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(which resembles equation (15.4) of Schiff [29]). The extra potential term (1/r2) represents the contribution of the angular momentum to the energy, see [29], p. 81. This current section has shown a possible way forward for deducing non-spherical eigenstates from a general spherically symmetric potential, as may be encountered in a quantum dot. Clearly, this work is incomplete and remains left open for the interested reader. 8.9 APPROACHES TO PYRAMIDAL DOTS There has been a drive to simplify the fabrication of quantum dots, and in a particular kind of material system this has been achieved with the aid of a process known as self-assembly. This occurs when a thin (perhaps even sub-mono) layer of one material is deposited on top of a substrate which has a quite different lattice constant. In such a system, the strain energy in the bonds forms a substantial fraction of the total energy, and in an effort to minimize this the deposited atoms rearrange themselves, moving from a thin layer into 'clumps'. This method has become known as the 'StranskiKrastanov' growth mode [166]. Microscopy has shown that these small groups of atoms form pyramidal dots, as illustrated earlier in Fig. 8.4. It appears that the pyramidal dot requires a fully three-dimensional solution of Schrodinger's equation, as given by Cusack et al. [167]. However, a little thought

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The Delete Style column indicates either Track embeddings or User speci ed Track embeddings means that the Propagate box is checked if this is the source of an embedding relationship, and unchecked otherwise It is mostly easy to test delete propagation Set Propagate on the Subclasses property of Generalization, for example; click Transform All Templates, build, and run; and draw some components and several generalization connectors between them When you delete one component, you will see its subclasses in your diagram vanish too, along with all their subclasses And of course any ports on the components also vanish because of the default setting of Propagate on the embedding relationship, ComponentHasPorts

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allows the problem to be simplified into a combination of a series of decoupled Schrodinger equations and some perturbation theory. For example, a pyramidal dot could be thought of as a finite-barrier cuboid dot, but with four additional perturbing potentials, as shown in Fig. 8.19 (right) (shaded area). These also take the shape of square based pyramids, with the base, in this example, vertical. However, these pyramids are not symmetric, having the apex centred on the upper edge rather than over the centre of the base. If the finite dot was solved with decoupled one-dimensional wave functions, then for relatively small systems (< 100 A) where the unperturbed wave function penetrates considerably into the barrier of the cuboid quantum dot, there is the possibility of apartial cancellation of the ' 2V and ' V potentials at the corners and along the edges (see Section 8.6), with these '+V non-symmetric pyramidal potentials. Such a partial cancellation may be vital to ensure convergence of the perturbed energy levels. Recent work employing this approach has derived connection rules between the energy of a pyramidal dot with a cubic quantum dot of a different volume [164]. As mentioned above, lattice mismatch is what drives the self-assembly process, hence the atoms constituting self-assembled quantum dots are under the influence of significant amounts of strain. The effect of this strain is to alter the conduction and valence band profiles so they are no longer constant either inside or outside the quantum dot. The method for determining the strain field and the effect on the conduction and valence band edges has been summarised by Califano and Harrison [168] for both In 1 _ x Ga x As on GaAs and Sii_xGex on silicon quantum dots. 8.10 MATRIX APPROACHES

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Alternative solutions are being sought to this problem by using an extension of the method of Gangopadhyay and Nag cited earlier [165]. This involves expanding the wave function as a Fourier series of infinite quantum well solutions and has proved itself to be a workable and reliable technique [169,170]. The basic approach is to expand the two- or three-dimensional wave function of the quantum wire or quantum dot as a linear combination of some basis functions. The basis functions are usually sine or plane waves of a larger box that encompasses the quantum wire or dot and has an infinite exterior potential. This leads to a formulation

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