Semi-classical approximation of electron-phonon scattering beyond Fermi’s Golden Rule * C. Ringhofer, † M.Nedjalkov, ‡ H. Kosina and S. Selberherr § January 5, 2004 Abstract We derive a quantum mechanical correction to the semiclassical Fermi Golden Rule operator for scattering of electrons in a crystal. This correction takes into account the fact that electron phonon interaction is not instantaneous in quantum mechanics. The corrective term is derived via an oscillatory, i.e. weak, limit in the Levinson equation for large time scales. KEYWORDS: Asymptotic analysis, quantum mechanics, Levinson equation, Wigner functions, Fermi’s Golden Rule, Boltzmann equation. AMS CLASSIFICATION: 65N35, 65N05 1 INTRODUCTION It is generally accepted that the dominant collision mechanism for electron transport in crystals is scattering of electrons with phonons, i.e. with vibrations of the crystal lattice. In a semiclassical description this collision mechanism is described by the Fermi Golden Rule. In the absence of an electric field and in the spatially homogeneous case, the evolution of the effective single electron density function is then given by the Boltzmann equation (a) ∂ t f (p, t)= Q F GR [f ](p, t) :=  dp  [S F GR (p, p  )f (p  ,t) - S F GR (p  ,p)f (p, t)], (1) * This work was supported by NSF grant DECS-0218008. † Department of Mathematics, Arizona State University, Tempe, AZ 85287-1804, USA (ringhofer@ asu.edu). ‡ Department of Electrical Engineering, Arizona State University. § Institute for Microelectronics, Technical University of Vienna, Gusshausstrasse 27-29, A1040 Vienna, Austria (kosina@iue.tuwien.ac.at). 1