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7.3 STRAIN BALANCING In order to ensure pseudomorphic growth of multiple layers of quantum wells a technique called 'strain-balancing' is commonly employed. This implies a careful choice of the epitaxial layer compositions and widths as well as in the choice of the substrate (or buffer layer) parameters if possible, with the aim of engineering the strain state of the entire structure, and therefore preventing the appearance of misfit dislocations and other defects. However, the available margins for such parameter manipulation are quite small as the requirements of the quantum well structure enforce tight constraints. The general method for evaluating a 'strain-balancing condition' is based on the fact that the stable strain state corresponds to the minimum of the elastic energy in each layer of the quantum well structure. However, when the multilayer system is considered such a state needs to be reached across the entire structure and not only in

The computational complexity of a NN is directly influenced by: 1. The network architecture: The larger the architecture, the more feedforward calculations are needed to predict outputs after training, and the more learning calculations are needed per pattern presentation. 2. The training set size: The larger the training set size, the more patterns are presented for training. Therefore, the total number of learning calculations per epoch is increased. 3. Complexity of the optimization method: As will be discussed in Section 7.3, sophisticated optimization algorithms have been developed to improve the accuracy and convergence characteristics of

A P = Qxd r }

a single layer so that a favourable strain distribution is reached overall and prevents the relaxation process. Such a condition can be met by minimising the average elastic energy with respect to the in-plane strain (known as the 'zero-stress' condition) [154] given as:

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{ 0 ^ r < Q A P = Qxd + r } .

where Ik is the width and Wk the strain energy density of the k-th layer. Recalling the elastic energy density definition stated in equation (7.14) the energy density of the k-th layer can be written as

NNs. The sophistication comes, however, at the cost of increased computational complexity to determine the weight updates. Training time is usually quantified in terms of the number of epochs to reach specific training or generalization errors. When different learning algorithms are compared, the number of epochs is usually not an accurate estimate of training time or computational complexity. Instead, the total number of pattern presentations, or weight updates are used. A more accurate estimate of computational complexity is to count the total number of calculations made during training.

Figure 15.1 Elementary remainder and quotient computation (for natural number P). variable d still has a role in the assertions that establish the algorithm's correctness; it becomes an existentially quantified variable, as shown below. { remainder computation elementary algorithm }

where Cz- is the elastic stiffness constant and e\ is the strain component (in matrix notation) of the k-th layer. For the commonly used cubic and hexagonal semiconductors the elastic energy densities are given by equations (7.15) and (7.16). If the nature of the strain in the epitaxial layers is assumed to be due to a biaxial stress, then the equations can be written in a simpler form by substituting equation (7.12) or (7.13) and equation (7.21) into equation (7.14) for the cubic and equivalently for the hexagonal crystals. The elastic energy density then reads:

{ (KP A 0 < Q }

where: for the cubic and: for the hexagonal crystal geometries. With respect to equation (7.28), equation (7.29) can then written as:

The convergence characteristics of a NN can be described by the ability of the network to converge to specified error levels (usually considering the generalization error). The ability of a network to converge to a specific error is expressed as the number of times, out of a fixed number of simulations, that the network succeeded in reaching that error. While this is an empirical approach, rigorous theoretical analysis has been done for some network types.

The pseudomorphic condition implies that the modified lattice constants (a/t and ak 1) of adjacent layers must be equal to that of the substrate. The in-plane strains in each of the layers can then be written as:

r := P ;

and:

{ Invariant: 0 ^ r A (3d :: P = Qxd + r) Bound function:

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