Derivation of the depletion width of an a abrupt PN homojunction
\psi_{\text{potential}},\; \quad -\frac{\partial \psi}{\partial x} = E\begin{cases}
\text{Slope} \quad \frac{\partial E}{\partial x} = -\frac{q}{\epsilon_s} n(x) = -\frac{q N_a}{\epsilon_s} \\\text{Slope} \quad \frac{\partial E}{\partial x} = \frac{q}{\epsilon_s} n(x) = -\frac{q N_d}{\epsilon_s}
\end{cases}\begin{aligned}
\begin{cases}
E(x) = -\frac{q N_a}{\epsilon_s} x + k_p &&&&&&&&&&&&&\\
E(x) = -\frac{q N_d}{\epsilon_s} x + k_n
\end{cases}&&&&&&&&&&&& \\
\begin{cases}
E(x) = -\frac{q N_a}{\epsilon_s} (-x - x_p) \; ,\;&& E(x_p) = 0&&&& \\\\
E(x) = -\frac{q N_d}{\epsilon_s} (x - x_n) \; ,\; &&E(x_n) = 0
\end{cases}
\end{aligned}\begin{aligned}
V_{bi} &= -\int_{x_p}^{0} \frac{-q N_a}{\epsilon_s} (x + x_p) \, dx - \int_{0}^{x_n} \frac{q N_d}{\epsilon_s} (x - x_n) \, dx \\
&= \frac{q N_a}{\epsilon_s} \left[ \frac{x^2}{2} - x x_p \right]_{-x_p}^{0} - \frac{q N_d}{\epsilon_s} \left[ \frac{x^2}{2} - x x_n \right]_{0}^{x_n} \\
&= \frac{q N_a}{\epsilon_s} \cdot \frac{x_p^2}{2} + \frac{q N_d}{\epsilon_s} \cdot \frac{x_n^2}{2}
\end{aligned}\begin{aligned} &W = x_p + x_n \quad \text{with} \quad x_p N_a = x_n N_d \Rightarrow x_p = x_n \frac{N_d}{N_a} \\ &W = x_n \left(1 + \frac{N_d}{N_a} \right) \Rightarrow x_n = \frac{W}{1 + \frac{N_d}{N_a}}, \quad x_p = \frac{W N_d}{N_a + N_d}
\end{aligned}V_{bi}=\frac{qN_{a}W^{2}}{2\epsilon_{s}}\frac{N_{d}^{2}}{\left(N_{a}+N_{d}\right)^{2}}+\frac{qN_{d}W^{2}}{2\epsilon_{s}}\frac{N_{a}^{2}}{\left(N_{a}+N_{d}\right)^{2}}\\V_{bi} = \frac{q W^2}{2 \epsilon_s} \left( \frac{N_a N_d}{(N_a + N_d)^2} \cdot (N_a + N_d) \right)
V_{bi} = \frac{q W^2}{2 \epsilon_s} \cdot \frac{N_a N_d}{N_a + N_d}
\quad \Rightarrow \quad
W = \sqrt{ \frac{2 \epsilon_s V_{bi}}{q \cdot \frac{N_a N_d}{N_a + N_d}} }
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