Abstract
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.
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Communicated by P. Constantin
After the first submission of the manuscript Naoufel Ben Abdallah unexpectedly left us. I would like to express here my feelings of gratitude for his precious friendship and for the extraordinary scientific enrichment he offered me at every occasion (L.B.).
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Barletti, L., Ben Abdallah, N. Quantum Transport in Crystals: Effective Mass Theorem and K·P Hamiltonians. Commun. Math. Phys. 307, 567–607 (2011). https://doi.org/10.1007/s00220-011-1344-4
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DOI: https://doi.org/10.1007/s00220-011-1344-4