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Figure 7.12 On the left hand side is a schematic diagram of typical induced electric fields in a quantum well based on piezoelectric materials and on the right hand side is the band profile of a GaN/AIN quantum well structure with wavefunction moduli of the first two bound states.
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Figure 7.13 Illustration of the influence of the piezoelectric field on the electronic and optical properties of GaN/AIN quantum wells, (a) the energy difference and (b) the dipole matrix element, between the first two bound states with and without the piezoelectric field.
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interfaces is usually much larger than the quantum well doping limit this screening effect is generally not pronounced in applications. The change of the band profile due to a piezoelectric field can have a strong influence on the energies of intersubband transitions. This is illustrated in Fig. 7.13(a) which shows a comparison of the transition energies between the lowest two subbands in GaN/AlGaN quantum wells with and without the internal electric field included. A substantial increase (blue shift) of the intersubband transition energies is clearly observed and has been observed experimentally [158].
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The piezoelectric field can also considerably modify the localisation properties of the wavefunctions, increasing or decreasing the overlap between the ground and other excited states. This can reflect on the dipole matrix element (defined as Zij = (wi | z | Wj;)) and consequently on the intersubband optical properties of quantum well structures such as the intersubband absorption (Aij ~ |zij |2). For GaN/AlGaN quantum wells the relative change in the dipole matrix element z12 is shown in Fig. 7.13(b). Clearly, the estimated decrease of up to 50% in zij is enough to significantly deteriorate the performance of a device in a potential application. A similar problem is present in quantum well lasers based on interband transitions between the conduction and the valence bands (see [159]). The previously mentioned effects on optical transition energies and absorption strengths highlight the importance of a thorough understanding of the piezoelectric field properties for the modelling and design of optoelectronic devices.
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8.1 FURTHER CONFINEMENT It has already been shown that the reduction in dimensionality produced by confining electrons (or holes) to a thin semiconductor layer leads to a dramatic change in their behaviour. This principle can be developed by further reducing the dimensionality of the electron's environment from a two-dimensional quantum well to a one-dimensional quantum wire and eventually to a zero-dimensional quantum dot. In this context, of course, the dimensionality refers to the number of degrees of freedom in the electron momentum; in fact, within a quantum wire, the electron is confined across two directions, rather than just the one in a quantum well, and, so, therefore, reducing the degrees of freedom to one. In a quantum dot, the electron is confined in all three-dimensions, thus reducing the degrees of freedom to zero. If the number of degrees of freedom are labelled as "Df and the number of directions of confinement are labelled as Dc, then clearly:
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Quantum Wells, Wires and Dots, Second Edition. P. Harrison 2005 John Wiley & Sons, Ltd.
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for all solid state systems. These values are highlighted for the four possibilities shown in Table 8.1. Tradition has determined that the reduced-dimensionality systems are labelled by the remaining degrees of freedom in the electron motion, i.e. T>f, rather than the number of directions with confinement T>c.
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Table 8.1 The number of degrees of freedom X>f in the electron motion, together with the extent of the confinement T>c, for the four basic dimensionality systems
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System Bulk Quantum well Quantum wire Quantum dot
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Vc 0 1 2 3
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Fig. 8.1 gives a simple outline of how quantum wires might be fabricated, although note that there is more than one method and the interested reader should refer to a specialist growth treatise for further details. A standard quantum well layer can be patterned with photolithography or perhaps electron-beam lithography, and etched to leave a free standing strip of quantum well material; the latter may or may not be filled in with an overgrowth of the barrier material (in this case, Gai-^ AlxAs ). Any charge carriers are still confined along the heterostructure growth (z-) axis, as they were in the quantum well, but in addition (provided the strip is narrow enough) they are now confined along an additional direction, either the x- or the y-axis, depending on the lithography.
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