Investigation of the room temperature figure of merit of Pb0·7Sn0·3Te

Energy Conversion.

Vol. 10, pp. 113-117.

Pergamon Press, 1970. Printed in Great Britain

Investigation of the Room Temperature Figure of Merit of Pbo.Sno,3Te H, L, LIVINGSTONt, R. N, TAUBER:]:and I, B, CADOFF§ (Received 20 October 1969)

Introduction The isomorphous semiconducting alloy system Pb~Snl-~Te has been the subject of numerous recent investigations [1]. It is known that these alloys are good thermoelectric materials. The thermoelectric figure of merit of a material is defined as: ~'°a (deg) -1 Z = ~

or

ZT--

a°'aT Ke + Kz,'

(1)

where T is the average temperature for a small AT; c~ is the thermoelectric power, a is the electrical conductivity; Kz is the total thermal conductivity, consisting of Kz, the lattice contribution, and Ke, the electronic contribution. Machonis [2] found that the alloy Pb0.TSno.aTe had the highest room temperature figure of merit (Z=300°K); the value obtained was 0.4. The figure of merit, however, is sensitive to carrier concentration, but no attempt was made by Machonis to maximize Z. A theoretical analysis by Joffe [3], and extended by Chasmar and Stratton [4] and Wasscher et al. [5] provided a means of predicting the maximum Z that could be obtained if the carrier concentration were optimized. As in the case of PbTe, PbTe-SnTe alloys were found to exist over a narrow range of composition, and for alloys up to Pb0.ssSn0.1~Te the Pb/Sn-rich solidus exists on the excess metal rich side [6]. For compositions greater than 15 per cent the Pb/Sn-rich solidus occurs in the range of excess Te compositions. In addition, retrograde solubility of the Pb/Sn-rich solidus was found for alloys up to 15 per cent [6], enabling control of carrier concentrations by equilibrating single crystal specimens in a Pb/Sn-rich atmosphere. These data suggested that similar behavior could be expected in the Pbo.TSn0.aTe alloy and that the retrograde solubility effect could be used to optimize the figure of merit.

Experimental Methods ShTgle crystal preparation

The materials used for single crystal growth were ultra high purity (99.999 per cent) Pb, Sn and Te.

Charges of the desired stoichiometry were prepared by co-melting weighed amounts of Pb, Sn and Te in vacuum sealed quartz tubes at 1000°C, for 24 hr. The capsules were shaken frequently to insure homogeneity. Single crystals were grown in carbon coated, evacuated, tipped Vycor tubes 20 mm in diameter. The capsules containing the pre-homogenized charges were lowered through a Bridgman-Stockbarger furnace at a rate of 1 in. per day. Machonis [7] found that only the front inch of the resulting crystal could be used, since the longitudinal segregation (as determined from an X-ray fluorescence analysis) is minimal here. Since the first solid precipitating out was lower in per cent SnTe than the charge, the composition Pb0.TSno.aTe required a charge composition of Pbo.raSn0.arTe. The seed end of the grown crystals were cut into 0.3 cm slices. Standard metallographic examination sufficed to determine if grain boundaries were present. The slices were then cut into rectangular parallelepipeds of about 1.2 cm × 0.3 cm × 0.3 cm. Composition control

To control the Te/(Pb/Sn) ratio the specimens were diffused with (Pb0.TSn0.a)o.55Te0.45 charges. In this fashion Pbo.TSn0.3Te specimens were obtained lying closest to the stoichiometric composition. The diffusion was carried out in 13 mm Vycor capsules each ,-~ 2 in. in length. They were prepared so that the diffusion charge was on the bottom of the capsule and the sample lay horizontally on a ledge, about one inch above the charge. This permitted a physical separation between the charge and specimen, which was necessary because the charge exists in the two-phase liquid plus compound region at the diffusion temperature. Equilibrium is established through the vapor phase. The capsules were kept at temperature for sufficient time to approach equilibrium. The times necessary to approach equilibrium were taken from the Brebrick and Allgaier [8] work on PbTe and the Tauber [6] work on the alloys up to 15 per cent SnTe; they varied from 4 hours to 30 days. After diffusion, the capsules were brine quenched.

Electronic and thermoelectric measurements Standard Hall and resistivity measurements were t Present address: Department of Metallurgy, Polytechnic Institute of Brooklyn, Brooklyn, New York. made utilizing chopped d.c. to eliminate any thermo**Present address: Bell Laboratories, Allentown, Pennsylvania. electric and thermomagnetic effects that might arise § Department of Metallurgy and Materials Sciences, New York during the measurements. University, Bronx, New York. 12 113

114

H.L.

L I V I N G S T O N , R. N. T A U B E R and L B. C A D O F F

Thermoelectric parameters were measured on the Z-meter using the apparatus developed by Harmon [9] and Putley [10]; the specific procedure used being outlined by Zalar [11].

800

/

P - type

/

/

/

Results

Equilibration and carrier concentration control 700

The specimens were all P-type and near degenerate. The carrier concentration, p, was determined from the Hall coefficient, Ra by the following relationship:

1

d

P = Raq"

(2) ~"

The room temperature dependence of the Hall coefficient, Rtt and hole concentration on annealing temperature (temperature at which the single crystal specimens were diffused in the Pb/Sn-rich atmosphere) is shown in Fig. 1. Retrograde solubility exists on the metal-rich side of the solidus, thus enabling the carrier concentration to be varied for the maximization of the figure of merit.

600

500

!

2.5

~

4

o -R H

450

2.0

-~

i 50.02 % "re

50.03

50.04

Atomic,

2

'0

Fig. 2. Pb/Sn-rich solidus.

I0 '9

c"

,.5

9

'-~ o

-r,

Lo

7'

8

0~

I50

=o

o.s

a

c aa

125

o

I00

E

r 5o.0~

50,00

160

¢ w

ls

50

500

550 600 65'0 Annealin~ lempetoture,

700 "C

750

t 800

3 XlO,S

_~.

Fig. 1. Hall coefficient and carrier concentration dependence

~ 75

on annealing temperature. 50

All specimens were P-type so that the solidus lay completely on the Te-rich side of the stoichiometric composition. By assuming that each excess Te atom added one hole, or: N(Te) -- N(Pb/Sn) ----p cm -3 (3) the carrier concentration was converted into per cent deviation from stoichiometry. The calculated solubility limit of the Pb/Sn-rich side of the solidus as a function of temperature is shown in Fig. 2.

25

-

....

;

2XlO '~ 3

,,

r

,,'

4

f

~

'

$ 6 ? ~ 9 10'9

2

Corriet c Onc~ttofiOfl,

9

,

,

,r

~

3

4

5

6 7

c~ 7'

eM1-3

Fig. 3. Thermoelectric power and resistivity dependence on

carrier concentration.

of c~ with the logarithm of p will be discussed subsequently. Analysis of Z-meter data yielded the total thermal conductivity which is listed in Table 1 along with other pertinent data.

Figure of merit parameters Figure 3 shows a plot of resistivity (p) and thermoelectric power (=) vs. hole concentration (p) for Pb0.rSno.3Te diffused in a Pb/Sn-rich atmosphere. The discontinuity in the slope of the two linear segments

Optimization of figure of merit By neglecting degeneracy and the electronic contribution to the total thermal conductivity in Equation [1] and maximizing Z with respect to p (number of holes),

Table 1. Some electronic properties of Pbo.TSno.3Te Specimen

KT(W/cm°K)

e (Q-cm)

I/KL (era°K/W)

a(/~V/°K)

Z j (300°K)

1 2 3 4

1 . 8 4 x 10-2 1 ' 8 0 × 1 0 -8 1-96.'< 10 -2 I ' 9 6 × 10 -2

0 . 8 6 × 10-3 l ' 1 5 x 1 0 -3 1 - 3 2 × 10 -3 1 . 4 9 × 10 -3

97 86 71 68

71 91.1 128-8 155

0.29 0.31 0-30 0-35

Investigation of the Room Temperature Figure of Merit of Pbo.TSn0.3Te

115

Joffe [3] obtained an expression which was simplified by Wasscher et al. [5]. The final result was: \eq/

~

exp

(deg)-L

By substituting the values for k (Boltzmann constant), e (2.718), and q (electron charge) and multiplying Equation (4) by 300°K, the temperature at which all measurements were performed, we obtain: Zj(300°K) = 1 . 2 0 6 ×

10-6 (~LL) exp {11,600~}.

(5)

From this equation it is apparent that maximization of Zj(300°K) involves obtaining the optimum values of a, KL, and ~. In the theoretical figure of merit Z j T , Equation (1), the lattice contribution to the total thermal conductivity is needed. The total conductivity in semiconductors is: K z = KL + K , + K,, (6) where Ka the ambipolar contribution to the total thermal conductivity which need only be considered in near intrinsic material and was neglected in this study. The term Ke is the electronic contribution to the total thermal conductivity and was obtained from the WiedemannFranz Law, LeT K, -(7) p where ~ is the Lorentz ratio. The value of ~ for degenerate semiconductors and metals is 2.44 × 10-s VZ/ °K2. At 300°K, the numerator in the WiedemannFranz Law, LeT, is 7.32 × 10-6VZ/°K. The lattice thermal conductivity may therefore be calculated from the relation: KL = KT -- Ke. (8) The reciprocal of KL (lattice thermal resistivity) is listed in Table 1 and in Fig. 4, it is plotted vs. per cent deviation from stoichiometry. liC' ""

JO0 -

t

9oi so

~

~F

(4)

2.

~-

iC ""~

I

IM

=E "6

,p 4

~

3

'i

IC ~ 2

3

4

5

Cc:rier

6

7

concenlrotion,

5

9

10 ~

2

Cm ~-

Fig. 5. Figure of merit as a function of carrier concentration.

and the value obtained for Zmax experimentally was 0.81 × 10-3 °K -a, corresponding t o a value of Zmax (300°K) of 0"243. To obtain correlation with the theoretical value of the figure of merit, the values of ~, p, and KL were substituted into Equation (5) for Z j (300°K). The values obtained for each specimen were expected to yield the same value as Z : (300°K) as this is the maximum value that may be obtained for this composition, neglecting degeneracy and the electronic contribution to the total thermal conductivity. The results are shown in Table 1. An average value for Z j (300°K) of 0.31 was obtained. In their analysis of the maximization of the figure of merit Wasscher et al. [5], presented a plot of Z j T as calculated from Equation (4) vs. ZmaxT calculated including degeneracy and the electronic contribution to the thermal conductivity for three values of r; 0, 1 and 2. Their curves are reproduced in Fig. 6. In order for our experimental value of 0-243 for Zmax (300°K) to agree with the calculated result of 0.31 for Z j (300°K), r = 0 must be used. This value corresponds to scattering by acoustical modes and is usually used to describe the scattering of PbTe [12]. Thus it is seen that the theoretical evaluation of Z j T using Equation (4), when corrected, agrees very well with the experimental determination of ZraaxT.

70 -

Discussion -~

6° 1

Stoichiometry and figure of" merit

-

5ol

5000

50 0 ;

50.02

A~om[c,

50.03

50.04

%Te

Fig. 4. Lattice thermal resistivity vs. atomic per cent Te.

Figure o f merit

With the measurements of ~, p and K~,, the figure of merit, Z ( T ) can now be calculated from Equation (1) at each carrier concentration. This is shown in Fig. 5

As in the case of PbTe [8] and PbxSnl-xTe (x ~ 0.15) [6], Pb0.vSn0.3Te was found to exist over a narrow range of composition. Furthermore, the Pb/Sn-rich solidus exhibits retrograde solubility so that equilibrating as grown single crystal specimens of Pb0.vSn0.3Te in a Pb/Sn-rich atmosphere can be used to control the carrier concentration [6]. The observed retrograde solubility is at variance with a report by Calawa et al. [16] but consistent with the results of Tauber [6] for alloys of

116

H. L, LIVINGSTON, R. N. TAUBER and I. B. C A D O I ~

lower Sn contents. These alloys are all P-type, since all compositions of Pb0.vSn0.aTe lie completely on the Te-rich side of the stoichiometric composition. For comparison of the value for Zmax (300°K) obtained in this study with Machonis' [2] data on Pb0.vSn0.sTe, we again consider Equation (5) for Z,r (300°K). The value Machonis obtained on undiffused specimens was 0.4 and upon substituting his values of a, a, and KL into Equation (5) a value of 1.2 was obtained. Using this value and r = 0, we obtain from Fig. 6 a value of 0.8 for ZmaxT, for his data. I0

5

////I

carrier concentration as given by the following equation: = k (r + 2 + In N v - - Inp). q

(9)

The scattering parameter was shown to be r = 0, and the term Nv is the density of states in the valence band, given by:

Nv

=

2 (2rtmokT~ 3/2-'(mn~S/2

~-~--/

= 2"5

× 101'

~~ /

(toni 3/2,-

(10)

\ mo/

where mn is the effective hole mass and m0 is the free hole mass. Using the appropriate constants, Equation (10) reduces to: { 3 m~ }/xV e~=8.615 × I0 -s 46.616 + ] In - - -- l n p 1710

~"

(11)

5N , 0.5

r=2

0.2

r=O

0.~

~

,

0.2

,

i

The number 46.616 corresponds to (0 + 2 + In 2.5 + 19 In I0). Equation (I I) indicates that a plot of ~ vs. In p should yield a straight line. This data, as plotted in Fig. 3, shows non-linear behavior, indicating that mh is not a constant. ,

0.5

I

i 2

=

i I I i' ; , 5 IO

Fig. 6. Zm=xT, calculated including degeneracy and electronic thermal eomtuefi~W, as a function of ZJT for r = 0, 1 and 2.

This theoretical value of ZmaxT is approximately three times higher than the experimental result of ZmaxT obtained in this study. The difference may be accounted for by considering the variation of Za (300°K) on composition. The plot of Z j (300°K) vs. per cent SnTe in PbTe, using Machonis' undiffused data as shown in Fig. 7 indicates the sensitivity near the 30 per cent alloys. The Pb/Sn ratio is obviously critical in the maximization of the figure of merit. t.3

/

0.52~-

Zj'i"

Ol 5 0 ~

0.48, 0.461 0.44 0.42 0.40 "~

0.38~-0 34[-"

030 f 028

t,2

0.26 i. 0

0.24~ 1

0.9 - 0.8 N~ 0.7 0.6 E 0.5 0.4

0.22~-" 0.201 J Z XIO ~

I 5

; 6

i ' l i E ; 7 8 9 I0 ~ Z 3 Carrier concentration, cm "]

l 4

I 5

Fig. 8. Effective mass dependence on hole concentration.

& o.3 E

I 4

0 .2

0.1 0

lI

~

io

20

Fig. 7.

~

I

~

t

30 40 50 60 70 Mole % SnTe in PbTe

Z j T vs. mole per cent SnTe

80

i

90

~00

in PbTe.

Dependence of the thermoelectric power on .carrier concentration For the P-type semiconductor using the spherical band approximation, the thermoelectric power depends on the

From Equation (1 I) the ratio m~,/mo can be determined at each carrier concentration. The results are shown in Fig. 8. At hole concentrations between 3 × 10 Is and 6 × 10 ~8 cm -3 it is apparent that mh/mo is approximately constant, i.e. mn,/mo ----- 0.218. Thus, at low hole concentrations, one valence band is operating (due to the constancy of ran~too). For hole concentrations in excess of 6 × 101Scm -a, mn/mo increases with increasing carrier concentration.

Investigation of the Room Temperature Figure o f M e r i t o f Pb0 r S n 0 - 3 T e

band

117

This increase in the lattice thermal resistivity with increasing number of defects is analagous to the increase in lattice thermal resistivity observed arising from other lattice imperfections. Conclusions

Votence

L o w m ,~

High m h

E

-, K

Fig. 9. Proposed band structure at room temperature.

A doubly degenerate valence band similar to that proposed by numerous authors [13-15] explains this variation in mh/mo and is indicated schematically in Fig. 9. At low hole concentrations valence band (V1) with a low effective mass acts. However, as more states become vacated, valence band (Vz) with a higher effective mass becomes operative, and the effective mass reflects contributions from both bands.

Thermal conductiviO'

In Fig. 4 it was seen that the lattice contribution to the total thermal conductivity is a function o f the deviation from stoichiometry or equivalently the atomic per cent Te. The number o f carriers are a direct indication o f the number o f defects in this system (excess Pb/Sn, n-type; excess Te, P-type) and therefore the deviation from stoichiometry. The closer we are to stoichiometry the more perfect the lattice, and the lower the lattice thermal resistivity.

(1) The phenomenon of retrograde solubility is found to exist on the Pb/Sn-rich side of the solidus of Pb0.TSno.aTe, enabling the control of carrier concentration. (2) The maximum r o o m temperature figure of merit, Zmax (300°K) for this alloy was found to be 0.243. (3) Because of the pronounced dependence of the figure of merit of this alloy on the Pb/Sn ratio, the alloys immediately surrounding this composition should be investigated. (4) The anomalous behavior of the thermoelectric power with hole concentrations is consistent with a two-valence band model. References

[1] A. J. Strauss, Trans. AIME 242, 354 (1968). [2] A. A. Machonis, Trans. ~lIME 230, 333 (1964). [3] A. F. Joffe, Semiconductor Thermoelements and Thermoelectric Cooling. lnfosearch, London (1957). [4] R. P. Chasmar and R. Stratton, J. Electron. Control 7, 52 (1959). [5] J. D. Wasscher, W. Alberts and C. Haas, Solid-St. Electron. 6, 261 (1963). [6] R. N. Tauber, Optical and electronic properties of some Pb=Snz-zTe alloys. Thesis, New York University (1966). [7] A. A. Machonis, Investigation of alloys of the system PbTeSnTe. Thesis, New York University (1964). [8] R. F. Brebrick and R. S. Allgaier, J. Chem. Phys. 32, 1826 (1960). [9] T. C. Harmon, J. appl. Phys. 29, 1373 (1958). [10] E. H. Putley, Proe. phys. Soc. 68B, 35 (1955). [11] S. M. Zalar, Scmiconducting compounds of the chalcopyrite structure AzBmC~v. Thesis, New York University (1959). [12] R. S. ALIgaier,J. appl. Phys. 32, Suppl. 2185 (1961). [13] R. F. Brebrick and A. J. Strauss, Phys. Rev. 131, 104 (1963). [14] J. R. Dixon and H. R. Reidl, Phys. Rev. 138A, 873 (1965). [15] R. N. Tauber, A. A. Macho,~is and I. B. Cadoff, J. appl. Ph)s. 37, 4855 (1966). [16] A. R. Calawa, T. C. Harman, M. Finn and P. Youtz, Trans. AIME 242, 374 (1968).