From 6663f277ba9b158af02a2955086223675e25be54 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 16:21:43 +0200 Subject: [PATCH] add some diode magic --- 06_pn_junction_diode.tex | 5 ++ 07_diode_applications.tex | 129 ++++++++++++++++++++++++++++++ imgs/esaki_tunnel_diode.png | Bin 0 -> 300105 bytes imgs/zener_diode_band_bending.png | Bin 0 -> 143985 bytes semiconductor_summary.tex | 1 + 5 files changed, 135 insertions(+) create mode 100644 07_diode_applications.tex create mode 100644 imgs/esaki_tunnel_diode.png create mode 100644 imgs/zener_diode_band_bending.png diff --git a/06_pn_junction_diode.tex b/06_pn_junction_diode.tex index 6d879fa..965334f 100644 --- a/06_pn_junction_diode.tex +++ b/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. diff --git a/07_diode_applications.tex b/07_diode_applications.tex new file mode 100644 index 0000000..ce3bf4b --- /dev/null +++ b/07_diode_applications.tex @@ -0,0 +1,129 @@ +\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 diff --git a/imgs/esaki_tunnel_diode.png b/imgs/esaki_tunnel_diode.png new file mode 100644 index 0000000000000000000000000000000000000000..1d175462e60b18976dd6491a766c7d2fab30581a GIT binary patch literal 300105 zcmc$`hd<SO{0H8YkR&8zC4`Wy5G5fyd+)vX9+j0&hos{eA?c)Jb_m%?#xb*ZGLDlS zve)nB-uwN2f4{%r=h35kPv;!xbKd9ue!ZUSbz(Hs6wZ+`kR3aA?A${|In86oNNkQB zJH<+R0{$h*qw^B{=cJeHLv2!0(vb=EUvTrX_XB-zEjL?lKMPNrV|K1?E;cv3tUPUO zT)phwyq8bbN*_DMdhDUxJ#GIt3u6K9#seW-J6=Jx)yu!n-4!cs>>l)UALnREyw>_E ze{l7?T1$c_b(4;ORsg=Yjm3=zb-i}wJ@vr%&Wl{W&RnI=3Kt(xU-RI)M|CVRWV~zl z=kJip>RK88KIh}C4>Z|vzc;tTPrqBP4pAHLIX5a0N;J;X;AVx({NEp0QnnUTvH!h! z?AYUwQ>#M%uiHycswn>N=Hnn%fs_At^A~e->;K(4_T|j~_o<nSKL0(~f08Wno2l6G 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b/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} -- GitLab