Complementary error function erfc

Exposure of bulk GaAs Si wafers to a capacitively coupled rf deuterium plasma at different temperatures generates deuterium diffusion profiles as shown in Fig. 1. These profiles are close to a complementary error function (erfc) profile. At 240°C, the effective diffusion coefficient is 3 x 10 12 cm2/s. The temperature dependence of the hydrogen diffusion coefficient is given by (Jalil et al., 1990) ... 

Again we find the complementary error function, erfc(y), with the same argument as in Eq. 18-20. This time the solution applies to both sides of the interface x < 0 for system B, x > 0 for system A. The interface is located at x = 0, where C (0,t) is always equal to (1/2) (Cg -CA), since erfc(0) = 1. Note that the solution is symmetrical around x = 0 in the sense that the losses and gains are equal at equal distances from the boundary (Fig. 18.5e). The transformation back to the original concentrations (see Eq. 18-26) yields ... 

Thus, for the boundary conditions just described, the concentration of impurity as a function of space and time is given by a complementary error function (erfc) whose argument is x/ Dt. The complementary error function is a tabulated function. 

In test programs, where the numerical solution is compared with the analytical solution, the latter often involves the error function erf or the complementary error function erfc. The latter could be obtained simply by subtracting erf from unity but a better approximation is obtained by the direct algorithm. The two routines, ERF and ERFC, were given to the author by a colleague, who probably obtained them from an IBM collection. They have been adapted to Fortran 90/95 by the author, and coupled to the above module. The comments in capitals are the original comments. 

This is written more conveniently in terms of the complementary error function, erfc<77) = 1 - erfCrj) hence. 

In this equation, erfc(z) is the complementary error function erfc(z) = 1 -erf(z). 

The two requirements on the sphtting function / mentioned above leave a large freedom of choice [6,33,34]. The traditional selection is the complementary error function erfc(r) = dt e. This results in the well-... 

Error Function (erffm]) and Complementary Error Function (erfc[< ])... 

The error function can be viewed as a partial area under the Gaussian distribution curve (2/n )exp -u ). It has the value 0 at x = 0, and a value of 1 at X = oo. The latter case corresponds to the full area under the distribution curve. A related expression is the complementary error function erfc x, which is defined as... 

Fig. 9.9. The complementary error function erfc(x). On the left, we have plotted erfcW on a linear scale. On the right, we have plotted the function erfc(A )/jc of (9.8.28) and (9.8.29) (grey line) and its upper bound exp(-A- )/(V x ) (black line) on a logarithmic scale. The function erfc(x)/x represents the error incurred by calculating interactions between Gaussian charge distributions using point-charge models.

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