Metallic Nanosphere in a Magnetic Field: an Exact Solution

TL;DR

This paper provides an exact analytical solution for an electron gas on a spherical surface under a magnetic field, revealing distinct regimes and magnetic responses depending on field strength and flux quanta.

Contribution

It introduces an exact solution using oblate spheroidal functions for the electron gas on a sphere in a magnetic field, covering weak and strong field regimes.

Findings

Magnetic susceptibility exhibits jumps at half-integer flux quanta.

Landau levels form and electron states localize near poles in strong fields.

Green's functions are derived for limiting cases.

Abstract

We consider the electron gas moving on the surface of a sphere in a uniform magnetic field. An exact solution of the problem is found in terms of oblate spheroidal functions, depending on the parameter $p= \Phi/\Phi_0$, the number of flux quanta piercing the sphere. The regimes of weak and strong fields are discussed, the Green's functions are found for both limiting cases in the closed form. In the weak fields the magnetic susceptibility reveals a set of jumps at half-integer $p$. The strong field regime is characterized by the formation of Landau levels and localization of the electron states near the poles of the sphere defined by a direction of the field. The effects of coherence within the sphere are lost when its radius exceeds the mean-free path.