(a) Constant surface concentration diffusion : The impurity source is present throughout diffusion and the impurity concentration at the surface C_s is thus held at a fixed value. C_s is taken as solid solubility limit in silicon at the diffusion temperature for the impurity in question (see Figure 5). The intitial condition (t = 0) is C(x,0) = 0, while the boundary conditions for all subsequent times are C(0,t) = C_s and C(,t) = 0; i.e. the concentration at the surface remains at the solid solubility limit while at large distances below the surface, where the impurities are unlikely to penerate during realistic diffusion times, the concentration is zero. The solution for constant surface concentration diffusion is;

where erfc is the complimentary error function and is the diffusion length. If the units of D and t are µm² / hr and hours respectively, then the diffusion length is in µm. It is common to plot the curve of equ. 2 in "normalised", universal form as C(x,t) / C_s vs x (or even ) which can be applied to any semiconductor if D and C_s are known for a particular impurity diffusing in the semiconductor of interest (Figure 6).

Figure 5: Solid solubility (cm^-3) of various impurity atoms in silicon as a function of temperature (^oC).

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Figure 6: Plots of C(x,t)/Cs vs diffusion depth x(µm) under constant surface concentration conditions for three different values of . This could mean either a change of temperature (i.e D(T)) or time, t.

An important quantity is the total number of impurities Q(t) introduced into the semiconductor in time t under constant surface concentration conditions. This is simply the integral of C(x,t) over all values of x;

There will be frequent need to calculate Q(t) for a "predeposition" step (see next section) when the diffusion time is usually relatively short, the diffusion depth is very shallow and the impurity profile is assumed to be a delta-function.

(b) Constant total dopant (number) diffusion : A fixed number of impurities (cm^-2 or m^-2 of surface area) given by equ. 3 is introduced into the wafer in a short pre-deposition step under constant surface concentration diffusion conditions, and then diffused deeper into the wafer with the external source of impurity atoms removed during a second phase called drive-in. The boundary conditions are;

where Q is the total number of dopant atoms per unit area which is constant. The solution of Fick's equation under these conditions is;

This is a Gaussian distribution.

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Figure 7: Three impurity profiles carried out under constant total dopant diffusion conditions. Note the reduction in the surface concentration with time, and the corresponding rise in the bulk density.

The surface concentration (x = 0) is

which decreases with time since Q is constant; i.e. the impurities are initially located in a very thin layer close to the surface but spread out into the wafer during drive-in, reducing the surface volumetric concentration and increasing it elsewhere (Figure 7).