From c97579e94eea559adbf7e6300e97f6fcb2fb0577 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 13:53:54 +0200
Subject: [PATCH] add diode current
---
02_carrier_transport.tex | 4 +--
03_pn_junction_basics.tex | 12 +++----
05_pn_junction_bias.tex | 2 +-
06_pn_junction_diode.tex | 76 +++++++++++++++++++++++++++++++++++++++
semiconductor_summary.tex | 3 +-
5 files changed, 87 insertions(+), 10 deletions(-)
create mode 100644 06_pn_junction_diode.tex
diff --git a/02_carrier_transport.tex b/02_carrier_transport.tex
index 87ee148..51d5be0 100644
--- a/02_carrier_transport.tex
+++ b/02_carrier_transport.tex
@@ -90,8 +90,8 @@ 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}
+ J_n^{diff} & = qD_n\frac{\mathrm{d} n}{\mathrm{d} x} \label{label:eq:diff_current_n} \\
+ J_p^{diff} & =- qD_p\frac{\mathrm{d} p}{\mathrm{d} x} \label{label:eq:diff_current_p}
\end{align}
diff --git a/03_pn_junction_basics.tex b/03_pn_junction_basics.tex
index e4647a0..0a5d0a4 100644
--- a/03_pn_junction_basics.tex
+++ b/03_pn_junction_basics.tex
@@ -55,9 +55,9 @@ If the doping changes slowly with x:
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
+ n & =n_{ref}e^{q(\phi-\phi_{ref})/kT} \\
+ \phi_{ref} & =0 \\
+ n_{ref} & =n_i
\end{align}
And by extension
\begin{align}
@@ -65,10 +65,10 @@ And by extension
p & =n_ie^{-q\phi/kT}
\end{align}
-Rearranging the above, we find a rule of thumb for the potential:
+Rearranging the above, we find an expression for the potential:
\begin{align}
- \phi & =\frac{kT}{q}\ln\frac{n}{n_i} \\
- \phi & = -\frac{kT}{q}\ln\frac{p}{n_i}
+ \phi & =\frac{kT}{q}\ln\frac{n}{n_i} \label{label:eq:boltzman:phi_n} \\
+ \phi & = -\frac{kT}{q}\ln\frac{p}{n_i} \label{label:eq:boltzman:phi_p}
\end{align}
For Si at room temperature this is an increase of 60 mV per decade in doping.
\begin{equation}
diff --git a/05_pn_junction_bias.tex b/05_pn_junction_bias.tex
index 272179a..bd2312c 100644
--- a/05_pn_junction_bias.tex
+++ b/05_pn_junction_bias.tex
@@ -2,7 +2,7 @@
\subsection{Model}
We can see the junction as a series of resistors as follows.
-(Junction $V_{pn}>0$ is a forward bias.)
+(Junction $V_{pn}>0$ is a forward bias, QNR is a quasi-neutral region.)
\begin{center}
\begin{circuitikz}
\draw (0,0) to[R, l=$R_{mp}$] ++(2,0)
diff --git a/06_pn_junction_diode.tex b/06_pn_junction_diode.tex
new file mode 100644
index 0000000..6d879fa
--- /dev/null
+++ b/06_pn_junction_diode.tex
@@ -0,0 +1,76 @@
+\section{PN junction diode}
+\subsection{Carrier concentration under bias}
+Under forward bias, the net current is no longer zero.
+\begin{equation}
+ \left| J_{drift} \right|<\left| J_{diff} \right|
+\end{equation}
+Which causes injection of minority carriers into the QNR regions giving rise to `high' currents.
+
+
+\subsection{Diode current}
+To calculate the current, we \begin{enumerate}
+ \item Calculate concentration of minority carriers at the edges of SCR
+ \item Calculate minority carrier diffusion current in each QNR for $I_n$ and $I_p$
+ \item Sum the currents $I_n$ and $I_p$
+\end{enumerate}
+
+\subsubsection{Minority carrier conditions}
+We use the quasi-equilibrium equation to misuse equations for equilibrium.
+\begin{align}
+ \frac{n(x_1)}{n(x_2)} & \approx \exp{\frac{q(\phi(x_1)-\phi(x_2))}{kT}} \\
+ \frac{p(x_1)}{p(x_2)} & \approx \exp{\frac{-q(\phi(x_1)-\phi(x_2))}{kT}}
+\end{align}
+
+So by using $x_n$ and $x_p$ in the above equation we have the following:
+\begin{align}
+ \frac{n(x_n)}{n(-x_p)} & \approx \exp{\frac{q(\phi_B-V)}{kT}} \\
+ \frac{p(x_n)}{p(-x_p)} & \approx \exp{\frac{-q(\phi_B-V)}{kT}} \\
+ p(-x_p) & =N_a \\
+ n(x_n) & =N_d
+\end{align}
+
+And so we find what we needed:
+\begin{align}
+ n(-x_p) & \approx N_d\exp\frac{q(V-\phi_B)}{kT} \\
+ p(x_n) & \approx N_a\exp\frac{q(V-\phi_B)}{kT}
+\end{align}
+
+
+Then by using the Boltzman relations \eqref{label:eq:boltzman:phi_n} and \eqref{label:eq:boltzman:phi_p} we find
+\begin{align}
+ \phi_B & = \frac{kT}{q}\ln\frac{N_dN_a}{n_i^2} \\
+ \Rightarrow n(-x_p) & \approx \frac{n_i^2}{N_a}\exp\frac{qV}{kT} \\
+ \Rightarrow p(x_n) & \approx \frac{n_i^2}{N_d}\exp\frac{qV}{kT}
+\end{align}
+
+\subsubsection{Diffusion current in QNR}
+We assume a linear gradient between $n(-W_p)$ and $n(-x_p)$ to easily use \eqref{label:eq:diff_current_n} to find
+\begin{equation}
+ \begin{split}
+ J_n^{diff} &= qD_n\frac{n_p(-x_p)-n_p(-W_p)}{W_p-x_p}\\
+ &= qD_n \frac{\left(\frac{n_i^2}{N_a}\exp{\frac{qV}{kT}}\right)-\frac{n_i^2}{N_a}}{W_p-x_p}\\
+ &= q\frac{n_i^2}{N_a}\frac{D_n}{W_p-x_p}\left(\exp{\frac{qV}{kT}}-1\right)
+ \end{split}
+\end{equation}
+
+\subsubsection{Total diode current}
+\begin{equation}
+ \begin{split}
+ J & = J_n+J_p \\
+ & =q n_i^2 \left( \frac{1}{N_A}\frac{D_n}{W_p-x_p} + \frac{1}{N_D}\frac{D_p}{W_n-x_n} \right)\left(\exp
+ \frac{qV}{kT} - 1\right)
+ \end{split}
+\end{equation}
+Or so simplify
+\begin{align}
+ I & =I_0\left(\exp\frac{qV}{kT}-1\right) \\
+ I_0 & = A q n_i^2 \left( \frac{D_n}{L_n N_A} + \frac{D_p}{L_p N_D} \right)
+\end{align}
+
+
+\subsection{PN junction reverse bias}
+When applying a reverse bias, the depletion region gets wider and the electric field increases.
+There comes a point when the diode breaks down and destroys itself.
+\begin{equation}
+ W_{dep} = \sqrt{\frac{2\varepsilon}{q}\left(\frac{1}{N_A}+\frac{1}{N_D}\right)\left(V_0+V_R\right)}
+\end{equation}
\ No newline at end of file
diff --git a/semiconductor_summary.tex b/semiconductor_summary.tex
index 0697fde..6c92d28 100644
--- a/semiconductor_summary.tex
+++ b/semiconductor_summary.tex
@@ -42,5 +42,6 @@
\include{02_carrier_transport}
\include{03_pn_junction_basics}
\include{04_pn_junction}
-\include{05_pn_junction_bias.tex}
+\include{05_pn_junction_bias}
+\include{06_pn_junction_diode}
\end{document}
--
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