Diffusion-Segregation Equation for Simulation in Heterostructures
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[1]
EQUATION
The flux equation of diffusion-segregation (FEDS) and the DSE are respectively given by
J(x) = -D _cC ax
(1)
a_ a [Do C ca at
ax
(ax
c-l ax!J
(2)
where Ceq is the species' thermal equilibrium concentration at position x. These expressions are derived on thermodynamic bases [1]. Segregation phenomenon occurs because the Gibbs free energy of formation gf of a (point defect or impurity) species is different in the different material regions, i.e., it is a function of the spatial coordinate x, gf(x). Using gf(x) and the species' spatially varying concentration C(x), we can write an expression for the Gibbs free energy density at x, G(x). Subsequently, we obtain the chemical potential at x, ji(x)=aG(x)/aC(x), which in turn yields the force exerted on the species, f(x)=-ajg(x)/ax. f(x) accelerates the species to the velocity v(x)=M(x)f(x), where M(x) is the species' mobility which is related to the species' diffusivity D(x) by the Einstein relationship M(x)=D(x)/kBT. With v(x) available, we obtain Eqs. (1) and (2). The FEDS and DSE accounts for both the diffusion and the segregation processes of a species in a crystal by two terms on 31 Mat. Res. Soc. Symp. Proc. Vol. 318. @1994 Materials Research Society
have examined the validity of this empirical formulation by comparing it to our FEDS. Using the central-difference approximation for spatial derivatives, we obtain from Eq. (4) J =Fs =DmEl- 1./2 ( C1-m-C2) (7) where mi+l/,=(l+m 1-2 )/2. The factor (C1 -C2/m 1 -2 ) in Eq. (6) is in principle a valid description, which is the reason for its limited success. The parameter h, however, turned out to be not only proportional to local material properties D and m 1+11/2 but also dependent upon the computational grid size dx. Furthermore, Eq. (6) computes the flux across the transition region by only two grid points, wherein a great number of grid points are needed for ensuring accuracy. This will lead to large computational inaccuracies for cases with large segregation coefficient and/or diffusivity differences in the two regions. It is obvious that the approach [2] is unsatisfactory. The other formulation is that of Orlowski [3]. He treated the impurity diffusion-segregation problems using a set of coupled rate equations which contain absorption and emission coefficients. In his opinion, the segregation phenomenon of a two region heterostructure results from the existence of a narrow but non-vanishing interfacial region which constitutes a third phase. In this region he used absorption and emission coefficients which are functions of the impurity concentration, yielding a quadratic dependence of the ac/at term on C, which is rather aphysical. The existence of a different interfacial phase is a problem of a higher degree of complexity. Ignoring the presence of this third phase, we have obtained a correlation between Orlowski's model and our DSE. We have found that his absorption and emission coefficients are functions of material parameters D and m, as well as the computational grid size dx. In addition
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