Transport with Microscopic Inhomogeneities. Residual Resistivity Dipole, Landauer Formula, and the Three‐Dimensional Impurity Case

Abstract

A superposition method for the density matrix of stationary transport processes is developed where an arbitrary perturbation is superimposed on a weakly scattering background. Additional scatterers, areal perturbations of the grain boundary type and/or geometrical confinements in wires or films are candidates for applications. In the present paper an arbitrarily strong added scatterer is treated whose linear size remains small compared with the mean free path within the background. This problem is solved for a one-channel quantum wire and for the three-dimensional bulk. In both cases the evolving residual resistivity dipole (RRD) is determined. Already existing results are confirmed, generalized, and complemented by the inclusion of oscillatory density terms. A comparison of both the cases demonstrates the decisive role played by the topology of the problem. Within the scope of a completely formalized nonclassical transport theory, the internal mechanism producing the RRD is analysed. In this way it is shown that the voltage drop due to an added scatterer is given correctly, in the one-channel case, by an expression of the (R/T)-type quite similar to the original Landauer formula (LF). This derivation of a modified LF abandons the concept of dissipative reservoirs and ideal leads. The physical difference between ideal and resistive leads yields different results, too. These agree completely only for a sharp Fermi energy while, in the general case of finite temperatures, non-negligible energy dependences destroy the full equivalence of both approaches.

Electron transport through planar defects: a new description of grain boundary scattering

“…Since our m u t z p already yields the required density gradient the total density has to be asymptotically ( 1 2 1 S l ) a dipole distribution, namely = Cspn(z) (11) where Cis an unknown constant. We remark that in contrast to the ID and 3~ systems with an added scatterer [2], 6p (11) does not correspond to the induced density (9) without a damping factor. The evaluation of C via Fourier transformation is straightforward because the oscillatory term (Cf)*G; in the integral kernel in (2) can be neglected (cf discussion of equation (7)).…”

“…A stationary diffusion current in a homogeneous bulk material is related to a constant carrier density gradient. Under the combined action of current flow and additional perturbation, the density p(r) can be expressed according to [2] as p(r) = p(r) + 6p(r).…”

“…Finally we notice that in (2) and (4) the regions of the added defect can be neglected in the integrals over r'. This approximation is justified because the MFP determines the length scaleof the regions that contribute significantly.…”

“…Tosummarize, onecansay that thes~~dividestheprobleminto twoseparatedones: determination of the coherent density change (4) occurring in the vicinity of the defect and solution of a modified transport problem (2), which yields a delocalized diffusive behaviour of the total density redistribution 6p. The next two sections are devoted to them.…”

“…In section 2 a short summary of the SPM, adjusted to our purposes, is given based on a more detailed explanation in [2]. In sections 3 and 4 we construct, in a two-step procedure, the solution in the diffusion picture.…”