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There are two side conditions on this rule The first side condition is that expression R may not depend on any variable in the list js The reason for this is that the scope of the variables in the list js on the right side of the equality is delimited by the innermost angle brackets and, thus, does not extend to the range R of the bound variable j Were R to depend on variables in js, those variables would be released in the process of replacing the left side by the right side This is an example of avoiding the circumstance that a bound variable becomes free were the rule to be used from left to right when R does depend on some variable in the list js, that variable would be bound in the left-hand occurrence of R but free in the right-hand occurrence of R.

Making use of the ion-implantation-enhanced diffusion profile of Fig. 4.11 as a starting point, Fig. 4.12 shows the results of simulating annealing out the lattice damage with a time-dependent diffusion coefficient of the form shown in equation (4.18). In this case, the decay time T of the vacancy concentration was taken to be 100 s, and so the curves represent the points at which the vacancy concentration is a fraction 1, e 1/2, e 1 , and e 2 of its original value. Clearly the curves are converging to a point which represents the region at which the ion-implantation-enhanced diffusion has been frozen. 4.9 (5-DOPED QUANTUM WELLS In the previous sections, examples have been given of simulating diffusion for all of the various forms of diffusion coefficient that can exist. It is clear that the computational method can be extended to include combinations of all three dependencies, and indeed in the final example the diffusion coefficient had both depth and time dependency. It now serves a purpose to follow through an example of a diffusion problem which is of direct relevance to semiconductor heterostructures, and to calculate the subsequent effects on an observable, in this case, the quantum-confinement energies. Contemporary epitaxial growth techniques, such as molecular beam epitaxy (MBE) and chemical beam epitaxy (CBE) etc., allow for the possibility of growing very thin layers of semiconductor material. Another possibility, is the potential for these techniques to lay down very thin layers of dopant atoms. Fig. 4.13(a) represents bulk doping of a layer, as used in HEMTs and the majority of semiconductor heterostructure devices. Fig. 4.13(b) represents the dopant profile for a single 6(Mayer in a quantum

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[Hirose et al 1991] Y Hirose, K Yamashita, S Hijiya, Back-Propagation Algorithm which Varies the Number of Hidden Units, Neural Networks, Vol 4, 1991, pp 61 66 [Hole 1996] A Hole, Vapnik-Chervonenkis Generalization Bounds for Real Valued Neural Networks, Neural Computation, Vol 8, 1996, pp 1277 1299 [Holland 1975] JH Holland, Adaptation in Natural and Artificial Systems, Ann Arbor, MI: University of Michigan Press, 1975 [Holland et al 1986] JH Holland, KJ Holyoak, RE Nisbett, PR Thagard, Induction: Processes of Inference, Learning, and Discovery, MIT Press, 1986 [Holland 1990] JH Holland, ECHO: Explorations of Evolution in a Miniature World, in JD Farmer, J Doyne (eds), Proceedings of the Second Conference on Artificial Life, Addison-Wesley, 1990 [Holmstrom and Koistinen 1992] L Holmstrom, P Koistinen, Using Additive Noise in Back-Propagation Training, IEEE Transactions on Neural Networks, Vol 3, 1992, pp 24 38.

The right side would, thus, be an expression that depends on the value of this variable, whereas the left side does not The second side condition is that the list js may not include the variable j This is because 'jjs' in the left side of the equality would then include two occurrences of ' /, and it would not be possible to distinguish between related and unrelated occurrences of '/ in the range and term For example, a naive attempt to apply the nesting rule to < Z t : i = Ovi = l : < Z i : i = 2vi = 3 : i ) - 2xi> gives (Si,i : (i = O v i = l ) A ( i = 2 v i = 3) : i - 2xi) .

well, although of course, it could be in a barrier or at an interface. Such thin layers of dopant are called delta-layers, as the dopant profile resembles the Dirac 6-function, and the whole process is known as 6-doping; for a comprehensive treatise of this technique see Schubert [100].

This is meaningless because it is impossible to determine which occurrences of i are related, and which not It is always possible to avoid such complications by suitably renaming bound variables before using the nesting rule Using the renaming rule, the above summation equals.

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