From 85e216cad78b8bfd176d9038abf1386e7ed85b64 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Simon=20Th=C3=BCr?= <thuer.simon@hotmail.com>
Date: Sat, 8 Apr 2023 01:26:38 +0200
Subject: [PATCH] add intro to PN junction

---
 01_fundamentals.tex                     |   1 +
 02_carrier_transport.tex                |   5 +-
 03_pn_junction_basics.tex               |  76 ++++++++++++++++++++++++
 04_pn_junction.tex                      |  46 ++++++++++++++
 imgs/pn_carrier_profile_equilibrium.png | Bin 0 -> 32866 bytes
 imgs/pn_fermi_level_band_bending.png    | Bin 0 -> 104280 bytes
 semiconductor_summary.tex               |  16 +++--
 7 files changed, 138 insertions(+), 6 deletions(-)
 create mode 100644 03_pn_junction_basics.tex
 create mode 100644 04_pn_junction.tex
 create mode 100644 imgs/pn_carrier_profile_equilibrium.png
 create mode 100644 imgs/pn_fermi_level_band_bending.png

diff --git a/01_fundamentals.tex b/01_fundamentals.tex
index 1587fd9..0952adb 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 cccd15b..87ee148 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 0000000..e4647a0
--- /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 0000000..b84b162
--- /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
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diff --git a/semiconductor_summary.tex b/semiconductor_summary.tex
index e446ff8..43a65c0 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}
-- 
GitLab