Semiclassical quantization of electrons in magnetic fields: the generalized Peierls substitution

TL;DR

This paper develops a generalized Peierls substitution incorporating Berry phase effects, crucial for accurate semiclassical equations and quantization of electrons in magnetic fields, with applications to Bloch and Dirac electrons.

Contribution

It introduces a new generalized Peierls substitution that accounts for Berry phase, improving semiclassical quantization methods for electrons in magnetic fields.

Findings

Derived a general expression for cross-sectional area.

Applied the method to calculate energy levels of Bloch and Dirac electrons.

Highlighted the importance of Berry phase in semiclassical quantization.

Abstract

A generalized Peierls substitution which takes into account a Berry phase term must be considered for the semiclassical treatment of electrons in a magnetic field. This substitution turns out to be an essential element for the correct determination of the semiclassical equations of motion as well as for the semiclassical Bohr-Sommerfeld quantization condition for energy levels. A general expression for the cross-sectional area is derived and used as an illustration for the calculation of the energy levels of Bloch and Dirac electrons.