Thermal transport in disordered systems

I. Phys. C: Solid State Phys. 21 (1988) L26SL270. Printed in the UK

LETTER TO THE EDITOR

Thermal transport in disordered systems M J Kearney and P N Butcher Department of Physics, University of Warwick, Coventry CV4 7AL, UK

Received 19 January 1988, in final form 8 February 1988

Abstract. An exact result due to Chester and Thellung is extended to discuss thermal transport in disordered systems in general. The results are non-perturbative and are consequently true for arbitrary disorder. We conclude that, for low T , the thermal conductivity will scale as the electrical conductivity and thus the Wiedemann-Franz law is a very general result. In addition, we find weak localisation corrections to the thermopower will be present. The nature of fluctuations in the thermal transport coefficients are also discussed and compared with the phenomenon of universal conductance fluctuations.

Chester and Thellung (1961) were able to derive an exact relationship between the correlation functions governing transport in disordered systems. Starting from the Kubo formula, the proof rests upon assuming that the electrons are non-interacting and that the scattering is elastic, but is otherwise quite general. In particular, it does not depend upon any assumption about the strength of the disorder. As a result, they were able to conclude that the Wiedemann-Franz law is valid no matter how strong the scattering in the limit T-. 0. Although Chester and Thellung only considered an isotropic system, SmrEka and Stieda (1977) subsequently extended the theorem to show its validity even for anisotropic systems and magnetic fields, provided the transport parameters are regarded as second-rank tensors. On the other hand, studies of the localisation of electrons in disordered systems might appear to indicate that the proof is only applicable to systems where the electronic states are extended. This would restrict the validity of the result to three-dimensional systems above the metal-insulator transition. In this Letter, we wish both to argue that the result is more general than this and to extend it to discuss not only the thermal conductance but also the thermopower of disordered systems. Given the recent interest in thermal transport (Ting et a1 1982, Sivan and Imry 1986, Kaiser 1987, Strinati and Castellani 1987, Esposito et a1 1987) we feel the results offer useful insight into the phenomenon of heat transport in general disordered systems. We begin by outlining the derivation of the Chester-Thellung theorem. Consider a system subject to an electric fieldE and temperature gradient V T , and assume the system is isotropic (the extension to the anisotropic case follows naturally from the work of SmrEka and Stieda (1977)). To linear order ( h = 1 throughout),

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where J is the electric current and Q the heat current. The Lapare given by the general Kubo formula (Mahan 1981)

Lap= -

lox los d r e-”

d p ’ Tr[pojp(- t - ip’)j,(O)]

pa = exp[p(n - x ) ]

(2)

where pois the equilibrium density matrix, p = l/kT, Q is the thermodynamic potential and at is the total Hamiltonian of the system, including disorder. The operator j l is the electric current operator and j 2 is the heat current operator. (As defined here, the Lap are slightly different to the S, used by Chester and Thellung, although of course each set leads to the same expressions for the physical observables.) Of crucial importance to the argument is that the total Hamiltonian at is separable into a sum of single-electron Hamiltonians H . This corresponds to the assumption that the electrons are non-interacting. If we define the complete set of one-electron eigenfunctions In) (subject to periodic boundary conditions) such that, Hln) = En In)

then it is clear that the separability of at implies that pois diagonal in this representation. This leads to significant simplifications in the formal manipulations of equation (2). The main steps in the argument are as follows. Using the set In) as the basis of the representation, the operators may be cast into second quantised form,

The integrals over t and p’ in equation ( 2 ) are trivial, as is the evaluation of the trace since po is diagonal in this representation. Consequently, T O OC i C m , C,+Cns 1 = f m (1 -f n ) d n , m d m ’ n

f m = Tr(p0 CA C m )

(4) where fm is the usual Fermi-Dirac distribution. The term with m = m‘,n = n’ is not considered; this is necessary to ensure that the transport coefficients are finite, and does not result in any serious loss of generality (Greenwood 1958, Chester and Thellung 1961). Combining equations ( 2 ) , (3) and (4) gives

As defined here, the electric current operatorj, = - lelj (where j is the particle current operator) and the heat current operatorj, = ( j E - p j ) where the energy current j E = f ( H j + jH).

Using this result, and letting the size of the system L + 00 so that the summation may be replaced by an integral over the density of states gives

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L267 (6c)

where P ( E ) is the density of states. Notice that at no point in the derivation of these results has it been necessary to assume anything about the strength of the disorder. The , in general will be very complicated, important point is that the function G ( E ) which occurs in the integrand of all the Lap. Thus without actually knowing the exact form of G ( E )it, is possible to draw general conclusions about the relationship between the L,,, and hence about the thermal conductivity and the thermopower. In addition, since G ( E ) is simply related to the electrical conductivity of the system; ( 0 = LI1so a(T = 0) = G ( E ~ )it) is , also possible to make quantitative predictions about thermal transport in the limits where the perturbative forms of a are known. It is also important to realise that the above results do not depend on the nature of the eigenstates of the disordered system. Consequently they are true whether the states are extended or localised, and thus apply not only to three-dimensional systems (both above and below the metal-insulator transition), but also to lower-dimensional systems as well. The type of transport resulting in systems where the electron states are localised depends upon the relative magnitudes of the system size L to the electron localisation length f . If L < f the system will be weakly localised, and we would expect the thermal transport coefficients to exhibit weak localisation corrections similar to the conductivity. In the opposite limit where L % 5 the localisation is strong and the transport is said to be activated. This latter limit may not however be consistent with the neglect of inelastic processes if, for example, hopping is important in a particular system. To illustrate the ideas we shall consider the form of the thermal transport coefficients in some of the limits which have received interest, and which have been analysed in a variety of ways. All the other limits may be obtained just as easily from the general results (equations (6a)-(6c)). We begin by considering the thermal conductivity K. In terms of the L,,

) a slowly varying function on the scale of kBT, If the temperature is low, so that G ( E is we can make use of a well known expansion theorem to write

where the prime denotes a derivative with respect to E . Within this approximation,

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This result implies that as T-, 0, K/Tscales as the electrical conductivity U . In particular, the Wiedemann-Franz law

is valid whatever the strength of the disorder (Chester and Thellung 1961, SmrEka and Stieda 1977). This is of course subject to the condition that G(E) is slowly varying as outlined before, and that the scattering is predominantly elastic. Such an expansion is applicable to the weakly localised regime where B k g T , indicating that there are no weak localisation corrections to the Wiedemann-Franz law. Similarly, in the region of a metal-insulator transition where the conductivityon the metallicside a(&) (E - EJ’, the expansion will still be valid provided - E, B kBT.In this case,

-

K

lim T-0 T

- ( E -~ E ~ ) ’

and so the thermal conductivity has the same critical exponent as the electrical conductivity. These results were given recently by Strinati and Castellani (1987) who used diagrammatic perturbation theory. The proof given here seems simpler and also more general, in that no assumption about the disorder being weak is required. The consequences for the thermopower, defined by

are equally as interesting. Within the approximation of G(E) slowly varying we have the well known Mott formula (see e.g. SmrEka and Stieda 1977),

The validity of this result has been demonstrated in a different and elegant way by Sivan and Imry (1987) who used a generalised Landauer formula. They point out a particularly interesting consequence for the thermopower in the vicinity of a 3D metal-insulator transition. For - E, k g T ,equations (60) and (66) readily give

A and B are constants that depend upon the value of the critical exponent Y and B is of order unity. This result implies that if the Fermi energy is varied so that it crosses the mobility edge E,, then Swill vary smoothly across the mobility edge. In particular, and in sharp contrast to the conductivity U , no structure is expected for = E,. This would be interesting to test experimentally. The problem would be separating the diffusive component (calculated here) from the phonon-drag component of the thermopower. If the experimentswere performed in very disordered samples, the phonon mean free path would be severely limited and a direct comparison may be possible. For a 2D sample, in the limit of weak disorder, it is well known from perturbation theory that (Lee and Ramakrishnan 1985)

n(E)e2t e* D(E) = 7 - -1nm 231.2

t

(E) t

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where t is the elastic scattering time (independent of E in 2D) and T J E ) is the phase relaxation time. Typically in a 2~ system rv T’ (Lee and Ramakrishnan 1985) so there is a temperature-dependent weak localisation correction to the Boltzmann conductivity. This correction term is also energy dependent. If S= kBT,the scattering will be mainly elastic and the inelastic correction term will vary slowly with energy over a scale of order k,Tabout E ~ In. such a situation the above formulae may justifiably be used (despite the fact that they are strictly valid only for elastic scattering) with the result that

-

where 1 is the mean free path, kF1% 1 and Tois a reference temperature. Hence there should be a weak localisation correction to the thermopower in 2D systems. The same conclusion may also be drawn in 3D systems which supports the recent conjecture of Kaiser (1987). Ting et a1 (1982) on the other hand performed a perturbation calculation using diagrammatic techniques for 2~ systems and concluded that there are no weak localisation corrections. It is not clear exactly how this discrepancy arises in their result, since they present only an outline of the calculation. We have independently carried out their calculation and find that the in T term in equation (14) is present. Finally, another interesting feature of the above theory is that no ensemble average over different configurations of impurities has had to be taken. The results are thus applicable to small samples which are not self averaging. Such non-averaging effects are known to lead to sample specific fluctuations in the conductance which, in the metallic regime at T = 0, are of order e2/h(for a comprehensive review of the so-called universal conductance fluctuations see Lee et a1 (1987)). Equations ( 6 ) ,(7) and (11) lead us to the tempting conclusion that the analogue of these fluctuations may also be present in the thermal transport coefficients. Numerical studies reveal that sample specific fluctuations should indeed be present in the thermal conductivity and the thermopower (at sufficiently low temperatures) as a function of the chemical potential. However, the amplitude of these fluctuations is not a simple constant; rather it depends upon the precise values of the chemical potential and of the temperature. The fluctuations are most pronounced in the thermopower which may even change sign at low temperatures. As the temperature is raised, the fluctuations decrease until the ‘average’ results given above are obtained. ideas similar to these have very recently been proposed by Esposito et a1 (1987). Such effects should be observable in very narrow quasi-one-dimensional MOSFETSfor example, though the experiments would of course be very difficult to perform. One of us (MJK) would like to thank the Science and Engineering Research Council of Great Britain and the General Electric Company Hirst Centre for financial support.

References Chester G V and Thellung A 1961 Proc. Phys. Soc. 77 1005 Greenwood D A 1958 Proc. Phys. Soc. 71 585 Esposito F P, Goodman B and Ma M 1987 Phys. Rev. B 36 4507 Kaiser A B 1987 Phys. Rev. B 35 2480 Lee P A and Ramakrishnan T V 1985 Reo. Mod. Phys. 57 287 Lee P A , Stone A D and Fukuyama H 1987 Phys. Reo. B 35 1039

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Mahan G D 1981 Many Particle Physics (New York: Plenum) Sivan U and Imry Y 1986 Phys. Reu. B 33 551 SmrEka Land Stieda P 1977J . Phys. C: Solid Stare Phys. 10 2153 Strinati G and Castellani C 1987 Phys. Reu. B 36 2270 Ting C S , Houghton A and Senna J R 1982 Phys. Reo. B 25 1439