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

Abstract

Employing a superposition representation of the density matrix the perturbation of the statiouary current flow by a fixed point Scatterer is treated approximately. The current is driven by a weak constant force. The problem, the method, and the result parallel closely those of the corresponding force-free diffusion case discussed previously. For all current-proportional perturbation terms, the Einstein equivalence between a force and a density gradient is established for the inhomogeneous case under study. This equivalence, however, is broken by an additional current-free term of the density matrix. This term can be identified as a relict of the inhomogeneous equilibrium state in the given static force field. This equilibrium state itself is also studied. In the ensemble the limitations of the concept of a local chemical potential become evident, even in the current-free state.Unter Benutzung einer Superpositionsdarstellung fiir die Dichtematrix wird die Storung des durch eine konstante Kraft angetriebenen stationaren Stromflusses durch einen fixierten Punktstreuer naherungsweise behandelt. Problemstellung, Methode und Ergebnis laufen parallel zu denen des entsprechenden kraftfreien Diffusionsfalls, der in einer vorangegangenen Arbeit diskutiert worden ist. Fur alle stromproportionalen Storterme wird die Einstein-Aquivalenz zwischen einer Kraft und einem Dichtegradienten fur den untersuchten speziellen inhomogenen Fall bestatigt. Ein zusatzlicher stromfreier Term in der Dichtematrix bricht jedoch diese Aquivdenz, er wird als ein Relikt des inhomogenen Gleichgewichtszustands im gegebenen statischen Kraftfeld identifiziert. Dieser Gleichgewichtszustand selbst wird ebenfalls untersucht. Im Ensemblebild werden die Grenzen des Konzepts eines lokalen chemischen Potentials deutlich, und zwar im Gleichgewicht wie im Nichtgleichgewicht.

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

“…Now the Einstein equivalence holds not only for the current itself but for any current-proportional quantity, too [4]. This means especially that in the force case a definite asymptotic density difference between both sides of the obstacle corresponds to just the value (29) we have found in the diffusion case.…”

“…Equation (4) states that the characteristic attenuation quantity Im k2(r) can be defined as a local function and that it is proportional to the local density of states z-l Im G(v, r).…”

“…While the older versions of this approach comprised some intuitive elements, we have put it on a sound mathematical basis 121 and demonstrated also its ability to produce results which are beyond the scope of a classical transport theory as for the first-order quantum corrections in the 3d bulk [12]. Then we applied the method to the residual resistivity dipole (RRD) [3,4] for an idealized impurity in the 3d bulk, a problem which will be revisited now in the last section.…”

“…In our previous papers [3,4] on the RRD problem, the impurity has been treated explicitly as an additional scatterer, complementing the disordered ensemble of background scatterers. This method has some drawbacks, however.…”

“…The superposition method can be applied to the diffusion case with a given carrier density gradient or to the force-driven mobility case. For the point-like impurity we have treated the force case parallel to diffusion [4] demonstrating in a very detailed manner that and how and with what restrictions the Einstein equivalence works for that inhomogeneous problem. Within the superposition method, the diffusion case is mathematically much simpler than the force case and therefore we choose it in the following to start with.…”