Transport properties of some bismuth-antimony alloys

TRANSPORT PROPERTIES OF S O M E BISMUTH-ANTIMONY ALLOYS

S. D. PROBERT

Cranfield Institute ~! Technology, Cranfield, Bertlord MK43 OAL (Great Britain) and C. B. THOMAS

Uni~'ersiO' o]' Bradfbrd, Brarlfbrd, West Yorkshire BD7 I DP (Great Britain)

SUMMARY

The thermal and electrical conducti~'ities, as well as the thermoelectric powers./~," single crystals of Bi-Sb alloys, in the temperature range 60 to 90 K hare been measured. The thermoelectric' power behat,iour is attributed to a change J'rom extrinsic to intrinsic conduction occurring near 80 Kjbr alloys with between 9 and 32 atomic "//oSb. For alloys having less than 24 atomic % Sb, the phonons suffer a large strain field scattering whereas,for higher concentrations, bipolar d([fusion probably ensues.

INTRODUCTION

As part of our energy conservation programme, improvements in the efficiency of transforming energy directly between heat and electricity are important. 1 The same knowledge can also be applied profitably in the design of miniaturised thermoelectric compensators for mechanically strong, thermally insulating supports. 2,3 The maximum thermoelement efficiency, q mayassuming constant temperatures of the hot (Th) and cold (Tc) thermoelement junctions, is given according to |oiTe 4'5 as:

+5 (L 127 Applied Energy (5) (1979)--© Applied Science Publishers ktd, England, 1979 Printed in Great Britain

128

S. D. PROBERT, C. B. THOMAS

where the intrinsic property of the thermoelectric material for this function, namely the figure of merit: O(2o-

Z-

(2)

/,

being the thermoelectric power, ~r the electrical conductivity and k the thermal conductivity. It may be deduced that other conditions remaining invariant, as Z increases ~/..... also increases. It is desirable to determine the variations of ~, a, k and hence Z with temperature in order to assess the temperature dependence of the efficiency of energy conversion which may be achieved with that material and the temperature at which the maximum effectiveness ensues. For the Bi Sb system of alloys considered in the present study, the component concentration dictates whether semi-metallic or semi-conducting behaviour will ensue, o'v F r o m the thermomagnetic and galvanomagnetic properties s of the alloy system, a model has been formulated of the band structure of Bi containing 6 atomic "/,, ~v.qh 9 Further details of the band structure have been revealed by measurement of the galvanomagnetic coefficients of Te doped Bi containing 15 atomic .... ,o Sb. lO

TERMINOLOGY

The transport properties were measured in either the P or the S direction, i.e. respectively parallel with, or perpendicular to, the trigonal axis of the considered single crystals. The number following the directional letter P or S in Table 1 TABLE 1 THt" Bi Sb SPECIMENS

Specimen P0 SO $6 P9 S16 P18 $23 $29 $32 P36 $43

Length ~[ ~s7~ecimen in P direction (cm) 1-256 2-474 0-735 0'820 1"185 1"318 1"495 1-060 1"103 1"461 1"819

Cross-sectional area in S plane (cm 2) 0.247 0"239 0-341 0"447 0"215 0'213 0-074 0"169 0"154 0.122 0"232

represents the atomic percentage concentration of Sb in that particular Bi Sb crystal.

T R A N S P O R T P R O P E R T I E S OF SOME B I S M U T H - - A N T I M O N Y A L L O Y S

129

EXPERIMENTAL DETAILS

The electrical c o n d u c t i v i t y was m e a s u r e d with a c h o p p e r circuit ~1 a n d the t h e r m a l c o n d u c t i v i t y a n d t h e r m o e l e c t r i c p o w e r d e t e r m i n e d s i m u l t a n e o u s l y using a steadystate t h e r m a l flux f r o m a heater fixed to the end o f each specimen. 12 The m a x i m u m p r o b a b l e errors in the values o f a, k a n d ~ so d e t e r m i n e d were e s t i m a t e d to be _+ 6 o~,, + 10 % a n d _+ 4 %, respectively. T h e r e c t a n g u l a r single crystal specimens exhibited m i c r o s c o p i c cellular structures resulting f r o m the differing segregation rates o f the Bi and Sb and also f r o m c o n s t i t u t i o n a l supercooling. 13. la The specimen c o m p o s i t i o n s were d e t e r m i n e d from density m e a s u r e m e n t s ~5 a n d are presented in the first c o l u m n o f Table 2. The TABLE 2 MEASURED THERMOELECTRIC CHARACTERISTICS OF THE Bi Sb ALLOYS AT 80 K

Atomic % concentration~)fSh

c~(P) (103ohm-lcm 1)

~(S) (103ohm lcm t}

0 6 9 16 18 23 29 32 36 43

24 3.8 2-2 4-4 5-1 6.0 6.2 6.3 7.0

26-5 3.7 1-6 4.1 5.0 5.4 5.9 6.2 8.0

:~(P) (pVK l ) - 77 - 132 -142 - 124 - 120 - 105 -65 -55

:t(S) (pVK t} - 36 - 103 -118 - 121 - 102 -84 -60 -46 -21

samples were cut so that their longest sides were either parallel with, or p e r p e n d i c u l a r to, their trigonal axes, which were a s c e r t a i n e d from the direction o f the principal cleavage plane, parallel o p p o s i t e faces being achieved by l a p p i n g the surfaces with a 280 mesh c a r b o r u n d u m powder. The crystals were a n n e a l e d s u b s e q u e n t l y at 200 °C for a p p r o x i m a t e l y twelve h o u r s a n d finally etched in a 1 : 1 : 2 p a r t s by v o l u m e solution o f H N O 3, HCI a n d distilled H 2 0 . The final m e a n d i m e n s i o n s o f the specimens, m e a s u r e d with a m i c r o m e t e r screw gauge, are shown in T a b l e 1.

Thermoelectric power A n interesting feature o f the e x p e r i m e n t a l curves o f Figs. 1 a n d 2, as indicated by the d a s h e d lines, is the presence o f m a x i m u m c~ values occurring in the c o n c e n t r a t i o n range from 9 to 36 a t o m i c % Sb. This implies that a c h a n g e from extrinsic to intrinsic c o n d u c t i o n o c c u r r e d in the region o f 8 0 K a n d this is consistent with Jain's c o n c l u s i o n 6 t h a t an energy g a p (-~ 6 meV) exists for such a c o n c e n t r a t i o n range. F o l l o w i n g the analysis o f G o l d s m i d , 9 the b a n d structure for b i s m u t h f o r m u l a t e d

130

S. D. PROBERT, C. B. THOMAS

- 150

9

g

/

../,,'/

-13o/

9

"~.

/..%~- ~,,

..-110 • o o #,

L) -90

-EXPERIMENTAL VALUES FOR P9 P18 P0 _

P36

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~D

0 .''''°'''''° .....

-70

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....... 6

.o" d'"'"" -50

...d'

"~.. hs

-3o do Fig. 1.

40

~o

Go

TEMPERATURE (K)

Comparison of the experimental and predicted values for the thermoelectric power of Bi Sb alloys in the P-direction -110

o - so - S/.3 x - $23

/ fX"XXxX . ~ , ~ l /x

-gO

/x x ~ . ; . .

/'

~.

d~-so ~A

'~

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• - $29 o - $32

~, ~.

~" ; , ,

~

,

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~o ~ ~O~ o # jO I

-,c

Fig. 2.

~0

I0

&

TEMPERATURE (K)

GO

T h e r m o e l e c t r i c p o w e r in the S - d i r e c t i o n f o r various Bi Sb alloys. - observed behaviour.

- -

Predicted behaviour.

Experimentally

T R A N S P O R T P R O P E R T I E S OF SOME B I S M U T H - A N T I M O N Y

131

ALLOYS

by Abeles and Meiboom ~6 has been applied to the Bi-Sb alloys. The conduction band has three equal ellipsoidal energy surfaces with axes parallel to the principal ones, and the valence band has isoenergy surfaces which are ellipsoids of revolution about the trigonal axis for the heavy holes. The absolute thermoelectric powers parallel with, and perpendicular to, the trigonal axis, namely ct(P) and ~(S), respectively, can be expressed as a function of the partial electron and hole thermoelectric powers (~, and ctp, respectively) and conductivities (a, and ~p, respectively).17 However, the solution of these equations requires a knowledge of ct(P), ct(S) and the corresponding conductivities, a(P) and a(S), for each sample. The variations o f a ( P ) and a(S) with temperature for the tested samples are shown in Figs. 3 and 4. The thermoelectric powers and electrical conductivities in directions perpendicular to those listed in Table 1 were determined at 60 and 80 K. Such measurements (see Tables 2 and 3) were performed on specimens cut from the TABLE 3 T H E R M O E L E C T R I C PARAMETERS OF

Atomic '~, concentration o f S b 0 6 9 16 18 23 29 32 36 43

o( P) (103ohm 30"2 3-2 1-8 3'7 4"5 5"2 5"4 5"7 6"8

icm-I)

Bi Sb

ALLOYS AT

a(S) (103ohm 31"7 3"0 l'0 3'5 4"2 4"6 5"3 6"1 11"0

l c m ~ i)

60 K ~( P) (ItV K i) -47 - 84 - 122 - 132 104 - 53 -50 -45

~(S) (ItV K 1) -21 - 64 - 108 - 128 - 40 -59 -46 - 15

original samples and, as the dimensions of these were correspondingly smaller, the estimated maximum possible errors in the thermoelectric powers and electrical conductivities rose to + 8 ~o and _+ 16 ojjo, respectively. For 60 and 80 K, ~, and % were calculated assuming: (i) the presence of equal electron and hole concentrations and (ii) that the ratios between the three electron mobilities and the two hole mobilities are the same for the alloys considered as those determined for bismuth.l~ The values of %, %, ~, and ~p for pure Bi and for Bi containing 6 atomic % Sb are shown in Table 4. However, for higher concentrations the calculations failed since they implied a negative value for %. Using a parabolic model for the band structure, Brown and Silverman 1° found, for a Te-doped Bi-Sb alloy containing 15 atomic ')0 Sb, that the ratios between the three electron mobilities were a factor of two larger than those for bismuth. The ratio of the hole mobilities for the present specimens were calculated from Ertl's 15 and the authors' measurements of the Hall coefficients and electrical conductivities, respectively assuming the existence of a single spheroidal heavy hole band. With this new data, ~,, and % were recalculated, but it was again found that %, was negative for

Co to

TABLE 4 SPECIMEN PROPERTIES

~o concentration of Sb

Temperature (K)

a.(P) (1030hm 1on 1 t)

ap(P) (1030hm lcm-a )

~r.(S) (1030hm-.tcm i)

0 6 0 6

80 80 60 60

22"81 3"72 27-60 3"12

2"26 0'15 2"73 0.12

19'5 0"15 23'5 2-66

ap(S) (1030hm-lcm 8"38 0"55 10"1 0"46

i)

"o lm © rio t"rl

~t,, (llVK-l)

(t~V~-t)

-91'6 -142'1 --57"4 --82-3

92"4 121"8 63"1 52"0

¢3 -I ©

TRANSPORT

PROPERTIES

OF SOME

BISMUTH-ANTIMONY

133

ALLOYS

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r

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_> if)

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W

o

f-'

, _ , - ~

3

/X

~

X __.._._..._~ X / x ~ x ~

_x~X~x~-X~X------

•

- PO

- P36 o - P18

x - P9 1

60

76

80

TEMPERATURE

F i g . 3.

90

(K)

Experimentally measured values for the electrical conductivity in the P-direction.

concentrations exceeding 6 atomic % Sb. This failure to correctly predict ~, and % for higher Sb concentrations probably arises from using an incorrect model of the band structure. For the semi-metal Bi, Engeler 18 has shown that a second valence band containing light holes is located at 24 meV below the conduction band, having an effective mass equal to that of the conduction band. Also, the addition of Sb to Bi reduces the overlap of the conduction and hole bands: eventually an energy gap will occur comparable in magnitude with that between the conduction and light hole band for high concentrations (_> 9 atomic % Sb). 6 Thus a change from extrinsic to intrinsic conduction is expected. Below 80 K the variation of the thermoelectric power with temperature may be described if the relaxation time z is related to the electronic energy E by the equation: r = E"

(3)

A value of - 0 . 5 has been used for the index h as this is the measured ratio of the

134

S. D. PROBERT, C. B. THOMAS 40 30 w

1

20 tA "7

E

~o 9

>-

8

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•

-

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do Fig. 4.

7~ TEMPERATURE IKI

do

9b

Experimentally measured values for the electrical conductivity in the S-direction.

Nernst and Hall coefficients. 8 The thermoelectric power (in/~V K - ~) for single-band conduction is then given by: ~= 86'2{[2~]-

4}

(4)

where Fo({) and F1(~) are the Fermi integrals of orders zero and unity respectively and { is the reduced Fermi energy measured from the edge of" the conduction band. Equation (4) implies that the absolute thermoelectric powers of the Bi specimens containing more than 9 atomic '~, Sb are isotropic. This condition is satisfied within the limits of experimental error by the measured values of a(S) and :~(P) listed in Table 2. The values of a calculated using eqn. (4) with Fermi energies of 14, 33, 19, 24 and 31 meV for samples P9, PIS, P36, $23, $29 and $32, respectively (deduced by fitting eqn. (4) to the appropriate experimental curve near 70 K) are shown as continuous lines in Figs. 1 and 2. Examination of these curves indicates good agreement between experiment and theory in the temperature range from 60 to 80 K

TRANSPORT PROPERTIESOF SOME BISMUTH-ANTIMONYALLOYS

135

except for sample $23. This exception is attributed to the presence of a higher concentration of impurities than originally suspected and which was discovered subsequently by spectrographic analysis. Thus it is meaningful to describe Bi containing between 9 and 36 atomic O//oSb as an extrinsic conductor below 80 K.

Electronic thermal conductivity It is assumed that the total thermal conductivity, k, can be expressed as the sum of its electronic and lattice components--k~ and k L, respectively. Since Bi containing up to 6 atomic % Sb behaves as an intrinsic conductor there is, for these alloys, a bipolar contribution to the thermal conductivity. Thus k,, can be described by: k,. = k. + kp + k.p

(5)

where k., kp and k.p are the thermal conductivities due respectively to the electrons alone, to the holes alone and to bipolar diffusion. From Gallo et al.,19 k,, can be described by: k~ _ 1 ( ~ ) 2 {Y(~.)a. + Y(¢p)ap} + \( a "a2a P /) (% - %, 2 aT a

(6)

where: 3F2(¢) Y ( ~ ) - Fo(~)

4[F,(¢)] 2

L~J

(7)

and Fj(~) is the Fermi integral of order j, ~ the reduced Fermi energy of the electrons or holes, e the electronic charge, 1~the Boltzmann constant and cr equals either a(P) or a(S) according to the direction considered. The values of ~ were calculated from those for c~, and % given in Table 4 via eqn. (4) and thus Y(~,) and Y(¢p), evaluated from eqn. (7), were found to be equal. Hence (ke/aT) can be treated as a pseudoLorenz number for comparing the calculated electronic conductivities with those from previous studies. The Lorenz numbers parallel with, or perpendicular to, the trigonal axis, L(P) or L(S), respectively, were determined from eqn. (6) using the appropriate information from Tables 2 and 3. The deductions, shown in Table 5, TABLE 5 THE LORENZ NUMBERS AND BIPOLAR CONTRIBUTIONS 1N DIRECTIONS PARALLEL WITH, AND PERPENDICULAR TO, THE TRIGONAL AXIS

Atomic % Temperature L(P) concentrationofSb (K) (10-8V2K-2) 0 0a 6 6b 0 6

80 100 80 80 60 60

a Data from reference 19 b Data from reference 8.

2.21 2.30 2-1 2.23 2.14

L(S) B(P) B(S) (10-8V2K 2) (10 8V~K-Z) (IO-8VZK z) 2.69 2-98 2-76 2.66 2-41 2-32

0.28 0.15 0.26

0.76 1.14 0.97

0.11 0-07

0.33 0.25

136

s . D . PROBERT, C. B. THOMAS

indicate that for both Bi and Bi containing 6 atomic ~o Sb, the electronic conductivity is anisotropic. The bipolar contributions, B(P) and B(S), have also been included in Table 5 since they are responsible for the anisotropy. The quantitative agreement between the value of L(S) determined from eqn. (6) and that obtained by extrapolation from thermomagnetic data 1s for Bi containing 6 atomic ~o Sb provides corroboration for the inclusion of the bipolar expression in eqn. (6). In the previous section it was shown that Bi-Sb alloys with 9 or more atomic ~o Sb behave as extrinsic n-type conductors below 80 K and so the Lorenz numbers of such alloys are assumed to equal the degenerate value of 2-45 x 10- 8 V - 2 K 2 over the temperature range from 60 to 90 K. Then the electronic thermal conductivities can be calculated from these Lorenz numbers and the appropriate electrical conductivities (as given in Tables 2 and 3).

Lattice thermal conductivity The use ofeqn. (5) and the appropriate values of the Lorenz numbers from Table 5 enable the lattice components of the thermal conductivities of the alloys to be evaluated from the experimental data (see Figs. 5 and 6). Also shown in Fig. 7 are data determined directly from thermomagnetic measurements, 15 A notable 18 •

-

P0

o - P36

16[

a. - P0 'e - P18

E•.lZ

1(:

8 u

8

/.,

I

TEMPERATURE

Fig. 5.

(K)

Experimental values of the thermal conductivity of the various specimens in the P-direction.

TRANSPORT

~

PROPERTIES

OF

SOME

BISMUTH-ANTIMONY

a

ALLOYS

137

• - SO

20

18 •

o

- 543

a •

- 523 - $29

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16

v

14

E

>-

_>

b 10 O Z O ~D

j

8

r~ LU T

G

4

2

TEMPERATURE (K) F i g . 6.

Experimental values of the thermal conductivity in the S-direction.

characteristic is the sharp decrease of the lattice thermal conductivity between zero and 5 atomic % Sb. Analysis of the data shown in Fig. 7 has been accomplished using the phenomenological theory of Abeles 2° for phonon scattering by other phonons, mass defects and strain fields. In this way, the lattice thermal conductivity can be described as: tan- ~ U 1

(8) l 4-~fl

U4-31 U2 + I

where:

tanu-1U

i

U=

I 10 6 ~ 3"64 x

-k o- F - -~

(9)

138

S. D. P R O B E R T , C. B. T H O M A S

---o

....

5 DIRECTION (PRESENT WORK)

FROM THERMONAGNETIC ....... x ......... MEASUREMENTS (15} 22

cl) # =0. E = O PREDICTED,~(b) ,8 = 2. ~ = 25 ~c1,8 =0, E =25

2C E

I~

C 21 8 z

o,¢ s ¢,

8 ~

6

o:

l.

(o) icl

2

Ib)

w ~

.

~

×

..............................

~s ~

o

1'2

Fig. 7.

16

2JO 2'/., ATOMIC % Sb

218

3'2

3'6

~'o

s

~.'~

Lattice thermal conductivity in the S-direction at 80 K.

F = x(1 - x )

+ a

(10)

and: 5 ' 7 M¢~O3

k0 -

~/T 3

(11)

Also M is the mean atomic mass of the alloy and AM the difference between the values for Bi and Sb: 6 is the cube root of the lattice unit cell which, for the alloys, is evaluated using Vegard's rule: A~ is the appropriate difference for Bi and Sb. A linear interpolation according to concentration x of Sb in the alloy is assumed in order to evaluate the Debye temperature, 0, from those for Bi and Sb, namely 117 and 140 K, respectively. The ratio of the normal and umklapp processes when they both have the same frequency dependence is denoted by/~, and ~ is a parameter describing the relative importance of strain fluctuations to mass fluctuations for point-defect scattering. From eqn. (11) it can be deduced that the Griineisen parameter ~/must equal 2.2 for Bi if k o is to agree with the experimentally determined value of k L. A similar value of ~ applies for Sb, as may be appreciated by using the White and Woods zl thermal conductivity measurements. Hence ~ equal to 2.2 is adopted for all the Bi-Sb alloys considered.

TRANSPORT PROPERTIES OF SOME BISMUTH-ANTIMONY ALLOYS

139

The lattice thermal conductivity, k L, was evaluated from eqns. (8) to ( ! 1) (using a computer) for 0 < fl < 2 and 0 < e < 100. With fl and e both equal to zero, eqn. (8) reduces to the Klemens 22 expression for k L which is shown as curve (a) of Fig. 7. Although it describes qualitatively the shape of the thermal conductivity variation up to about 20 atomic ~ Sb, quantitatively it leads to erroneously large values. The best fit ofeqn. (8) to the present data--shown in Fig. 7 as curve (b)--is obtained with fl and ~ equal to 2 and 25, respectively, but even then this closest correlation only applies for concentrations up to about 25 atomic ~o Sb. Although the position and shape of the theoretical thermal conductivity curve could be varied considerably by altering e, the calculation was relatively insensitive to changes in [t. (This is demonstrated by curve (c) of Fig. 7, for which fl and ~: equal zero and 25, respectively). Such a conclusion suggests that strain field scattering makes a much larger contribution to the thermal resistance than that arising from normal (i.e. Ntype) processes. Above 25 atomic ~o Sb, the experimental values o f k L increase with concentration whereas those calculated from eqn. (8) continue to decrease. Such a disagreement implies the presence of another mode of heat transfer. This could be bipolar diffusion since above concentrations of 25 atomic ''/ /,, Sb, the energy gap decreases rapidly until a semi-metallic alloy is formed. 6 At these higher Sb concentrations it is desirable to perform thermomagnetic experiments to eliminate all electronic forms of heat conduction.

CONCLUSION

From a cross-plot of the experimental data it is suggested that the maximum figure of merit for the Bi-Sb system of alloys would occur for Bi with 16 ( + 2) atomic ";i Sb at 80 ( + 1) K. This corroborates closely the conclusion of Smith and W o l f e . 23

ACKNOWLEDGEMENTS

This paper is published with the permission of the Managing Director of the Reactor Group of the United Kingdom Atomic Energy Authority to whom we are indebted for financial assistance. We also wish to thank the Science Research Council for supporting this project.

REFERENCES 1. A. P. |VANJUK,A. S. OKHOTINand A. S. PUSHKARSKY,Energ.v Cont'ersion, 17, (1977), pp. 19 21. 2. C. B. THOMAS and S. D. PROBE~T, Brit. J. Appl. Phys., 15, (1964), pp. 1120 3. 3. S. D. PROBI~RT, Thermal insulation in relation to cryogenics, HMSO. 1968.

140 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

S. D. PROBERT, C. B. THOMAS A. S. OKHOTIN and A. S. PUSHKARSKY,Dokl. Akad. Nauk., 203(2), (1972). A. S. OKHOTIN, Thermoelectric generators, Atomizdat, Moscow, 1971. A. L. JAIN, Phys. Rev., 114, (1959), p. 1518. S. TANUMA, J. Phys. Soe., Japan, 14, (1959), p. 1246. M. E. ERTL, D. W. HAZELDENand H. J. GOLDSMIO,Proc. Semiconductor ConJerence, Exeter, 1962, p. 777. H. J. GOLDSMIO, Phil. Mag., 8, (1963), p. 1225. D. M. BROWN and S. J. SILVERMAN,Phys. Bey., A136, (1964), p. 290. R. T. BATE, Thermoelectric Properties oJ' Bi2Te 3, Batelle Memorial Institute, Ohio, Technical Report No. 3, 1960. C. B. THOMAS, M.Sc. Thesis, University of Wales, 1965. D. M. BROW~ and F. K. HEUMANN,J. Appl. Phys., 35, (1964), p. 1947. W. YIM, Trans. Metall. Sot'. AIME., 236, (1966), p. 474. M. E. ERTL, Ph.D. Thesis, University of London, 1964. B. ABELESand S. MEIBOOM,Phys. Rev., 101, (1956), p. 544. B. S. CHANDRASEKHAR,Phys. Chem. Solids, 11, (1959), p. 268. W. ENGELER, Phys. Rev., 129, (1963), p. 1509. C. F. GALLO, B. S. CHANDRASEKHARand P. H. SETTER, d. Appl. Phys., 34, (1963), p. 144. B. ABELES, Phys. Rev., 31, (1963), p. 1906. G. K. WHITE and S. B. WOODS, Phil. Mag., 3, (1958), p. 342. P. G. KLEMENS,Phys. Rev., !19, (1960), p. 507. G. E. SMITH and R. WOLFE, J. Appl. Phys., 33, (1962), p. 841.