Added figures for q1

PS1/doc.tex

@@ -4,29 +4,41 @@
 \usepackage{setspace}
 \usepackage{listings}
 \usepackage{color}
-\usepackage{circuitikz}
+\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,%
+%\lstset{language=Matlab,%
-    %basicstyle=\color{red},
+%    %basicstyle=\color{red},
-    breaklines=true,%
+%    breaklines=true,%
-    morekeywords={matlab2tikz},
+%    morekeywords={matlab2tikz},
-    keywordstyle=\color{blue},%
+%    keywordstyle=\color{blue},%
-    morekeywords=[2]{1}, keywordstyle=[2]{\color{black}},
+%    morekeywords=[2]{1}, keywordstyle=[2]{\color{black}},
-    identifierstyle=\color{black},%
+%    identifierstyle=\color{black},%
-    stringstyle=\color{mylilas},
+%    stringstyle=\color{mylilas},
-    commentstyle=\color{mygreen},%
+%    commentstyle=\color{mygreen},%
-    showstringspaces=false,%without this there will be a symbol in the places where there is a space
+%    showstringspaces=false,%without this there will be a symbol in the places where there is a space
-    numbers=left,%
+%    numbers=left,%
-    numberstyle={\tiny \color{black}},% size of the numbers
+%    numberstyle={\tiny \color{black}},% size of the numbers
-    numbersep=9pt, % this defines how far the numbers are from the text
+%    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=[1]{for,end,break},emphstyle=[1]\color{red}, %some words to emphasise
-    %emph=[2]{word1,word2}, emphstyle=[2]{style},    
+%    %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}
@@ -45,27 +57,70 @@
     \singlespacing
     \pagenumbering{arabic}
 
-    \section{Question 1}
+    \section*{Question 1}
-        \subsection{(a)}
+        \subsection*{(a)}
-
+        Beginning with the following two equations:
         \begin{equation}
-            N = $\int_{-\infty}^{\infty}$
+            \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:
 
 
-    \section{Design Section}
+        \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}
 
-    In order to design the desired systems, the Xilinx Vivado software was
+            \caption{Number of electrons and current versus drain voltage.}
-    used to write VHDL code that described the operation of each circuit.
+        \end{figure}
-
-    \newpage
-    \paragraph{MUX / DEMUX Circuit}
-    To implement the multiplexing/demultiplexing system, the following circuit had to be written in VHDL.
-
-    The VHDL architecture below was written to implement this circuit in hardware.
 
+            Below is our code. Note that some variable names are different from those in the example code.
             \begin{lstlisting}[language=Matlab]
+            
 clear all;
 
 %% Constants
@@ -144,7 +199,7 @@ for n = 1:length(voltages)
 
         % 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)
+        %U_P = U_P + 0.1 * (tmpU_P - U_P);
     end
 
     % Calculate current based on solved potential.
@@ -173,4 +228,56 @@ 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}

PS1/q1c.m

@@ -41,6 +41,7 @@ D = D ./ (dE*sum(D));
 N_0 = 0;
 
 voltages = linspace(0, 1, 101);
+dV = voltages(2) - voltages(1);
 
 % Terminal Voltages
 V_G = 0;
@@ -80,13 +81,65 @@ for n = 1:length(voltages)
 
         % 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)
+        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);
 
+    if (abs(V_D-0.0) <= dV/2) 
+        figure(3); title('VD = 0.0 V');
+        subplot(2,3,1); plot(f_1,E,'k-'); axis([-0.1 1.1 -1 1]);
+        xlabel('f1(E+U)'); ylabel('ENERGY  [eV]'); title('VD = 0.0 V');
+        subplot(2,3,2); plot(D/100,E,'k-'); axis([-0.1 1.1 -1 1]);
+        xlabel('D(E)/100'); ylabel('ENERGY  [eV]'); title('VD = 0.0 V');
+        subplot(2,3,3); plot(f_2,E,'k-'); axis([-0.1 1.1 -1 1]);
+        xlabel('f2(E+U)'); ylabel('ENERGY  [eV]'); title('VD = 0.0 V');
+        subplot(2,3,5); plot(f_1-f_2,E,'--',D/100,E,'k-'); axis([-0.1 1.1 -1 1]);
+        xlabel('f1(E+U)-f2(E+U), D(E)/100'); ylabel('ENERGY  [eV]'); title('VD = 0.0 V');
+    elseif (abs(V_D-0.2) <= dV/2) 
+        figure(4); title('VD = 0.2 V');
+        subplot(2,3,1); plot(f_1,E,'k-'); axis([-0.1 1.1 -1 1]);
+        xlabel('f1(E+U)'); ylabel('ENERGY  [eV]'); title('VD = 0.2 V');
+        subplot(2,3,2); plot(D/100,E,'k-'); axis([-0.1 1.1 -1 1]);
+        xlabel('D(E)/100'); ylabel('ENERGY  [eV]'); title('VD = 0.2 V');
+        subplot(2,3,3); plot(f_2,E,'k-'); axis([-0.1 1.1 -1 1]);
+        xlabel('f2(E+U)'); ylabel('ENERGY  [eV]'); title('VD = 0.2 V');
+        subplot(2,3,5); plot(f_1-f_2,E,'--',D/100,E,'k-'); axis([-0.1 1.1 -1 1]);
+        xlabel('f1(E+U)-f2(E+U), D(E)/100'); ylabel('ENERGY  [eV]'); title('VD = 0.2 V');
+    elseif (abs(V_D-0.5) <= dV/2) 
+        figure(5); title('VD = 0.5 V');
+        subplot(2,3,1); plot(f_1,E,'k-'); axis([-0.1 1.1 -1 1]);
+        xlabel('f1(E+U)'); ylabel('ENERGY  [eV]'); title('VD = 0.5 V');
+        subplot(2,3,2); plot(D/100,E,'k-'); axis([-0.1 1.1 -1 1]);
+        xlabel('D(E)/100'); ylabel('ENERGY  [eV]'); title('VD = 0.5 V');
+        subplot(2,3,3); plot(f_2,E,'k-'); axis([-0.1 1.1 -1 1]);
+        xlabel('f2(E+U)'); ylabel('ENERGY  [eV]'); title('VD = 0.5 V');
+        subplot(2,3,5); plot(f_1-f_2,E,'--',D/100,E,'k-'); axis([-0.1 1.1 -1 1]);
+        xlabel('f1(E+U)-f2(E+U), D(E)/100'); ylabel('ENERGY  [eV]'); title('VD = 0.5 V');
+    elseif (abs(V_D-0.8) <= dV/2) 
+        figure(6); title('VD = 0.8 V');
+        subplot(2,3,1); plot(f_1,E,'k-'); axis([-0.1 1.1 -1 1]);
+        xlabel('f1(E+U)'); ylabel('ENERGY  [eV]'); title('VD = 0.8 V');
+        subplot(2,3,2); plot(D/100,E,'k-'); axis([-0.1 1.1 -1 1]);
+        xlabel('D(E)/100'); ylabel('ENERGY  [eV]'); title('VD = 0.8 V');
+        subplot(2,3,3); plot(f_2,E,'k-'); axis([-0.1 1.1 -1 1]);
+        xlabel('f2(E+U)'); ylabel('ENERGY  [eV]'); title('VD = 0.8 V');
+        subplot(2,3,5); plot(f_1-f_2,E,'--',D/100,E,'k-'); axis([-0.1 1.1 -1 1]);
+        xlabel('f1(E+U)-f2(E+U), D(E)/100'); ylabel('ENERGY  [eV]'); title('VD = 0.8 V');
+    elseif (abs(V_D-1.0) <= dV/2) 
+        figure(7); title('VD = 1.0 V');
+        subplot(2,3,1); plot(f_1,E,'k-'); axis([-0.1 1.1 -1 1]);
+        xlabel('f1(E+U)'); ylabel('ENERGY  [eV]'); title('VD = 1.0 V');
+        subplot(2,3,2); plot(D/100,E,'k-'); axis([-0.1 1.1 -1 1]);
+        xlabel('D(E)/100'); ylabel('ENERGY  [eV]'); title('VD = 1.0 V');
+        subplot(2,3,3); plot(f_2,E,'k-'); axis([-0.1 1.1 -1 1]);
+        xlabel('f2(E+U)'); ylabel('ENERGY  [eV]'); title('VD = 1.0 V');
+        subplot(2,3,5); plot(f_1-f_2,E,'--',D/100,E,'k-'); axis([-0.1 1.1 -1 1]);
+        xlabel('f1(E+U)-f2(E+U), D(E)/100'); ylabel('ENERGY  [eV]'); title('VD = 1.0 V');
+    end
+
 end