diff --git a/01_fundamentals.tex b/01_fundamentals.tex
index 1587fd95cb21a26bb2c0718e37bf3f58464516df..0952adb19827c5131870e5b5c4da3f42abaef31f 100644
--- a/01_fundamentals.tex
+++ b/01_fundamentals.tex
@@ -15,6 +15,7 @@ It then leaves behind a hole.
\begin{figure}[h]
\centering
\includegraphics[width=.6\textwidth]{imgs/band_gap_electorn_holes.png}
+ \caption{Band-gap electrons/holes}
\end{figure}
\subsection{Generation and recombination}
diff --git a/02_carrier_transport.tex b/02_carrier_transport.tex
index cccd15b46982b4fae513f32b3d1d11cfe3214ea5..87ee1485e2c7db03b9a660f099299bde0b672587 100644
--- a/02_carrier_transport.tex
+++ b/02_carrier_transport.tex
@@ -96,13 +96,14 @@ ergo the double negative for electron diffusion.)
\subsection{Einstein relation between mobility and diffusion coefficient}
+\label{label:sss:einstein_rel_mob_diff}
\begin{equation}
\frac{D_n}{\mu_n} = \frac{D_p}{\mu_p} = \frac{kT}{q^2}
\end{equation}
\subsection{Total current}
\begin{alignat}{2}
- J_{total} & =J_n+J_p & & \\
- J_n & =J_n^{drift}+J_n^{diff} & & =qn\mu_nE+qD_n\frac{\mathrm{d} n}{\mathrm{d} x} \\
+ J_{total} & =J_n+J_p & & \\
+ J_n & =J_n^{drift}+J_n^{diff} & & =qn\mu_nE+qD_n\frac{\mathrm{d} n}{\mathrm{d} x}\label{label:eq:electron_current_density} \\
J_p & =J_p^{drift}+J_p^{diff} & & =qp\mu_pE-qD_p\frac{\mathrm{d} p}{\mathrm{d} x}
\end{alignat}
\ No newline at end of file
diff --git a/03_pn_junction_basics.tex b/03_pn_junction_basics.tex
new file mode 100644
index 0000000000000000000000000000000000000000..e4647a0f0595a4697d963c61a5d1f62a9a905a89
--- /dev/null
+++ b/03_pn_junction_basics.tex
@@ -0,0 +1,76 @@
+\section{PN junction basics}
+\subsection{Uniformly doped semiconductor}
+Assuming n type (i.e. lots of electrons, few holes) semiconductor with a uniform doping profile,
+we have the following volume charge density.
+\begin{align}
+ n_0 & = N_d \\
+ \rho & = q\left(N_d-n_0\right)=0 \ \left[C/cm^{3}\right]
+\end{align}
+
+\subsection{Non-uniformly doped semiconductor}
+At thermal equilibrium, the total current must be 0.
+Because we have non-uniform doping,
+we need the drift current to balance the diffusion current.
+(For both electrons and holes.)
+\begin{equation}
+ J_n(x)=J_n^{drift}(x)+J_n^{diff}(x)=0
+\end{equation}
+Which implies
+\begin{equation}
+ n(x)\neq N_d(x)
+\end{equation}
+Which gives us the space charge density
+\begin{equation}
+ \label{label:eq:electron_space_charge_density}
+ \rho(x)=q\left[N_d(x)-n(x)\right]\neq 0
+\end{equation}
+
+This results in a potential difference (quick electromag recap):
+\begin{align}
+ \frac{\mathrm{d}E}{\mathrm{d}x} & = -\frac{\rho}{\varepsilon} \label{label:eq:def_electric_field_differential} \\
+ E(x)-E(0) & =\frac{1}{\varepsilon}\int_{0}^{x}\rho(x')\,\mathrm{d}x' \label{label:eq:def_electric_field_integral}
+\end{align}
+
+Since there is an electric field, there's a potential.
+\begin{align}
+ \frac{\mathrm{d}\phi}{\mathrm{d}x} & = -E \\
+ \phi(x)-\phi(0) & =-\int_{0}^{x}E(x')\,\mathrm{d}x'
+\end{align}
+
+By combining eq \eqref{label:eq:electron_current_density} ,
+\eqref{label:eq:electron_space_charge_density},
+and \eqref{label:eq:def_electric_field_differential} we find
+\begin{equation}
+ \frac{\mathrm{d}^2}{\mathrm{d}x^2}\ln{n(x)} = \frac{q^2}{\varepsilon kT}\left(n(x)-N_d(x)\right)
+\end{equation}
+
+\subsection{Quasi-neutral approximation}
+If the doping changes slowly with x:
+\begin{equation}
+ n(x)\approx N_c(x)
+\end{equation}
+
+
+\subsection{Boltzman relation between $n$ and $\phi$}
+We saw in \autoref{label:sss:einstein_rel_mob_diff} the relation between mobility and diffusion coefficients.
+From this we find
+\begin{align}
+ n & =n_{ref}e^{q(\phi-\phi_{ref})/kT} \\
+ \phi_ref & =0 \\
+ n_{ref} & =n_i
+\end{align}
+And by extension
+\begin{align}
+ n & = n_i e^{q\phi/kT} \\
+ p & =n_ie^{-q\phi/kT}
+\end{align}
+
+Rearranging the above, we find a rule of thumb for the potential:
+\begin{align}
+ \phi & =\frac{kT}{q}\ln\frac{n}{n_i} \\
+ \phi & = -\frac{kT}{q}\ln\frac{p}{n_i}
+\end{align}
+For Si at room temperature this is an increase of 60 mV per decade in doping.
+\begin{equation}
+ \phi\approx(60\,\mathrm{mV})\log_{10}\frac{n}{10^{10}}
+\end{equation}
diff --git a/04_pn_junction.tex b/04_pn_junction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..b84b162596581406beb5f00729d3593e0d7249b0
--- /dev/null
+++ b/04_pn_junction.tex
@@ -0,0 +1,46 @@
+\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}.
+
diff --git a/imgs/pn_carrier_profile_equilibrium.png b/imgs/pn_carrier_profile_equilibrium.png
new file mode 100644
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diff --git a/imgs/pn_fermi_level_band_bending.png b/imgs/pn_fermi_level_band_bending.png
new file mode 100644
index 0000000000000000000000000000000000000000..1d4fccda220aaf1ff4bb72fd3ef18c9d86d31e38
Binary files /dev/null and b/imgs/pn_fermi_level_band_bending.png differ
diff --git a/semiconductor_summary.tex b/semiconductor_summary.tex
index e446ff87eed033bce2132cc7933d73b42dbd0e2a..43a65c0eefce58819bda154dae90781b94869a75 100644
--- a/semiconductor_summary.tex
+++ b/semiconductor_summary.tex
@@ -2,7 +2,7 @@
-\title{Micro and nanoelectronic devices \\ PHYS-201(d)}
+\title{Micro and nanoelectronic devices \\ EE-333}
\author{Simon Thür}
\date{ \today \\ \small{ Version 0.1 } }
@@ -13,7 +13,7 @@
\thispagestyle{empty}
This summary for
- \href{https://gitlab.epfl.ch/sthuer/semiconductors_summary}{Micro and nanoelectronic devices}
+ \href{https://gitlab.epfl.ch/sthuer/semiconductors_summary}{micro and nanoelectronic devices}
© 2023 by
\href{https://gitlab.epfl.ch/sthuer}{Simon Thür}
is licensed under
@@ -21,11 +21,18 @@
To view a copy of this license, visit
\url{http://creativecommons.org/licenses/by/4.0/}
-
+ To see the source code, visit \href{https://gitlab.epfl.ch/sthuer/semiconductors_summary}{https://gitlab.epfl.ch/sthuer/semiconductors\_summary}
\section*{Introductory remarks}
This is a summary of Prof Ionescu's course on semiconductor devices.
+ It follows the course structure but does not retain the same enumeration.
+ Since the reference material is in english, this summary is also in english.
+
+ The aim of this summary is to provide a denser overview of the course material.
+ Specifically, it is intended to be used as a reference for the exam,
+ which is why it has only few remarks and focuses mainly on equations with little regard for how they were derived.
+
\end{titlepage}
@@ -33,5 +40,6 @@
\include{01_fundamentals}
\include{02_carrier_transport}
-
+\include{03_pn_junction_basics}
+\include{04_pn_junction}
\end{document}