Semi-classical Limit in a Semiconductor Superlattice

In this paper, we perform a mathematical study of a semiconductor superlattice. Since the thickness of the layers is very small, quantization plays an important role. The modelling is therefore given by a Schr6dinger equation with a periodic potential. The scaled lattice thickness is denoted by a small parameter e which is of the same order of magnitude as the Planck constant. When this parameter tends to zero, i.e., the semi-classical limit, we obtain classical transport of the charge carriers described by a Vlasov equation.


INTRODUCTION
Superlattices technology represent a modern tech- nology used also in the design of semiconductors.In order to gain insight in the complex physical behaviour of these devices it is sometimes neces- sary to have very precise models which give a de- scription on the atomic level.In this work we start with such a fundamental model that takes in account the quantum effects and investigate its semi-classical approximation in a mathematically rigorous way.
A superlattice can be described as material which consists of a stack of layers, those will be of two kinds: a layer A and a layer B as it is shown on Figure 1.This is the model of Si/Ge, GaAs/A1As or InAsSb/InSb superlattices for instance.
In the first part of this paper, we give a quantum description of such superlattices describing struc- tures which are periodic in one direction x but whose properties vary slowly in other directions Xz, X3.This means that in this model we regard layers which have a slowly varying doping profile.
The motion of a charged particle in a stratified medium can be described by the Schr6dinger equation: Axe(X, T) + qV(X)(X, T) where h is the Planck constant, q the charge of the particle and m its mass.In our "stratified medium" the crystal is periodic with period A in the direction X1 such that the potential V verifies: We will also assume that the potential is bounded, i.e., VEL(R ) (1.3) Let us define the quantum scale: A is the length of the period of the medium in the direction X1 -is its time unit such that we have (e.g.[10]) mA 2   Now if we denote by L and To the macroscopic length of the crystal, we can define a scaled potential by: qV(X)---T-v & L' L (1.4) Let us introduce new variables x, t, e: X T A X--L, o' =L We now scale the wave function.
We recall that has to be normalized in such a way that" I (x, T)I 2tin 1.So we put: )(x, t) L 3/2q(Lx, Tot (1.5)We remark that is also normalized.
Let us choose To and L such that: h 7 -2 h2 7_2 2 00.mA---2 e and 2mL--5"mA ---2 (1.6)With this scaling Eq. (1.1) becomes: (1.7) We want to study rigorously the semi-classical limit of 1.7 when e goes to zero.In particular we are concerned with the limit behavior of the density" le(x, t)l: Semi-classical limits of Schr6dinger operators have been intensively studied.In the case of perturbed periodic potential we refer to [2].In this works the limit behaviour of the Schr6dinger operator is given.Our purpose is to study the limit behaviour of the particle dynamics.This problem has been already been investigated in the case of: v periodic [8, 6]v E C [4, 7, 6]-v + u with v periodic and u a small perturbation [9,1].The result in the periodic case is that the limit concentration is given by the sum of the concentrations np(X, t) p 1, oo corresponding to the eigenvalue problems e.g.[8].and the limit concentration n is given by: and the concentration are given by: np(x, t) fp(x,k, t) & /" f(x, k, t) dk p (.10) We remark in Eq. (1.9) that the band velocity in the first variable Ok, Ep (kl, x2, x3)   where the non negative distribution fp solves the transport equations.
In this work the main tools are the Wigner trans- form of wave functions and the Bloch decomposition of Schr6dinger operators.
We overcome this difficulty by using asymptotic expansions of Wigner transforms given in [6].
In the following of this paper, we will make the assumption that the energy Bands don't cross.In fact it is possible to choose the eigenvectors of the Bloch decomposition such that the energy Bands do not cross.Such an choice can always been done for a Bloch decomposition in one dimension, which is the case here, but it has been proved that it is false in higher dimensions (e.g.[5,3]).
The result is that the distribution function solves the transport equation: depends on the two parameters X2, X 3 and that in the other variable the velocity corresponds to free transport.
This paper is organized as follows: In the next section we introduce the Bloch decomposition of the wave operators but only in the x-direction and examine the x2, x dependence of the projectors and the energy bands.Then we derive a Wigner equation by using Wigner series in the direction x and Wigner transforms in the directions x2, x.In the last section we pass to the semi-classical limit and give the main results.
In this paper we will assume that the initial data is e-ocillating; more precisely we assume: C for I1 4   IDz(x)l 2dX 3 (1.11) Remark 1.1 As in [6], this assumption can be weaken to the usual condition of e-ocillation of [4].

WIGNER TRANSFORMS
In this paper since we have periodicity in the direction xl, we will associate Wigner series in the direction x, and classical Wigner transforms in the direction x2, x3.
Let us now mention a property which will be very useful in the following of this paper.This is an expansion of Wigner transform with a pseudo differential operator p(x, eD).DEFINITION 3.2 Let us consider a symbol m p(x, ) s( x x then for f E wq'() m P(x, z))f(x) p(x, eel)f( y)e i(x-y)4dCdy (2) ( (3.4) DEFINITION 3.4 If the notation {p, q} denotes the Poisson bracket of p p(x, e) and q q(x, ) defined by Remark 3.5 We can weaken the regularity ofp if we take a more regular function f. {p, q} (x, ) V(x, ).Vxq(X, ) Vxp(x, ).Vq(x, ) Let p C( rn X Rm) satisfy for w (w m x W m)'lOx, p(x, )1 <_ C ( + I1)'" Iff and y lie in a bounded set of L 2([)m, we have the expansion we(P w (x, eD)f, g) pwe(f, g) + {p(x, ), we(f,g)} + 2r e (3.7 where rc is bounded in S'(mx gm) uniformly w.r.t.e.
This property is proved in [6].
The proof of Theorem (3.7) and Theorem (3.8) is almost the same as in [6]: we make a Taylor expansion of the Fourier transform of the Wigner function and we get the desired result.
By a little computation using Lemma (4.1) and the asymptotic expansion of Theorem (3.7), we have We are done.

3
If Okl is the Hamiltonian which acts on k-periodic functions i.e., are all analytical.The Cp(.,kl) are kl-periodic and Cp L 2([0, 1] x DEFINITION 2.2 functions: Let us define now the Wannier p( yl x2, x3)

DEFINITION 3 . 6
Classical Wigner transform ex- pansion with a pseudo differential operator.The classical Wigner transform of (f,g) EL2(