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plot_polarizability_bilayer.m
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% MATLAB R2015a function
% S. M. Farzaneh, [email protected]
% Created: September 10, 2017
% Title: Plot polarizability of bilayer graphene versus momentum
function [] = plot_polarizability_bilayer()
%% Initialize necessary constants, variables, and functions
init_constant; % initialize physical constants and parameters
init_variable; % initialize variables
init_function; % initialize functions
%% Plot for gapless bilayer graphene (delta = 0 and multiple Ef's)
% We use set delta to 0.01 avoid integral divergence.
% Prepare the figure
figure('Units', 'inches', ...
'PaperSize', [2.33, 2.33], ...
'PaperPosition', [0, 0, 2.33, 2.33], ...
'visible', 'off')
ax = gca;
hold(ax, 'on')
grid(ax, 'on')
axis(ax,[0 0.3 0 10])
% Set parameters and plot
delta = 0.01*qe; % energy asymmetry [J]
Ef = 0:0.1:0.5; % Fermi level [eV]
for i = 1:length(Ef)
% Load data from file
filename = strcat('data/polarizability/polrz', ...
'_Ef', num2str(Ef(i)), ...
'_del', num2str(delta/qe), ...
'_T', num2str(T), ...
'.csv');
M = csvread(filename);
q = M(:, 1); % first column: momentum (q) [1/m]
polrz = M(:, 2); % second column: polarizability [1/Jm^2]
% Normalize vectors
q = q/(pi/b); % with respect to cut-off momentum, pi/b.
polrz = polrz/N0; % with respect to density of states at zero energy.
% Define finer grid for plotting
q_fine = linspace(0, 0.3, 100);
polrz_fine = spline(q, polrz, q_fine); % interpolation
% Plot
plot(ax, q_fine, polrz_fine, 'LineWidth', 1)
end
% Set axes properties
set(ax, ...
'Units', 'normalized', ...
'FontUnits', 'points', ...
'FontWeight', 'normal', ...
'FontSize', 9, ...
'FontName', 'Times New Roman')
title(ax, '$\Delta$ = 0 eV, $T$ = 300 K', 'FontSize', 9, 'Interpreter', 'latex')
xlabel(ax, 'Wavevector $q \times b/ \pi$', 'FontSize', 9, 'Interpreter', 'latex')
ylabel(ax, 'Polarizability $\Pi(q) /N_0$', 'FontSize', 9, 'Interpreter', 'latex')
% Annotations
annotation('arrow', [0.6, 0.6], [0.4, 0.8], 'HeadWidth', 6, 'HeadLength', 6)
str1 = '$$E_f$$ = 0 eV';
str2 = '$$E_f$$ = 0.5 eV';
text(0.13, 2.5, str1, 'FontSize', 9, 'Interpreter', 'latex')
text(0.13, 9, str2, 'FontSize', 9, 'Interpreter', 'latex')
% Save the plot
print -dpdf 'polrz_vs_momentum_delta0.pdf'
message = strcat('polrz_vs_momentum_delta0.pdf', ' was saved.');
disp(message)
%% Plot for gapped bilayer graphene (delta = 0.8 and multiple Ef's)
% Prepare the figure
figure('Units', 'inches', ...
'PaperSize', [2.33, 2.33], ...
'PaperPosition', [0, 0, 2.33, 2.33], ...
'visible', 'off')
ax = gca;
hold(ax, 'on')
grid(ax, 'on')
axis(ax,[0 0.3 0 10])
% Set parameters and plot
delta = 0.8*qe; % energy asymmetry [J]
Ef = 0:0.1:0.5; % Fermi level [eV]
for i = 1:length(Ef)
% Load data from file
filename = strcat('data/polarizability/polrz', ...
'_Ef', num2str(Ef(i)), ...
'_del', num2str(delta/qe), ...
'_T', num2str(T), ...
'.csv');
M = csvread(filename);
q = M(:, 1); % first column: momentum (q) [1/m]
polrz = M(:, 2); % second column: polarizability [1/Jm^2]
% Normalize vectors
q = q/(pi/b); % with respect to cut-off momentum, pi/b.
polrz = polrz/N0; % with respect to density of states at zero energy.
% Define finer grid for plotting
q_fine = linspace(0, 0.3, 100);
polrz_fine = spline(q, polrz, q_fine); % interpolation
% Plot
plot(ax, q_fine, polrz_fine, 'LineWidth', 1)
end
% Set axes properties
set(ax, ...
'Units', 'normalized', ...
'FontUnits', 'points', ...
'FontWeight', 'normal', ...
'FontSize', 9, ...
'FontName', 'Times New Roman')
title(ax, '$\Delta$ = 0.8 eV, $T$ = 300 K', 'FontSize', 9, 'Interpreter', 'latex')
xlabel(ax, 'Wavevector $q \times b/ \pi$', 'FontSize', 9, 'Interpreter', 'latex')
ylabel(ax, 'Polarizability $\Pi(q) /N_0$', 'FontSize', 9, 'Interpreter', 'latex')
% Annotations
annotation('arrow', [0.6, 0.6], [0.4, 0.8], 'HeadWidth', 6, 'HeadLength', 6)
str1 = '$$E_f$$ = 0 eV';
str2 = '$$E_f$$ = 0.5 eV';
text(0.13, 2.5, str1, 'FontSize', 9, 'Interpreter', 'latex')
text(0.13, 9, str2, 'FontSize', 9, 'Interpreter', 'latex')
% Save the plot
print -dpdf 'polrz_vs_momentum_delta08.pdf'
message = strcat('polrz_vs_momentum_delta08.pdf', ' was saved.');
disp(message)
%% Plot literature's analytic results vs. our numerical results
% Prepare the figure
figure('Units', 'inches', ...
'PaperSize', [2.33, 2.33], ...
'PaperPosition', [0, 0, 2.33, 2.33], ...
'visible', 'off')
ax = gca;
hold(ax, 'on')
grid(ax, 'on')
axis(ax,[0 0.3 0 10])
% Set parameters and plot
delta = 0.01*qe; % energy asymmetry [J]
Ef = 0.1*qe; % Fermi level [eV]
% Load data from file
% numerical polarizability
filename = strcat('data/polarizability/polrz', ...
'_Ef', num2str(Ef/qe), ...
'_del', num2str(delta/qe), ...
'_T', num2str(T), ...
'.csv');
M = csvread(filename);
q = M(:, 1); % first column: momentum (q) [1/m]
polrz = M(:, 2); % second column: polarizability [1/Jm^2]
% Normalize vectors
q = q/(pi/b); % with respect to cut-off momentum, pi/b.
polrz = polrz/N0; % with respect to density of states at zero energy.
% Define finer grid for plotting
q_fine = linspace(0, 0.3, 100);
polrz_fine = spline(q, polrz, q_fine); % interpolation
% Plot
plot(ax, q_fine, polrz_fine, 'LineWidth', 1)
% analytic polarizability
% Find Fermi momentum (kF)
kF = an_plrz_kf(Ef);
polrz_an = an_plrz(q_fine*pi/b, kF);
% Normalize vectors
polrz_an = polrz_an/N0; % with respect to density of states at zero energy.
% Plot
plot(ax, q_fine, polrz_an, 'LineWidth', 1)
% Set axes properties
set(ax, ...
'Units', 'normalized', ...
'FontUnits', 'points', ...
'FontWeight', 'normal', ...
'FontSize', 9, ...
'FontName', 'Times New Roman')
title(ax, '$\Delta$ = 0 eV, $k_BT \ll E_f$', 'FontSize', 9, 'Interpreter', 'latex')
xlabel(ax, 'Wavevector $q \times b/ \pi$', 'FontSize', 9, 'Interpreter', 'latex')
ylabel(ax, 'Polarizability $\Pi(q) /N_0$', 'FontSize', 9, 'Interpreter', 'latex')
% Annotations
str1 = 'Numerical';
str2 = 'Analytical';
text(0.15, 5.6, str1, 'FontSize', 9, 'Rotation', 45, 'Interpreter', 'latex')
text(0.15, 1.9, str2, 'FontSize', 9, 'Interpreter', 'latex')
% Save the plot
print -dpdf 'polrz_vs_momentum_analytic_numerical.pdf'
message = strcat('polrz_vs_momentum_analytic_numerical.pdf', ' was saved.');
disp(message)
end