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The two-dimensional nature of the Sigma and Pi notations makes them very readable because it is easy to identify the constituent components of the notation. There is a quantifier ^ or 0 in the two examples which identifies the operation to be carried out (addition in the case of and multiplication in the case of 0)- There is also a so-called bound variable (k in both examples above) which has a certain range (from 0 to n in the first example and from 1 to n in the second example). Finally, there is a term defining a function of the bound variable which is to be evaluated at each point in the range (k2 in the first example and k in the second example). The bound variable is always to the left of the equals sign in the expression below the quantifier, and it ranges over a consecutive sequence of numbers, where the lower bound is given to the right of the equals sign and the upper bound is placed above the quantifier. The function of the bound variable is written immediately to the right of the quantifier. The general form of the Sigma and Pi notations is thus

1. Compare fuzzy sets and rough sets to show their similarities and differences. 2. Discuss the validity of the following two statements: (a) two-valued sets form a subset of rough sets (b) two-valued sets form a subset of fuzzy sets (c) fuzzy sets are special kinds of rough sets. 3. Discuss how rough sets can be used as classifiers.

This phenomenon of an apparently increased 'band edge' to absorption is known as band filling. This can also occur in undoped systems under very high excitation intensities which produce large numbers of both electrons and holes, which in turn can fill all of the available states and thus increase the minimum required photon energy for absorption. It can be accounted for under the present scheme by simply calculating the quasi-Fermi energy, EF, as described in Section 2.4.

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where 0 is the quantifier, bv is an identifier denoting the bound variable, Ib and ub are expressions denoting the lower and upper bounds of the range of the quantification, respectively, and E is an expression denoting the function of the bound variable that is to be evaluated at each point in the range of the bound variable.

4.1 INTRODUCTION Any substance will attempt to diffuse from an area where it is present in high concentrations to an area of low concentration. For example, obvious though it seems, if the curve x(z] in Fig. 4.1 represented the concentration of water in a trough then the water would fall very rapidly from the region of high concentration to the region of low! Crude though it may seem, this is an example of diffusion. It occurs also for gaseous systems, e.g. smoke gradually disperses in an enclosed room; however the point of interest in this present work is diffusion of material species in solids. In the context of semiconductors, and in particular semiconductor heterostructures, it is clear that diffusion of material species could be important as their very nature derives from discontinuous changes in materials. Fig. 4.1 could therefore represent a dopant, in either a bulk semiconductor, i.e. a homo junction, or an alloy component, e.g. Al at a GaAs/Ga1_xAlxAs heterojunction. Diffusion at such boundaries is a strong possibility, particularly during the elevated temperatures often used during growth, or the lower but more prolonged heating that may occur during normal device operation. Any movement of material, e.g. Al at a

Because of their readability, the Sigma and Pi notations are widely used. A major drawback, however, is that they are limited to quantifications over a consecutive sequence of numbers. Problems arise if the notation is to be used for quantifications over non-consecutive numbers. In some cases it is possible to get around the problems. For example, if we want an expression denoting the sum of all odd numbers in, say, the range l..n, we can write

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GaAs/Ga1-xAlxAs heterojunction will 'blur' the interface, i.e. the change from one material to the other will occur via a range of intermediatary alloys. Such a process is known as interface mixing and is represented schematically in Fig. 4.2. The change in profile of the junction will inevitably alter the electronic properties of the system, which will in turn affect the device characteristics and therefore the operating lifetime. The motivation behind modelling diffusion is to understand such time dependency of the properties of quantum well systems, in order to be able to control or prevent it, design a device in which its impact will be minimised, or predict the lifetime of a device. Besides these detrimental effects, controlled diffusion is sometimes initiated deliberately in order to tailor the optical and electronic properties of quantum wells systems [59]. For a comprehensive treatment of utilizing quantum well interdiffusion for photonics see the review by Li [74].

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