Abstract
We study a semi-classical Schrodinger equation which describes the dynamics of an electron in a crystal in the presence of impurities. It is well-known that under suitable assumptions on the initial data, the wave function can be approximated in the semi-classical limit by the solution of a simpler equation, the effective mass equation. Using Floquet-Bloch decomposition and with a non-degeneracy condition on the critical points of the Bloch bands, as it is classical in this subject, we establish effective mass equations for more general initial data. Then, when the critical points are degenerated (which may occur in dimension strictly larger than one), we prove that a similar analysis can be performed, leading to a new type of effective mass equations which are operator-valued and of Heisenberg form. Our analysis relies on Wigner measure theory and, more precisely, to its applications to the analysis of dispersion effects.
Highlights
The dynamics of an electron in a crystal in the presence of impurities is described by a wave function Ψ(t, x) that solves the Schrödinger equation: (1.1)
Effective Mass Theory consists in showing that, under suitable assumptions on the initial data ψ0ε, the solutions of (1.2) can be approximated for ε small by those of a simpler Schrödinger equation, the effective mass equation, which is of the form: (1.3)
The analysis of Schrödinger operators with periodic potentials has a long history that has its origins in the seminal works by Floquet [Flo83] on ordinary differential equations with periodic coefficients, and by Bloch [Blo28], who developed a spectral theory of periodic Schrödinger operators in the context of solid state physics
Summary
The dynamics of an electron in a crystal in the presence of impurities is described by a wave function Ψ(t, x) that solves the Schrödinger equation:. The potential Vper is periodic with respect to a fixed lattice in Rd, which, for the sake of definiteness will be assumed to be Zd. Effective Mass Theory consists in showing that, under suitable assumptions on the initial data ψ0ε, the solutions of (1.2) can be approximated for ε small by those of a simpler Schrödinger equation, the effective mass equation, which is of the form:. We focus here on initial data which are structurally related with one of the Bloch mode in a sense that we will make precise later, we assume that this Bloch mode is of constant multiplicity and we introduce a new method for deriving rigorously the equation (1.3) The advantage of this method is that it allows to treat the case where the critical points of the considered Bloch modes are degenerate, leading to the introduction of a new family of Effective mass equations which are of Heisenberg type. Note that different scaling limits for equation (1.1) have been studied in the literature: the interested reader can consult, among many others, references [AP06, BMP01, CS12, DGR06, Gér91a, GMMP97, HST01, PR96, PST03]
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