From 8c10f7a818242a35b5762ee06a2047c14d403879 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Simon=20J=2E=20Th=C3=BCr?= <thuer.simon@hotmail.com>
Date: Sat, 8 Apr 2023 09:46:50 +0200
Subject: [PATCH] add depletion region width and metal junction potential
---
04_pn_junction.tex | 43 +++++++++++++++++++++++++++++++++----------
1 file changed, 33 insertions(+), 10 deletions(-)
diff --git a/04_pn_junction.tex b/04_pn_junction.tex
index b84b162..eee6d98 100644
--- a/04_pn_junction.tex
+++ b/04_pn_junction.tex
@@ -30,17 +30,40 @@ 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} \\
+ 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}
+ 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}.
+\subsection{Electrostatic potential - (Width of the depletion zone)}
+Because of overall charge neutrality
+\begin{equation}
+ qN_ax_p = qN_cx_n
+\end{equation}
+and continuity of the potential at the junction interface
+\begin{equation}
+ \phi_p+\frac{qN_a}{2\varepsilon}x_p^2 = \phi_n-\frac{qN_d}{2\varepsilon}x_n^2
+\end{equation}
+we can find $x_n$ and $x_p$:
+\begin{align}
+ x_n & = \sqrt{\frac{2\varepsilon\phi_BN_a}{qN_d(N_a+N_d)}} \\
+ x_p & = \sqrt{\frac{2\varepsilon\phi_BN_d}{qN_a(N_a+N_d)}}
+\end{align}
+Where $\phi_B$ is the built-in potential, which is the potential over the junction.
+Also, it is the less heavily doped region that defines the junction width.
+It is also in the less heavily doped region that the depletion zone extends farther.
+
+\subsection{Contact potential}
+Although there is a potential accross the diode, it cannot be measured because there are the metal semi-conductor junctions for both p and n regions.
+\begin{equation}
+ \phi_B = \phi_mn+\phi_mp
+\end{equation}
\ No newline at end of file
--
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