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Figure 4.10 Gaussian fit to the T.R.I.M. code data of Fig. 4.9 together with an exaggerated depth dependence
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Figure 4.10 displays the Gaussian fit to the T.R.I.M. data of Fig. 4.9, described by the parameter o=1700 A, together with the modified data for the purpose of this example, given by a=600 A. In real systems where intensive investigations for a particular semiconductor multilayer with a particular ion implantation dosage are carried out, it would be necessary to relate the diffusion coefficient to the absolute vacancy concentration; however, in these present demonstrator examples of the numerical solution to the diffusion equation, it suffices to say let D be proportional to pvacancy, and furthermore, choose the constant of proportionality to be 10 A2s-1, i.e.
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Given this, Fig. 4.11 displays both the initial and final alloy concentration profile after 200 s of diffusion described by equation (4.16) for a generic 150 A AC/50 A A 1-x B x C superlattice/multiple quantum well. As expected, the central wells have diffused considerably more than those near the edges where the diffusion coefficient is lower. The exaggerated z-dependence of V has fulfilled its goal in this illustration of producing a much clearer depth dependence than previously published for a realistic system (see for example [98]). Photoluminescence measurements on diffused systems such as those shown in Fig. 4.11 would exhibit a broadened emission line, as the photogenerated carriers in the central wells would have a different energy to those in the outer wells. This
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Figure 4.11 Alloy concentration profile x after 200 s of diffusion described by the depth dependent coefficient of equation (4.16)
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phenomenon of line broadening in a diffused superlattice has been observed by Elman et al. [90] and subsequently modelled theoretically [98]. Alternatively, the structure, including the depth dependence of the diffusion, can be mapped directly by using double-crystal X-ray diffraction (DCXRD) [99], a technique which has showed itself to be very valuable in monitoring the progress of interface intermixing [89]. 4.8 TIME DEPENDENT DIFFUSION COEFFICIENT Vacancy-enhanced diffusion will continue for as long as the vacancies are present, but one way to control their lifetime and 'freeze' the diffusant profile is to anneal out the radiation damage. The annealing process thermally activates the interstitials back into vacancies and so restores order to the crystal. This subsequent annealing process could be described by a time-dependent diffusion coefficient, perhaps of the form:
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thus presenting the opportunity to complete the examples of the functional dependencies listed in Section 4.2. D ( z ) is the initial (time-independent) depth-dependent diffusion coefficient due to the vacancy distribution. Using the form in equation (4.16), the diffusion coefficient would then be fully specified by:
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Figure 4.12 Annealing out radiation damage an example of time-dependent diffusion, with annealing times of 0, 50, 100, and 200 s giving the concentration profiles of decreasing amplitude
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Making use of the ion-implantation-enhanced diffusion profile of Fig. 4.11 as a starting point, Fig. 4.12 shows the results of simulating annealing out the lattice damage with a time-dependent diffusion coefficient of the form shown in equation (4.18). In this case, the decay time T of the vacancy concentration was taken to be 100 s, and so the curves represent the points at which the vacancy concentration is a fraction 1, e 1/2, e 1 , and e 2 of its original value. Clearly the curves are converging to a point which represents the region at which the ion-implantation-enhanced diffusion has been frozen. 4.9 (5-DOPED QUANTUM WELLS In the previous sections, examples have been given of simulating diffusion for all of the various forms of diffusion coefficient that can exist. It is clear that the computational method can be extended to include combinations of all three dependencies, and indeed in the final example the diffusion coefficient had both depth and time dependency. It now serves a purpose to follow through an example of a diffusion problem which is of direct relevance to semiconductor heterostructures, and to calculate the subsequent effects on an observable, in this case, the quantum-confinement energies. Contemporary epitaxial growth techniques, such as molecular beam epitaxy (MBE) and chemical beam epitaxy (CBE) etc., allow for the possibility of growing very thin layers of semiconductor material. Another possibility, is the potential for these techniques to lay down very thin layers of dopant atoms. Fig. 4.13(a) represents bulk doping of a layer, as used in HEMTs and the majority of semiconductor heterostructure devices. Fig. 4.13(b) represents the dopant profile for a single 6(Mayer in a quantum
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