Relation between conductivity and transmission matrix

Some time ago Landauer' proposed that the dc conductance I' of noninteracting (spinless) electrons in a disordered medium in strictly one dimension is given by I' = (e'/2vrh) ~ r ~'/~ r (' where t and r are the transmission and reflection amplitudes. A relation of this kind, especially if it can be generalized to higher dimensions, is of great interest for at least two reasons. First, the cost of numerical computation may be greatly reduced compared with the conventional use of the Kubo formula. '~ Second, such a relation emphasizes the fundamental role of the conductance which is assumed to be the only relevant variable in a recent scaling theory treatment of the localization problem. " This point of view was discussed in a recent analysis of the conductivity of a one-dimensional (ID) chain9 in which Landauer's expression was used. The argument given by Landauer is a heuristic one and not easily generalized to higher dimensions. Recently Economou and Soukoulis' derived from the Kubo formula an expression for the conductance of a one-dimensional chain, I' = (e'/2mt) ~ t ~'. This result is in agreement with Landauer for long chains for which ~r ~2 —I, but in disagreement with his result for short chains. Their derivation is, however, also restricted to 1D. In this paper a completely general relation between the conductance and the transmission matrix is derived, and some implications for numerical calculations are discussed. In the derivation of Landauer' and Anderson et al. the conductance is obtained by dividing the current I by the chemical potential difference, Ap„ between the left and the right of the sample. (To avoid additional complications of long-range Coulomb interaction, we consider the transport of neutral particles. ) Thus some inelastic scattering mechanism must be introduced outside the sample to allow the regions to the left and right to reach local thermal equilibrium. The interface between the nonequilibrium region in the sample and the outside regions is difficult to treat, particularly in the multichannel case. However, even in the absence of a precise tre'atment, it is clear that in the limit of a perfeet sample ( ~ t ~ = 1), d p, must vanish and the conductance will be infinite, a feature satisfied by the Landauer formula. Traditionally the nonequilibrium problem is bypassed using the.Kubo formula, which relates via linear response the conductance to the equilibrium properties of the system. However, in applying the Kubo formula to a finite sample, we are faced with some ambiguity. If the sample is isolated, its energy levels are discrete and a. (ru) is a sum of closely spaced 5 functions. Some procedure for averaging over these 8 functions is then necessary in order to obtain a useful result. Alternatively, we can embed the disordered region of interest in an infinite system with no disorder, and probe the system with an electric field (or potential gradient) of frequency cu which exists only in the vicinity of the disordered region. The energy spectrum for the whole system will be continuous and o.(cu) will be smooth with a well-defined limit as cv 0. This latter procedure was used by Economou and Soukoulis, and we shall adopt it here. The result, however, has the feature that the conductance is finite in the limit of a perfect conductor. This is because in the presence of a finite electric field, there is no mechanism for establishing thermal equilibrium outside the sample, a situation that can be realized if the frequency is high compared with the inelastic scattering rate. While a fully satisfactory expression for the conductance based on the chemical potential difference is not yet available, the expectation9 is that in the limit of a large number of channels, I' = (ez/2rrt) Tr(ttr), a result that we shall derive here from the Kubo formula. In most cases of interest (2D, 3D, and t « I in ID) the difference between the different definitions of conductance will be small. To be specific we consider a disordered sample of length L in the z direction (0 & z & L) and crosssectional area A. Perfect regions of cross section 3 are attached to the + z directions so that the total length of the system is A (to be taken to infinity), and we assume periodic boundary conditions in the transverse directions [ p = (x,y) ] for the entire system. We apply an electric field in the z direction with