\documentclass{article} \usepackage{graphicx} \usepackage{setspace} \usepackage{listings} \usepackage{color} \usepackage{amsmath} \usepackage{float} \usepackage{caption} \usepackage{subcaption} \usepackage[margin=0.75in]{geometry} \renewcommand{\thesubsection}{\indent(\alph{subsection})} \definecolor{dkgreen}{rgb}{0,0.6,0} \definecolor{gray}{rgb}{0.5,0.5,0.5} \definecolor{mauve}{rgb}{0.58,0,0.82} %\lstset{language=Matlab,% % %basicstyle=\color{red}, % breaklines=true,% % morekeywords={matlab2tikz}, % keywordstyle=\color{blue},% % morekeywords=[2]{1}, keywordstyle=[2]{\color{black}}, % identifierstyle=\color{black},% % stringstyle=\color{mylilas}, % commentstyle=\color{mygreen},% % showstringspaces=false,%without this there will be a symbol in the places where there is a space % numbers=left,% % numberstyle={\tiny \color{black}},% size of the numbers % numbersep=9pt, % this defines how far the numbers are from the text % emph=[1]{for,end,break},emphstyle=[1]\color{red}, %some words to emphasise % %emph=[2]{word1,word2}, emphstyle=[2]{style}, %} \lstset{basicstyle=\small, keywordstyle=\color{mauve}, identifierstyle=\color{dkgreen}, stringstyle=\color{gray}, numbers=left, xleftmargin=5em } \title{ECE 456 - Problem Set 1} \date{2021-02-06} \author{David Lenfesty \\ lenfesty@ualberta.ca \and Phillip Kirwin \\ pkirwin@ualberta.ca} \setcounter{tocdepth}{2} % Show subsections \begin{document} \doublespacing \pagenumbering{gobble} \maketitle \newpage \singlespacing \pagenumbering{arabic} \section*{Question 1} \subsection*{(a)} Beginning with the following two equations: \begin{equation} \label{eq:N_old} N = \int_{-\infty}^{\infty}\frac{\gamma_1 f_1(E) + \gamma_2 f_2(E)}{\gamma_1 + \gamma_2} D(E-U) dE, \end{equation} \begin{equation} \label{eq:I_old} I = \frac{q}{\hbar}\frac{\gamma_1 \gamma_2}{\gamma_1 + \gamma_2} \int_{-\infty}^{\infty}[f_1(E) - f_2(E)] D(E-U) dE, \end{equation} and changing the variable of integration to $E' = E - U$: \begin{equation*} N = \int_{-\infty}^{\infty}\frac{\gamma_1 f_1(E' + U) + \gamma_2 f_2(E' + U)}{\gamma_1 + \gamma_2} D(E') dE', \end{equation*} \begin{equation*} I = \frac{q}{\hbar}\frac{\gamma_1 \gamma_2}{\gamma_1 + \gamma_2} \int_{-\infty}^{\infty}[f_1(E' + U) - f_2(E' + U)] D(E') dE'. \end{equation*} Replacing $E' \rightarrow E$, we obtain equations \ref{eq:N_new} and \ref{eq:I_new}. \begin{equation} \label{eq:N_new} N = \int_{-\infty}^{\infty}\frac{\gamma_1 f_1(E + U) + \gamma_2 f_2(E + U)}{\gamma_1 + \gamma_2} DE dE, \end{equation} \begin{equation} \label{eq:I_new} I = \frac{q}{\hbar}\frac{\gamma_1 \gamma_2}{\gamma_1 + \gamma_2} \int_{-\infty}^{\infty}[f_1(E + U) - f_2(E + U)] DE dE, \end{equation} \subsection*{(b)} For the provided constants, the plots of the number of channel electrons and the channel current follow: \begin{figure}[H] \centering \begin{subfigure}{0.5\textwidth} % Mess around with widths later \centering \includegraphics[width=\textwidth]{q1b_electrons.png} \caption{Plot of channel electrons vs. drain voltage.} \label{fig:q1b_electrons} % Note that the comment after \end{subfigure} is required for side by side figures \end{subfigure}% \begin{subfigure}{0.5\textwidth} % Mess around with widths later \centering \includegraphics[width=\textwidth]{q1b_current.png} \caption{Plot of channel current vs. drain voltage.} \label{fig:q1b_current} \end{subfigure} \caption{Number of electrons and current versus drain voltage.} \end{figure} Below is our code. Note that some variable names are different from those in the example code. \begin{lstlisting}[language=Matlab] clear all; %% Constants % Physical constants hbar = 1.052e-34; % Single-charge coupling energy (eV) U_0 = 0.25; % (eV) kBT = 0.025; % Contact coupling coefficients (eV) gamma_1 = 0.005; gamma_2 = gamma_1; gamma_sum = gamma_1 + gamma_2; % Capacitive gate coefficient a_G = 0.5; % Capacitive drain coefficient a_D = 0.5; a_S = 1 - a_G - a_D; % Central energy level mu = 0; % Energy grid, from -1eV to 1eV NE = 501; E = linspace(-1, 1, NE); dE = E(2) - E(1); % TODO name this better cal_E = 0.2; % Lorentzian density of states, normalized so the integral is 1 D = (gamma_sum / (2*pi)) ./ ( (E-cal_E).^2 + (gamma_sum/2).^2 ); D = D ./ (dE*sum(D)); % Reference no. of electrons in channel N_0 = 0; voltages = linspace(0, 1, 101); % Terminal Voltages V_G = 0; V_S = 0; for n = 1:length(voltages) % Set varying drain voltage V_D = voltages(n); % Shifted energy levels of the contacts mu_1 = mu - V_S; mu_2 = mu - V_D; % Laplace potential, does not change as solution is found (eV) % q is factored out here, we are working in eV U_L = - (a_G*V_G) - (a_D*V_D) - (a_S*V_S); % Poisson potential must change, assume 0 initially (eV) U_P = 0; % Assume large rate of change dU_P = 1; % Run until we get close enough to the answer while dU_P > 1e-6 % source Fermi function f_1 = 1 ./ (1 + exp((E + U_L + U_P - mu_1) ./ kBT)); % drain Fermi function f_2 = 1 ./ (1 + exp((E + U_L + U_P - mu_2) ./ kBT)); % Update channel electrons against potential N(n) = dE * sum( ((gamma_1/gamma_sum) .* f_1 + (gamma_2/gamma_sum) .* f_2) .* D); % Re-update Poisson portion of potential tmpU_P = U_0 * ( N(n) - N_0); dU_P = abs(U_P - tmpU_P); % Unsure why U_P is updated incrementally, perhaps to avoid oscillations? %U_P = tmpU_P; %U_P = U_P + 0.1 * (tmpU_P - U_P); end % Calculate current based on solved potential. % Note: f1 is dependent on changes in U but has been updated prior in the loop I(n) = q * (q/hbar) * (gamma_1 * gamma_1 / gamma_sum) * dE * sum((f_1-f_2).*D); end %%Plotting commands figure(1); h = plot(voltages, N,'k'); grid on; set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]); xlabel('Drain voltage [V]'); ylabel('Number of electrons'); figure(2); h = plot(voltages, I,'k'); grid on; set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]); xlabel('Drain voltage [V]'); ylabel('Current [A]'); \end{lstlisting} \subsection*{(c)} \begin{figure}[H] \centering \begin{subfigure}{0.5\textwidth} % Mess around with widths later \centering \includegraphics[width=\textwidth]{q1c_1.png} \caption{Plot of channel electrons vs. drain voltage.} \label{fig:q1c_1} % Note that the comment after \end{subfigure} is required for side by side figures \end{subfigure}% \begin{subfigure}{0.5\textwidth} % Mess around with widths later \centering \includegraphics[width=\textwidth]{q1c_2.png} \caption{Plot of channel electrons vs. drain voltage.} \label{fig:q1c_1} % Note that the comment after \end{subfigure} is required for side by side figures \end{subfigure} \begin{subfigure}{0.5\textwidth} % Mess around with widths later \centering \includegraphics[width=\textwidth]{q1c_3.png} \caption{Plot of channel electrons vs. drain voltage.} \label{fig:q1c_1} % Note that the comment after \end{subfigure} is required for side by side figures \end{subfigure}% \begin{subfigure}{0.5\textwidth} % Mess around with widths later \centering \includegraphics[width=\textwidth]{q1c_4.png} \caption{Plot of channel electrons vs. drain voltage.} \label{fig:q1c_1} % Note that the comment after \end{subfigure} is required for side by side figures \end{subfigure} \begin{subfigure}{0.5\textwidth} % Mess around with widths later \centering \includegraphics[width=\textwidth]{q1c_5.png} \caption{Plot of channel electrons vs. drain voltage.} \label{fig:q1c_1} % Note that the comment after \end{subfigure} is required for side by side figures \end{subfigure}% \caption{Visual representation of the fermi functions of the contacts and channel.} \end{figure} % TODO appendix for Part C code \end{document}