From f707b3138b9b6441cbded37d5453b0c5d5943792 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Simon=20Th=C3=BCr?= <thuer.simon@hotmail.com>
Date: Fri, 7 Apr 2023 23:03:16 +0200
Subject: [PATCH] add chap2
---
02_carrier_transport.tex | 53 +++++++++++++++++++++++++++++++++++++--
semiconductor_summary.tex | 2 +-
2 files changed, 52 insertions(+), 3 deletions(-)
diff --git a/02_carrier_transport.tex b/02_carrier_transport.tex
index ff129f4..eb80b64 100644
--- a/02_carrier_transport.tex
+++ b/02_carrier_transport.tex
@@ -52,7 +52,56 @@ Average drift velocity:
\end{equation}
\subsubsection{Mobility}
+For the sake of simplicity, let's define mobility for both holes and electrons.
+(These values are usually found in diagrams.)
\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
+ v_{dn} & =-\mu_nE \\
+ v_{dp} & = \mu_pE \\
+ \mu_n & >_mu_p
+\end{align}
+
+\subsubsection{Drift current}
+For the net drift current density slap together velocity, density and charge.
+\begin{equation}
+ label{eq:drift_current}
+ J^{drift} = J_n^{drift}+J_p^{drift} = q(n\mu_n+p\mu_p)E
+\end{equation}
+From which we can find Ohm's law:
+\begin{alignat}{2}
+ J & =\sigma E & & = \frac{E}{\rho} \\
+ \rho & =\frac{1}{\sigma} & & = \frac{1}{q \left(n\mu_n+p\mu_p\right)}
+\end{alignat}
+Which gives us different resistances for n and p type semiconductors.
+\begin{align}
+ \rho_n & \approx \frac{1}{qN_d\mu_n} \\
+ \rho_p & \approx \frac{1}{qN_a\mu_p}
+\end{align}
+
+
+\subsubsection{Diffusion current}
+If there is a concentration gradient, the carriers will diffuse to equalize the concentration. Here flux $F \ [cm^{-2}s^{-1}]$ is the number of electrons/holes per unit area per unit time.
+\begin{align}
+ F_n & = -D_n\frac{\mathrm{d} n}{\mathrm{d} x} \\
+ F_p & = -D_p\frac{\mathrm{d} p}{\mathrm{d} x}
+\end{align}
+
+Which gives us the diffusion current density:
+(Defined as density times charge, ergo the double negative for electron diffusion.)
+\begin{align}
+ J_n^{diff} & = qD_n\frac{\mathrm{d} n}{\mathrm{d} x} \\
+ J_p^{diff} & =- qD_p\frac{\mathrm{d} p}{\mathrm{d} x}
+\end{align}
+
+
+\subsubsection{Einstein relation between mobility and diffusion coefficient}
+\begin{equation}
+ \frac{D_n}{\mu_n} = \frac{D_p}{\mu_p} = \frac{kT}{q^2}
+\end{equation}
+
+\subsubsection{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_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/semiconductor_summary.tex b/semiconductor_summary.tex
index 4f507d3..e446ff8 100644
--- a/semiconductor_summary.tex
+++ b/semiconductor_summary.tex
@@ -14,7 +14,7 @@
This summary for
\href{https://gitlab.epfl.ch/sthuer/semiconductors_summary}{Micro and nanoelectronic devices}
- © 2021 by
+ © 2023 by
\href{https://gitlab.epfl.ch/sthuer}{Simon Thür}
is licensed under
\href{http://creativecommons.org/licenses/by/4.0/}{CC BY 4.0}.
--
GitLab