B6. Pearson Function

For a realistic description of dopant profiles the Pearson Type IV distribution is well established today. The Pearson function family is based on the differential equation

$$\frac{df(y)}{dy} = \frac{y - a}{b_0 + b_1 y + b_2 y^2} f(y), \quad y=x-R_p .$$ (B9)

with the coefficients

$$a = - \frac{\gamma_1 \sigma_p (\beta_2 + 3)} {10 \beta_2 - 12 \gamma_1^2 - 18}$$ (B10)

$$b_0 = - \frac{\sigma_p^2 (4 \beta_2 - 3 \gamma_1^2)} {10 \beta_2 - 12 \gamma_1^2 - 18}$$ (B11)

$$b_1 = - \frac{\gamma_1 \sigma_p (\beta_2 + 3)} {10 \beta_2 - 12 \gamma_1^2 - 18}$$ (B12)

$$b_2 = - \frac{2 \beta_2 - 3 \sigma_1^2 - 6)} {10 \beta_2 - 12 \gamma_1^2 - 18}$$ (B13)

where R_p, $\sigma$, $\gamma$, and $\beta$ are the projected range, standard deviation, skewness and kurtosis, respectively. The solution for the Pearson Type IV function is

$$f(y) = K \left ( - \left ( b_0 + b_1 y + b_2 y^2 \right )... ... b_2 y - b_1}{\sqrt{4 b_2 b_0 - b_1^2}} \right ) } \right ) }$$ (B14)

$$0 < \gamma_1^2 < 32$$ (B15)

$$\beta_2 > \frac{39 \gamma_1^2 + 48 + 6(\gamma_1^2 + 4)^{\frac{3}{2}}} {32 - \gamma_1^2}$$ (B16)

The maximum value is reached for x = R_p. The projected range R_p is the average depth of the distribution. Skewness $\gamma$ measures the asymmetry of the distribution, a positive value places the peak closer to the surface than the projected range. The kurtosis $\gamma$ describing the flatness of the top of the distribution.