save changes

PS1/doc.tex

@@ -11,12 +11,22 @@
 \definecolor{gray}{rgb}{0.5,0.5,0.5}
 \definecolor{mauve}{rgb}{0.58,0,0.82}
 
-\lstset{basicstyle=\small,
+\lstset{language=Matlab,%
-        keywordstyle=\color{mauve},
+    %basicstyle=\color{red},
-        identifierstyle=\color{dkgreen},
+    breaklines=true,%
-        stringstyle=\color{gray},
+    morekeywords={matlab2tikz},
-        numbers=left
+    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},    
+}
 
 \title{ECE 456 - Problem Set 1}
 \date{2021-02-06}
@@ -35,29 +45,14 @@
     \singlespacing
     \pagenumbering{arabic}
 
-    \section{Introduction}
+    \section{Question 1}
+        \subsection{(a)}
 
-    The purpose of this lab was to design control circuits according to the provided
+        \begin{equation}
-    specifications, and then verify their operation using a simulation or an FPGA.
+            N = $\int_{-\infty}^{\infty}$
+        \end{equation}
 
-    In the first part of the lab, a series of boolean expressions
-    were designed to implement a Multiplexer/Demultiplexer circuit,
-    intended to route data from one of three radio recievers to one
-    of three engineers, and signal which engineer was currently
-    recieving data.
-    First, Xilinx Vivado Software was used to produce a circuit
-    to fulfill this objective. Then, using the same software,
-    the circuit was simulated against input combinations which
-    would be encountered during normal use, for verification.
    
-    For the second part of the lab, an Access Control circuit was to be designed,
-    allowing lab entry only if a valid ID was provided alongside a proper keypad
-    combination. Otherwise, an alarm signal was to be sent out.
-    The method of designing this circuit was very similar to the method in part one:
-    Again using Xilinx Vivado, the circuit was designed and simulated against inputs
-    to verify if the outputs matched those in the specification.
-    However, for this section, the design was also uploaded to a physical FPGA board
-    where various could be manually tested and validated.
 
     \section{Design Section}
 
@@ -70,30 +65,112 @@
 
     The VHDL architecture below was written to implement this circuit in hardware.
 
-    \begin{lstlisting}[language=MATLAB]
+    \begin{lstlisting}[language=Matlab]
-U0 = 0.25;
+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;
 
-% Capacitance parameters
+% 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));
 
-alpha_G = 0.5;
+% Reference no. of electrons in channel
-alpha_D = 0.5;
+N_0 = 0;
-alpha_S = 1 - alpha_G - alpha_D;
 
-% Energy grid in eV, from -1 eV to 1 eV
+voltages = linspace(0, 1, 101);
 
-NE = 501;
+% Terminal Voltages
-E = linspace(-1,1,NE);
+V_G = 0;
-dE = E(2) - E(1);
+V_S = 0;
 
-% Gamma parameters, in eV
+for n = 1:length(voltages)
+    % Set varying drain voltage
+    V_D = voltages(n);
 
-gamma_1 = 0.005;
+    % Shifted energy levels of the contacts
-gamma_2 = 0.005;
+    mu_1 = mu - V_S;
-gamma = gamma_1 + gamma_2;
+    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}
-    \newpage
 
 \end{document}

PS1/q1b.m

@@ -4,10 +4,6 @@ clear all;
 
 % Physical constants
 hbar = 1.052e-34;
-q = 1.602e-19;
-%epsilon_0 = 8.854e-12;
-%epsilon_r = 4;
-%mstar = 0.25 * 9.11e-31;
 
 % Single-charge coupling energy (eV)
 U_0 = 0.25;

PS1/q1c.m

@@ -0,0 +1,109 @@
+clear all;
+
+%% Constants
+
+% Physical constants
+hbar = 1.052e-34;
+q = 1.602e-19;
+%epsilon_0 = 8.854e-12;
+%epsilon_r = 4;
+%mstar = 0.25 * 9.11e-31;
+
+% 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]');