL
COEFFICIENT
II
FACTOR
simplified expression for r,,
0.. B-O.05 . ... 8=0.5 13-
(11)
----
In this case aZ and OL, are solutions equations
of the
i g
!wlr (
at c >
=aa
2’
ref (6)
0=0.05
Vhf
N,= N,;O 64
,5 -3 B=DSVs/m xl0 cm 1
N,=O
12.A’
@ z z 0
VS/d vs/m2 I
.I
lb, -
(12)
In this limit mZZ is independent of the magnetic field and cZU is proportional to the magnetic field. Consequently rH is independent of the magnetic field. In the limit of high magnetic fields, on the other hand, o-!,~ is small compared with o-&,. It is easily shown that
r=+ Fig. 1. Dependence of the scattering factor on y for several magnetic fields and impurity concentrations. The material parameters used in the calculation are those given by Wolfe et al. [ 171. Polar optical, piezoelectric, and deformation scattering and nonparabolicity are taken into account.
deviation exists in the lattice temperature range where the maximum of the scattering factor occurs. Concerning this point we want en 1 to emphasize that Hall measurements using ‘“=x.--&=1, van der Pauw methods are extremely difficult at low magnetic fields and are usually less accurate than those at larger magnetic fields. where % is given by At magnetic fields B 2 0.05Vs/m’ p X B is large compared with 1 in the lattice tempera(14) ture range where piezoelectric and deformation potential scattering predominate [ 141. and % is independent of the scattering Consequently, piezoelectric and deformation mechanisms under consideration. potential scattering have no noticable effect on the scattering factor as shown in Fig. 1. 3. RESULTS The general behavior of r, can be underWe applied the theory as outlined in Section stood as follows: The deviation of r, from 1 2 to re-analyse the experiments made by at vanishing magnetic fields is a consequence Stillman et a1.[6]. Equations (9) are solved of the energy dependence of the scattering numerically (the method used is described in mechanisms. For simplicity we consider Ref. [ 131) and the results are given in Fig. 1. scattering by polar optical modes only. The For lattice (polar optical, acoustic deformamobility Al. is expected to be nearly energy tion potential, and piezoelectric) scattering independent at lattice temperatures T- 8 and with a magnetic field of 0.5 Vs/m2 our [ 151 and T -G 8, [ 163 (0 is the optical Debye theoretical results agree quantitatively (in temperature defined by 8 = fi~~/k~). Accordthe whole lattice temperature range con- ing to Ehrenreich [ 151 for T < 8 p is at first sidered) with the experimental values[6] of inversely and then directly proportional to sample a (Ni = 6.9 X 1013cm-3). At lower some small positive power of the electron magnetic fields (B = 0.05 Vs/m’), however, a energy as the temperature is decreased. For
12
D. KRANZER
T > 0 p~ is proportional to E "s. The scattering factor at low magnetic fields reflects this behavior; that is, rn becomes approximately 1 when /z is nearly energy independent and deviates appreciably from 1 w h e n / z depends on the c a r d e r energy (Fig. 2). If, in addition, scattering by ionized impurities is taken into account, the combined mobility changes noticeably below a certain lattice temperature To only. According to the dominant scattering mechanism/x is proportional to eats far below To and approaches the energy dependence due to polar optical scattering above To. Consequently, a minimum of rn exists at a lattice temperature where the effect of the ionized impurity scattering on the energy dependence of the combined mobility is partly compensated by polar optical scattering. It has been shown previously[l] that a maximum of the /x vs. T curve exists. T h e corresponding temperature depends on the total ionized impurity concentration. F o r Nt = 6.3 × 10]Scm-a (sample e of Ref. [6]) the maximum of /~ appears at T = 85°K. This temperature is close to the temperature where
1.3
3 I
I ..ref./I /
2..xef./3/ 3...ref,/4/and
~
work
present
.--GoAs
l/I" \~ 1"2
I
""'~
---"
lnAs
\
o
the minimum of the rz vs. y curve (y = O/T) occurs in Fig. 1. Below this temperature rn increases as T decreases and approaches the value due to scattering by ionized impurities. A larger amount of impurity scattering (as compared with Ni = 6.9 × l0 ]3 cm -a) tends to smooth out the rH vs. y curve as has been suggested [6]. The effect of a nonparabolic conduction band on the rn vs. y curve is shown in the limit of vanishing magnetic fields in Fig. 2 for scattering by polar optical modes only (InAs, Eo = 0.43 eV). Results obtained previously [1,3,4] using a parabolic energy-momentum relation are included in Fig. 2. In this case our results for a parabolic band are in complete agreement with those of Fortini et a/.[4] in the whole lattice temperature range considered. In addition results for perfectly pure n - G a A s which include scattering by polar optical modes, piezoelectric and acoustic deformation potential scattering are also shown in Fig. 2. In the calculation we used the material parameters given by Wolfe et al. [17]. A noticeable calculated difference is found between parabolic and nonparabolic energy bands even in the case of n - G a A s where the effect of the nonparabolic conduction band on the low field mobility is negligible [14] for T < 0. Considering Fig. 1 the agreement between theory and experiment is improved substantially when the nonparabolic conduction band is taken into account. 4. CONCLUSION
.E o
I
2
3
4
5
6
7
8
Fig. 2. Dependence of the scattering factor on 7 in the limit of vanishing magnetic fields. 1, 2, 3 parabolic band, polar optical scattering only; . . . . . . nonparabolic band, polar optical, piezoelectric, and deformation potential scattering.
We want to emphasize that in our analysis any serious assumption is avoided. T h e measured and the theoretical scattering factor evaluated from the Boltzmann equation agree very well. T h e nonparabolic conduction band affects the value of rz much more than it affects the low field mobility. T h e contribution of ionized impurity scattering to the scattering factor is large at low temperatures and can be completely explained by considering the c a r d e r energy dependence of the mobility. F o r the special case of a parabolic conduction
HALL C O E F F I C I E N T F A C T O R
band and very small magnetic fields our results are in complete agreement with those of Fortini et al.[ 14]. Acknowledgements--We wish to thank Prof. Dr. H. W. PiStzl for his interest in our work and many valuable discussions. We are very thankful to the Institute fiJr Numerische Mathematik, Technische Hochschule Wien for the use of their computer.
REFERENCES
1. LEWIS B. F. and S O N D H E I M E R E. H. Proc. R. Soc. A227, 241 (1954). 2. DELVES R. T. Proc. Phys. Soc. 73, 572 (1959). 3. DELV1N S. S. Physics and Chemistry of I I - V I Compounds (Edited by M. Aven and J. S. Prener), p. 551, North-Holland, Amsterdam (1967). 4. FORTINI A., D I G U E T D. and L U G A N D J., J. appl. Phys. 41, 3121 (1970).
13
5. E H R E N R E I C H H., J. Phys. Chem. Solids 9, 129 (1959). 6. STILLMAN G. E., W O L F E C. M. and D I M M O C K J. O.J. Phys. Chem. Solids 31, 1199 (1970). 7. K A N E E. O.J. Phys. Chem. Solids 1,249 (1959). 8. G R I G O R E V N. N., D Y K M A N I. M. and TOMC H U C K P. M., Soviet Phys.-solidSt. 10, 837 (1968). 9. F R O H L I C H H. and P A R A N J A P E B. V. Proc. Phys. Soc. B69, 21 (1956). 10. MEIJER H. J. G. and POLDER D., Physica 19, 255 (1953). 11. BARDEEN J. and SHOCKLEY W., Phys. Rev. 80, 72 (1950). 12. BROOKS H., Adv. Electron. Electron Phys. 7, 85(1955). 13. K R A N Z E R D., Phys. Status Solidi 46, Nr. 2 (1971). 14. RODE D. L., Phys. Rev. B 2, 1012 (1970). 15. E H R E N R E I C H H.,J. appl. Phys. 32, 2155 (1961). 16. HOWARTH D. J. and S O N D H E I M E R E. H., Proc. R. Soc. A219, 53 (1953). 17. W O L F E C. M., STILLMAN G. E. and L I N D L E Y W. T.J. appl. Phys. 41, 3088 (1970).