Narrow electronic bands in high electric fields: Static properties

Abstract

We present a gauge-invariant formalism for studying the motion of electrons in uniform electric and magnetic fields. This paper treats static properties, mainly the spectral function and density of states. These results will be exploited in a second paper devoted to transport, with the aim of extending the results of Khan, Davies, and Wilkins [Phys. Rev. B 36, 2578 (1987)] to include periodic bands as well as parabolic bands. In particular, we define a gauge-invariant density of states that can be used in the presence of applied electric and magnetic fields. This density of states can be interpreted as a function of kinetic energy in the case of parabolic bands. It reduces smoothly to the usual results in the absence of applied fields and gains a tail into negative kinetic energies when an electric field is applied—a result well known from the theory of electroabsorption. Structure due to Landau levels is seen when a magnetic field is applied. The results are more striking in the case of a periodic energy band. Here the continuous density of states is split into a discrete ‘‘Stark ladder’’ when an electric field is applied, with the levels separated by the energy of Bloch oscillations. It is not necessary to introduce the Stark levels explicitly through the use of a scalar potential; they emerge naturally within the gauge-invariant formalism. Again the density of states shows no unphysical discontinuous behavior as the field goes to zero. The implications of these results for transport are briefly discussed.