Bloch oscillations in uniform magnetic and electric fields

Abstract

In the present paper, by using the method of the effective Hamiltonian, we study Bloch's electrons in the presence of a uniform electric and magnetic field. We show that all wave functions are periodic in time. This result is the fully quantum-mechanical analog of the well-known Bloch oscillations predicted by quasi-classical dynamics. Furthermore, using the githt-binding method for a single band of an electron in a simple cubic lattice in the presence of a uniform electric and magnetic field, we obtain an equation of Harper's form. From this equation we take exactly the points at which Bragg reflection occurs for the subbands of Landau.