Semiclassical quantization of skipping orbits

We propose a simple and pedagogical description of the spectrum of edge states in the quantum Hall regime, in terms of semiclassical quantization of skipping orbits along hard wall boundaries, ${\cal A}$ =2 π(n + γ) $\ell$ B 2 , where ${\cal A}$ is the area enclosed between a skipping orbit and the wall and $\ell$ B is the magnetic length. Remarkably, this description provides an excellent quantitative agreement with the exact spectrum. We discuss the value of γ when the skipping orbits touch one or two edges, and its variation when the orbits graze the edges and the semiclassical quantization has to be corrected by diffraction effects. The value of γ evolves continuously from 1/2 to 3/4. We calculate the energy dependence of the drift velocity along the different Landau levels. We find that when the classical cyclotron orbit is sufficiently close to the edge, the drift velocity which is zero classically, starts to increase and its value is directly related to the variation of γ when approaching the edge. We compare the structure of the semiclassical cyclotron orbits, their position with respect to the edge, to the wave function of the corresponding eigenstates. Copyright EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2011