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We will not go into the full details of the proof of (137) Briefly, (131) provides the basis of the inductive proof, whilst (134) establishes the induction step (Equation (131) states that Q is guaranteed to hold if the loop body is executed zero times, and (134) enables the proof that, if the loop body T is executed at least once, postcondition Q will still hold on termination) The case that bf^O needs to be considered separately Design Steps The design of a loop is a non-trivial process, because there are several aspects to keep in mind With practice, however, it becomes second nature In order to reinforce this introduction, let us summarize the different items and their roles We assume that the precondition P and the postcondition Q are given.

Figure 5.37 The proportion of ionised donors from a doping concentration Nd as a function of temperature for a two-dimensional (quantum well) system, for several different dopant densities. The material system is GaAs with a donor energy 5.8 meV below the conduction band edge and an effective mass of 0.067 mo.

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The task is to construct an initialization S and a loop with termination condition done and body T to meet the specification { P } S ; do --done T od { Q } (138) We do this with the aid of an invariant, inv, and a bound function (measure of progress), bf The invariant and termination condition, done, are chosen so that, in all states, their conjunction implies the postcondition, Q: [inv A done => Q] (139) The invariant should also guarantee that, in all states, the bound function is greater than zero, unless the termination condition is true:.

when is the ionisation (binding) energy of the isolated impurity and Nd is the impurity (donor) concentration. The constant a depends on the impurity species and the host and was deduced experimentally by Pearson and Bardeen for boron and phosphorus in silicon.

(4-17)

[inv =* bf>0vdone] . (13.10) (13.11)

6.1 EXCITONS IN BULK If photons of energy comparable to the band gap are incident on a semiconductor, then they can be absorbed by the electrons forming atomic bonds between neighbouring atoms, and so provide them with enough energy to break free and move around in the body of the crystal. Within the band theory of solids, this would be described as 'exciting an electron from the valence band across the band gap into the conduction band'. If the energy of the photon is larger than the band gap, then a free electron is created and an empty state is left within the valence band (see 'high energy excitation' in Fig. 6.1). The empty state within the valence band behaves very much like an air bubble in a liquid and rises to the top the lowest energy state. This 'hole' behaves as though it were positively charged and hence often forms a bond with a conductionband electron (see 'exciton formation' in Fig. 6.1). The attractive potential leads to a reduction (by an amount Exo ) in the total energy of the electron and hole. This bound electron-hole pair is known as an 'exciton'. Photons of energy just below the band gap can by absorbed, thus creating excitons directly (see 'resonant excitation').

As the hole mass is generally much greater than the electron mass, then the twobody system resembles a hydrogen atom, with the negatively charged electron orbiting the positive hole. The exciton is quite stable and can have a relatively long lifetime, of the order of hundreds of ps to ns. Exciton recombination is an important feature of low temperature photoluminescence, although as the binding energies are relatively low, i.e. a few meV to a few tens of meV, they tend to dissociate at higher temperatures.

The initialization statement, 5, is constructed to establish the invariant:

9k(t) = 9k(t~l) + P(ok,p - gk(t - 1)) (4.18) In the above, dkjp is the Euclidean distance as defined in equation (4.14), / is the total number of input units, and <7jt(0) = 0. Thus, bk(0) = j, which initially gives each output unit an equal chance to be the winner; bk(t) is the conscience factor defined for each output unit. The more an output unit wins, the larger the value of gk(t) becomes, and bk(t) becomes larger negative. Consequently, a factor \bk(t)\ is added to the distance dk,p. Usually, = 0.0001 and 7 = 10.

Figure 6.2 Schematic diagram of an exciton in bulk material, with the hole (filled circle near centre) and electron 'separated' by a Bohr radius A, orbiting around the centre of mass

{ P } S {inv} .

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