\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}
(for $phi_p$ see \autoref{label:eq:boltzman:phi_p}) and so finally
\begin{align}
\begin{split}
\phi_B &= V_B + V_{ox}\\
&= \frac{q N_A x_D^2}{2\varepsilon_{sc}}+\frac{q 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}
\begin{figure}[H]
\caption{Contact potentials}
\label{fig:mos_contactpotentials}
\centering
\includegraphics[width=.95\textwidth]{imgs/contact_potentials_mos.png}
\end{figure}
\begin{figure}[H]
\label{fig:mos_workfunctiondifferences}
\centering
\includegraphics[width=.8\textwidth]{imgs/MOSFET_workfunction.png}
\caption{a) Constant $E_0$ convention used in the MOS structure in this course. b) Constant $E_f$ convention used in the MOS structure.}
\end{figure}
%$\Phi_M$ is the work function we build up to have a constant fermi-level (blue).
\subsection{Bias}
\begin{figure}[H]
\caption{MOS under bias (1)}
\label{fig:mos_under_bias01}
\centering
\includegraphics[width=.95\textwidth]{imgs/mos_under_bias_SCR.png}
\end{figure}
Applying voltage increases niveau of the dotted line at $-t_{ox}$, which increases the SCR.
\begin{figure}[H]
\caption{Mos under bias (2)}
\label{fig:mos_bias02}
\centering
\includegraphics[width=.95\textwidth]{imgs/MOS_under_bias.png}
\end{figure}
\begin{equation}
x_d(V_{GB}) = \frac{\varepsilon_{sc}}{C_{ox}}\left(\sqrt{1+\frac{4\left(\phi_B+V_{GB}\right)}{\gamma^2}}-1\right)
\end{equation}
\subsection{Depletion regime}
In all previous equations, replace $\phi_B \to \phi_B+V_{GB}$.
\begin{align}
x_d(V_{GB}) & = \frac{\varepsilon_s}{C_{ox}}\left( \sqrt{1 + \frac{4\left( \phi_B + V_{GB} \right)}{\gamma^2}} - 1 \right) \\
V_B(V_{GB}) & = \frac{q N_A \left(x_d(V_{GB})\right)^2}{2\varepsilon_{sc}} \\
V_{ox}(V_{GB}) & = \frac{q N_a x_d(V_{GB})t_{ox}}{\varepsilon_{ox}}
\end{align}
\subsection{Flatband regime}
At a certain voltage, the SCR disappears.
If the voltage is decreased further,
we find ourselves in the \emph{accumulation regime} which only acts as a capacitor.
\begin{equation}
V_{FB} = -\phi_B
\end{equation}
\subsection{Threshold voltage}
At a certain $V_{GS}$, we find that $n(0) = N_a$.
At this point we can no longer neglect minority carriers for electrostatics.
\begin{align}
n(0) & = n_ie^{q\phi(0)/kT} \\
\left.\phi(0)\right|_{V_T} & = \left.\frac{kT}{q}\ln\frac{n(0)}{n_i}\right|_{V_T} = \frac{kT}{q} \ln\frac{N_a}{n_i} \\
V_B(V_T) & = -2\phi_p \\
V_B(V_T) & = \frac{q N_A \left(x_D(V_T)\right)^2}{2\varepsilon_{sc}} = - 2\phi_p \\
x_d(V_T) & =x_{d,max}=\sqrt{\frac{-4\varepsilon_{sc}\phi_p}{qN_A}} \\
V_{ox}(V_T) & =E_{ox}(V_T)t_{ox}=\frac{qN_Ax_{d,max}t_{ox}}{\varepsilon_{ox}}=\gamma\sqrt{-2\phi_p} \\
V_T & = V_{FB} - 2\phi_p+\gamma\sqrt{-2\phi_p}
\end{align}
Notes:
\begin{enumerate}
\item Higher inner doping level, higher $V_T$.
\item Thinner oxide, lower $V_T$.
\end{enumerate}
\subsubsection{Body effect}
In the case of a non-zero $V_{SB}$, we have to adjust the threshold voltage:
\begin{equation}
V_{TH} = V_{TH0} + \gamma\left(\sqrt{|2\phi_F + V_{SB}|} - \sqrt{|2\phi_F|}\right) \qquad \text{where}\quad \Phi_F = \frac{k T}{q} \ln\left(\frac{N_{sub}}{n_i}\right)
\end{equation}
\subsection{Strong inversion}
\begin{equation}
V_{GB} > V_T
\end{equation}
By applying higher voltage, we increase inversion charge.
\begin{align}
Q & =CV \\
Q_n & = -C_{ox}\left(V_{GB}-V_T\right) \\
\end{align}
\begin{figure}[H]
\caption{MOS charge summary}
\label{fig:mos_chargesummary}
\centering
\includegraphics[width=.5\textwidth]{imgs/mos_charge_summary.png}
\end{figure}