Local Density of States in Mesoscopic Samples from Scanning Gate Microscopy

TL;DR

This paper explores how scanning gate microscopy can relate the local density of states to conductance changes in mesoscopic systems, proposing an analytical model and validating it through numerical simulations.

Contribution

It introduces a generalized Kramers-Kronig relation linking LDOS and conductance variation, applicable to complex mesoscopic structures with multiple channels.

Findings

Conductance shift is proportional to the Hilbert transform of LDOS.

The LDOS-$G$ relation holds under specific physical conditions.

Numerical simulations confirm the theoretical relationship in realistic Aharonov-Bohm rings.

Abstract

We study the relationship between the local density of states (LDOS) and the conductance variation $\Delta G$ in scanning-gate-microscopy experiments on mesoscopic structures as a charged tip scans above the sample surface. We present an analytical model showing that in the linear-response regime the conductance shift $\Delta G$ is proportional to the Hilbert transform of the LDOS and hence a generalized Kramers-Kronig relation holds between LDOS and $\Delta G$. We analyze the physical conditions for the validity of this relationship both for one-dimensional and two-dimensional systems when several channels contribute to the transport. We focus on realistic Aharonov-Bohm rings including a random distribution of impurities and analyze the LDOS-$\Delta G$ correspondence by means of exact numerical simulations, when localized states or semi-classical orbits characterize the wavefunction of the system.