Density of one-particle states for 2D electron gas in magnetic field

TL;DR

This paper analyzes the density of states for a 2D electron gas in a magnetic field, revealing distinct energy regions, discrete levels at low energies, and oscillatory behavior of Fermi energy with magnetic field strength.

Contribution

It provides a detailed calculation of the 2D density of states in a magnetic field, including discrete and continuous spectra, and derives the magnetic field dependence of the Fermi energy.

Findings

Density of states has three energy regions: discrete, mixed, and continuous.

Fermi energy oscillates with magnetic field and increases proportionally to the square of the magnetic induction.

Total energy of the electron gas also oscillates and increases monotonically with magnetic field.

Abstract

The density of states of a particle in a 2D area is independent both of the energy and form of the area only at the region of large values of energy. If energy is small, the density of states in the rectangular potential well essentially depends on the form of the area. If the bottom of the potential well has a potential relief, it can define the small eigenvalues as the discrete levels. In this case, dimensions and form of the area would not have any importance. If the conservation of zero value of the angular momentum is taken into account, the effective one-particle Hamiltonian for the 2D electron gas in the magnetic field in the circle is the Hamiltonian with the parabolic potential and the reflecting bounds. It is supposed that in the square, the Hamiltonian has the same view. The 2D density of states in the square can be computed as the convolution of 1D densities. The density of one-particle states for 2D electron gas in the magnetic field is obtained. It consists of three regions. There is a discrete spectrum at the smallest energy. In the intervening region the density of states is the sum of the piecewise continuous function and the density of the discrete spectrum. At great energies, the density of states is a continuous function. The Fermi energy dependence on the magnetic field is obtained when the field is small and the Fermi energy is located in the region of continuous spectrum. The Fermi energy has the oscillating correction and in the average it increases proportionally to the square of the magnetic induction. Total energy of electron gas in magnetic field also oscillates and increases when the magnetic field increases monotonously.