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state diffusion [81]:
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where t is the time and the diffusion coefficient V could have temporal t, spatial z and concentration x dependencies, i.e. V = D(x, z, t). Given this, then the derivative with respect to z operates on both factors, resulting in the following:
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which is a second-order ( 8 2 / 8 z 2 ) non-linear ( ( d / d z ) 2 ) differential equation. In any given problem it is likely that two of the following unknowns will be known (!): The initial diffusant profile, x(z, t 0); Thefinaldiffusant profile, x(z,t); The diffusion coefficient, D = D ( x , z , t ) . The problem will be to deduce the third unknown. This could manifest itself in several ways: a. given the initial diffusant profile and the diffusion coefficient, predict the diffusant profile a certain time into the future; b. given the initial and final diffusant profiles, calculate the diffusion coefficient; c. given the final diffusant profile and the diffusion coefficient, calculate the initial diffusant profile. Knowing the versatility achieved by the numerical shooting method solution to Schrodinger's equation as discussed in the previous chapter, then clearly a numerical solution would again be favourable. Learning from the benefits of expanding the derivatives in the Schrodinger equation with finite differences, this would then appear to offer a promising way forward. Recall the finite difference approximations to first and second derivatives, i.e.
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Then equation (4.6) can be expanded to give:
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Notice that the derivative with respect to time has been written as:
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as would be expected from the expansion in equation (4.7). In fact, in this case where this is the only time derivative, the two are equivalent. Assuming the most common class of problem, as highlighted above, as point (a), namely that the function x(z, t] is known when t = 0, i.e. it is simply the initial profile of the diffusant, and the diffusion coefficient D is fully prescribed, then it is apparent from equation (4.9) that the concentration x at any point z can be calculated a short time interval 8t into the future, provided that the concentration x is known at small spatial steps 6z either side of z. This approach to the solution of the differential equation is known as a numerical simulation. It is not a mathematical solution, but rather a computational scheme which has been derived to mirror the physical process. It has already been mentioned that the diffusion coefficient V could be a function of x, z, and t, with the form of D being used to define the class of diffusion problem, e.g. i. D= Do, a constant, for simple diffusion problems. ii. V = D(x), a function of the concentration as encountered in non-linear diffusion problems [86]. Note, that as x = x (z) then D. is intrinsically a function of position too. iii. D D(z), a function of position only, as could occur in ion implantation problems [87]. Here the diffusion coefficient could be linearly dependent on the concentration of vacancies for example, where the latter itself is depth dependent. iv. D = D(t], a function of time, as could occur during the annealing of radiation damage. For example, ion implantation can produce vacancies which aid diffusion [88,89]. During an anneal, the vacancy concentration decreases as the lattice is repaired, which in turn alters the diffusion coefficient.
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4.3 BOUNDARY CONDITIONS Thus, given the initial diffusant profile and a fully prescribed diffusion coefficient, everything is in place for predicting the profile of the diffusant at any time in the future, except for the conditions at the ends of the system. These can not be calculated with the iterative equation (equation (4.9)), as the equation requires points which lie outside the z-domain. For diffusion from an infinite source, it may be appropriate to fix the diffusant concentration x at the two end points, e.g. x(z 0,t) = x(z = 0,t = 0). Alternatively, the concentrations x at the limits of the z.-domain could be set equal to the adjacent points which can be deduced from equation (4.9). Physically this defines the semiconductor structure as a closed system, with the total amount of diffusant remaining the same. It is these latter 'closed system' boundary conditions which will be employed exclusively in the following examples. 4.4 CONVERGENCE TESTS Figure 4.3 shows the result of allowing the diffusant profile in Fig. 4.1 to evolve to equilibrium, using the closed-system boundary conditions as described above. Clearly the 'closed' nature of the system can be seen the total amount of diffusant remains the same, and ultimately as would be expected for the 'water step', the concentration reaches a constant value. In the case of water, this could be looked upon as minimising the potential energy. If the diffusion process can be described by a constant diffusion coefficient, D = DO, then the general diffusion equation, equation (4.5), and its equivalent computation form, (equation (4.9)) simplifies to the following:
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which as mentioned above has error-function solutions for the case of diffusion at the interface (z=0) of a semi-infinite slab of concentration X3 [81], i.e.
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where erfc is the complementary error function see reference [38], p. 295. This technique can be imposed on multiple heterojunctions by linearly superposing solutions [77]. Fig. 4.4 compares just such an error function solution* with the numerical solution for a 200 A single GaAs quantum well surrounded by 200 A Gao.9Alo.1 As barriers after 100 s of diffusion described by a constant coefficient Do=10 A2s-1. Clearly, the numerical method advocated here exactly reproduces the analytical solution.
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