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As shown in figure 1, A is the depletion width on the heavily-doped side of the junction, B is the depletion width on the lightly-doped side of the junction, N(A) and N(B) are the respective net dopant densities at distances A and B from the junction, No is the dopant density at the surface of the diffused layer, and N, is the estimated background dopant density in the diffused layer. The characteristic Pength of the Gaussian diffusion is represented by L, and I is the area of either of the two shaded regions in figure 1. The lower case w and a are used as variables of integration; whereas, upper case W and A represent specific values of w and a. The quantities W, A and B are related by eq (11).
In computer program cvi, the integral on the left in eq (9) is calculated by numerical integration of the N(W) vs. W data. Equation (7) is used in calculating the first step in the numerical integration and the trapezoidal approximation is used in subsequent steps. The value of this integral, I, is used to solve eq (10) for A. Knowing A, N(A) is calculated from eq (5), N(B) is calculated from eq (8) and B is calculated from eq (11). The desired profile is a plot or listing of N(B) vs. B.
Equation (12) below is used to calculate a true Gaussian profile for comparison with the experimentally determined profile.
N(B) = No
Experimental profiles sometimes show a decrease in N(B) for small B. A plot of the true Gaussian profile can be helpful in determining whether such an experimental observation is caused by the diffusion tail or by some other phenomenon. The true Gaussian plot is also helpful in evaluating the adequacy of the assumed N, value. If the experimental plot and the true Gaussian plot do not asymptotically approach the same background density, the calculation should perhaps by repeated using a different assumed N, value.
The solution of eqs (7) and (10) calls for calculating the complementary error function and its inverse. The calculation of the complementary error function employs equations given by Stegun and Zucker (14) as
Figure 1. Schematic representation of the depletion region of a p-n junction diode diffused in an epitaxial semiconductor. (The distances A and B have their origins at the junction. Note that the actual background dopant density, represented by the irregular line to the right of the junction, is assumed to be constant within the diffused layer to the left of the junction.)
The reader is advised that eqs (13) to (15) and the software that implements them (see sec. 3.3.) have been thoroughly proven within the range of applicability. Other approximations to the error function and its complement should be used only with extreme caution. gram cvi, the power series representation of eq (13) is used when x < 1, and the continued fraction of eq (15) is used when x > 1.
The inverse complementary error function is calculated by interpolation in a table of values. The equation to be solved is
(17) ln (10-7) to yield a value of z from 0 to 100 corresponding to a value of x from 10-7 to 1. The transformation is illustrated in figure 2.
The 10-7 lower limit was chosen so that eq (10) could be solved for all practical ratios of No to N: The table of values (see listing in lines 150 to 188 of Appendix A) lists those values of y corresponding to integer values of 2 from 1 to 100. For example, the first value in the table y 3.74580, corresponds to z = 1 or
1.17490 x 10-7. To calculate y from a given x, z is first calculated and three data pairs (21, yı), (22, y2), and (23, y3) are chosen from the table such that
This is illustrated in figure 3. A parabolic fit to these three data pairs is made by sirultaneously solving the equations
2, and 23 - 22 = 1.
Equations (20) and (21) can be simplified since 22 - 21 = 1, 23 - 21
yi + y 3
Figure 2. Semilog plot of the function x = erfc (y) showing the transformed variable
(The discontinuity which occurs at z 2, and the fact that y = 3.76656 for 0 <z < 2 reflect the value of y calculated by the INERF subroutine rather than the true value of y as explained in sec. 3.4.)