The charge carriers in semiconductors are electrons ($n$ from "negative" with charge $-e$) and holes ($p$ from "positive" with charge $e$), whose chaotic thermal motion is slightly distorted by external fields. The elementary charge $e=1.60 \cdot 10^{-19}~\text{C}$. To describe the transfer processes, the carrier mobility $\mu$ is introduced, which determines the dependence of its average velocity $v$ on the applied external electric field $\mathcal{E}$:
\[ |v| = \mu |\mathcal{E}|.\]
The mobility depends on the temperature and the concentration of defects in the crystal lattice. In this case, the volume concentration of electrons $n$ (units $1/\text{m}^3$) and the volume concentration of holes $p$ (units $1/\text{m}^3$) under conditions of thermodynamic equilibrium are related to each other by the relation $np = n_i^2$, where $n_i$ is the concentration of carriers of both types in a semiconductor without impurities.

If there is spatial inhomogeneity in the distribution of charge carriers in a semiconducting sample, diffusion currents with corresponding volume densities flow in addition to conduction currents:
\[j_{Dp} = - eD_p \frac{dp}{dx}, \quad j_{Dn} = eD_n \frac{dn}{dx}.\]
And from Einstein's relation $D = \mu k_\text{B} T/e$, so the total current density inside the semiconductor is
\[ j =\mu_p \left[ \mathcal{E} e p - k_\text{B} T \frac{dp}{dx} \right] + \mu_n \left[ \mathcal{E} e n + k_\text{B} T \frac{dn}{dx} \right]. \tag{1}\]

Let us denote $\mu_p/ \mu_n=\alpha$, for silicon we will consider this value to be $3.1$ and independent of impurities. The concentration of impurity-donors increasing the number of electrons will be called $N_D$ and the concentration of impurity-acceptors increasing the number of holes will be called $N_A$.

We will consider a one-dimensional problem, for inhomogeneously doped silicon at temperature $T=300~\text{K}$, then $n_i=1.1 \cdot 10^{10}~\text{cm}^{-3}$.

1. In the region $x<0$ a p-silicon is created with $N_A = N_p \gg n_i$, $N_D=0$.
2. In the region $x>0$ an n-silicon is created with $N_D = N_n \gg n_i$, $N_A=0$.

We will also assume that all processes are stationary, i.e., charge does not accumulate anywhere and therefore $dj/dx=0$. The electric potential is denoted by the letter $\varphi$. For nondimensionalization, let's work in terms of relative concentrations $u=p/n_i$, $v=n/n_i$. The relative permittivity of a semiconductor is $\varepsilon$.

Part А. Equilibrium in PN junction

First, let us consider an equilibrium PN junction with no current flowing. In such a case, holes and electrons are in thermodynamic equilibrium at every point, i.e. $v=1/u$.

A1 Substitute $j=0$ and $v=1/u$ into the equation $(1)$ and get the relationship between $\varphi'$, $u'$, and $u$.

A2 Integrate the equation obtained in A1 and find the potential difference $V_{bi}=\varphi(+\infty) - \varphi(-\infty)$ of the electric field at the pn junction. Express the answer in terms of $N_p$, $N_n$, $n_i$

A3 Write down Gauss' theorem and express $\varphi'$ by $u$ and $N_D-N_A$. Get the differential equation for $\Psi$ where $\Psi = \ln u$ as
\[L^2 \Psi'' = 2 \sinh \Psi - \frac{N_A-N_D}{n_i} \tag{2}\]

We have obtained a nonlinear differential equation on $\Psi$ and will solve it numerically.To set the initial conditions, we consider the behavior of $\Psi$ away from the pn junction, where $\Psi = \operatorname{asinh} \frac{N_A-N_D}{2n_i} + \delta \Psi$ and $|\delta \Psi| \ll |\Psi|$.

A4 Find the analytic expression for $\delta \Psi$ of the previous equation in regions far from the pn junction.

A5 Find the analytic expression for $\Psi$ for the domain, where $n_i|2 \sinh \Psi| \ll |N_A - N_D|$.

A6 Use the program pn-junction.py to solve equation (2) numerically for $N_p=2 \cdot 10^5$, $N_n=10^5$. Collect some data and plot a graph $\Psi(0)/\ln(N_nN_p)$ vs. $\ln N_n/N_p$.

Part B. PN junction capacitance

Inside the pn junction there is a region where $p,n \ll N_p, N_n$. To obtain the analytical result, we will assume that the charge density $\rho=e(p - N_A - n + N_D)$ has the form shown in the graph below. That is

1. in the region $x<-x_p$ we have equilibrium p-silicon with $p=N_p$;
2. in the region $-x_p < x < x_n$ we have $N_p, N_n \gg p ,n$;
3. in the region $x>x_n$ we have equilibrium n-silicon with $n=N_n$.

B1 Find the dependences of the electric field $\mathcal{E}(x)$ in the chosen model and draw its graph. What relation between $x_p$, $x_n$, $N_p$, and $N_n$ must be satisfied so that the field $\mathcal{E}$ goes to zero at the boundaries $-x_p$ and $x_n$ and to is continuous at the point $x=0$?

If we connect the pn junction to a source of voltage $U$ (plus to $p$ and minus to $n$) and infinite internal resistance, the potential difference across at the pn junction becomes $V_{bi} - U$.

The capacitance of the system $C$ is defined as $C=dQ/dU$, where $dQ$ is the change in charge on each side of the pn junction (they have different signs). In our model, when voltage $U$ is applied, the size of the spatial charge distribution region decreases from $[-x_p,x_n]$ to $[-x_p',x_n']$. In particular, when voltage $U$ is applied, the value of $u$ (relative hole concentration) in the vicinity of the junction itself changes slightly, so these changes do not affect the appearance of the charge and field distributions.

B2 Express $x_p'$ through $x_p$, $V_{bi}$ and $U$. Substitute $dQ = -eSN_p dx_p'$, where $S$ is the area of the pn junction, into the expression for the capacitance $C$ and find the dependence of $C(U)$. Express the dependence $C(U)$ in terms of $S$, $N_{n,p}$.

Thus, a pn junction can be used as a capacitor whose capacitance depends on the applied voltage - such devices are called varicaps.

Часть С. ВАХ pn перехода.

When an external voltage is applied, the charge carrier distribution picture changes significantly only at the edges of the pn junction, where the concentration of non-main carriers (i.e., electrons in p-silicon and holes in n-silicon) changes significantly.

In the domain $[-x_p, x_n]$ the expression from A1 remains valid, since the current density $j$ is a small correction relative to the other terms. We denote the recombination rate (the rate of the process of mutual annihilation of holes and electrons) by $R$ and the formula for it follows from the consideration that the case $nv=1$ must correspond to the equilibrium:
\[R = \frac{uv-1}{\tau_p (u+1) + \tau_n(v+1)} n_i,\]where $\tau_p$ and $\tau_n$ are the characteristic lifetimes of the corresponding particles.

C1 Suppose that due to the presence of the voltage $U$, the concentrations of holes and electrons have changed with respect to the equilibrium $U=0$ as follows $u \to u+ \delta u$, $v \to v + \delta v$. Express the current density $j$ through $\delta u$, $\delta v$, their derivatives, and the field $\mathcal{E}$.

C2 Write the stationarity equations for $\delta u$, $\delta v$ through their values and the values of their derivatives, considering that changes in these quantities over time can only be due to currents and recombination.

C3 Transform the stationarity conditions for points where $\mathcal{E}=0$ and find the form of the functions $\delta v$ and $\delta u$ in the range $x<-x_p$. Take into account $N_p, N_n \gg n_i$ and the smallness of $\delta u$, $\delta v$.

You should find that $\delta v$ and $\delta u$ are of the same order of magnitude, so the majority carrier density is approximately unchanged from equilibrium.

C4 Assuming that the equation from A1 is satisfied in the region $x \in [-x_p', x_n']$, find the constants for $\delta u$ and $\delta v$ for the results of C3.

C5 Obtain the I-V characteristic of the pn junction in the form $j = f(U)$.