Added matlab listings and fixed matlab colouring
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PS1/doc.tex
298
PS1/doc.tex
@ -16,29 +16,35 @@
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\definecolor{gray}{rgb}{0.5,0.5,0.5}
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\definecolor{mauve}{rgb}{0.58,0,0.82}
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%\lstset{language=Matlab,%
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% %basicstyle=\color{red},
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% breaklines=true,%
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% morekeywords={matlab2tikz},
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% keywordstyle=\color{blue},%
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% morekeywords=[2]{1}, keywordstyle=[2]{\color{black}},
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% identifierstyle=\color{black},%
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% stringstyle=\color{mylilas},
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% commentstyle=\color{mygreen},%
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% showstringspaces=false,%without this there will be a symbol in the places where there is a space
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% numbers=left,%
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% numbersty_sumle={\tiny \color{black}},% size of th_sume numbers
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% numbersep=9pt, % this defines how far the numbers are from the text
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% emph=[1]{for,end,break},emphstyle=[1]\color{red}, %some words to emphasise
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% %emph=[2]{word1,word2}, emphstyle=[2]{style},
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%}
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\lstset{basicstyle=\small,
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keywordstyle=\color{mauve},
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identifierstyle=\color{dkgreen},
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stringstyle=\color{gray},
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numbers=left,
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xleftmargin=5em
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}
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\definecolor{mygreen}{RGB}{28,172,0} % color values Red, Green, Blue
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\definecolor{mylilas}{RGB}{170,55,241}
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\lstset{language=Matlab,%
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%basicstyle=\color{red},
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breaklines=true,%
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morekeywords={matlab2tikz},
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keywordstyle=\color{blue},%
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morekeywords=[2]{1}, keywordstyle=[2]{\color{black}},
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identifierstyle=\color{black},%
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stringstyle=\color{mylilas},
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commentstyle=\color{mygreen},%
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showstringspaces=false,%without this there will be a symbol in the places where there is a space
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numbers=left,%
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numberstyle={\tiny \color{black}},% size of th_sume numbers
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numbersep=9pt, % this defines how far the numbers are from the text
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emph=[1]{for,end,break},emphstyle=[1]\color{red}, %some words to emphasise
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%emph=[2]{word1,word2}, emphstyle=[2]{style},
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xleftmargin=5em
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}
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%\lstset{basicstyle=\small,
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% keywordstyle=\color{mauve},
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% identifierstyle=\color{dkgreen},
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% stringstyle=\color{gray},
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% numbers=left,
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% xleftmargin=5em
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% }
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\title{ECE 456 - Problem Set 1}
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\date{2021-02-06}
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@ -230,6 +236,8 @@ ylabel('Current [A]');
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\subsection*{(c)}
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Code for this section can be found in appendix A.
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\subsubsection*{(i)}
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\begin{figure}[H]
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\centering
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@ -443,6 +451,77 @@ ylabel('Current [A]');
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Below is our code. Note that some variable names are different from those in the example code.
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\begin{lstlisting}[language=Matlab]
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% Thermo-electric current
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% Physical constants
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hbar = 1.054e-34;
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q = 1.602e-19;
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% Parameters (eV)
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kBT_1 = 0.025;
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kBT_2 = 0.026;
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mu = 0;
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gamma_1 = 0.005;
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gamma_2 = gamma_1;
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gamma_sum = gamma_1 + gamma_2;
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% Channel energy levels, varying between -0.25eV and 0.25eV
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epsilon = linspace(-0.25, 0.25, 101);
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depsilon = epsilon(2) - epsilon(1);
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% Energy grid
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E = linspace(-1, 1, 501);
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dE = E(2) - E(1);
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% Contact fermi functions
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f_1 = 1 ./ (1 + exp((E - mu)./kBT_1));
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f_2 = 1 ./ (1 + exp((E - mu)./kBT_2));
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% Iterate through channel energy levels
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for n = 1:length(epsilon)
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% Compute energy level density functions - integral normalized to unity
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D = (gamma_sum./(2*pi))./((E-epsilon(n)).^2+((gamma_sum./2).^2));
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D = D./(dE*sum(D));
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% Compute number of channel electrons
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N(n) = dE*sum( ((gamma_1./gamma_sum).*f_1 + (gamma_2./gamma_sum).*f_2).*D );
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% Compute the current in Amps; factor of q to resolve units
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I(n) = q*(q/hbar)*dE*sum((f_1 - f_2).*D.*gamma_1.*gamma_2./gamma_sum);
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%plot f_1 - f_2 and D/2
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if (abs(epsilon(n) + 0.05) <= depsilon / 2) & (epsilon(n) <= 0)
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figure(3);
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h = plot(f_1-f_2, E, 'x', D/2500, E, 'k-');
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set(gca, 'Fontsize', [18]);
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axis([-0.01 0.02 -1 1]);
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xlabel('f1(E) - f2(E), D(E)/2500');
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ylabel('ENERGY [eV]');
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legend('f1-f2', 'D(E)/2500');
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title('CHANNEL LEVEL = -0.05 eV');
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elseif (abs(epsilon(n)) <= depsilon / 2) & (epsilon(n) <= 0)
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figure(4);
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h = plot(f_1-f_2, E, 'x', D/2500, E, 'k-');
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set(gca, 'Fontsize', [18]);
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axis([-0.01 0.02 -1 1]);
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xlabel('f1(E) - f2(E), D(E)/2500');
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ylabel('ENERGY [eV]');
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legend('f1-f2', 'D(E)/2500');
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title('CHANNEL LEVEL = 0 eV');
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end
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end
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% Final plots
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figure(1);
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\end{lstlisting}
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\subsection*{(b)}
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\begin{figure}[H]
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@ -700,9 +779,178 @@ ylabel('Current [A]');
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\end{equation*}
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%
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We conclude there are 3 levels.
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% TODO appendix for Part C code
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\appendix
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\newpage
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\section{Question 1b Code}
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\begin{lstlisting}[language=Matlab]
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clear all;
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%% Constants
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% Physical constants
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hbar = 1.052e-34;
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q = 1.602e-19;
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%epsilon_0 = 8.854e-12;
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%epsilon_r = 4;
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%mstar = 0.25 * 9.11e-31;
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% Single-charge coupling energy (eV)
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U_0 = 0.25;
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% (eV)
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kBT = 0.025;
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% Contact coupling coefficients (eV)
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gamma_1 = 0.005;
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gamma_2 = gamma_1;
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gamma_sum = gamma_1 + gamma_2;
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% Capacitive gate coefficient
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a_G = 0.5;
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% Capacitive drain coefficient
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a_D = 0.5;
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a_S = 1 - a_G - a_D;
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% Central energy level
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mu = 0;
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% Energy grid, from -1eV to 1eV
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NE = 501;
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E = linspace(-1, 1, NE);
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dE = E(2) - E(1);
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% TODO name this better
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cal_E = 0.2;
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% Lorentzian density of states, normalized so the integral is 1
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D = (gamma_sum / (2*pi)) ./ ( (E-cal_E).^2 + (gamma_sum/2).^2 );
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D = D ./ (dE*sum(D));
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% Reference no. of electrons in channel
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N_0 = 0;
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voltages = linspace(0, 1, 101);
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dV = voltages(2) - voltages(1);
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% Terminal Voltages
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V_G = 0;
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V_S = 0;
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for n = 1:length(voltages)
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% Set varying drain voltage
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V_D = voltages(n);
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% Shifted energy levels of the contacts
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mu_1 = mu - V_S;
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mu_2 = mu - V_D;
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% Laplace potential, does not change as solution is found (eV)
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% q is factored out here, we are working in eV
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U_L = - (a_G*V_G) - (a_D*V_D) - (a_S*V_S);
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% Poisson potential must change, assume 0 initially (eV)
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U_P = 0;
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% Assume large rate of change
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dU_P = 1;
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% Run until we get close enough to the answer
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while dU_P > 1e-6
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% source Fermi function
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f_1 = 1 ./ (1 + exp((E + U_L + U_P - mu_1) ./ kBT));
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% drain Fermi function
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f_2 = 1 ./ (1 + exp((E + U_L + U_P - mu_2) ./ kBT));
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% Update channel electrons against potential
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N(n) = dE * sum( ((gamma_1/gamma_sum) .* f_1 + (gamma_2/gamma_sum) .* f_2) .* D);
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% Re-update Poisson portion of potential
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tmpU_P = U_0 * ( N(n) - N_0);
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dU_P = abs(U_P - tmpU_P);
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% Unsure why U_P is updated incrementally, perhaps to avoid oscillations?
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%U_P = tmpU_P;
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U_P = U_P + 0.1 * (tmpU_P - U_P);
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end
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% Calculate current based on solved potential.
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% Note: f1 is dependent on changes in U but has been updated prior in the loop
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I(n) = q * (q/hbar) * (gamma_1 * gamma_1 / gamma_sum) * dE * sum((f_1-f_2).*D);
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if (abs(V_D-0.0) <= dV/2)
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figure(3); title('VD = 0.0 V');
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subplot(2,3,1); plot(f_1,E,'k-'); axis([-0.1 1.1 -1 1]);
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xlabel('f1(E+U)'); ylabel('ENERGY [eV]'); title('VD = 0.0 V');
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subplot(2,3,2); plot(D/100,E,'k-'); axis([-0.1 1.1 -1 1]);
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xlabel('D(E)/100'); ylabel('ENERGY [eV]'); title('VD = 0.0 V');
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subplot(2,3,3); plot(f_2,E,'k-'); axis([-0.1 1.1 -1 1]);
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xlabel('f2(E+U)'); ylabel('ENERGY [eV]'); title('VD = 0.0 V');
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subplot(2,3,5); plot(f_1-f_2,E,'--',D/100,E,'k-'); axis([-0.1 1.1 -1 1]);
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xlabel('f1(E+U)-f2(E+U), D(E)/100'); ylabel('ENERGY [eV]'); title('VD = 0.0 V');
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elseif (abs(V_D-0.2) <= dV/2)
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figure(4); title('VD = 0.2 V');
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subplot(2,3,1); plot(f_1,E,'k-'); axis([-0.1 1.1 -1 1]);
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xlabel('f1(E+U)'); ylabel('ENERGY [eV]'); title('VD = 0.2 V');
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subplot(2,3,2); plot(D/100,E,'k-'); axis([-0.1 1.1 -1 1]);
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xlabel('D(E)/100'); ylabel('ENERGY [eV]'); title('VD = 0.2 V');
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subplot(2,3,3); plot(f_2,E,'k-'); axis([-0.1 1.1 -1 1]);
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xlabel('f2(E+U)'); ylabel('ENERGY [eV]'); title('VD = 0.2 V');
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subplot(2,3,5); plot(f_1-f_2,E,'--',D/100,E,'k-'); axis([-0.1 1.1 -1 1]);
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xlabel('f1(E+U)-f2(E+U), D(E)/100'); ylabel('ENERGY [eV]'); title('VD = 0.2 V');
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elseif (abs(V_D-0.5) <= dV/2)
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figure(5); title('VD = 0.5 V');
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subplot(2,3,1); plot(f_1,E,'k-'); axis([-0.1 1.1 -1 1]);
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xlabel('f1(E+U)'); ylabel('ENERGY [eV]'); title('VD = 0.5 V');
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subplot(2,3,2); plot(D/100,E,'k-'); axis([-0.1 1.1 -1 1]);
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xlabel('D(E)/100'); ylabel('ENERGY [eV]'); title('VD = 0.5 V');
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subplot(2,3,3); plot(f_2,E,'k-'); axis([-0.1 1.1 -1 1]);
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xlabel('f2(E+U)'); ylabel('ENERGY [eV]'); title('VD = 0.5 V');
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subplot(2,3,5); plot(f_1-f_2,E,'--',D/100,E,'k-'); axis([-0.1 1.1 -1 1]);
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xlabel('f1(E+U)-f2(E+U), D(E)/100'); ylabel('ENERGY [eV]'); title('VD = 0.5 V');
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elseif (abs(V_D-0.8) <= dV/2)
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figure(6); title('VD = 0.8 V');
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subplot(2,3,1); plot(f_1,E,'k-'); axis([-0.1 1.1 -1 1]);
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xlabel('f1(E+U)'); ylabel('ENERGY [eV]'); title('VD = 0.8 V');
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subplot(2,3,2); plot(D/100,E,'k-'); axis([-0.1 1.1 -1 1]);
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xlabel('D(E)/100'); ylabel('ENERGY [eV]'); title('VD = 0.8 V');
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subplot(2,3,3); plot(f_2,E,'k-'); axis([-0.1 1.1 -1 1]);
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xlabel('f2(E+U)'); ylabel('ENERGY [eV]'); title('VD = 0.8 V');
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subplot(2,3,5); plot(f_1-f_2,E,'--',D/100,E,'k-'); axis([-0.1 1.1 -1 1]);
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xlabel('f1(E+U)-f2(E+U), D(E)/100'); ylabel('ENERGY [eV]'); title('VD = 0.8 V');
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elseif (abs(V_D-1.0) <= dV/2)
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figure(7); title('VD = 1.0 V');
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subplot(2,3,1); plot(f_1,E,'k-'); axis([-0.1 1.1 -1 1]);
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xlabel('f1(E+U)'); ylabel('ENERGY [eV]'); title('VD = 1.0 V');
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subplot(2,3,2); plot(D/100,E,'k-'); axis([-0.1 1.1 -1 1]);
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xlabel('D(E)/100'); ylabel('ENERGY [eV]'); title('VD = 1.0 V');
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subplot(2,3,3); plot(f_2,E,'k-'); axis([-0.1 1.1 -1 1]);
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xlabel('f2(E+U)'); ylabel('ENERGY [eV]'); title('VD = 1.0 V');
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subplot(2,3,5); plot(f_1-f_2,E,'--',D/100,E,'k-'); axis([-0.1 1.1 -1 1]);
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xlabel('f1(E+U)-f2(E+U), D(E)/100'); ylabel('ENERGY [eV]'); title('VD = 1.0 V');
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end
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end
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%%Plotting commands
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figure(1);
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h = plot(voltages, N,'k');
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grid on;
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set(h,'linewidth',[2.0]);
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set(gca,'Fontsize',[18]);
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xlabel('Drain voltage [V]');
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ylabel('Number of electrons');
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figure(2);
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h = plot(voltages, I,'k');
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grid on;
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set(h,'linewidth',[2.0]);
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set(gca,'Fontsize',[18]);
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xlabel('Drain voltage [V]');
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ylabel('Current [A]');
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\end{lstlisting}
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\end{document}
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