fixed calc, redid graphs, writing discussion and part a

2D-Conductivity/conductivity_calc_surface.m

@@ -32,33 +32,33 @@ f_vals = f_vals .* (2*pi); % rads-1
 
 % Carrier Density
 %%%%%%%
-carrier_vals = logspace(0, MAX_Y, Y_TOTAL); % m-2
+% carrier_vals = logspace(0, MAX_Y, Y_TOTAL); % m-2
-
+% 
-% below turns turns carrier densities into Fermi energies
+% % below turns turns carrier densities into Fermi energies
-fermi_vals = zeros(1, length(carrier_vals));
+% fermi_vals = zeros(1, length(carrier_vals));
-for carr=1:length(carrier_vals)
+% for carr=1:length(carrier_vals)
-    fermi_vals(carr) = fermi_from_carrier_density(carrier_vals(carr), ev_to_j(t));
+%     fermi_vals(carr) = fermi_from_carrier_density(carrier_vals(carr), ev_to_j(t));
-end
+% end
-
+% 
-% CALCULATE SHEET CONDUCTIVITY
+% % CALCULATE SHEET CONDUCTIVITY
-cond = zeros(length(f_vals),... % frequency
+% cond = zeros(length(f_vals),... % frequency
-             length(fermi_vals),... % fermi
+%              length(fermi_vals),... % fermi
-             2); % intra/inter
+%              2); % intra/inter
-for freq=1:length(f_vals)
+% for freq=1:length(f_vals)
-    for y=1:length(fermi_vals)
+%     for y=1:length(fermi_vals)
-        
+%         
-        cond(freq, y, :) = sheet_conductivity(f_vals(freq),... % omega (rads-1)
+%         cond(freq, y, :) = sheet_conductivity(f_vals(freq),... % omega (rads-1)
-                                              fermi_vals(y),... % fermi_level (J)
+%                                               fermi_vals(y),... % fermi_level (J)
-                                              300,... % temp (K)
+%                                               300,... % temp (K)
-                                              5e-12); % scatter_lifetime (s)
+%                                               1e-12); % scatter_lifetime (s)
-    end
+%     end
-end
+% end
 
 % Temperature
 %%%%%%%
 % temp_vals = linspace(0, 2230, Y_TOTAL); % K
 % 
-% CALCULATE SHEET CONDUCTIVITY
+% % CALCULATE SHEET CONDUCTIVITY
 % cond = zeros(length(f_vals),... % frequency
 %              length(temp_vals),... % fermi
 %              2); % intra/inter
@@ -74,21 +74,21 @@ end
 
 % Scatter Lifetime
 %%%%%%%
-% scatt_vals = logspace(-11, -14, Y_TOTAL); % s-1
+scatt_vals = logspace(-11, -13, Y_TOTAL); % s-1
-% 
+
-% % CALCULATE SHEET CONDUCTIVITY
+% CALCULATE SHEET CONDUCTIVITY
-% cond = zeros(length(f_vals),... % frequency
+cond = zeros(length(f_vals),... % frequency
-%              length(scatt_vals),... % fermi
+             length(scatt_vals),... % fermi
-%              2); % intra/inter
+             2); % intra/inter
-% for freq=1:length(f_vals)
+for freq=1:length(f_vals)
-%     for y=1:length(scatt_vals)
+    for y=1:length(scatt_vals)
-%         
+        
-%         cond(freq, y, :) = sheet_conductivity(f_vals(freq),... % omega (rads-1)
+        cond(freq, y, :) = sheet_conductivity(f_vals(freq),... % omega (rads-1)
-%                                               fermi_from_carrier_density(1.3e13*10000, ev_to_j(t)),... % fermi_level (J), ttf = 1.3e13*10000, cocp2 = 2.2e13*10000
+                                              fermi_from_carrier_density(1.3e13*10000, ev_to_j(t)),... % fermi_level (J), ttf = 1.3e13*10000, cocp2 = 2.2e13*10000
-%                                               300,... % temp (K)
+                                              300,... % temp (K)
-%                                               scatt_vals(y)); % scatter_lifetime (s)
+                                              scatt_vals(y)); % scatter_lifetime (s)
-%     end
+    end
-% end
+end
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%       RENDER
@@ -98,7 +98,8 @@ if DISPLAY_HZ % divide radians back to hertz
     f_vals = f_vals ./ (2*pi);
 end
 
-y_vals = carrier_vals;
+y_vals = scatt_vals;
+cond = cond * 1e3;
 
 % cond = sign(cond).*log10(abs(cond));
 
@@ -124,11 +125,11 @@ set(gca, 'yscale', 'log')
 
 set(gca, 'ColorScale', 'log')
 
-ylabel('Net Carrier Density (m^{-2})');
+% ylabel('Net Carrier Density (m^{-2})');
-% ylabel('Temperature (K)');
+% ylabel('Temperature (K)');carrier_vals
-% ylabel('Scatter Lifetime (s)');
+ylabel('Scatter Lifetime (s)');
 
-zlabel('Conductivity (S)');
+zlabel('Conductivity (mS)');
 if DISPLAY_HZ
     xlabel('Frequency (Hz)');
 else
@@ -157,11 +158,11 @@ set(gca, 'yscale', 'log')
 
 set(gca, 'ColorScale', 'log')
 
-ylabel('Net Carrier Density (m^{-2})');
+% ylabel('Net Carrier Density (m^{-2})');
 % ylabel('Temperature (K)');
-% ylabel('Scatter Lifetime (s)');
+ylabel('Scatter Lifetime (s)');
 
-zlabel('Conductivity (S)');
+zlabel('Conductivity (mS)');
 if DISPLAY_HZ
     xlabel('Frequency (Hz)');
 else

2D-Conductivity/conductivity_calculations.m

@@ -15,8 +15,8 @@ MAX_F = 15;
 F_TOTAL = 1e2; % number of points to generate
 
 % EXCITATION_TYPE = 'intra';
-% EXCITATION_TYPE = 'inter';
+EXCITATION_TYPE = 'inter';
-EXCITATION_TYPE = 'all';
+% EXCITATION_TYPE = 'all';
 
 MULTIPLE_SERIES = true; % for comparing two dopants
 
@@ -31,8 +31,8 @@ x_vals = x_vals .* (2*pi); % rads-1
 cond = zeros(length(x_vals), 2);
 for x=1:length(x_vals)
     cond(x, :) = sheet_conductivity(x_vals(x),... % omega (rads-1)
-                                    fermi_from_carrier_density(1.3e13*100*100, ev_to_j(3)),... % fermi_level (J)
+                                    fermi_from_carrier_density(1.3e17, ev_to_j(3)),... % fermi_level (J)
-                                    300,... % temp (K)
+                                    10,... % temp (K)
                                     1e-12); % scatter_lifetime (s)
 end
 
@@ -40,18 +40,18 @@ if MULTIPLE_SERIES
     cond2 = zeros(length(x_vals), 2);
     for x=1:length(x_vals)
         cond2(x, :) = sheet_conductivity(x_vals(x),... % omega (rads-1)
-                                        fermi_from_carrier_density(2.2e13*100*100, ev_to_j(3)),... % fermi_level (J)
+                                        fermi_from_carrier_density(1.3e17, ev_to_j(3)),... % fermi_level (J)
                                         300,... % temp (K)
                                         1e-12); % scatter_lifetime (s)
     end
     
-%     cond3 = zeros(length(x_vals), 2);
+    cond3 = zeros(length(x_vals), 2);
-%     for x=1:length(x_vals)
+    for x=1:length(x_vals)
-%         cond3(x, :) = sheet_conductivity(x_vals(x),... % omega (rads-1)
+        cond3(x, :) = sheet_conductivity(x_vals(x),... % omega (rads-1)
-%                                         fermi_from_carrier_density(1.3e13*10000, ev_to_j(3)),... % fermi_level (J)
+                                        fermi_from_carrier_density(1.3e17, ev_to_j(3)),... % fermi_level (J)
-%                                         300,... % temp (K)
+                                        2230,... % temp (K)
-%                                         1e-12); % scatter_lifetime (s)
+                                        1e-12); % scatter_lifetime (s)
-%     end
+    end
 end
 
 if DISPLAY_HZ % divide radians back to hertz
@@ -65,9 +65,9 @@ end
 RE_COLOUR = 'r-';
 IM_COLOUR = 'r--';
 MAG_COLOUR = 'r:';
-RE_COLOUR2 = 'b-';
+RE_COLOUR2 = 'g-';
-IM_COLOUR2 = 'b--';
+IM_COLOUR2 = 'g--';
-MAG_COLOUR2 = 'b:';
+MAG_COLOUR2 = 'g:';
 RE_COLOUR3 = 'b';
 IM_COLOUR3 = 'b--';
 MAG_COLOUR3 = 'b:';
@@ -77,6 +77,11 @@ figure(1);
 hold on;
 % INTRA
 if strcmp(EXCITATION_TYPE, 'intra')
+    cond = cond * 1e3;
+    cond2 = cond2 * 1e3;
+    cond3 = cond3 * 1e3;
+    ylabel('Conductivity (mS)');
+    
     plot(x_vals, real(cond(:, 1)), RE_COLOUR, 'LineWidth', LW);
     plot(x_vals, imag(cond(:, 1)), IM_COLOUR, 'LineWidth', LW);
     plot(x_vals, abs(cond(:, 1)), MAG_COLOUR, 'LineWidth', LW);
@@ -86,14 +91,19 @@ if strcmp(EXCITATION_TYPE, 'intra')
         plot(x_vals, imag(cond2(:, 1)), IM_COLOUR2, 'LineWidth', LW);
         plot(x_vals, abs(cond2(:, 1)), MAG_COLOUR2, 'LineWidth', LW);
         
-%         plot(x_vals, real(cond3(:, 1)), RE_COLOUR3, 'LineWidth', LW);
+        plot(x_vals, real(cond3(:, 1)), RE_COLOUR3, 'LineWidth', LW);
-%         plot(x_vals, imag(cond3(:, 1)), IM_COLOUR3, 'LineWidth', LW);
+        plot(x_vals, imag(cond3(:, 1)), IM_COLOUR3, 'LineWidth', LW);
-%         plot(x_vals, abs(cond3(:, 1)), MAG_COLOUR3, 'LineWidth', LW);
+        plot(x_vals, abs(cond3(:, 1)), MAG_COLOUR3, 'LineWidth', LW);
     end
     title('2D Intraband Sheet Conductivity');
 
 % INTER
 elseif strcmp(EXCITATION_TYPE, 'inter')
+    cond = cond * 1e6;
+    cond2 = cond2 * 1e6;
+    cond3 = cond3 * 1e6;
+    ylabel('Conductivity (\muS)');
+    
     plot(x_vals, real(cond(:, 2)), RE_COLOUR, 'LineWidth', LW);
     plot(x_vals, imag(cond(:, 2)), IM_COLOUR, 'LineWidth', LW);
     plot(x_vals, abs(cond(:, 2)), MAG_COLOUR, 'LineWidth', LW);
@@ -103,14 +113,19 @@ elseif strcmp(EXCITATION_TYPE, 'inter')
         plot(x_vals, imag(cond2(:, 2)), IM_COLOUR2, 'LineWidth', LW);
         plot(x_vals, abs(cond2(:, 2)), MAG_COLOUR2, 'LineWidth', LW);
         
-%         plot(x_vals, real(cond3(:, 2)), RE_COLOUR3, 'LineWidth', LW);
+        plot(x_vals, real(cond3(:, 2)), RE_COLOUR3, 'LineWidth', LW);
-%         plot(x_vals, imag(cond3(:, 2)), IM_COLOUR3, 'LineWidth', LW);
+        plot(x_vals, imag(cond3(:, 2)), IM_COLOUR3, 'LineWidth', LW);
-%         plot(x_vals, abs(cond3(:, 2)), MAG_COLOUR3, 'LineWidth', LW);
+        plot(x_vals, abs(cond3(:, 2)), MAG_COLOUR3, 'LineWidth', LW);
     end
     title('2D Interband Sheet Conductivity');
     
 % COMPLEX
 else
+    cond = cond * 1e3;
+    cond2 = cond2 * 1e3;
+    cond3 = cond3 * 1e3;
+    ylabel('Conductivity (mS)');
+    
     plot(x_vals, real(sum(cond, 2)), RE_COLOUR, 'LineWidth', LW);
     plot(x_vals, imag(sum(cond, 2)), IM_COLOUR, 'LineWidth', LW);
     plot(x_vals, abs(sum(cond, 2)), MAG_COLOUR, 'LineWidth', LW);
@@ -120,9 +135,9 @@ else
         plot(x_vals, imag(sum(cond2, 2)), IM_COLOUR2, 'LineWidth', LW);
         plot(x_vals, abs(sum(cond2, 2)), MAG_COLOUR2, 'LineWidth', LW);
         
-%         plot(x_vals, real(sum(cond3, 2)), RE_COLOUR3, 'LineWidth', LW);
+        plot(x_vals, real(sum(cond3, 2)), RE_COLOUR3, 'LineWidth', LW);
-%         plot(x_vals, imag(sum(cond3, 2)), IM_COLOUR3, 'LineWidth', LW);
+        plot(x_vals, imag(sum(cond3, 2)), IM_COLOUR3, 'LineWidth', LW);
-%         plot(x_vals, abs(sum(cond3, 2)), MAG_COLOUR3, 'LineWidth', LW);
+        plot(x_vals, abs(sum(cond3, 2)), MAG_COLOUR3, 'LineWidth', LW);
     end
     title('2D Sheet Conductivity');
 end
@@ -130,15 +145,18 @@ end
 set(gca,'Xscale','log')
 % set(gca,'Yscale','log')
 axis tight
+% ylim([-inf 225])
 
 if MULTIPLE_SERIES
-    legend('Re(TTF)', 'Im(TTF)', '|TTF|', 'Re(CoCp_2)', 'Im(CoCp_2)', '|CoCp_2|');
+%     legend('TTF Re(\sigma)', 'TTF Im(\sigma)', 'TTF |\sigma|', 'CoCp_2 Re(\sigma)', 'CoCp_2 Im(\sigma)', 'CoCp_2 |\sigma|');
-%     legend('Re(1x10^{8}m^{-2})', 'Im(1x10^{8}m^{-2})', '|1x10^{8}m^{-2}|', 'Re(1x10^{15}m^{-2})', 'Im(1x10^{15}m^{-2})', '|1x10^{15}m^{-2}|', 'Re(1.3x10^{17}m^{-2})', 'Im(1.3x10^{17}m^{-2})', '|1.3x10^{17}m^{-2}|');
+    legend('1x10^{8}m^{-2} Re(\sigma)', '1x10^{8}m^{-2} Im(\sigma)', '1x10^{8}m^{-2} |\sigma|', '1x10^{15}m^{-2} Re(\sigma)', '1x10^{15}m^{-2} Im(\sigma)', '1x10^{15}m^{-2} |\sigma|', '1.3x10^{17}m^{-2} Re(\sigma)', '1.3x10^{17}m^{-2} Im(\sigma)', '1.3x10^{17}m^{-2} |\sigma|');
+%     legend('10K Re(\sigma)', '10K Im(\sigma)', '10K |\sigma|', '300K Re(\sigma)', '300K Im(\sigma)', '300K |\sigma|', '2230K Re(\sigma)', '2230K Im(\sigma)', '2230K |\sigma|');
+%     legend('5x10^{-12} s Re(\sigma)', '5x10^{-12} s Im(\sigma)', '5x10^{-12} s |\sigma|', '1x10^{-12} s Re(\sigma)', '1x10^{-12} s Im(\sigma)', '1x10^{-12} s |\sigma|', '1x10^{-13} s Re(\sigma)', '1x10^{-13} s Im(\sigma)', '1x10^{-13} s |\sigma|');
 else
     legend('Real', 'Imaginary');
 end
 grid;
-ylabel('Conductivity (S)');
+
 if DISPLAY_HZ
     xlabel('Frequency (Hz)');
 else

2D-Conductivity/sheet_conductivity.m

@@ -27,7 +27,7 @@ term_2_term_2 = (1/pi) * ...
                     / ...
                 (2*kb*temp));
 
-term_2_term_3 = (1i/2*pi) * ...
+term_2_term_3 = (1i/(2*pi)) * ...
             log((hbar*omega + 2*fermi_level)^2 ...
                     / ...
                 ((hbar*omega - 2*fermi_level)^2 + 4*((kb*temp)^2)));

README.md

@@ -1,3 +1,6 @@
 # Graphene Coursework
 
 Post-grad nanolectronics and devices coursework, terahertz graphene applications and 2D sheet conductivity modelling
+
+![Conductivity Magnitude](Resources/david-recreation-mag.png)
+![Interband Conductivity](Resources/david-recreation-inter-mag.png)

Report/references.bib

@@ -110,3 +110,50 @@
 	year = {2016}
 }
 
+@article{geim-04,
+	abstract = {We describe monocrystalline graphitic films, which are a few atoms thick but are nonetheless stable under ambient conditions, metallic, and of remarkably high quality. The films are found to be a two-dimensional semimetal with a tiny overlap between valence and conductance bands, and they exhibit a strong ambipolar electric field effect such that electrons and holes in concentrations up to 1013 per square centimeter and with room-temperature mobilities of \~{}10,000 square centimeters per volt-second can be induced by applying gate voltage.},
+	author = {Novoselov, K. S. and Geim, A. K. and Morozov, S. V. and Jiang, D. and Zhang, Y. and Dubonos, S. V. and Grigorieva, I. V. and Firsov, A. A.},
+	doi = {10.1126/science.1102896},
+	eprint = {https://science.sciencemag.org/content/306/5696/666.full.pdf},
+	issn = {0036-8075},
+	journal = {Science},
+	number = {5696},
+	pages = {666--669},
+	publisher = {American Association for the Advancement of Science},
+	title = {Electric Field Effect in Atomically Thin Carbon Films},
+	url = {https://science.sciencemag.org/content/306/5696/666},
+	urldate = {2021-04-26},
+	volume = {306},
+	year = {2004}
+}
+
+@article{gnrfet-structure-image,
+	abstract = {In this paper, we present a physics-based analytical model of GNR FET, which allows for the evaluation of GNR FET performance including the effects of line-edge roughness as its practical specific non-ideality. The line-edge roughness is modeled in edge-enhanced band-to-band-tunneling and localization regimes, and then verified for various roughness amplitudes. Corresponding to these two regimes, the off-current is initially increased, then decreased; while, on the other hand, the on-current is continuously decreased by increasing the roughness amplitude.},
+	article-number = {11},
+	author = {Banadaki, Yaser M. and Srivastava, Ashok},
+	doi = {10.3390/electronics5010011},
+	issn = {2079-9292},
+	journal = {Electronics},
+	number = {1},
+	title = {Effect of Edge Roughness on Static Characteristics of Graphene Nanoribbon Field Effect Transistor},
+	url = {https://www.mdpi.com/2079-9292/5/1/11},
+	urldate = {2021-04-26},
+	volume = {5},
+	year = {2016}
+}
+
+@article{gnrfet-applications,
+	abstract = {The dimension down scaling capability of the silicon based transistors has produced significant developments in the electronic industry. The channel length reduction has been accompanied by many limitations and challenges in the performance of the transistor. According to the ITRS and Moore law silicon based technology is near to its end, consequently the novel material innovations are needed in the near future. The graphene based material is the promising candidate for silicon channel replacement in conventional transistor. In this paper, graphene, graphene nanoribbons and their fundamental properties such as mechanical, electrical and electronic specifications are introduced. Then, graphene nanoribbon field effect transistor and its modeling and simulation methods are investigated. The best method for device simulation is the self-consistent solving of Poisson and Schr{\"o}dinger equations under non-equilibrium green's function with the tight binding approximation. In order to investigate the effect of down scaling on the transistor performance, parameters such as drain induced barrier lowering, sub-threshold swing, ION/IOFFratio, and transconductance are studied. Moreover, utilizations of the graphene nanoribbon field effect transistor including circuit-based, high frequency, and biosensors applications are introduced. The results show that graphene based transistors are an excellent replacement to silicon based transistors.},
+	author = {Radsar, Tahereh and Khalesi, Hassan and Ghods, Vahid},
+	doi = {10.1016/j.spmi.2021.106869},
+	issn = {0749-6036},
+	journal = {Superlattices and Microstructures},
+	keywords = {Graphene; Graphene nanoribbon field effect transistor; GNRFET; Graphene bio application},
+	pages = {106869},
+	title = {Graphene nanoribbon field effect transistors analysis and applications},
+	url = {https://www.sciencedirect.com/science/article/pii/S0749603621000677},
+	urldate = {2021-04-26},
+	volume = {153},
+	year = {2021}
+}
+