Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice

Multiphase semiclassical approximation of the one-dimensional harmonic crystal: I. The periodic case

“…We have proposed an original two-scale WKB technique adapted to electronic conduction in 1-D harmonic crystals, see (7)- (9)- (10). A numerical processing has been presented further together with a validation against direct Schrödinger computations.…”

“…In case one assumes translational invariance in two directions, three dimensional objects can be modelled to some extent by means of a one-dimesional mean-field equation. We shall follow these ideas which have already been studied in former works, [10,11,12]. However, we go now one step further considering that atoms vibrate as "coupled harmonic oscillators".…”

“…Without V ions , ions would arrange periodicallyX α = 2πα, α ∈ Z, like in [10]; now, let's call U α (t) := X α (t) − 2πα the displacement around these abscissae "at rest". The cores' equation is decoupled thanks to our third hypothesis,…”

“…We assume the reader to be familiar with Bloch's theory; a comprehensive presentation is given in [1], see also [7,8,14,15,10]. In this section, we shall show that for simple enough displacements (2) (typically with few vibration modes), it is possible to derive a WKB approximation for the wave function ψ solution of (3) for smooth V per .…”

“…Let us also note that the inclusion of the Poisson self-consistent potential wouldn't be easy within a ray-tracing approach. We don't plan to recall it completely instead we refer to [10,11] where it has been explained how to extend it to the case where semi-classical dynamics are driven by energy bands obtained from Bloch's decomposition. The inclusion of the Poisson term for systems endowed with translational invariance in two directions has been described in [12] and will be used in §4.3.…”