add some diode magic

06_pn_junction_diode.tex

@@ -67,6 +67,11 @@ Or so simplify
     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}
 
+Note that sometimes a non-ideality factor $n$ is used:
+\begin{equation}
+    I  = I_0\left(\exp\frac{qV}{nkT}-1\right)
+\end{equation}
+
 
 \subsection{PN junction reverse bias}
 When applying a reverse bias, the depletion region gets wider and the electric field increases.

07_diode_applications.tex

+\section{Diode applications}
+\subsection{Small signal}
+If $i\ll I$ and $v\ll V$, then we can use the small signal approximation.
+\begin{equation}
+    \begin{split}
+        I+i &= I_0\left(\exp\frac{q(V+v)}{kT}-1\right) \\
+        &=I_0\left(\exp\frac{q(V)}{kT}\exp\frac{q(v)}{kT}-1\right)\\
+        &\approx I_0\left(\exp\frac{q(V)}{kT}\left(1+\frac{qv}{kT}\right)-1\right)\\
+        &= I_0\left( \exp\frac{qV}{kT} - 1 \right) + I_0 \left( \exp\left(\frac{qV}{kT}\right)\frac{qv}{kT} \right)
+    \end{split}
+\end{equation}
+
+Which gives us the smallsignal current
+\begin{align}
+    i   & = \frac{q\left(I+I_0\right)}{kT}v = g_d v \\
+    g_d & = \frac{q\left(I+I_0\right)}{kT}
+\end{align}
+
+
+\subsubsection{Capacitances of small-signal model}
+\begin{figure}[h]
+    \centering
+    \caption*{Small signal model for diode}
+    \begin{circuitikz}
+        \draw (-3,2) to [Do] (-3,0);
+        \draw[->] (-2,1) -- (-1,1);
+        \draw (0,0) to [R,l=$g_d$] (0,2);
+        \draw (2,0) to [C,l=$C_j$] (2,2);
+        \draw (4,0) to [C,l=$C_d$] (4,2);
+        \draw (0,0) to [short] (4,0);
+        \draw (0,2) to [short] (4,2);
+        \draw (2,2) to [short] (2,2.5);
+        \draw (2,-0.5) to [short] (2,0);
+    \end{circuitikz}
+\end{figure}
+\begin{align}
+    C_j    & = \frac{C_{j0}}{\sqrt{1-\frac{V}{\phi_B}}}        \\
+    C_d    & = \frac{q}{kT}\tau_T I                            \\
+    \tau_T & \equiv \text{equivalent transit time of carriers}
+\end{align}
+Where $C_j$ dominates in reverse bias and small forward bias,
+and $C_d$ dominates in large forward bias ($V>\frac{\phi_B}{2}$).
+
+
+\subsection{Large signal}
+\subsubsection{Rectifier}
+This is pretty much trivial.
+\begin{center}
+    \begin{circuitikz}
+        \draw (0,-2) to [sV,l=$V_{in}$] ++(0,4)
+        to [short] ++(2,0)
+        to [Do,v=$V_D$] ++(0,-2)
+        to [R,l=$R_L$,v=$V_o$] ++(2,0)
+        to [Do] ++(0,-2)
+        to [short] (0,-2);
+        draw (4,0) to [Do] ++(0,2)
+        to [short] ++(-2,0);
+        \draw (2,-2) to [Do] ++(0,2);
+        \draw (4,0) to [Do] ++(0,2)
+        to [short] ++(-2,0);
+    \end{circuitikz}
+\end{center}
+\begin{equation}
+    V_o = V_{in}-2V_D
+\end{equation}
+
+\subsubsection{Voltage regulator}
+\begin{center}
+    \begin{circuitikz}
+        \draw (0,0) to [V,v<=$10\pm1\ V$] (0,8)
+        to [short] ++(2,0)
+        to [R,l=$R_1$] ++(0,-2)
+        to [short] ++(0,-2)
+        to [Do] ++(0,-1)
+        to [Do] ++(0,-1)
+        to [Do] ++(0,-1)
+        to [short] ++(0,-1)
+        to [short] (0,0);
+        \draw (2,5) to [nos] ++(2,0)
+        to [R,l=$R_2$] ++(0,-5)
+        to [short] (0,0);
+    \end{circuitikz}
+\end{center}
+
+How to go about it:
+\begin{align}
+    I          & = \frac{V_{in}-\sum V_D}{R_1}                                            \\
+    r_d        & =\left[\frac{\mathrm{d}I_D}{\mathrm{d}V_d}\right]^{-1} =\frac{nV_t}{I_0} \\
+    r          & = \sum r_d                                                               \\
+    \Delta V_o & = \Delta V_{in}\frac{r}{r+R_2}                                           \\[1em]
+    n          & \equiv\text{non-ideality factor}                                         \\
+    V_t        & = \frac{kT}{q}
+\end{align}
+
+Once the load is connected and draws current, we have a further small variation:
+\begin{align}
+    I_{load}   & = \frac{\sum V_D}{R_2} \\
+    \Delta V_o & = I_{load}r
+\end{align}
+
+\subsection{Special diode types}
+\subsubsection{Zener}
+Is heavily doped making the depletion layer extremely thing, and thus allowing for QM tunneling in reverse biased diode.
+(In this case known as band-to-band tunneling.)
+
+\begin{figure}[h]
+    \centering
+    \caption[short]{Band-bending zener diode}
+    \includegraphics[width=.75\textwidth]{imgs/zener_diode_band_bending.png}
+\end{figure}
+The voltage at which the diode starts conducting is called the zener voltage $V_Z$.
+The diode then has a low resistance $R_Z$.
+
+\subsubsection{Esaki}
+Heavily doped with the tunneling effect in forward bias.
+\begin{figure}[h]
+    \centering
+    caption{Esaki tunnel diode}
+    \includegraphics[width=.75\textwidth]{imgs/esaki_tunnel_diode.png}
+\end{figure}
+
+\subsubsection{Schottky}
+A schottky diode has $I_0$ $10^3$ to $10^8$ times bigger than a PN diode.
+Preferred in low voltage high current applications.
+
+\subsubsection{Photodiodes}
+\begin{equation}
+    I = I_0\left( e^{\frac{qV}{kT}} - 1\right)-I_{photo}
+\end{equation}
\ No newline at end of file

semiconductor_summary.tex

@@ -44,4 +44,5 @@
 \include{04_pn_junction}
 \include{05_pn_junction_bias}
 \include{06_pn_junction_diode}
+\include{07_diode_applications.tex}
 \end{document}