ALu

aL being the lattice constant. Then,

where we have taken into account Eqs. (10) and (21). Introducing

and substituting Eqs. (29) and (31) in Eq. (28) we obtain en(-iV) + U(7, t) - ih -

where it is clear that f = l'. /n(r, t) is formally tried as a continuous function that varies slowly with the position around f [according to Eq. (30)].

Indeed, let us compare Eq. (13) with Eq. (5) [considering Eq. (8)].

If a weak external field, which varies slowly in distances of the order of lattice constant, is applied, the electron wave-function in a determined band is expressed as [according to Eq. (13)]

where /n(l, t) is defined from (33). We have considered that the following relation must be satisfied:

and where we formally have passed from the discrete variable l to the continuous one r, which is acceptable if fn(l, t) varies slowly in distances of the order of the lattice constant, around the ll node.

In absence of external field [U(r, t) = 0], we simply obtain f°n(l, t) = exp i[k • l_1

which, if is substituted in Eq. (34), leads to the equation of the stationary states.

Notice that instead of the original Eq. (12), now we have a Schrodinger equation for the envelope function, in which the H + U Hamiltonian has been substituted by the effective Hamiltonian

that, as one can appreciate, depends on the dispersion relation Bn(k).

Thus, the envelope function can be understood like the effective wavefunction for the electron in the crystal.

Fortunately, in most of physical phenomena occurring in semiconductors and metals, the participating electrons are those situated in the bottom or the top of the bands (this is, in their extremes). For these electrons it is convenient to introduce the effective mass and the effective Hamiltonian, which can be written as follows:

h2 32 h2 d2 h2 d2

In the isotropic crystal case, the effective masses are equal in any direction and Eq. (38) is rewritten as

where m*n is the effective mass of the n band.

In this case we say that the band is parabolic since the dispersion relation is given by

h2 k2 2m*n

which is, besides, isotropic.

So for a constant potential (in the time) one can write the effective Schrodinger equation for a stationary state as

_m*nxdx2 m*ny3y2

which is widely used.

This method is known as the effective mass approximation.

2.5. Envelope Function Description of Quasi-Particle States for Nanostructured Systems

The envelope function model is particularly adequate to study quasi-particle properties of low-dimensional semiconductor structures. This is a dynamic method for the determination of energy bands of particles in solid state physics, and the nanostructures are considered dynamic systems since they are always under the action of either a barrier potential or interface potential. The envelope function scheme and the effective mass approximation were clearly discussed, for first time, by Bastard [61, 62].

In order to obtain the nanostructure energy states, in the envelope function scheme, our problem will be to find the boundary conditions which the slowly varying part of the nanostructure wavefunctions must fulfill at the heterointerfaces.

Other approaches to the nanostructure eigenenergies have been proposed which are more microscopic in essence than the envelope function description. The tight-binding model is nowadays successfully used for nanostructures of any size, although it may have difficulties in handling self-consistent calculations, which arise when charges are present in the nanostructures. Another microscopic approach is the pseudo-potential formalism, which is very successful for the bulk materials. The advantage of these microscopic approaches is their capacity to handle any nanostructure energy levels (i.e. those close to or far from the F edge). This occurs because these models reproduce the whole bulk dispersion relations. The envelope function approximation has no such generality. Basically, it is restricted to the vicinity of the high-symmetry points in the host's Brillouin zone (r, X, L). However, we feel that it is invaluable due to its simplicity and versatility. It often leads to analytical results and leaves the user with the feeling that he can trace back, in a relatively transparent way, the physical origin of the numerical results. Besides, most of the nanostructures energy levels relevant to actual devices are relatively close to a high symmetry point in the host's Brillouin zone.

In the following we shall assume that A and B materials constituting the nanostructure are perfectly lattice matched and they crystallize with the same crystallographic structure. In order to apply the envelope function model two key assumptions are made:

(1) Inside each material the wavefunction is expanded on the Wannier functions (with l = 0) of the edges under consideration

if r corresponds to an A material, and

if r corresponds to a B material, where k0 is the point in the Brillouin zone around which the nanostructure states are built. The summation over n runs over as many edges as are included in the analysis.

(2) The periodic parts of the Bloch functions are assumed to be the same in each kind of material, which constitutes the nanostructure

0 0