diff --git a/05_pn_junction_bias.tex b/05_pn_junction_bias.tex new file mode 100644 index 0000000000000000000000000000000000000000..272179aee5a5e67f617885a3d4bd57a2f2d32611 --- /dev/null +++ b/05_pn_junction_bias.tex @@ -0,0 +1,69 @@ +\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} + + diff --git a/format.tex b/format.tex index 7885945dbf17d79b181b805c8190943c87b9cd6e..f24f6aa72273100ebadbc9329111df990ab86a57 100644 --- a/format.tex +++ b/format.tex @@ -27,6 +27,11 @@ \usepackage{subcaption} \usepackage{graphicx, xcolor} + + +\usepackage[european]{circuitikz} +\usepackage{tikz} + \usepackage{url} \usepackage[pdfusetitle]{hyperref} \hypersetup{ diff --git a/semiconductor_summary.tex b/semiconductor_summary.tex index 43a65c0eefce58819bda154dae90781b94869a75..0697fde30de03ad29c994c8c124b7ef7a9a8213f 100644 --- a/semiconductor_summary.tex +++ b/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}