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sample_code.tex
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% EXAMPLE CODE %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% INCLUDE PDF %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \includepdf[pages=-]{null.PDF}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% SAMPLE IMAGE COMMAND %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \includegraphics[scale=0.15, center]{null.PNG}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% SUB-QUESTION FORMS %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{enumerate}[label=(\roman*)]
%
% \end{enumerate}
%
% \begin{enumerate}[label=(\alph*)]
%
% \end{enumerate}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% SCIENTIFIC NOTATION %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Number only: \num{1e-10}
% Number with units: \SI{1e-10}{\meter\per\second}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% SIGMA NOTATION %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Sum $\sum_{n=1}^{\infty} 2^{-n} = 1$ inside text
%
% \[ \sum_{n=1}^{\infty} 2^{-n} = 1 \]
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% STANDARD MATRIX FORM %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\left[\begin{matrix}
% 1 & 0 \\
% 0 & 1 \\
%\end{matrix}\right]
%
% \left[\begin{matrix}
% 1 & 0\\[0.25cm]
% 0 & 1
% \end{matrix}\right]
% \cdot
% \left[\begin{matrix}
% 1\\[0.25cm]
% 2\\
% \end{matrix}\right]
% =
% \left[\begin{matrix}
% 1\\[0.25cm]
% 2\\
% \end{matrix}\right]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% AUGMENTED MATRIX FORM %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\left[\begin{matrix}
%1 & 0 \\
%0 & 1 \\
%\end{matrix}\left| \, \begin{matrix} 0 \\ 0 \end{matrix} \right.\right]
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% PROOF SHELL %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\textbf{Claim. } null
%\begin{proof}
% \text{We will verify the claim by...}
%\end{proof}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% PIECEWISE FUNCTION %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\[ \boxed{\mathcal{E}(x) = \begin{cases}
% -\frac{q\;N_A}{\epsilon_{si}}\;\big( x_p + x \big) &,\; -x_p \leq x \leq 0 \\ \\
% -\frac{q}{\epsilon_{si}}\;\bigg(N_A \cdot x_p - \frac{N_D \cdot x}{2}\bigg) &,\; 0 \leq x \leq x_0 \\ \\
% -\frac{q\;N_D}{\epsilon_{si}}\;\big(x_n - x \big) &,\; x_0 \leq x \leq x_n
% \end{cases}}
%\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% INDUCTION SHELL %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\textbf{Claim. } null
%\begin{proof}
% \text{We will verify the claim by...}\\ \\
% \textbf{\underline{Base case}}\\ \\
% null\\ \\
% \textbf{\underline{Inductive hypothesis}}\\ \\
% null\\ \\
% \textbf{\underline{Inductive step}}\\ \\
% null\\ \\
% Therefore, by induction we have proven the claim.\\
%\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% ITEMIZED LIST %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{itemize}[topsep=8pt,itemsep=4pt,partopsep=4pt, parsep=4pt]
% \item{null}
% \end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% TABLE SHELL %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{displaymath}
% \begin{array}{|c|c|}
% \hline null & null\\
% \hline
% null & null\\
% null & null\\
% \hline
% \end{array}
%\end{displaymath}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% PHYSICS SYMBOLS %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \mathcal{E} % electric field
% \varepsilon % permittivity
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% CONSTANTS %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Charge:
% $\num{1.60217662e-19}C$
%
% Kelvin:
% $T=300^{\circ}K$
%
% Permittivity in Si
% $\epsilon_s = \num{1.03545e-10}\;F/m$
% $\epsilon_s = \num{1.03545e-12}\;F/cm}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% EQUATIONS %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Diffusion:
% $\vec{J}_{n_{diff}} &= q \cdot D_N \cdot \frac{dn}{dx}$
% $\vec{J}_{p_{diff}} &= -q \cdot D_P \cdot \frac{dp}{dx}$
%
% Built-in Voltage
% $\phi_{bi} = \frac{kT}{q} \cdot ln\;\bigg( \frac{N_D \cdot N_A}{{n_i}^2} \bigg)$
%
% Total Depletion Region Width
% $W_{dep} = \sqrt{\frac{2\epsilon_s \phi_{bi}}{q} \cdot \Bigg( \frac{1}{N_A} + \frac{1}{N_D} \Bigg)}$
%
% Individual Depletion Widths for Applied Bias
% x_p = \sqrt{\frac{2\epsilon_s}{q} \cdot \Bigg( \frac{N_D}{N_A(N_A + N_D)} \bigg) \cdot \Bigg( \phi_{bi} - V_{applied} \bigg)}
% x_n = \sqrt{\frac{2\epsilon_s}{q} \cdot \Bigg( \frac{N_A}{N_D(N_A + N_D)} \bigg) \cdot \Bigg( \phi_{bi} - V_{applied} \bigg)}
%
% Integral Forms
% \large{y} = \int_{0}^{L} x} \,dx\
% \[ \oint_V f(s) \,ds \]
%
% Maximum Electric Field
% $\lvert \mathcal{E}_{MAX} \rvert = \lvert x_n \rvert \cdot \frac{q \cdot N_D}{\epsilon_s} = \lvert x_p \rvert \cdot \frac{q \cdot N_A}{\epsilon_s}$
%
% Electric Field in Terms of Potential
% $\large{\vec{\mathcal{E}}(x)} = -\frac{dV}{dx} = -\int_{-x_p}^{x_n} \frac{\rho}{\epsilon_{si}} \,dx\$
%
% Poisson's Equation is defined as:
% $\nabla \cdot \vec{\mathcal{E}} = \Large{\frac{\rho}{\epsilon}}$