c

CHINESE JOURNAL OF PHYSICS VOL. 18, NO. 2 SUMMER, 1980

Iteration Method for the Calculation of Phonon-LimitedThermal Conductivity of Normal Metals

W. C. C H A N (@$$i~.)

Departmenr of PhyksTamkang College of Arrs and Sciences

Tamsui, Taipei Hsien, Taiwan 2S1, China

(Received February 29, 1978)

A simple linear Boltzmann equation for the calculation of phonon-limitedthermal conductivity of normal metals has been obtained. It has been solved withthe generalized relaxation time approximation. At high temperature, it gives theWiedemann-Franz law. At low temperature, it gives a Tz dependence for thethermal resistivity.

1. INTRODUCTION

u.JNTIL now, thermal conductivity of nornian metals has been calculated most frequently by the

use of the variational methodî). But there are always doubts about the use of the first trialfunction, especially in the low temperature region(2y3). Another approach is io solve the complexBoltzmann equation numerically(4t5). It is intent here to reduce the complex Boltzmann equation intoa simple form and then solve it algebraically.

2. GENERAJ, CONSIDERATIONS AND THE BOLTZMANN EQUATION

The electrical and electronic thermal currents in a metals are given, respectively, by(l*a)

J,-aE--usVT, ( 1 )J, - SUTE + LTTVT, ( 2 )

where G, s and LTT are the electric conductivity, thermalpower and thermalelectric coefficient respec-tively. For thermal conductivity (K) measurements, J,=O, and so

K== -(LTT+s%T). (3)

Since &T/n. is of the order lo-’ or less for most metals, one can take approximately

K== - Ly*. (4)

The linearized Boltzmann equation with zero external electric and magnetic fields can be writtenî)

.

(1)

(2)( 3 )( 4 )( 5 )( 6 )( 7 )

-_____ .._-- ~~~ ..-See, for example, J. M. Zimam, Electrons and Phonons (Oxford U. P. Oxford, England, 1960). Chap.7 and Chap. 9.E. II. Sondheimer, Proc. Roy. Sot. A203, 75 (1950).J. W. Ekin, Phys. Rev. BG, 371 (1972).P. G. Klemens, Aust. J. Phys. 7, 64 (19S4).C. 17. Leavens, J. Phys. F: Metal Phys. 7, 163 (1977).

*Ii. S. Newrock and B. W. Maxficld, Phys. Rev. B7, 1233 (1973):F. J. Blatt, Physics of Electronic Conduction in Solids (New York, blcGraw-Hill, 1968) p. 183.

61

62 ITERATION METHOD FOR THE CALCULATION

- 1 gj(k, kí) 1’ Ií $l&/,(e) [l--fo(E+~~dl (E+hwqj-ri)}+ (analogous terms involving emission processes), (5)

where gj(k, kf) is the matrix element for the scattering of an electron from k to kf through theabsorption or emission of a phonon with wave vector q, polarization vector e,,j and energy Aw,j, nrand &(E) are the equilibrium distribution functions of the phonons and electrons respectively, 0 is theFermi energy and R is the mean displacement travelled by the electrons between collisions. All in-tegrals performed over the energy surface Str are subjected to the constraints Eí-E+F2O,j for absorp-tion processes and eí=E-/?ti,j for emission processes.

Just for the sake of convenience, let us define two new sets of functions Ij(k) and I;(k) such that

I,(k) = J / gl(k, kí) I3 R -$’

I;(k) -.I

*I gj(k, kí) jz R’ $;’

(6)

( 7 )

In order to eliminate the rapidly ,.,;;:+ing functions (af0/8e) and fo(e)[l--f,(eí)], we integrate bothsides of Eq. (5) with respect to E. \sslme v and all I,(k), I;(k) are smooth functions of e in com-paring with (afo/&) and fO(e)[l --/o(Eí)] and expand them by Taylor series in terms of (E-T) up tofirst order. Then, we have,

*(l/3) (zkJí)2 @v/L%>

J Ij(k>fdt) Cl -fo(~+h~qj)l (E-VW

=. Jpj(k,)+$m (E-_i]f~(t)[l-f~(~+fiO,jl(E--O)dt

S(kBT)’ F,(X) Ij(k,)+(k,T)3 Fz(x)(aIj/ac)

JI;(k) f&J [l-~(~+h~,j)l(~+h~,j-~)~~= (kBT)Z [-F,(r)] I;@,) + (k,TY [F,(x) +xFdx)l -FE’

and for emission processes, we have

J Ij(k)f~(~)[1--f~(~-h~,j)l('-_77)"

= (kBT)” [ - F,(X)] e-’ Ii + (kB7î)’ F,(X) e-I -a,ë;

(8)

( 9 )

(10)

(11)

I'I;(k)fo(~) [l-fo(~-h~aj)l (~-h@qj-V)d~

= (k,T)2 FL(x) e-s I;(k,) + (kBZí)” [F2(x) +xF~(x)] e-I q> (12)

where x represents hWqj/k,<T andsr

(13) ot,.N yídy x(x2+ 7rJ) (14)I WkJ__ (e;+~l)(,+e-ë-s) = --3( 1 -e-x)

TtAfter putting Eq. (S), Eq. (9), Eq. (IO), Eq. (ll), Eq. (12), Eq. (13),and Eq. (1-l) into Eq. (5). we have -..

1 8” ,’ .::, ?n

I~~-+$ Jlgj(k,k')1*~(x)[(~-~~)+(~-)2(~+-:-~~)]~~~-

--zF - ii(19

(1

t

’ )

,th,m-b to

3)

,lO)

(11)

(13)

(14)

(5), we have

(15)

W. C. CIIAN 63

where the factor F(x) is given by(a)

F(x)=x/(e=- 1)(1-P) (16)

The sum the first order terms in (6-q) on the right hand side of Eq. (5) is zero after integratingover E because absorption and emission processes cancel each other exactly. Integrating both sidesof Eq. (15) with respect to E once more, one finally obtains the simple Boltzmann equation

(17)

Here, the integration constant is chosen to be zero because the Boltzmann equation is supposed tohave a unique solution.

3. RELAXATION TIME AND THERMAL CONDUCTIVITY CALCULATIONS

To solve Eq. (17), we apply the generalized relaxation time method as used by Huntington andChan(g~lo). The generalized relaxation time T)(k), defined by i,,/v, for a particular direction P, isdetermined from a series

idk)- iz rdk) (18)

by an iteration scheme based on the assumption that Tp(k) is not a rapidly varying function of k.The leading term is given by

rpo(k) - -ëhî- T Jpas. IgLk,kí) IíF(x)[ (I--$--) + ($)ë(I+ ;ë$)I ëz:ë- (19)

and successively

x L.rpn-l(kí) - z- pn-l(k)1 -$;ë-}From the microscopic transport equation for the electronic thermal current

(20)

J,-sEVR l [eE-(E--)V In T] (L?fO/Gíe)&/c

one can easily get the thermal conductivity tensor

i=(?kZ,T/3) s, s (vRjv)dS,. .

(21)

(22)

By combining Eqs. (18), (19), (20) and (22), one can obtain the thermal conductivity tensor up toany order.

4. DISCUSSION AND CONCLUSION

We now discuss the principal features of our result.(i) At high temperature, the second term inside the square bracket in Eq. (17) ëbccomes~negligiblc

small and one recovers the Boltamann equation with external electric field on!y. From Eq. (22), oneobtains

;=(7r2kî,T/3eî)b

This is just the Wiedemann-Franz law. -

(8) W.C. Chan, J. Phys. P: Metal Phys. S, X59 (1978).(9) 1l.B. Huntington and W. C. Chan, Phys. Rev. 012, 5423 (1975).(10) W.C. Ghan, Chinese J. Phy:, 16, 24 (1978).

(23)

64 ITERATION METHOD FOR THE CALCULATION

(ii) At low temperature, in order to compare the result with the variational method, we changethe thermal conductivity tensor into thermal resistivity tensor Iv. Using the first order solution forthe generalized relaxation time, one approximately has

W,, i(3/nZkZ,T) X;; <rp)-I

=(3/n2k2,T) X;; (T;ë)

= -,.&Z,T6x;; -~~~,~,~~g,(k,kí),z~(.)~(,.;:~-) (I-$-)

+ _?_ (25) 2 ( X) ] S;’ -A?

where ,? is the velocity factor defined by

.Y== I,.*, (vvI4 d&I.%

X (24)

(25)

and <rp) is the average value of the relaxation time over the Fermi surface. The double integrationin Eq. (24) is very similar to the one obtained by the variational method(l) except for the factorinside the square bracket. For normal process, one approximately has

(26)

and

v;/vp= 1 (27)

So, at low temperature, Eq. (24) gives a T2 dependence for the termal resistivity of metals. Sincethe ?ë? term has a factor of 3/2 instead of 3, one expects that its coefficient B(z WT-2-AT-3) shouldbe roughly one half of the value BJ obtained by the variational method, in better agreement withRlemens’ numerical calculation for pure metals (0.67 &).

Finally, we like to point out that in obtaining Eq. (24), the expression for thermal resistivity,we have made many approximations. The most serious approximation is the assumption that therelaxation time is a slowly varying function of k on the Fermi surface. Fortunately, both electricaland thermal resistivity can be written as an average of the relaxation time over the Fermi surface.So, they are rather insensitive to the anisotropy of r)(k). Futhermore, from Eq. (20), one has

approximately

<rpn(k)> s 0 for all n>I

That is

<~Jk)>~<~po(k)>

(30)

(31)

So, Eq. (24) is approximately good to any order.

ACKNOWLEDGEMENT

The author wishes to thank Dr. H. B. Huntington for intially suggesting this investigation. Also,

this work was sponsored by the National Sciences Council of the Republic of China.