First‐Order Quantum Corrections to the Current Driven by a Force and Einstein Equivalence

Abstract

Employing a superposition representation of the density matrix for a fixed energy the first order quantum corrections to the current driven by a weak constant force are calculated for independent charge carriers scattered by uncorrelated point-like scatterers. This complements a preceding treatment of the corresponding diffusion problem and demonstrates the Einstein equivalence as a property of the density matrix itself. The changes in kinetic energy caused by the force field lead to a reformulation of the correction terms which resemble now the known vertex structure of the Bethe-Salpeter equation. The influence of the force is considered explicitly in the propagator construction.Unter Benutzung einer Superpositionsdarstellung fur die Dichtematrix bei fixierter Energie werden fur unabhangige Ladungstriger Quantenkorrekturen erster Ordnung zu dem durch eine achwache konstante Kraft hervorgerufenen Strom berechnet. Die Streuzentren werden als punktformig und unkorreliert angenommen. Die Arbeit erganzt eine vorhergegangene Behandlung des entsprechenden Diffusionsproblems und weist die Einstein-Aquivalenz als eine Eigenschaft der Dichtematrix selbst nach. Die vom Kraftfeld bewirkten Anderungen der kinetischen Energie legen eine Umformulierung der Korrekturterme nahe, wodurch gleichzeitig die von der BetheSalpeter-Gleichung her bekannten mathematischen Vertex-Bildungen entstehen. Der EinfluB der Kraft wird explizit in der Ausbreitungsfunktion erfaBt.

Theory of the Residual Resistivity Dipole. Flow of Non‐Interacting Particles under a Weak Driving Force around a Fixed Point Scatterer

“…The force-induced part of the bulk density matrix has been calculated in Section 3 of [2] in the so-called classical approximation as a 1 ar -R e K , ;…”

“…With a weak force applied, the graphs (2) now have to be evaluated with the systematic inclusion of the influence of the force on the propagators. The corresponding technique has been developed, summarized in a graphical scheme and applied in [2], cf. especially Sections 2 and 4 of [2].…”

“…Now we solve the same problem for the force-driven case. Thus we do not feel the need to repeat the general considerations, the reader is referred to [2] for the method in the force case and to [3] for the RRD problem itself, and the corresponding quotations can be found there, too.…”

“…Similar as in [2], where the force-driven bulk transport has been treated, the force case implies also in the treatment of the flow around an impurity some intricacy in comparison with the mathematically simpler diffusion case but without giving us, at the end, much more information. indeed, as regards the current itself, there is no result which could not be obtained in a simpler way via the well-known Einstein equivalence between a force and a density gradient.…”

“…We note that it is sensible that the equivalence (10) in the inhomogeneous case is mediated by the density of states n(E) = (2m/h2) (k'/4n2) of the homogeneous bulk because it is only in the bulk that the asymptotic selection of a sharp energy value becomes possible. This intricacy of the force case has been discussed in more detail in [2], and it will be essential here again in the next section.…”