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work giving rise to new science something that could not have been predicted from theory. In calculating the self-consistent exciton energy in series of multiple quantum wells, Piorek et al. [150] found that a 4-period multiple quantum well could have a larger exciton energy than the equivalent 3-period system. This seemed a contradiction as a 4-well system can be thought of as a 3-well system but with some of the potential barrier removed. Therefore it should have a lower exciton energy. This seemed to be a general result, i.e. that 2n-period multiple quantum wells had a larger energy than 2n 1 systems. Inspection of the self-consistently iterated electron and hole wave functions, 7/4 and whm , respectively, showed that in systems with odd numbers of wells, the electron and hole localised in the central well leading to a large binding energy. However, in systems with an even number of wells, the electron and hole localised in the two central wells, thus giving a smaller binding energy, and so leading to a higher total energy. However, for certain calculations, exceptions to the above rule were identified. It was found that if there was the slightest asymmetry in the original electron and hole wave functions, we and wh, then the self-consistent iteration repeatedly increased it. Such asymmetries in we and wh needed only to be minute in the first instance, and could arise merely from a coarse finite difference mesh; in fact, a difference in the wave function maxima of 1 part in 1010 was enough to have an effect. Given these initial conditions, the iteration forcibly broke the symmetry of the electron and hole, thus forcing them to localise in the second well of a four-well system, for example. A systematic study of the original multiple quantum wells found that if a very small asymmetry was introduced into the potential profile of the heterostructure (perhaps just one monolayer within the well, having an alloy concentration of 10 5 rather than zero), then it was always enough to spontaneously break the symmetry of the exciton [150], with the result being that the exciton energy always decreased as the number of wells in the system increased. A point to note here is the size of the asymmetry necessary in the potential to bring about localisation. This is very small and will always be present in even the highest quality semiconductor layers. However, in addition, there are many other mechanisms which could bring about the small asymmetry, e.g. the presence of a single impurity atom, a phonon, or another charge carrier, will all be sufficient to induce a tiny perturbation in either the electron or the hole wave function, which will then influence the final exciton state. Once the exciton begins to form, it 'pulls itself up by its own bootstraps' until it is fully localised within one well. Therefore, in conclusion, excitons within multiple quantum well systems will (in general) spontaneous localise into just one of the wells. It must be noted for the record, however, that the spontaneous symmetry-breaking of excitons in multiple quantum wells has recently been disputed in the literature [151].
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In a similar manner to both the cases of the hydrogen atom and impurities, excitons can also have excited states. While the electron and hole one-particle states remain unchanged, the corresponding choice for the relative motion factor of the 2s excited state is given by:
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where A2S is stated specifically to imply the Bohr radius of the 2s state and a is a (as yet undefined) parameter, chosen to ensure orthogonality of the Is and 2s eigenstates, say, for example, as follows:
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In the spirit of the above, this can be written as:
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where, in a similar manner to equation (6.18):
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Using the specific forms forw1srsandw2sras above, then:
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where I/A' = l/Ai s + l/A 2S .Writing this as L(a) = L\ aL-2, then obtain:
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The remaining functions, F(a), G(a), J(a) and K(a), can all be derived by using similar procedures to the 1s case, and after lengthy manipulation are given by:
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