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that not only is the potential infinite outside the well, but in addition the Schrodinger equation is not defined in these regions a slight contradiction with the deduction of the first boundary condition. This point, i.e. that there is still ambiguity in the choice of boundary conditions for commonly accepted solutions, will be revisited later in this chapter. 2.2 IN-PLANE DISPERSION

for all j = 2, , J - 1 and k = 2, , K - 1.

{ Invariant: ( K f c ^ N A sxXk = Bound function: k }

effect arises because the electron-hole separation can vary quite considerably between heterostructures. Fig. 6.3 illustrates this schematically; clearly the electron and hole separation is much smaller in the Type-I quantum well where both particles are localised in the same layer of semiconductor, than in the Type-II system, where they are localised in different layers. Hence, the exciton binding energy will be larger in the former. 6.3 EXCITON BINDING ENERGIES The Hamiltonian representing the interacting two-body electron-hole complex can be considered to be the sum of three terms

{ 5 = (Zi:Q^ where S, the value to be assigned to 5 in the body of the loop, is the only missing element. We calculate the appropriate value of S using the assignment axiom. The specification of the assignment in the body of the loop is given by the Hoare triple:

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where He and Hh are the one-particle Hamiltonians appropriate to the conduction and valence bands, respectively, of the particular microstructure of interest (as in 2). However, in the context of the exciton a constant effective mass along the growth (z-) axis must be assumed, i.e. the one-particle Hamiltonians are written:

The standard training algorithm for SOMs is stochastic, where codebook vectors are updated after each pattern is presented to the network. For each neuron, the associated codebook vector is updated as Wkj (t + l)= wkj (t) + hmn,kj (t) [zp - wkj (t)] (4.19)

A sxXk = ( I . i : k ^ i < N : a [ i ] x X i )

The third term, He-h represents the electron-hole interaction, and is itself composed of two terms. One of these corresponds to the kinetic energy of the relative motion of the electron and hole in the x-y plane (perpendicular to the growth axis), while the

1 90 Clearly,

where P is the quantum mechanical momentum operator for the in-plane component of the relative motion. Now:

where mn is the row and column index of the winning neuron. The winning neuron is found by computing the Euclidean distance from each codebook vector to the input vector, and selecting the neuron closest to the input vector. That is,

A k > 0 => (Kk-

and r is simply the electron-hole separation, given by:

So, it is indeed only the appropriate value of 5 that needs to be determined. We now use the summation rules to calculate the appropriate value of the unknowns: SxXk~l = (Zi:k-l^ = { SxXk~l { SxX ~ <=

Note, that as the main motivation is concerned with the optical properties of excitons in quantum well systems, the Hamiltonian then contains no term for the motion of the centre of mass of the exciton in the x-y plane, i.e. the exciton is assumed to be at rest within the plane of the well. The problem, as always, is to find the eigenfunctions * and eigenvalues E of the Schrodinger equation: which describes the system. Following standard procedures, the two-body exciton wave function \P is chosen to be a product of three factors, as follows:

The function h m n , k j ( t ) in equation (4.19) is referred to as the neighborhood function. Thus, only those neurons within the neighborhood of the winning neuron mn have their codebook vectors updated. For convergence, it is necessary that hmn,kj (t) > 0 when t -> oo.

+ a[k-l]xXk~l }

where wr represents the electron and hole relative motion. Following Hilton et al [128,129] and Harrison et al. [130,131] wr will be a variational wave function employed to minimize the total energy E of the system. The specific form of wr will be discussed later. The other two factors, w e ( z e ) and ^h(zh) are simply the eigenfunctions of the one-particle Hamiltonians of the heterostructure:

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