From b57a5f52d0b7a2d8dc99409f301dd891b01b9711 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Simon=20Th=C3=BCr?= <thuer.simon@hotmail.com> Date: Thu, 1 Jun 2023 22:43:33 +0200 Subject: [PATCH] add sum cmos --- 10_mos.tex | 110 +++++++++++++++++++++++- format.tex | 124 ++++++++++++++------------- imgs/nmos_electrostaticpotential.png | Bin 0 -> 161936 bytes 3 files changed, 170 insertions(+), 64 deletions(-) create mode 100644 imgs/nmos_electrostaticpotential.png diff --git a/10_mos.tex b/10_mos.tex index f7ded36..8a14cbb 100644 --- a/10_mos.tex +++ b/10_mos.tex @@ -1,3 +1,107 @@ -\section{MOS structure} - -IDKF, lecture hasnt been given yet \ No newline at end of file +\section{MOS structure} + +\begin{figure}[h] + \centering + \caption{Gate structure} + \begin{tikzpicture} + \draw (0,0) rectangle ++(1,2) + rectangle ++(2,-2) + rectangle ++(2,2) + rectangle ++(3,-2) + ; + \node at (0.5,1) () {$M$}; + \node at (2,1) () {$SiO_2$}; + \node at (4,1) () {$SCR$}; + \node at (6,1) () {$p-Si$}; + \end{tikzpicture} +\end{figure} + + +Where $M$ is the metal, $SiO_2$ is the gate oxide, $SCR$ is the space charge region and $p-Si$ is the p doped substrate. +(note that in for the SCR, there is short between the metal and the bulk.) + + +\subsection{Electrostatic analysis} +\begin{figure}[h] + \centering + \caption{Charge distribution at thermal equilibrium} + \begin{tikzpicture} + \draw (0,0) rectangle ++(1,2) + rectangle ++(2,-2) + rectangle ++(2,2) + rectangle ++(3,-2) + ; + \node at (0.5,1) () {$M$}; + \node at (2,1) () {$SiO_2$}; + \node at (4,1) () {$SCR$}; + \node at (6,1) () {$p-Si$}; + + \draw[red] (0,0.2) node[left](){$0$} -- ++(1,0) + -- ++(0,1.5) + -- ++(0,-1.5) + -- ++(2,0) + -- ++(0,-0.5) + -- ++(2,0) + -- ++(0,0.5) + -- ++(3,0) + ; + \node[red] at (4,0.2) () {$Q_M$}; + \end{tikzpicture} +\end{figure} + +Where at the interface of the metal and the oxide, we have continuity of the electric displacement: + +\begin{align} + \varepsilon_{ox} E_{ox} & = Q_M \\ + \Rightarrow E_{ox} & = \frac{Q_M}{\varepsilon_{ox}} \\ + D_{ox} & = D_{sc,int} \\ + \Rightarrow \varepsilon{ox}E_{ox} & = \varepsilon{sc}E_{sc,int} \\ + \Rightarrow \frac{E_{ox}}{E_{sc,int}} & = \frac{\varepsilon_{r,sc}}{\varepsilon_{r,ox}}\qquad\approx 3 +\end{align} + +And at the interface of the SCR and the substrate we have the following displacement continuity: +\begin{align} + \varepsilon_{sc} E_{sc}(x) & = -qN_A\left(x_D - x\right) \\ + \Rightarrow E_{sc}(x) & = -\frac{qN_A}{\varepsilon_{sc}}\left(x-x_D\right) +\end{align} + + +\subsection{Electrostatic potential} +Nothing new, just integrate over the electric field. +(see \autoref{fig:mos_electrostaticpotential}) +\begin{figure}[h] + \centering + \caption{Electrostatic potential of MOS} + \label{fig:mos_electrostaticpotential} + \includegraphics[width=.95\textwidth]{imgs/nmos_electrostaticpotential.png} +\end{figure} + +\begin{equation} + \phi(x) = \begin{cases} + \phi_p & : x_D < x \\ + \phi_p + \frac{q N_A}{2\varepsilon_{sc}}\left(x - x_D\right)^2 & : 0 < x < x_D \\ + \phi_p + \frac{q N_A}{2\varepsilon_{sc}}x_D^2 - \frac{q N_A x_D}{\varepsilon_{sc}}x & : -t_{ox}<x<0 \\ + \phi_{n+} & : x<t_{ox} + \end{cases} +\end{equation} + +and so finally + +\begin{align} + \begin{split} + \phi_B &= V_B + V_{ox}\\ + &= \frac{q N_A x_D^2}{2\varepsilon_{sc}}+\frac{1 N_A x_D t_{ox}}{\varepsilon_{ox}} + \end{split} \\ + \begin{split} + x_D & =\frac{\varepsilon_{sc}}{\varepsilon_{ox}}t_{ox}\left( + \sqrt{1 + \frac{2\varepsilon_{ox}^2 \phi_B}{\varepsilon_{sc} q N_A t_{ox}^2}} - 1 + \right)\\ + &=\frac{\varepsilon_{sc}}{C_{ox}} \left( \sqrt{1+\frac{4\phi_B}{\gamma^2}}-1 \right) + \end{split} \\ + C_{ox} & = \frac{\varepsilon_{ox}}{t_{ox}} \\ + \gamma & = \frac{1}{C_{ox}}\sqrt{2\varepsilon q N_A} +\end{align} + + + +\subsection{Contact potential} diff --git a/format.tex b/format.tex index 45caad0..e47d062 100644 --- a/format.tex +++ b/format.tex @@ -1,61 +1,63 @@ -\documentclass[11pt, a4paper]{article} - -\usepackage[utf8]{inputenc} -\usepackage[T1]{fontenc} -\usepackage{lmodern} -\usepackage{fourier} - -% even-odd -% \setlength{\textwidth}{146.8mm} % = 210mm - 37mm - 26.2mm -% \setlength{\oddsidemargin}{11.6mm} % 37mm - 1in (from offset) -% \setlength{\evensidemargin}{0.8mm} % = 26.2mm - 1in (from offset) -% \setlength{\topmargin}{-2.2mm} % = 0mm -1in + 23.2mm -% \setlength{\textheight}{221.9mm} % = 297mm -29.5mm -31.6mm - 14mm (It works, trust me) -% all same: (for even odd, comment out geometry) -\usepackage[a4paper,top=22mm,bottom=28mm,inner=30mm,outer=30mm]{geometry} - - -% \usepackage[french, nswissgerman, english]{babel} -\usepackage[english]{babel} - -\usepackage[parfill]{parskip} -\usepackage{microtype} -\usepackage{amsmath} -\usepackage{amsfonts} -\usepackage{amssymb} -\usepackage{mathtools} -\usepackage{subcaption} -\usepackage{graphicx, xcolor} - - - -\usepackage[european,straightvoltages]{circuitikz} -\usepackage{tikz} - -\usepackage{url} -\usepackage[pdfusetitle]{hyperref} -\hypersetup{ - pdfborder={0 0 0}, - colorlinks=true, - linkcolor=black, - citecolor=black, - urlcolor=black, - bookmarksopen=false, -} -\urlstyle{same} -% \usepackage{comment} -\setlength{\headheight}{14pt} - -%Fancy is fancy :) -\usepackage{fancyhdr} -\pagestyle{fancy} -\fancyhf{} -\fancyfoot{} -\fancyhead{} -\rhead{\today} -\lhead{} -\cfoot{\thepage} - - - -\numberwithin{equation}{subsection} +\documentclass[11pt, a4paper]{article} + +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage{fourier} + +% even-odd +% \setlength{\textwidth}{146.8mm} % = 210mm - 37mm - 26.2mm +% \setlength{\oddsidemargin}{11.6mm} % 37mm - 1in (from offset) +% \setlength{\evensidemargin}{0.8mm} % = 26.2mm - 1in (from offset) +% \setlength{\topmargin}{-2.2mm} % = 0mm -1in + 23.2mm +% \setlength{\textheight}{221.9mm} % = 297mm -29.5mm -31.6mm - 14mm (It works, trust me) +% all same: (for even odd, comment out geometry) +\usepackage[a4paper,top=22mm,bottom=28mm,inner=30mm,outer=30mm]{geometry} + + +% \usepackage[french, nswissgerman, english]{babel} +\usepackage[english]{babel} + +\usepackage[parfill]{parskip} +\usepackage{microtype} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{mathtools} +\usepackage{subcaption} +\usepackage{graphicx, xcolor} + + + +\usepackage[european,straightvoltages]{circuitikz} +\usepackage{tikz} + +\usepackage{url} +\usepackage[pdfusetitle]{hyperref} +\hypersetup{ + pdfborder={0 0 0}, + colorlinks=true, + linkcolor=black, + citecolor=black, + urlcolor=black, + bookmarksopen=false, +} +\urlstyle{same} +% \usepackage{comment} +\setlength{\headheight}{14pt} + +%Fancy is fancy :) +\usepackage{fancyhdr} +\pagestyle{fancy} +\fancyhf{} +\fancyfoot{} +\fancyhead{} +\rhead{\today} +\lhead{} +\cfoot{\thepage} + + + +\numberwithin{equation}{subsection} +\numberwithin{figure}{subsection} +\numberwithin{table}{subsection} diff --git 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