Quantum transport in crystals: effective mass theorem and KP Hamiltonians

In this paper the effective mass approximation and the k.p multi-band models, describing quantum evolution of electrons in a crystal lattice, are discussed. Electrons are assumed to move in both a periodic potential and a macroscopic one. The typical period epsilon of the periodic potential is assumed to be very small, while the macroscopic potential acts on a much bigger length scale. Such homogenization asymptotic is investigated by using the envelope-function decomposition of the electron wave function. If the external potential is smooth enough, the k.p and effective mass models, well known in solid-state physics, are proved to be close (in the strong sense) to the exact dynamics. Moreover, the position density of the electrons is proved to converge weakly to its effective mass approximation.