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analysis and adjusting A to give agreement with experiment, for any well characterised quantum well system. For a particular material system and acceptor atom, the screening length A should be constant. Therefore it can be used in further predictive calculations for different well widths, barrier heights, dopant positions, etc. Acceptor levels within quantum wells have been calculated successfully, as discussed by Bastard [18]. Masselink et al. [110] have achieved good agreement with the experimental measurements of Miller et al. [111 ] for the binding energy of carbon acceptors in GaAs-Ga1_xAlxAs quantum wells. 5.8 EFFECTIVE MASS AND DIELECTRIC MISMATCH A great deal of attention has been paid in the literature to the role of effective mass mismatch at interfaces in semiconductor heterostructures (see Bastard [18]) and to the effect that this may have on donor energies. In addition, Fraizzoli et al. [112] have studied in detail the role of a dielectric constant mismatch at interfaces between dissimilar materials and its effect on shallow donor impurity levels in GaAs-Ga1_xAlxAs quantum well structures. Such considerations can be readily incorporated in the present approach. At first sight it might be thought that this can be achieved by making allowance for the fact that both m* and e, which appear in the original Schrodinger equation, are a function of z. In as far as constructing the Schrodinger equation, this is true for the latter. However, as discussed in detail in 3, the change to a variable effective mass necessitates substantial alterations to the quantum mechanical kinetic energy operator. Again, as before the first term in equation (5.4) would need to be substituted with:

(4-7)

The problem is to find a closed formula for T.n. In this case, by writing down a few values, r.O = 1 , T.I = 2 , T.2=4 , T.3 = 8 , TA = 16 ,

In principle the analysis could be followed through again, with re-evaluation of the integrals affected by this substitution. The generalisation to include the material dependency of the permittivity e is simpler to deal with. It can be achieved by absorbing this function into the integral I4 which arises from the Coulombic term, i.e. with the appropriate change in the coefficient, I4 would become:

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it is plausible to guess that T.n = 2n. This we prove using the principle of strong induction. The induction hypothesis P.n is thus T.n = 2n. Although the principle of strong induction does not specifically mention a base case, we have to treat the case n = 0 separately because the definition of T.O is a special case. For n = 0, T.n = I by definition. But 1 = 2 . So, for n = 0, T.n = 2n as required. Also, for n > 0,

as e depends on the electron position and not simply the electron donor separation as usual. It is clear, however, that evaluation of this integral is non-trivial. To the author's knowledge, neither of these generalisations to the theoretical approach in this chapter have been implemented. While some authors deem it necessary to include such dependencies in calculations [103], it must be remembered that often the interest lies in systems of weak alloys

(4.8)

{ definition } { =

such as Ga1 xA1x As and Cdi-^MnxTe where the effective mass change between the well and barrier is relatively small. Nonetheless, the mechanisms by which the theory presented here can be extended have been mapped out and here lies an opportunity for the interested reader to explore such systems further. 5.9 BAND NON-PARABOLICITY Small well widths and large potential barriers could require the inclusion of the nonparabolicity of the conduction band [106]. Ekenberg [113] described the inclusion of non-parabolicity on the subband structure of quantum wells. This method can account accurately for a variety of physical phenomena, but is analytically complicated. Simpler procedures have been proposed by various authors for the more complex problem of a donor in a quantum well. For example, Chaudhuri and Bajaj [106] have used the following simple replacement:

2n .

where an represents a series of constants. Given the values of these constants, the effects of non-parabolicity can be incorporated into equation (5.17) simply by making the effective mass a function of the energy E. This extension is certainly much more straightforward than the two described in the previous section. In relation to the calculations below, it should be noted that Chaudhuri and Bajaj [106] showed that, even with relatively large potential barriers, band non-parabolicity was only significant for wells narrower than half the Bohr radius of the neutral donor, which, for the case of would be <50 A, and for Cd 1 x Mn x Te, would

Another variant of the Hebbian learning rule uses the correlation in the changes in activation values over consecutive time steps. For this learning rule, referred to as differential Hebbian learning, &uki(t) = ri&Zi(t)&ok(t - 1) where Azj(t) = Zif(t) - ziip(t - 1)

assume that ( V f e : 0 ^ k < n : T.k = 2 k ) . }

Just as there are many solutions to the Schrodinger equation for an isolated hydrogen atom, there are also many more solutions representing excited energy states of a donor in a heterostructure. Recalling the hydrogen-atom solutions [4]:

and similarly for 2py and 2pz The corresponding eigenenergy involves only the ground state and the principle quantum number, i.e. so very simply etc. In bulk semiconductors, the neutral donor does also exhibit these states (see [2], p. 314). The situation is more complex in semiconductor heterostructures as the onedimensional potential due to the layer structure breaks the symmetry of the spherical potential, and hence the wave function is more complex than for the hydrogen atom. Much detail has been given as to solving the corresponding ground state under these

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