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The process is repeated until the energy eigenvalues converge; at this point the wave functions are simultaneously solutions to both Schrodinger's and Poisson's equations the solutions are described as self-consistent, rather like Hartree's approach to solving many electron atoms (see e.g. reference [3] p. 396). Figure 3.36 shows the result of adding the potential due to the charge distribution Vp, as displayed in Fig. 3.34, to the original band-edge potential VCB for the single quantum well of the previous section. The perturbation, even at this relatively high carrier density of 2x 10 12 cm 2 , is rather small compared to the barrier height, for instance. Nonetheless it is important to calculate the effect of this perturbation on

More information on cyclic codes can be found in (for example) Blahut (1983).

Figure 3.36 The sum of the band-edge potential VCB and Poisson's potential Vp for the single quantum well of Fig. 3.32

Minimal Basis. A minimal basis for the prepositional calculus comprises three operators ('logical connectives'), listed below in ascending order of precedence, equivalence (=) , disjunction (v) , negation (-1) , and the following laws. Associativity of =: Symmetry of =: Unit of =: Negation: ((p = q) = r) = (p = (q = r)) . p=q=q=p .

B_x = {x e u\n$(x) = 1} BX = {x e u\p$(x) > 0}

the electron energy levels by continuing with the iterative process and looking for convergence of the resulting energy solutions. This process is illustrated in Fig. 3.37.

Distributivity of ->: Symmetry ofv: Associativity ofv: Idempotence of v: Distributivity ofv: Excluded middle:

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Figure 3.37 The ground-state energy of the single quantum well of Fig. 3.32 as a function of the number of iterative steps

Additional Operators. The remaining logical connectives are conjunction (A) ,

BNB(X) = {x e U\Q< jig(ar) < 1}

The first iterative loop produces the majority of the change in the energy level from the single carrier system to the doped system. Subsequent iterations produce only minor refinements to the energy level, which has certainly converged by the eighth loop. Altogether the ground state energy has changed by about 2 meV when accounting for this finite charge density. 3.19 COMPUTATIONAL IMPLEMENTATION As this work is about how to actually implement a computational method to provide quantitative predictions, in addition to deriving the theoretical equations in the first place, it is important to be aware that the self-consistent Schrodinger-Poisson solution presented here, can at times diverge. The doping density in the last series of examples in the previous two sections was chosen such that it produced a noticeable perturbation to the total potential profile, as illustrated in Fig. 3.36. In fact, this total carrier density of 2x 1012cm 2 is fairly high when contained within just one quantum well. The majority of calculations later in this text, on active regions of unipolar lasers for example, will have carrier densities much lower than this. However, in order to illustrate the intricacies of the computational method, consider now an even higher doping density, i.e. double that of the previous.

if(<=) ,

Figure 3.38 The ground state energy of the single quantum well of Fig. 3.32, but doped to 4x 1018cm 3 in this case, thus giving a total carrier density of 4x 1012cm 2

only if (=>) , inequivalence (^) . These are defined in terms of equivalence, disjunction and negation in the following laws. The precedence convention is that conjunction has the same precedence as disjunction, and inequivalence has the same precedence as equivalence. 'If and 'only if have the same precedence, which is less than conjunction and disjunction, and more than equivalence and inequivalence. false: false = p /\q = p = q = pv q . (p q) = ~^(p = q) .

The above shows that vagueness can, in fact, be defined in terms of uncertainty. Some properties of the rough membership function are summarized below:

The first energy calculated in the process is that of the ground state for an undoped system, which is obviously the same as before. However, the higher doping density

Golden rule: Inequivalence: If:

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