From 1834093ccab7b34627a277413eea95cdbfa0fd13 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Simon=20Th=C3=BCr?= <thuer.simon@hotmail.com>
Date: Fri, 7 Apr 2023 08:07:38 +0200
Subject: [PATCH] add 1st part of chap 2
---
02_carrier_transport.tex | 59 ++++++++++++++++++++++++++++++++++++++-
semiconductor_summary.tex | 2 +-
2 files changed, 59 insertions(+), 2 deletions(-)
diff --git a/02_carrier_transport.tex b/02_carrier_transport.tex
index 445514e..ff129f4 100644
--- a/02_carrier_transport.tex
+++ b/02_carrier_transport.tex
@@ -1 +1,58 @@
-\section{Carrier transport}
\ No newline at end of file
+\section{Carrier transport}
+\subsection{Fermi distribution}
+\label{label:sec:fermi}
+Fermions are weird particles, see QM II.
+Not sure if needed for this course, but heres the probability distribution:
+\begin{equation}
+ f(E) = \frac{1}{1+e^{(E-E_F)/kT}}
+\end{equation}
+
+Electron concentration in conductance band:
+\begin{equation}
+ n=N_ce^{-(E_c-E_f)/kT}
+\end{equation}
+Hole concentration in valence band:
+\begin{equation}
+ p=N_ve^{-(E_f-E_v)/kT}
+\end{equation}
+
+In intrinsic silicon ($n=p=n_i$) we have ($E_i$ somewhere in the middle of the bandgap)
+\begin{equation}
+ E_i=E_f=\frac{E_c+E_v}{2}-\frac{kT}{q}\ln{\frac{N_c}{N_v}}
+\end{equation}
+
+Which gives us the useful relation:
+\begin{equation}
+ n_i = \sqrt{N_cN_v}e^{-(E_c-E_v)/2kT} = \sqrt{N_cN_v}e^{-E_g/2kT}
+\end{equation}
+
+\subsubsection{Temperature dependance}
+Looking at the equations in \autoref{label:sec:fermi} we see that the higher the temperature or the lower the band-gap, the more electrons and holes are created.
+
+\subsection{Carrier transport}
+\subsubsection{Thermal equilibrium}
+\begin{align}
+ \lambda & \equiv \text{mean free path} [cm] \\
+ \tau_c & \equiv \text{mean time between collisions} [s^{-1}] \\
+ v_{th} & \equiv \text{thermal velocity} [cm/s] \\[1em]
+ \lambda & = v_{th}\cdot\tau_c
+\end{align}
+
+
+\subsubsection{Drift velocity}
+Quick electromag recap: (for holes use + and $m_p$)
+\begin{align}
+ F & = -qE \\
+ v(t) & =-\frac{qE}{m_n}t
+\end{align}
+
+Average drift velocity:
+\begin{equation}
+ v_d = \pm \frac{qE\tau_c}{2m_{n,p}}
+\end{equation}
+
+\subsubsection{Mobility}
+\begin{align}
+ \mu_{n,p} & = \frac{q\tau_c}{2m_{n,p}} \equiv \text{mobility}\ [cm^2/Vs] \\
+ amhere
+\end{align}
\ No newline at end of file
diff --git a/semiconductor_summary.tex b/semiconductor_summary.tex
index 0569140..4f507d3 100644
--- a/semiconductor_summary.tex
+++ b/semiconductor_summary.tex
@@ -32,6 +32,6 @@
\tableofcontents
\include{01_fundamentals}
-\include{02_carrier_transport.tex}
+\include{02_carrier_transport}
\end{document}
--
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