Parametric correlations of local density-of-states fluctuations in disordered pillars, wires and films

We present a theoretical analysis of correlation properties of the local density of states in a disordered emitter probed by resonant tunnelling through a localized impurity state. The emitter is considered to be a cylinder of length L and radius R with elastic mean free path l<<{L,R} and the effective dimensionality d⩽3 of the emitter is determined by the relation between the typical scale over which diffusion occurs, namely the quasi-particle relaxation length L_c, and the dimensions L and R. The differential conductance measured in asymmetric double-barrier structures has been used (see, e.g., Schmidt T, Haug R J, Fal'ko V I, von Klitzing K, Förster A and Lüth H 1996 Europhys. Lett. 36 61) to image local density-of-states fluctuations. We give analytic expressions for the variance and for correlations of the differential conductance with respect to voltage and applied magnetic field for the limits of a bulk three-dimensional emitter, a film, a wire and a pillar, and we determine the effect of magnetic anisotropy in lower dimensions. A numerical calculation, valid for arbitrary L_c, is performed in order to describe the crossovers between these limits where the correlation functions are sensitive to the shape of the emitter and the position of the resonant impurity.