Anisotropy of the seebeck coefficients of bismuth telluride

Adx.:~cedEnergyComersion. Vol. 1, pp. 99-105. PergamonPress. 1961. Printedin Great Brilain

ANISOTROPY

OF

THE

SEEBECK

BISMUTH

COEFFICIENTS

OF

TELLURIDE

JANE H. DENNIS* Summar.v--Anisotropy in the Seebeck effect is found to arise in two separate theoretical situations. It will arise in a single-carrier system when there are two competing scattering mechanisms. One of these (lattice scattering, for example) must be isotropic. The anisotropy will arise in a two-carrier system when there is only a single scattering mechanism. In this case, the anisotropy arises because the conduction and valence band structures are not identical; hence the ratio of the mobility of holes to the mobility of electrons is different in the two principal crystallographic directions. The single-carrier, multiple-scattering theory of the anisotropy, is confirmed experimentally on n-type iodine-doped Bi2Tea at low temperatures (100-300°K). The multiple-carrier, single-scattering theory is confirmed on stoichiometric undoped BizTe~ at high temperatures (30(O700°K), when mixed conduction has set in and lattice scattering predominates. INTRODUCTION MATERIALS with anisotropic crystal structures may in general be expected to have anisotropic electrical and thermal conductivities. However, the Seebeck coefficient tensor will be anisotropic only under two rather special conditions. One way to make the Seebeck effect anisotropic is to cause two scattering mechanisms to operate simultaneously. One of these mechanisms must be anisotropic, although the other may be isotropic. Another way is to cause comparable conduction to take place by electrons and holes simultaneously. We have obtained anisotropy in the Seebeck effect from both of these causes in the same material, Bi_oTe3, although over different temperature ranges and with different dopings. HISTORY Anisotropy in the Seebeck effect was first observed in 1927 by Boydston [1] in the semimetal bismuth. A preliminary search through the abstracts (in 1959) led us to believe that the theoretical calculation of the Seebeck coefficients of bismuth had not been carried out. so we proceeded with it ourselves. Later we discovered a paper published by B. S. Chandrasekhar [2] in October 1959, in which the calculation had been made. Our independent results for bismuth agree with his. We have also made a similar calculation for bismuth telluride. H. J. Goldsmid [3] in 1957 observed the kind of anisotropy in the Seebeck effect in Bi2Te3 which we now believe is caused by mixed scattering. Goldsmid however showed, by a very approximate treatment, only that it could be caused by a relaxation time which was a different function of energy in different crystal directions. His approximations also predicted a relationship between the anisotropy of the Seebeck coefficients and the conductivity ratio of the form

S 1 - - Sa = -- (~) d (crl/o3),/d in f T

(1)

* Now at Lincoln Laboratories, Massachusetts Institute of Technology, Lexington, Massachusetts. 99

100

JANE H. DENNIS

where the subscript 1 refers to directions parallel to the cleavage plane, and 3 to directions perpendicular to it. The plus sign refers to p-type conduction. We have shown that the necessary relaxation time tensor with anisotropic energy dependence can be obtained by the simultaneous action of two competing scattering mechanisms, and we have made a rather complete confirmatory analysis using the many-valleyed model of the band structure of Bi2Tea. This is the same model suggested and used successfully by Drabble [4, 5, 6] to interpret magnetoresistance data on this material. THEORY OF THE CONDUCT[ON

SEEBECK COEFFICIENT FOR AND A MIXED SCATTERING

SINGLE-CARRIER MECHANISM

The perturbation of the equilibrium particle distribution function arising from electricat and thermal gradients is found by means of Boltzmann's equation. The flux of particles due to these gradients is then computed from the equation

j. = / ~2 j f V v E d p

(2)

according to integration based on Drabble's many-valleyed model of the band structure of BizTe3, referred to before. The relation between the flux of particles and the gradients assumes the form J , = a • Vr'~ + la " V,T__ T

(3)

v.rhen

= E - e6

14)

is electro-chemical potential is the Fermi level 4' is the applied electrical potential The Seebeck coefficient of a material such as Bi2Ye3 is given by the limit of the Seebeck voltage measured across a thermocouple made of this material and another one (such as copper) whose Seebeck coefficient is much smaller, as the applied temperature difference of the two junctions approaches zero. Hence it is given by the gradient of potential per unit temperature gradient, with the current equal to zero. or Ot-I . [~ S --

(5)

eT

The et and [3 tensors are anisotropic, but their ratio is not unless the relaxation-time tensor has a non-factorable energy dependence. In an anisotropic crystal the lattice relaxation time will be of the form ~'lt = aE -1/'-

"ra3

:

bE -12

(6)

which has a factorable energy dependence. Hence lattice scattering alone, although anisotropic, will not give an anisotropic Seebeck effect. Suppose we postulate that another (possibly isotropic) form of scattering (by impurities, t\~r example) with a relaxation time of the form ~-=ce q

operates simultaneouslywith lattice scattering and independently of it.

.Anisotropy of the Seebeck Coefficients of Bismuth Telluride

101

-300 f

LId rr ¢.9 kU

__

::L -200

~

PARALLEL ORIENTATION

~

I--

PERPENDICULARORIENTATION

Z _W _t2 u. kl. t,, 0

0hi ID Ud Ld tO

-JO0,

, ~00

200

i

500

~

]

T(K")

FIG. 1. Seebeck coefficients of iodine

400

I

t

500

l

600 700 800 900 I00~

doped n-type Bi2Te3.

2oof ~J ~D LL/

Z L,J

L I

~00f

LU n~ PERPENDICULAR

•iO0t I00

200

500

400

500

600

T(K=) FIG. 2. Scebeck coefficients of undopedp-type Bi=T¢3.

i

~ i

=!T!

700 800 900 I000

102

JANE H. DENNIS

Then

1

1

I

7-

7" lattice

- = -

+ -

(7)

7" impurity

SO

1

1

'7"11

a E -a/"

+

I

--

{8)

cEq

and 1

1

r33

bE-l/2

1 ' cE q

T h u s the relaxation time tensor will have a non-factorable energy dependence. Hence an anisotropic lattice scattering acting simultaneously with even an isotropic impurity scattering will give an anisotropic Seebeck effect.

I0000

I

I

I

I

l

I

!

I

I

(

I

1

I00

m

m

¢J _o

\j

o >_

I000

OARALL L

i.-

ORIENTATION

ro

-

i.> i.(,.)

aE) z o °

z

p

~oo I

RATIO OF CONOUCTIVITtES__

"k

~ "

i 4

: [ IOOOK

i

I

,

I

;

!

o (,.)

I 1 IOOO"K

FIG. 3. Conductivity and conductivity ratio o f i o d i n e d o p e d n - t y p e Bi=Tea.

Anisotropy of the Seebeck Coefficients of Bismuth Telluride EXPERIMENTAL

103

VERIFICATION

The Seebeck coefficient of iodine-doped (n-type) Bi2Te3 is shown in Fig. 1 and is seen to be anisotropic by 25 out of 200 tzV/deg. This anisotropy is believed to be an indication that there is appreciable scattering by a mechanism other than lattice scattering in the iodine-doped material. This conclusion is consistent with the fact that the Seebeck coefficient of undoped p-type material, as shown in Fig. 2, is seen to be isotropic at temperatures below that at which mixed conduction begins. I0000

I

,

,

,

I

~

~

,

,

,

,

I00

0 lIENTATION

r~ I-

I0 >

I000

v

-

-

0

-

0

Z

RATIO O F O Z 0 o

I00

L

i

i

]

L

L

J

z

I 0 0 OK

I

I

i

J

IO00°K

FiG. 4. Conductivity and conductivity ratio of undoped p-type Bi2Te3

The conductivity ratio of n-type iodine doped material plotted against log AT is shown in Fig. 3 and is seen to have a slope of 0-25. From equation (1) the difference between the Seebeck coefficients should be

S z - - S3 = -- (~) d(al/crs)/d ln tT" =--(86"2tzV/deg) × 0 . 2 5 - - - 2 2 / ~ V / d e g

(9)

104

JANE H. DENhqS

which is very nearly that observed in Fig. 1. As mentioned previously, whereas this is the same calculation made earlier by H. J. Goldsmid, the present theory for the effect and the interpretation of the results is a good deal more complete than the earlier ones. The slope of the conductivity ratio of undoped p-type material as shown in Fig. 4 is seen to be zero. T H E O R Y OF T H E S E E B E C K C O E F F I C I E N T FOR T W O - C A R R I E R C O N D U C T I O N AND A S I N G L E S C A T T E R I N G M E C H A N I S M For the two-carrier case with a single scattering mechanism the current is given by T

(I0)

and S = (ah

~,)-i

. (13h -

i~,)

eT

(ll)

which assumes the form St.a = - -

'

(12)

'

C + D tm~!

~e l,~

where A, B, C, and D are functions only of the Fermi levels. In fact B and D are either very large or very small for one-carrier systems, and of intermediate value for two-carrier systems. It is seen from equation (12) that, if we assume only a single scattering process, the Seebeck coefficients will be anisotropic only if the ratio of the mobility of holes to the mobility of electrons is different in different directions, and if there is simultaneous comparable conduction by holes and electrons. EXPERIMENTAL

VERIFICATION

There is mixed conduction in BizTes beginning slightly above room temperature, and it is seen in Fig. 2 that the Seebeck effect does become anisotropic in this temperature range. We have also made a theoretical calculation of Seebeck coefficients vs. temperature using the mobility ratios of Drabble, Groves and Wolfe [5] (which we assumed to be constunt with temperature). We obtained the Fermi levels from the conductivity data of Shigitomi and Mori [7] and then compared the calculated Seebeck coefficient curve with Shigitomi and Mori's experimental Seebeck coefficient curve on the same sample from which the conductivity data was taken. The resulting plot is shown in Fig. 5 and is seen to follow their experimental points quite well. CONCLUSION It has been shown theoretically and verified experimentally, using as primary example the semiconductor Bi2Tea, that anisotropy of the Seebeck effect may be obtained basically in two ways. One way is to cause comparable conduction to take place simultaneously by holes and electrons, in the presence of a single scattering mechanism. Another way is to

Anisotropy of the Seebeck Coefficients of Bismuth Telluride

105

cause two scattering m e c h a n i s m s to operate simultaneously. One of these mechanisms must be anisotropic b u t the other m a y be isotropic. In general, a mixture of the above two conditions would also be expected to give an anisotropic Seebeck effect. 0

200

T,

--~ B +

t.,r ,,..b w

+

CALCULATED POINTS OF Sm C A L C U L A T E D POINTS OF S 3 E X P E R I M E N T A L POINTS OF G E T O M , ANO MOR, FOR S,

:& t-Z

IOO

I.L 0 v

\4- ~

Lid ~n

~÷

W

O

~- -'11pERPENDICULAR ~.... ORIENTATION

I

I

t

1

,oo

3oo

400

5oo

pA RA L L E L~--Q~Q "-~:) ORIENTATION + +

÷

+

T(*C)

+

FIo. 5. Calculated curves of the Seebeck coefficients of p-type Bi,-Te:;along with the experimental points of Shigitomi and Mori [7]. Acknowledgement--This work is based on a Ph.D. thesis completed in the Electrical Engineering Department of the Massachusetts Institute of Technology. The author wishes to express her gratitude to her thesis supervisor Prof. Richard B. Adler for his encouragement and many stimulating discussions. This work was supported in part by the U.S. Navy under Contract Nonr 1841(5 I), and by the Electronic Research Directorate of AFCRC under Contract AF-19(604)-4153. REFERENCES [I] R. W. BOYDSTON,Phys. Rev. 30, 911 (1927). [2] B. S. CI-IANDRASEKHAR,J. Phys. Chem. Solids 11,268 (1959). [3] H. J. GOLDSMID,Ph.D. Thesis, University of London (1957). [4] J. R. DRABBLEand R. WOLFE,Proc. Phys. Soc. 69, 1101 (19561. [5] J. R. DRABBLE,R. D. GROVESand R. WOL~, Proc. Ph.vs. Soc. 71,430 (1958). [6] J. R. DRABBLE,Proc. Ph)'s. Soc. 72, 380 (1958). [7] SHIGITOMIand MORI, Jr, Phys. Soc. Japan, I1, 915 (1956).