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Figure 13.14 The atomic positions for self-assembled pyramidal quantum dots formed from face-centred-cubic materials
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Thus far, only the zone-centre, i.e. k = 0 states have been looked at, but clearly it is possible to calculate transport or other properties which relate to larger momentum states. For example, an earlier figure, i.e. Fig. 13.9, showed the along-axis dispersion curve of a quantum wire. Transport through the dots is possible, and could be studied by using the empirical pseudopotential method, although given the necessary periodicity of the technique, solvable problems would be limited to arrays of dots.
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Figure 13.15 A schematic representation of how an array of cubic quantum dots embedded in a host crystal might look
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Fig. 13.15 illustrates a periodic array of cubic quantum dots, while Fig. 13.16 indicates the nature of possible solutions for such a system. Fig. 13.17 shows the calculated dispersion curves for the uppermost valence-band state and the lower most conduction-band state for a periodic array of cubic eightatom Ge quantum dots embedded in a Si host crystal. The dispersion curves along this [001] axis resemble those of a SiGe superlattice, with a zone-folded conduction-band minimum which is nearer the zone-centre than in the bulk material; the difference here though is that the [010] and [100] directions, i.e. the in-plane directions of the superlattices, also have this same dispersion curve. Thus, the outlying X-valleys in the conduction band are brought nearer the zone centre in all directions. This contrasts with those superlattices which retain valleys near the zone boundary for electron motion in the conduction band. It may be expected, therefore, that introducing periodic potentials in all directions, as occurs with arrays of quantum dots, might lead to a more direct bandgap in SiGe, and hence a better quantum efficiency for light emission.
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Figure 13.16 Possible transport modes through arrays of quantum dots
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Figure 13.17 Lowest conduction-band (CB) and highest valence band (VB) dispersion curves through a periodic array of Ge quantum dots embedded in a Si host crystal
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13.9 ANTI-WIRES AND ANTI-DOTS The 'reversing' of the materials from which the quantum dots were grown, e.g. depositing a thin wetting layer of Si on to a Ge substrate, could lead to the formation of Si quantum dots. However, as Si has a wider bandgap than Ge, these would repel carriers rather than attracting them, thus leading to use of the term anti-dots. Anti-wires represent a similar concept in one-dimensional systems. Fig. 13.18 shows possible transport trajectories for the former case.
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Figure 13.18 Possible transport modes through arrays of anti-dots
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The development of dot arrays is really still in its infancy, although work has been reported of transport through both dot [272] and anti-dot [273,274] structures. These systems of dots may have applications in future generations of logic [275,276] and Coulomb blockade based [277] devices.
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A detailed study of this book will have revealed that besides containing quite a considerable amount of information, there is also a fair bit missing, and indeed there remains one or two question marks over some of the content that has been presented. The reason for this is that the whole area is still a very active field. It is not 'all done and dusted', and there still remain some major pieces of theory that need to be developed. It is hoped that this present work will provide a stimulus to other workers in the field to fill the gaps gaps that may well have only become apparent upon reading this summary.
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The results of the calculations in the main text depend on the assumed material parameters, a brief summary of which are presented here. GaAs/Ga 1 x Al x As Bandgap, Eg = (1.426 + 1.247x) eV Band alignment: 33% of total discontinuity in valence band, i.e. AVvs =0.33; AVcB = 0.67 Electron effective mass, m* = (0.067 + 0.083x) m0 Heavy-hole effective mass, m* = (0.62 + 0.14x) mo Lattice constant A0 = 5.65 A Low frequency (static) dielectric constant es = 13.18eo High frequency dielectric constant COG = 10.89eo
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