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JOURNAL OF APPLIED PHYSICS

VOLUME 85, NUMBER 12

15 JUNE 1999

Enhancement of thermoelectric power factor in composite thermoelectrics David J. Bergmana) and Leonid G. Felb) School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel

~Received 1 December 1998; accepted for publication 4 March 1999! The analytical properties of macroscopic transport coefficients of two-component composites are first used to discuss the thermoelectric power factor of such a composite. It is found that the macroscopic power factor can sometimes be greater than the power factors of both of the pure components, with the greatest enhancement always achieved in a parallel slabs microstructure with definite volume fractions for the two components. Some interesting examples of actual mixtures are then considered, where the components are a ‘‘high quality thermoelectric’’ and a ‘‘benign metal,’’ leading to the conclusion that considerable enhancement of the power factor is often possible, with but a modest reduction in the thermoelectric figure of merit, compared to those of the high quality thermoelectric component. Two possibilities for fabricating real composites with such improved thermoelectric properties emerge from this study: a parallel slabs microstructure of benign metal and high quality thermoelectric, and a sintered collection of benign metal grains, each of them coated by a thin shell of high quality thermoelectric. © 1999 American Institute of Physics. @S0021-8979~99!02112-X# I. INTRODUCTION

factor compared with other microgeometries. We then set up a simple calculational procedure for determining, in practice, whether an enhancement is possible for a given set of two components, as well as what the optimum microstructure ~slabs or cylinders! and volume fractions are, and what the ~maximum! macroscopic power factor achieved in that microstructure is. In this connection, it should be mentioned that the well known zig-zag construction that is often used in thermoelectric devices, where p- and n-type conducting segments are connected in series electrically, but in parallel thermally @see Fig. 1~c!#, can also be viewed as a special kind of composite material. In this case, the special microstructure has the effect of enhancing the voltage or electric field produced by a given temperature difference or gradient. However, it is quite easy to verify that this particular microstructure does not enhance the power factor, and of course it cannot improve the figure of merit. The rest of this article is organized as follows. In Sec. II we briefly describe a decoupling transformation, which allows the thermoelectric properties of two-component composites to be discussed in terms of two simple, uncoupled conductivity problems with the same microstructure. In Sec. III we construct an exact, optimal bound for the macroscopic thermoelectric transport properties of a two-component composite, which is valid under a very mild assumption. This is then used in Secs. IV and V to develop a procedure for finding candidate microstructures that enhance the macroscopic power factor for a given set of two pure components: In Sec. IV we show that when an enhancement is possible, the strongest enhancement is obtained in a parallel slabs microstructure. In Sec. V we derive sufficient conditions for obtaining such an enhancement in the parallel slabs microstructure, and also in the coated spheres assemblage. In Sec. VI we consider some potentially interesting composites where one component is a ‘‘high quality thermoelectric,’’

The aim of making better thermoelectric devices has motivated many researchers to try and develop conducting materials with improved thermoelectric properties. Two material parameters that govern different aspects of performance of thermoelectric energy conversion devices are the thermoelectric ‘‘figure of merit’’ and the thermoelectric ‘‘power factor.’’ 1 Most efforts to increase those parameters have involved alloying or doping and, more recently, exploitation of quantum and other size effects in thin films and mesoscopic systems.2–7 Only rarely have classical composite mixtures been considered as a route to improved thermoelectric characteristics. In such materials the heterogeneity or granularity are on a large enough scale that the physical processes on the microscale, e.g., in each individual grain, are well described by classical continuum linear transport theory. For such materials it has been proven that the macroscopic thermoelectric figure of merit is never greater than its largest value among the different pure components.8,9 In contrast with that negative result, we have recently found that the macroscopic thermoelectric power factor of a two-component composite material can sometimes be considerably greater than both of the pure component values.10 In this article we expand that study to obtain an essentially complete theoretical solution to the problem of power factor enhancement in a two-component composite. Some very powerful bounding theorems for continuum classical conductivity problems are first used to show that, whenever such an enhancement is possible, the best microstructure is either parallel slabs or parallel cylinders @see Figs. 1~a! and 1~b!#. This proves a previous conjecture10 about a composite microgeometry that gives rise to the highest value of power a!

Electronic mail: [email protected] Electronic mail: [email protected]

b!

0021-8979/99/85(12)/8205/12/$15.00

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© 1999 American Institute of Physics

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J. Appl. Phys., Vol. 85, No. 12, 15 June 1999

D. Bergman and L. Fel

thermal and electrical transport; the 333 transport matrices, which appear when the material is anisotropic, will not be denoted by that symbol in this article# ~2.1!

ˆ E, J5Q

FIG. 1. Schematic drawings of some possible microstructures: ~a! parallel slabs, ~b! parallel cylinders, ~c! zig-zag microstructure often used in fabrication of thermoelectric devices, ~d! coated spheres assemblage or Hashin– Shtrikman microstructure, ~e! random dispersion of spherical inclusions of various sizes, ~f! realistic picture of a random dispersion of B-material inclusions in an A-material host. Note that the coated spheres in ~d! come in many different sizes, but they all have a B-material core and a concentric, spherical, A-material shell, and they all have the same core-to-shell volume ratio. Despite the differences, the solvable model shown in ~d! is often a good approximation to the more realistic composite structures shown in ~e! and ~f!.

i.e., it has a relatively large figure of merit, while the other component is a ‘‘benign metal,’’ i.e., it has large electrical and thermal conductivities, but the thermoelectric coupling is weak ~small Seebeck coefficient!, so that the figure of merit is very small. The results are quite surprising: It is possible to achieve a considerable improvement of the power factor over its value in the high quality thermoelectric component with only a moderate reduction in the figure of merit. Moreover, this is achieved either in a parallel slabs microstructure, or in a coated spheres assemblage, with a very small volume fraction of the high quality thermoelectric component. In Sec. VII we discuss the main conclusions from this study, and indicate how our calculated results might be implemented in the fabrication of composite thermoelectrics with improved properties. The Appendix includes some of the more technical aspects of the analysis of two functions which are defined and discussed in Sec. IV. We would like to point out that Secs. II–V of this article include some fairly intricate theoretical arguments and results regarding necessary and sufficient conditions for optimal enhancement of the power factor. For a reader who is interested mainly in the actual enhancements predicted for a variety of specific composites, along with a simple heuristic explanation, we recommend that those sections be skipped by proceeding directly to Sec. VI. II. BASIC THEORY OF TWO-COMPONENT COMPOSITE THERMOELECTRICS

In this section we rely heavily on Ref. 8, where many details can be found which are omitted here. Classical thermoelectric transport in a medium, which is isotropic at every point but may be heterogeneous, is described by the following equations @the hat symbol (^) is used exclusively to denote 232 matrices which couple the

S DS D S DS D

E[

E1 e¹ f [ ; E2 k B ¹T

J[

J1 2JE /e [ ; J2 2JS /k B

ˆ[ Q

S

DS S D

Q 11 Q 12 Q 12

[Q 12

[

Q 22

q 11

1

1

q 22

¹3Ei 50,

i51,2;

~2.2!

¹•Ji 50,

i51,2;

~2.3!

s /e 2

s a /ek B

a s /ek B

g /k 2B T

D ~2.4!

.

Here JE is the electric charge flux, JS is the entropy flux, f is the electric potential, T is the local temperature, s is the electrical conductivity at uniform T, g is the heat conductivity at uniform f, a is the thermoelectric coefficient or Seebeck coefficient or absolute thermopower, i.e., when the electric charge flux vanishes JE 50, then the relation between electric field 2¹f and temperature gradient ¹T is 2¹ f 5 a ¹T, e is the atomic unit of electric charge, and k B is Boltzmann’s constant. Note that these definitions imply that E has units of force, while J has units of particle flux. The thermoelectric power factor W and the thermoelectric figure of merit D are defined by W[ s a 2 5k 2B ~ Q 12! 2 /Q 11 ,

~2.5!

D[T a 2 s / g 5 ~ Q 12! 2 / ~ Q 11Q 22! .

~2.6!

Note that our definition of the figure of merit, which follows Ref. 8, differs from the more commonly used ‘‘ZT figure of merit.’’ 1 The relation between those two definitions is ZT5

D . 12D

~2.7!

Clearly, when D increases from its lowest possible physical value of 0 up to its highest possible physical value of 1, ZT increases from 0 up to `. In the case of a two-component composite, the coupled fields E1 ,E2 and fluxes J1 ,J2 can be decoupled by applying the following congruence transformation to E, J, and to the ˆ A ,Q ˆ B: transport matrices of the two components Q J 8 5Tˆ J,

E 8 5Tˆ 21 E,

ˆ a8 5Tˆ Q ˆ a Tˆ Q

for a5A,B,

~2.8!

ˆ 8a is a 232, real diagonal positive matrix, while Tˆ is where Q a 232, real symmetric matrix, given by Tˆ 5T 12

S

t 11

1

1

t 22

D

,

~2.9!

where T 12 is arbitrary and

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J. Appl. Phys., Vol. 85, No. 12, 15 June 1999

t 115

D. Bergman and L. Fel

1 $ q q 2q q 6 @~ q A22q B11 2 ~ q A112q B11! A22 B11 A11 B22 2q A11q B22! 2 14 ~ q A112q B11!~ q A222q B22!# 1/2% , ~2.10!

t 2252t 11

q A112q B11 . q A222q B22

~2.11!

Note that the ambiguity, inherent in the 6 sign which appears in Eq. ~2.10!, is more apparent than real: It corresponds to switching the roles of the two elements of the diagonal ˆ 8a . matrix Q In this way, the original coupled field problem is transformed into uncoupled, simple ‘‘conductivity’’ problems in two different composite media which have the same microstructure but different component ‘‘conductivities’’ ~the word conductivity appears in quotation marks in order to emphasize the fact that the uncoupled problems involve unphysical linear combinations of ¹f and ¹T, as well as unphysical linear combinations of JE and JS ). The relation between the transport coefficients of the coupled and uncoupled problems is given by ~here q ii ,Q 12 ,Q ii ,Q 8ii can refer to either one of the two components A, B; later we shall see that they can also stand for the macroscopic parameters of the composite!8

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8 ,Q B11 8 ,Q A22 8 ,Q B22 8 . If all the principal nent parameters Q A11 axes are fixed by symmetry, or if the properties are either spatially isotropic or isotropic in some plane, then this theory applies whenever the macroscopic fields lie in any of those directions. Once the macroscopic parameters Q 8e11 ,Q 8e22 are known, we can find the macroscopic physical transport parameters, i.e., the nondiagonal 232 macroscopic thermoelectric transˆ e , by performing the reverse congruence transport matrix Q ˆ 8e . This leads formation of Eq. ~2.8! on the diagonal matrix Q to the the same expressions as shown in Eqs. ~2.12!–~2.16!. We can now write an expression for the macroscopic power factor W e [ s e a 2e in terms of Q 8eii We

T 2 ~ t t 21 ! 2 5 k 2B 12 11 22

~ t 22Q 8e111t 11Q 8e22! 2

t 222Q 8e111Q 8e22

.

~2.18!

We also recall that the bulk effective Seebeck coefficient a e can attain all values between the two pure component values a A and a B , and only those values.8 Its value is simply related to the values of Q 8e11 ,Q 8e22 : 11

a e 52

S

D

kB 1 t 11t 2221 11 . e t 22 12t 22Q 8e11/Q 8e22

~2.19!

From this it is easily found that a e 50 whenever Q 8e22

S D

2

a B2 a A . g Aa Bs B2 g Ba As A

8 5Q 12T 212~ t 211q 1112t 111q 22! , Q 11

~2.12!

Q 8225Q 12T 212~ t 222q 2212t 221q 11! ,

~2.13!

Q 115 ~ t 222Q 8111Q 822! / @ T 12~ t 11t 2221 !# 2 ,

~2.14!

8 III. EXACT OPTIMAL BOUNDS FOR Q eii

Q 225 ~ Q 8111t 211Q 822! / @ T 12~ t 11t 2221 !# 2 ,

~2.15!

8 1t 11Q 22 8 ! / @ T 12~ t 11t 2221 !# 2 . Q 1252 ~ t 22Q 11

~2.16!

In this section we rely heavily on Ref. 12, where many details can be found which are omitted here. A general discussion of bounds for physical properties of composite media can be found in a review article published in 1992.14 In the absence of any information about the microstructure, the only restriction on possible values of Q 8eii is that 8 ,Q Bii 8 . they must lie between the two component values Q Aii However, because the two uncoupled ‘‘conductivity’’ problems i51,2 have the same microstructure, the values of Q 8e11 ,Q 8e22 are not independent. If one of these macroscopic parameters is known, that is tantamount to some implicit information vis-a-vis the microstructure, and this can be used to place tighter restrictions on the possible values of the other macroscopic parameter. A general theory for deriving and manipulating such restrictions or bounds appears in Ref. 12. It is based on the fact that Q 8e11 ,Q 8e22 are related in identical fashion to different values of the same analytic function F(s), which describes the dependence of an uncoupled macroscopic ‘‘conductivity’’ Q 8e on the uncoupled component ‘‘conductivities’’ Q A8 ,Q B8

As usual, one can now compute a macroscopic bulk effective ‘‘conductivity’’ Q 8eii for each of these problems, which relates the spatially averaged values of E8i and J8i :

^ J8i & 5Q 8eii • ^ E8i & ;

i51,2.

~2.17!

If the microstructure has a low symmetry, then the macro8 will usually be 333, real scopic ‘‘conductivities’’ Q 8e11 ,Q e22 symmetric positive matrices, whose precise values depend on the appropriate component parameters as well as on the microstructure. In order for the following discussion to apply, we will assume that at least some of the principal axes of Q 8e11 ,Q 8e22 are independent of the ~scalar! component param8 ,Q B11 8 ,Q A22 8 ,Q B22 8 . This happens whenever those eters Q A11 axes are determined by the symmetry of the microstructure, as is the case for parallel slabs, and also for parallel cylinders in a two-dimensional array. Under that mild assumption, we will confine our discussion to the case where the volume averaged or macroscopic vector fields ^ E81 & , ^ E82 & , ^ J81 & , ^ J82 & all lie along such an axis. We then only need to consider one diagonal element in each of the 333 matrices Q 8e11 ,Q 8e22 , i.e., the element which corresponds to the macroscopic ‘‘conductivity’’ along that axis. Henceforth the symbols Q 8e11 ,Q 8e22 , will always represent just one such element, and they will be treated as scalar quantities, just like the compo-

Q 8e11

s[

5T

kB e

Q A8 Q A8 2Q B8

,

s As B

F~ s ![

Q A8 2Q 8e Q A8

.

~2.20!

~3.1!

The precise form of F(s) depends on the microstructure, but it has some general properties that identify it as a Stieltjes function

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J. Appl. Phys., Vol. 85, No. 12, 15 June 1999

F~ s !5

F

(n s2 nf n ,

0,F n