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\section{PN junction}
\subsection{What are we even doing}
We stick together n and p doped regions, such that the doping effectively becomes a step function.
This causes majority carriers (electrons in n region, holes in p region) to diffuse the minority carrier side,
resulting in a new equilibrium (\autoref{label:fig:pn_carrier_profile_equilibrium}).
\begin{figure}[h]
\centering
\begin{subfigure}[b]{.45\textwidth}
\includegraphics[width=\textwidth]{imgs/pn_carrier_profile_equilibrium.png}
\caption{Resulting carrier profile in thermal equilibrium}
\label{label:fig:pn_carrier_profile_equilibrium}
\end{subfigure}
\hfill
\begin{subfigure}[b]{.45\textwidth}
\includegraphics[width=\textwidth]{imgs/pn_fermi_level_band_bending.png}
\caption{Resulting carrier profile in thermal equilibrium}
\label{label:fig:pn_fermi_level_band_bending}
\end{subfigure}
\end{figure}
As can be seen in \autoref{label:fig:pn_fermi_level_band_bending},
the energy levels for conduction and valence bands bend, whereas the fermi level remains constant.
\subsection{Depletion approximation}
We assume p and n regions quasi-neutral,
and the intermediate space charge region to be completely depleted of carriers.
We further assume all transitions are expressed as step-functions.
This allows the following simplified equations:
\begin{align}
\rho(x) & = \begin{dcases}
0 & x<-x_p \\
-qN_a & -x_p<x<0 \\
qN_d & 0<x<x_n \\
0 & x_n<x
\end{dcases} \\
E(x) & =\begin{dcases}
0 & x<-x_p \\
-\frac{qN_a}{\varepsilon}(x+x_p) & - x_p<x<0 \\
\frac{qN_d}{\varepsilon}(x-x_n) & 0<x<x_n \\
0 & x_n<x
\end{dcases}
\end{align}
Where $E$ is found using \eqref{label:eq:def_electric_field_integral}.