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Submitted on 1 Jan 1978
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ENERGY DEPENDENT RELAXATION TIMES IN
ELECTRONIC TRANSPORT PROPERTIES OF
METALS
G. Grimvall
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, supplPment au no 8, Tome 39, aotit 1978, page C6-1042
ENERGY DEPENDENT R E L A X A T I O N T I M E S I N E L E C T R O N I C TRANSPORT P R O P E R T I E S OF METALS
Department o f [PheoreticaZ Physics, RoyaZ i n s t i t u t e o f TeehnoZogy,S-ZOO 44 StockhoZm,Sweden.
Rdsum6.- On a consid&r& l'effet de la ddpendance en dnergie du temps de relaxation dans un cas spdcial intkressant ,dam lequel l'cquation de Boltzmann rcgissant le transport Clectronique dans les m&taux peut Stre rdsolue exactement.
Abstract.- The effect of energy dependence in the relaxation times is considered in an interesting special case, for which the Boltzmann equation of electronic transport in metals can be solved exac- tely.
INTRODUCTION.- It is well known that a solution of the Boltzmann equation for the electrons in metals leads to energy dependent relaxation times. However, the transport problem can be reformulated in terms of a variational principle, and approximate formu- lae for the conductivities can be obtained, with an average relaxation time which is independent of the electron energy and only varies with the temperatu- re. In this way, one arrives at the well known re- sults/l/, often referred to as Ziman's formulae, which have been the starting point of most numeri- cal calculations of the electrical and the thermal conductivities Q£ metals. The present paper discus- ses significant effects caused by the energy depen- dence of the relaxation time T(E). Since the full Boltzmann equation cannot be solved in a closed ma- thematical form, previous such discussions have in- volved complicated numerical computations. Sondhei- mer/2/, ~ e r r ~ / 3 / and others expanded T(&) in a sui- table set of basis functions and applied the varia- tional principle. Recently, ~eavens/4/ and Takega- hara and Wang/5/ solve the Fredholm equation for
-c (our equation 2) by direct numerical iteration for some alkali metals. In this paper, it is shown that there is an interesting special case for which the Boltzmann equation can be solved exactly, and the essential physical effects that result from the energy dependence of T(E) can be discussed in simple mathematical terms.
AN EXACT SOLUTION OF THE BOLTZMANN EQUATION.- The electrical resistivity p can be written as/5/
W
- = (ne2/my J-,(af/a~) T(E) d~
P(T) 1)
Here, ne2/m is in standard notation, f is the Fer- mi-Dirac function and T(E) satisfies the equation
(2) 'ci is a relaxation time for impurity scattering, a n d B = I/kBT. F 1 and F2 are electron-phonon coupling functions, i.e. they can be expressed as integrals over the electron-phonon interaction in momentum space, for a given energy transfer U. Formally F(-) = -F(w). The difference between F 1 and F2 lies in a geometrical factor, which is 1 for F1 and cos 8 for F2, where 8 is the angle between an initial and a final electron state in a scattering event on the Fermi surface. F1 = 2ra2~, with a 2 ~ being the well known coupling function that appears in the
theory of superconductivity. In the Ziman formula for the resistivity, one can introduce a similar transport function a2 F (see e.g. /6/), which dif-
t r
fers from a2F by an average of the geometrical fac- tor l-cos 8 . Except for the extreme low temperature limit, a2F and a* F have approximately the same
tr
shape as a function of energy. The special case that they are equal correspond? to F2 = 0. Then equation (2) reduces to an integral for r ( & ) ; i.e. we have an exact solution of the Boltzmann equation. In principle, there could be metals with F (U) = 0 for
2
all energies except extremelly small ones. In prac- tice, F = 0 is a reasonable approximation for real
2
metals, when the purpose is to discuss the effects of an energy dependent relaxation time at interme- diate and high temperatures. At very low temperatu- res, umklapp processes are frozen out, and cos 0
tends to 1 .
DISCUSSION.- As an illustration, consider the sim- plest possible case, with an Einstein spectrum for the phonons(characteristic temperature 8 ) and an
E impurity relaxation time -ci = I/Pi. Equations (1)
and (2) are solved by a straightforward application of the method of residues. We choose units such that p for the pure metals is T/BE when T>>BE. Fi- gure 1 shows the deviation from Matthiessen's rule (DMR) for three impurity concentrations.
Fig. 1 :
@(imp) = text for T/BE = 0
DMR normalized as DMR/p(impurity) for 0.05 (a), 0.1 (b) and 0.25 (c). See the units. Inset shows T(€/kBT)/T(0) for .5 (d), 0.2 (e) and 0.0 (f).
For a pure metal, the difference between the Ziman formula and the exact result is similar in shape to the DMR-curves, with a maximum of 0.028 (in our units) at T/OE = 0.33. The exact result (when Pi=O) corresponds to an average of T(E) as in equation(]) while Ziman's formula corresponds to the inverse of the averaged I/T(E). For karge Pi, the total resis- tivity is formally given by Matthiessen's rule,pro- vided that the resistivity of the pure metal is cal- culated from Ziman's formula. The inset in figure 1
shows -c(&). Its weighting factor in equation (1)
has the same energy dependence as T(E) /T(O) has in the limit of small T/OE.
The approach above can be repeated for the electronic part of the thermal conductivity of me- tals. One may also discuss other transport parame- ters. For instance, it can easily be shown that the inclusion of the energy dependence in -c(&) re- duces the thermopower, as given by Klemens/7/, by 60 % at intermediate and low temperatures in our Einstein model.
From a comparison between T(E) in figure 1 and the detailed numerical calculations of T(E)
for Rb by Takegahara and Wang/5/, it is found that our approximate treatment reproduces the essential features of T(E) down to very low temperatures. Be- low approximately 4 K (the Debye temperature
BD
ofRb is 56 K), the exact -c(&) of Rb rapidly becomes independent of the energy. The reason is the free- zing out of Umklapp processes. At these low tempera- tures, other complications, such as phonon drag, ma- ke a theoretical analysis difficult. Thus we can conclude that our simple approach well accounts for the effects of energy dependent relaxation times at those temperatures (T = O.lBD
-
0.5BD) for which such effects are important. Full details will be published elsewhere.References
/l/ Ziman,J.M., Electrons and phonons, (Oxford Uni- versity Press Oxford) 1960
/2/ Sondheimer,E.H., Proc. R. Soc. m ( 1 9 5 0 ) 75 /3/ Berry,R.J., Can. J. Phys.
48
(1970) 1441 /4/ Leavens,C.R., J. Phys. F.,7
(1977) 163/ 5 / Takegahara,K. and Wang,S., J. Phys. F.,
L
(1977) L293/6/ Hayman,B. and Carbotte,J.P., Can. J. Phys.
