Path-decomposition expansion and edge effects in a confined magnetized free-electron gas

Abstract

Path-integral methods can be used to derive a `path-decomposition expansion' for the temperature Green function of a magnetized free-electron gas confined by a hard wall. With the help of this expansion the asymptotic behaviour of the profiles for the excess particle density and the electric current density far from the edge is determined for arbitrary values of the magnetic field strength. The asymptotics are found to depend sensitively on the degree of degeneracy. For a non-degenerate electron gas the asymptotic profiles are essentially Gaussian (albeit modulated by a Bessel function), on a length scale that is a function of the magnetic field strength and the temperature. For a completely degenerate electron gas the asymptotic behaviour is again proportional to a Gaussian, with a scale that is the magnetic length in this case. The prefactors are polynomial and logarithmic functions of the distance from the wall, that depend on the number of filled Landau levels $n$. As a consequence, the Gaussian asymptotic decay sets in at distances that are large compared to the magnetic length multiplied by $\sqrt{n}$.