Landau levels in crossed electric and magnetic fields

A uniform electric field perpendicular to a uniform magnetic field shifts each cyclotron oscillator's guiding centre and makes its energy depend linearly on the conserved momentum. The formerly degenerate Landau levels therefore become tilted bands.

In uniform perpendicular electric and magnetic fields, a charged particle's guiding centre drifts with velocity ${\mathbf{v}}_D=\mathbf{E}\times \mathbf{B}/B^2$, independently of its charge and mass. For $\mathbf{E}=E{\mathbf{e}}_x$ and $\mathbf{B}=B{\mathbf{e}}_z$, this gives $v_y=-E/B$.

In Landau gauge with periodic boundary conditions of length $L_y$, adjacent guiding centres differ by $\Delta k_y=2\pi /L_y$. A perpendicular electric field tilts the energy by $-(E/B)\hslash k_y$, so adjacent states formerly in one degenerate Landau level are separated by $2\pi \hslash |E/B|/L_y$.