Eur. Phys. J. B 66, 41-49 (2008)
https://doi.org/10.1140/epjb/e2008-00404-6

Semiclassical analysis of edge state energies in the integer quantum Hall effect

¹ Department of Physics and Ilse Katz Center for Nanotechnology, Ben Gurion University, Beer Sheva, 84105, Israel
² RTRA – Triangle de la Physique, Les Algorithmes, 91190 Saint-Aubin, France
³ Institut de Physique Théorique, CNRS URA-2306, CEA Saclay, 91191 Gif-sur-Yvette Cedex, France
⁴ Laboratoire de Physique des Solides, CNRS UMR-8502 Université Paris Sud, 91405 Orsay Cedex, France

Corresponding author: ^a montambaux@lps.u-psud.fr

Received: 27 June 2008
Published online: 5 November 2008

Abstract

Analysis of edge-state energies in the integer quantum Hall effect is carried out within the semiclassical approximation. When the system is wide so that each edge can be considered separately, this problem is equivalent to that of a one dimensional harmonic oscillator centered at x = x_c and an infinite wall at x = 0, and appears in numerous physical contexts. The eigenvalues E_n(x_c) for a given quantum number n are solutions of the equation S(E,x_c)=π[n+ γ(E,x_c)] where S is the WKB action and 0 < γ < 1 encodes all the information on the connection procedure at the turning points. A careful implication of the WKB connection formulae results in an excellent approximation to the exact energy eigenvalues. The dependence of γ[E_n(x_c),x_c] ≡γ_n(x_c) on x_c is analyzed between its two extreme values as x_c ↦-∞ far inside the sample and as x_c ↦∞ far outside the sample. The edge-state energies E_n(x_c) obey an almost exact scaling law of the form and the scaling function f(y) is explicitly elucidated.