add pn bias and varicap

05_pn_junction_bias.tex

+\section{PN junction bias}
+\subsection{Model}
+
+We can see the junction as a series of resistors as follows.
+(Junction $V_{pn}>0$ is a forward bias.)
+\begin{center}
+    \begin{circuitikz}
+        \draw (0,0) to[R, l=$R_{mp}$] ++(2,0)
+        to [R, l=\textnormal{p-QNR}] ++(2,0)
+        to [R, l=\textnormal{SCR}] ++(2,0)
+        to [R, l=\textnormal{n-QNR}] ++(2,0)
+        to [R, l=$R_{mn}$] (10,0);
+        \draw (0,0) to [short] (0,2)
+        to [V, v=$V_{pn}$] (10,2)
+        to [short] (10,0);
+        \begin{scope}[opacity=.5]
+            \draw  (1,-1) rectangle (9,1);
+            \draw (4,-1) -- (4,1);
+            \draw (6,-1) -- (6,1);
+            \draw[dotted] (5,-1) -- (5,1);
+        \end{scope}
+        \node at (2.5,-1.5) {p-QNR};
+        \node at (5,-1.5) {SCR};
+        \node at (7.5,-1.5) {n-QNR};
+        \node at (4.5,-0.5) {$-$};
+        \node at (5.5,-0.5) {$+$};
+    \end{circuitikz}
+\end{center}
+
+Importantly, the SCR resistance ist the most important one and others can be neglected.
+
+\subsection{Space charge region (SCR)}
+In essence, applying a forward/reverse bias effects the depletion region:
+\begin{align}
+    \phi_B            & \rightarrow \phi_B-V_{pn}                                 \\
+    x_n(V)            & =\sqrt{\frac{2\varepsilon(\phi_B-V)N_a}{q(N_a+N_d)N_d}}   \\
+    x_p(V)            & =\sqrt{\frac{2\varepsilon(\phi_B-V)N_d}{q(N_a+N_d)N_a}}   \\
+    x_d(V)            & =\sqrt{\frac{2\varepsilon(\phi_B-V)N_dN_a}{q(N_a+N_d)}}   \\
+    \left|E(V)\right| & =\sqrt{\frac{2q(\phi_B-V)(N_aN_d)}{\varepsilon(N_a+N_d)}}
+\end{align}
+
+In the case of a strongly doped $p^+n$ junction,
+we can approximate the SCR since it exists only in the lesser doped region.
+\begin{equation}
+    x_n(V)=x_{n0}\sqrt{a-\frac{V}{\phi_B}}
+\end{equation}
+
+
+\subsection{PN small-signal capacitance}
+In reverse bias, the PN junction acts as a capacitor.
+\begin{equation}
+    C_{j0} = \frac{\varepsilon}{W_{dep}}
+\end{equation}
+
+So as a function of the bias voltage, we get
+\begin{equation}
+    \begin{split}
+        C_j(V) &= \frac{\varepsilon}{x_c(V)}\\
+        &=\sqrt{\frac{q\varepsilon N_aN_d}{2q(\phi_B-V)(N_a+N_d)}}\\
+        &=\frac{C_{j0}}{\sqrt{1-\frac{V}{\phi_B}}}
+    \end{split}
+\end{equation}
+
+In a strongly asymmetric junction $p^+n$
+\begin{equation}
+    \frac{1}{C_j^2} \approx \frac{2(\phi_B-V)}{q\varepsilon N_d}
+\end{equation}
+
+

format.tex

@@ -27,6 +27,11 @@
 \usepackage{subcaption}
 \usepackage{graphicx, xcolor}
 
+
+
+\usepackage[european]{circuitikz}
+\usepackage{tikz}
+
 \usepackage{url}
 \usepackage[pdfusetitle]{hyperref}
 \hypersetup{

semiconductor_summary.tex

@@ -42,4 +42,5 @@
 \include{02_carrier_transport}
 \include{03_pn_junction_basics}
 \include{04_pn_junction}
+\include{05_pn_junction_bias.tex}
 \end{document}