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In fact, Fig. 3.29 demonstrates that the effect is relatively small, but nevertheless significant, at narrower well widths. Indeed, energy level spacings in intersubband lasers are designed in relation to phonon energies which are typically a few 10s of meV, and hence it is debatable whether a 2 meV effect should be included. The effect is larger when the Ino.5sGao.47As wells are surrounded by strained AlAs barriers, so providing even more offset, and this material system has been the subject of several recent papers [64,67,68]. 3.17 POISSON'S EQUATION All of the techniques are now in place to be able to solve the Schrodinger equation for any heterostructure for which the band-edge potential profile defining the structure is known. However, all of the theoretical methods and examples described so far have concentrated solely on solving systems for a single charge carrier. In many devices such models would be inadequate as large numbers of charge carriers, e.g. electrons, can be present in the conduction band. In order to decide whether or not typical carrier densities would give rise to a significant additional potential on top of the usual band-edge potential terms (which will be labelled specifically as VCB or VVB), it then becomes necessary to solve the electrostatics describing the system.

O^ra A even.m A m+2 = M-^2 =*> rot.m = m+2 A m~2 - M~2 . But then, by the transitivity of equality, rot.m = m+2 A m+2 = M~2 => rot.m = M-f2 .

When considering the case of an n-type material, then (although obvious) it is worth stating that the number of 'free' electrons in the conduction band is equal to the number of positively charged ionised donors in the heterostructure. In the example shown in Fig. 3.30 this equates to six donors becoming ionised and supplying six electrons into the quantum well thus the system maintains charge neutrality. Fig. 3.30 is a

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Fuzzy set theory is the first to have a theoretical treatment of the problem of vagueness and uncertainty, and have had many successful implementations Fuzzy set theory is, however, not the only theoretical logic that addresses these concepts Pawlak developed a new theoretical framework to reason with vague concepts and uncertainty [Pawlak 1982] While rough set theory is somewhat related to fuzzy set theory, there are major differences Rough set theory is based on the assumption that some information, or knowledge, about the elements of the universe of discourse is initially available This is contrary to fuzzy set theory where no such prior information is assumed The information available about elements is used to find similar elements and indiscernible elements Rough set theory is then based on the concepts of upper and lower approximations of sets.

1 34

modulation-doped system in that the doping is located in a position where the free carriers it produces will become spatially separated from the ions. The additional potential term Vp(z) arising from this, or any other charge distribution p, can be expressed by using Poisson's equation:

So, by the transitivity of implication, O^ra A even.m A m+2 = M-4-2 => rot.m = M-f2 .) Substituting the calculated value for P in (10.16) and (10.15) gives us { O^ra A even.ra A m+2 = M-^-2 } m := rot.m { m = M-^2 } and the specification of S: (10.18) A even.m A In words, S must compute a positive, even number m which has the same integer division by 2 as M. The specification of S suggests that a conditional statement is appropriate. After all, m-f-2 = m-^2, so that, if m is even, nothing needs to be done. Formally, { 0 ^ m = M A even.m } skip { O ^ m A even.m A m-r2 = M-^2 } . (10.19)

where e is the permittivity of the material, i.e. e = e r eo. The solution is generally obtained via the electric field strength E; recalling that:

The lower approximation contains those elements that belong to the set with full certainty, while the upper approximation encapsulates elements for which membership is uncertain The boundary region of a set, which is the difference between the upper and lower approximations, thus contains all examples which cannot be classified based on the available information Rough sets have been shown to be fruitful in a variety of application areas, including decision support, machine learning, information retrieval and data mining What makes rough sets so desirable for real-world applications is their robustness to noisy environments, and situations where data is incomplete It is a supervised approach, which clarifies the set-theoretic characteristics of classes over combinatorial patterns of the attributes In doing so, rough sets also perform automatic feature selection by finding the smallest set of input parameters necessary to discern between classes The idea of discernibility is defined in Section 21.

(10.17)

the potential then would follow in the usual way [69]:

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