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induced Type-I to Type-II transition. It is generally accepted that such corrections to the one-particle wave functions are more important in Type-II systems, as a wellconfined electron may be able to 'suck in' the hole from the adjacent layer, thus having a large effect on the exciton binding energy. A fully self-consistent exciton model has been proposed by Warnock et al. [148] and later extended to include the variable symmetry relative motion term of this analysis [149]. The self-consistent correction to the one-particle eigenfunctions, ipe and ibh, is incorporated by solving a new Schrodinger equation:

where, from Section 3.2.1, z ( z , t ) is the stationary density according to which patterns are sampled, W describes the network weights, and z and t are respectively the input and target vectors. The function FNN is an approximation of the true underlying function. Since is generally not known, FNN is found through minimization of the empirical error function

'large'

where 7i is the total Hamiltonian describing the system, as in equation (6.7). This yields the first improved iteration to the electron wave function, t/4 say. The latter is then employed to improve the hole wave function, as follows:

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inequality K^l^N, together with the second universal quantification, does the same for the variable 1. The invariant is depicted in Figure 14.4. (The same warning about reliance on diagrams applies here as it did for the Dutch National Flag problem.) The computation is complete when it is the case that all values in the 'medium' segment are known to be equal. That is when, in addition to the invariant property, we have ( V i j :s^i<l A s ^ j < l : a [ i ] = a[j]) . We introduce a boolean value done whose function is to indicate when this property has been established. Formally, the function of done is captured by the invariant property

Following, this both and are utilized in the standard exciton-binding-energy calculation to give the new relative motion term, characterised entirely in terms of the parameters A and . The procedure is repeated until the total energy of the exciton is minimized and A and have converged. Piorek et al. [149] demonstrated that for Type-I systems, a variable-symmetry relative motion term is more important than self-consistency with simple 2D or 3D wave functions. The self-consistent correction proved itself to be important in flat-band (i.e. a zero offset in the valence band) and Type-II systems. The simple one-particle solutions are usually localised in different semiconductor layers, thus leading to a quite small exciton binding energy. However, the addition of self-consistency allowed the electron and hole eigenfunctions to move closer together, thus increasing the binding energy and, most importantly, reducing the total energy of the exciton. 6.10 SPONTANEOUS SYMMETRY BREAKING The aim of this book is to discuss the theoretical methods necessary to analyse semiconductor heterostructures. In addition, the computational implementation of these methods is paramount. This present section gives an example of purely computational

(7.2)

work giving rise to new science something that could not have been predicted from theory In calculating the self-consistent exciton energy in series of multiple quantum wells, Piorek et al [150] found that a 4-period multiple quantum well could have a larger exciton energy than the equivalent 3-period system This seemed a contradiction as a 4-well system can be thought of as a 3-well system but with some of the potential barrier removed Therefore it should have a lower exciton energy This seemed to be a general result, ie that 2n-period multiple quantum wells had a larger energy than 2n 1 systems Inspection of the self-consistently iterated electron and hole wave functions, 7/4 and whm , respectively, showed that in systems with odd numbers of wells, the electron and hole localised in the central well leading to a large binding energy.

done =* (V i,j : s^i<l A 5^ j <l : a[i] =a[j]) .

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