add sum cmos

10_mos.tex

-\section{MOS structure}
-
-IDKF, lecture hasnt been given yet
+\section{MOS structure}
+
+\begin{figure}[h]
\ No newline at end of file
+    \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}

format.tex

-\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}