adding graphs, beginning writing, added concentration/fermi level

2D-Conductivity/conductivity_calc_surface.m

@@ -8,20 +8,20 @@ close all; clear all; clc;
 %%    FLAGS & OPTIONS
 %%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-DISPLAY_HZ = true;
+DISPLAY_HZ = true; % convert rads back to Hz for presenting
 
-MIN_F = 0;
+MIN_F = 9;
 MAX_F = 15; % Hz
-F_TOTAL = 50;
+F_TOTAL = 50; % number of points to generate
 
-MAX_Y = 17; % carriers (m-2)
+MAX_Y = 18; % carriers (m-2)
-Y_TOTAL = 50;
+Y_TOTAL = 50; % number of points to generate
 
 % EXCITATION_TYPE = 'intra';
-EXCITATION_TYPE = 'inter';
+% EXCITATION_TYPE = 'inter';
-%EXCITATION_TYPE = 'all';
+EXCITATION_TYPE = 'all';
 
-t = 2.8; % eV
+t = 2.8; % eV, energy scale for Fermi velocity
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%     CALCULATE
@@ -30,30 +30,66 @@ t = 2.8; % eV
 f_vals = logspace(MIN_F, MAX_F, F_TOTAL); % hz
 f_vals = f_vals .* (2*pi); % rads-1
 
-carrier_vals = logspace(0, MAX_Y, Y_TOTAL); % m-2
+% Carrier Density
-%carrier_vals = carrier_vals + 273.15;
+%%%%%%%
-
+% carrier_vals = logspace(0, MAX_Y, Y_TOTAL); % m-2
-% below turns turns carrier densities into Fermi energies
+% 
-fermi_vals = zeros(1, length(carrier_vals));
+% % below turns turns carrier densities into Fermi energies
-for carr=1:length(carrier_vals)
+% fermi_vals = zeros(1, length(carrier_vals));
-    fermi_vals(carr) = fermi_from_carrier_density(carrier_vals(carr), ev_to_j(t));
+% for carr=1:length(carrier_vals)
-end
+%     fermi_vals(carr) = fermi_from_carrier_density(carrier_vals(carr), ev_to_j(t));
+% end
+% 
+% % CALCULATE SHEET CONDUCTIVITY
+% cond = zeros(length(f_vals),... % frequency
+%              length(fermi_vals),... % fermi
+%              2); % intra/inter
+% for freq=1:length(f_vals)
+%     for y=1:length(fermi_vals)
+%         
+%         cond(freq, y, :) = sheet_conductivity(f_vals(freq),... % omega (rads-1)
+%                                               fermi_vals(y),... % fermi_level (J)
+%                                               300,... % temp (K)
+%                                               5e-12); % scatter_lifetime (s)
+%     end
+% end
 
+% Temperature
+%%%%%%%
+temp_vals = linspace(0, 2230, Y_TOTAL); % K
 
 % CALCULATE SHEET CONDUCTIVITY
-cond = zeros(length(f_vals), % frequency
+cond = zeros(length(f_vals),... % frequency
-             length(fermi_vals), % fermi
+             length(temp_vals),... % fermi
              2); % intra/inter
 for freq=1:length(f_vals)
-    for y=1:length(fermi_vals)
+    for y=1:length(temp_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_from_carrier_density(1.3e13*10000, ev_to_j(t)),... % fermi_level (J)
-                                              300, % temp (K)
+                                              temp_vals(y),... % temp (K)
-                                              5e-12); % scatter_lifetime (s-1)
+                                              5e-12); % scatter_lifetime (s)
     end
 end
 
+% Scatter Lifetime
+%%%%%%%
+% scatt_vals = logspace(-11, -14, Y_TOTAL); % s-1
+% 
+% % CALCULATE SHEET CONDUCTIVITY
+% cond = zeros(length(f_vals),... % frequency
+%              length(scatt_vals),... % fermi
+%              2); % intra/inter
+% for freq=1:length(f_vals)
+%     for y=1:length(scatt_vals)
+%         
+%         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
+%                                               300,... % temp (K)
+%                                               scatt_vals(y)); % scatter_lifetime (s)
+%     end
+% end
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%       RENDER
 %%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -62,23 +98,37 @@ if DISPLAY_HZ % divide radians back to hertz
     f_vals = f_vals ./ (2*pi);
 end
 
+y_vals = temp_vals;
+
+% cond = sign(cond).*log10(abs(cond));
+
 figure(1)
-if EXCITATION_TYPE == 'intra'
+if strcmp(EXCITATION_TYPE, 'intra')
-    surf(f_vals, carrier_vals, transpose(real(cond(:, :, 1))));
+    surf(f_vals, y_vals, transpose(real(cond(:, :, 1))));
-elseif EXCITATION_TYPE == 'inter'
+    title('2D Real Intraband Sheet Conductivity');
-    surf(f_vals, carrier_vals, transpose(real(cond(:, :, 2))));
+elseif strcmp(EXCITATION_TYPE, 'inter')
+    surf(f_vals, y_vals, transpose(real(cond(:, :, 2))));
+    title('2D Real Interband Sheet Conductivity');
 else
-    surf(f_vals, carrier_vals, transpose(real(sum(cond, 3))));
+    surf(f_vals, y_vals, transpose(real(sum(cond, 3))));
+    title('2D Real Sheet Conductivity');
 end
-h = gca;
+
 rotate3d on
 grid;
 colorbar;
-set(h, 'xscale', 'log')
+axis tight;
-set(h, 'yscale', 'log')
+set(gca, 'xscale', 'log')
-title('2D Sheet Real Conductivity');
+% set(gca, 'yscale', 'log')
-ylabel('Net Carrier Density (m-2)');
+% set(gca, 'zscale', 'log')
-zlabel('Conductivity (S/m)');
+
+set(gca, 'ColorScale', 'log')
+
+% ylabel('Net Carrier Density (m^{-2})');
+ylabel('Temperature (K)');
+% ylabel('Scatter Lifetime (s)');
+
+zlabel('Conductivity (S)');
 if DISPLAY_HZ
     xlabel('Frequency (Hz)');
 else
@@ -86,23 +136,32 @@ else
 end
 
 figure(2)
-if EXCITATION_TYPE == 'intra'
+if strcmp(EXCITATION_TYPE, 'intra')
-    surf(f_vals, carrier_vals, transpose(imag(cond(:, :, 1))));
+    surf(f_vals, y_vals, transpose(imag(cond(:, :, 1))));
-elseif EXCITATION_TYPE == 'inter'
+    title('2D Imaginary Intraband Sheet Conductivity');
-    surf(f_vals, carrier_vals, transpose(imag(cond(:, :, 2))));
+elseif strcmp(EXCITATION_TYPE, 'inter')
+    surf(f_vals, y_vals, transpose(imag(cond(:, :, 2))));
+    title('2D Imaginary Interband Sheet Conductivity');
 else
-    surf(f_vals, carrier_vals, transpose(imag(sum(cond, 3))));
+    surf(f_vals, y_vals, transpose(imag(sum(cond, 3))));
+    title('2D Imaginary Sheet Conductivity');
 end
-surf(f_vals, carrier_vals, transpose(imag(sum(cond, 3))));
+
-h = gca;
 rotate3d on
 grid;
 colorbar;
-set(h, 'xscale', 'log')
+axis tight;
-set(h, 'yscale', 'log')
+set(gca, 'xscale', 'log')
-title('2D Sheet Imaginary Conductivity');
+% set(gca, 'yscale', 'log')
-ylabel('Net Carrier Density (m-2)');
+% set(gca, 'zscale', 'log')
-zlabel('Conductivity (S/m)');
+
+% set(gca, 'ColorScale', 'log')
+
+% ylabel('Net Carrier Density (m^{-2})');
+ylabel('Temperature (K)');
+% ylabel('Scatter Lifetime (s)');
+
+zlabel('Conductivity (S)');
 if DISPLAY_HZ
     xlabel('Frequency (Hz)');
 else

2D-Conductivity/conductivity_calculations.m

@@ -8,14 +8,17 @@ close all; clear all; clc;
 %%    FLAGS & OPTIONS
 %%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-DISPLAY_HZ = true;
+DISPLAY_HZ = true; % convert rads back to Hz for presenting
+
 MIN_F = 9;
 MAX_F = 15;
-F_TOTAL = 1e2;
+F_TOTAL = 1e2; % number of points to generate
 
 % EXCITATION_TYPE = 'intra';
-EXCITATION_TYPE = 'inter';
+% EXCITATION_TYPE = 'inter';
-%EXCITATION_TYPE = 'all';
+EXCITATION_TYPE = 'all';
+
+TWO_SERIES = true; % for comparing two dopants
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%     CALCULATE
@@ -27,10 +30,20 @@ x_vals = x_vals .* (2*pi); % rads-1
 % CALCULATE SHEET CONDUCTIVITY
 cond = zeros(length(x_vals), 2);
 for x=1:length(x_vals)
-    cond(x, :) = sheet_conductivity(x_vals(x), % omega (rads-1)
+    cond(x, :) = sheet_conductivity(x_vals(x),... % omega (rads-1)
-                                    fermi_from_carrier_density(2.2e17, ev_to_j(2.8)), % fermi_level (J)
+                                    fermi_from_carrier_density(1.3e13*10000, ev_to_j(3)),... % fermi_level (J)
-                                    300, % temp (K)
+                                    300,... % temp (K)
-                                    0.135e-12); % scatter_lifetime (s-1)
+                                    1e-12); % scatter_lifetime (s)
+end
+
+if TWO_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*10000, ev_to_j(3)),... % fermi_level (J)
+                                        300,... % temp (K)
+                                        1e-12); % scatter_lifetime (s)
+    end
 end
 
 if DISPLAY_HZ % divide radians back to hertz
@@ -41,24 +54,59 @@ end
 %%       RENDER
 %%%%%%%%%%%%%%%%%%%%%%%%%%%
 
+RE_COLOUR = 'r';
+IM_COLOUR = 'r--';
+RE_COLOUR2 = 'b';
+IM_COLOUR2 = 'b--';
+LW = 1.5;
+
 figure(1);
 hold on;
-%plot(x_vals, real(cond));
+% INTRA
-if EXCITATION_TYPE == 'intra'
+if strcmp(EXCITATION_TYPE, 'intra')
-    semilogx(x_vals, real(cond(:, 1)));
+    plot(x_vals, real(cond(:, 1)), RE_COLOUR, 'LineWidth', LW);
-    semilogx(x_vals, imag(cond(:, 1)));
+    plot(x_vals, imag(cond(:, 1)), IM_COLOUR, 'LineWidth', LW);
-elseif EXCITATION_TYPE == 'inter'
+    
-    semilogx(x_vals, real(cond(:, 2)));
+    if TWO_SERIES
-    semilogx(x_vals, imag(cond(:, 2)));
+        plot(x_vals, real(cond2(:, 1)), RE_COLOUR2, 'LineWidth', LW);
+        plot(x_vals, imag(cond2(:, 1)), IM_COLOUR2, 'LineWidth', LW);
+    end
+    title('2D Intraband Sheet Conductivity');
+
+% INTER
+elseif strcmp(EXCITATION_TYPE, 'inter')
+    plot(x_vals, real(cond(:, 2)), RE_COLOUR, 'LineWidth', LW);
+    plot(x_vals, imag(cond(:, 2)), IM_COLOUR, 'LineWidth', LW);
+    
+    if TWO_SERIES
+        plot(x_vals, real(cond2(:, 2)), RE_COLOUR2, 'LineWidth', LW);
+        plot(x_vals, imag(cond2(:, 2)), IM_COLOUR2, 'LineWidth', LW);
+    end
+    title('2D Interband Sheet Conductivity');
+    
+% COMPLEX
 else
-    semilogx(x_vals, real(sum(cond, 2)));
+    plot(x_vals, real(sum(cond, 2)), RE_COLOUR, 'LineWidth', LW);
-    semilogx(x_vals, imag(sum(cond, 2)));
+    plot(x_vals, imag(sum(cond, 2)), IM_COLOUR, 'LineWidth', LW);
+    
+    if TWO_SERIES
+        plot(x_vals, real(sum(cond2, 2)), RE_COLOUR2, 'LineWidth', LW);
+        plot(x_vals, imag(sum(cond2, 2)), IM_COLOUR2, 'LineWidth', LW);
+    end
+    title('2D Sheet Conductivity');
 end
 
+set(gca,'Xscale','log')
+% set(gca,'Yscale','log')
+axis tight
+
+if TWO_SERIES
+    legend('Real TTF', 'Imaginary TTF', 'Real CoCp2', 'Imaginary CoCp2');
+else
     legend('Real', 'Imaginary');
+end
 grid;
-title('2D Sheet Conductivity');
+ylabel('Conductivity (S)');
-ylabel('Conductivity (S/m)');
 if DISPLAY_HZ
     xlabel('Frequency (Hz)');
 else

2D-Conductivity/fermi_conc.m

@@ -0,0 +1,52 @@
+%% fermi_conc.m
+%%
+%% present fermi levels for different carrier concentrations
+
+close all; clear all; clc;
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%    FLAGS & OPTIONS
+%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+MIN_CONC = 0;
+MAX_CONC = 20;
+X_TOTAL = 1e2; % number of points to generate
+
+DISP_EV = true;
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%     CALCULATE
+%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+x_vals = logspace(MIN_CONC, MAX_CONC, X_TOTAL); % hz
+
+% CALCULATE SHEET CONDUCTIVITY
+energy = zeros(1, length(x_vals));
+for x=1:length(x_vals)
+    temp = fermi_from_carrier_density(x_vals(x), ev_to_j(2.8)); % scatter_lifetime (s)
+    if DISP_EV
+        temp = j_to_ev(temp);
+    end
+    energy(1, x) = temp;
+end
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%       RENDER
+%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+% INTRA
+plot(x_vals, energy, 'LineWidth', 1.5);
+
+title('Fermi level for differing carrier concentrations');
+
+set(gca,'Xscale','log')
+set(gca,'Yscale','log')
+% axis tight
+
+grid;
+xlabel('Carrier Concentration (m^{-2})');
+if DISP_EV
+    ylabel('Fermi Energy (eV)');
+else
+    ylabel('Fermi Energy (J)');
+end

2D-Conductivity/j_to_ev.m

@@ -1,4 +1,5 @@
 function eV = j_to_ev(j)
+%J_TO_EV Transform Joules into electron-volts
 eV = j / 1.602e-19;
 end
 

Report/references.bib

@@ -0,0 +1,58 @@
+@article{yao,
+	author = {Yao, Yu and Kats, Mikhail A. and Genevet, Patrice and Yu, Nanfang and Song, Yi and Kong, Jing and Capasso, Federico},
+	doi = {10.1021/nl3047943},
+	issn = {1530-6984},
+	journal = {Nano Letters},
+	note = {doi: 10.1021/nl3047943},
+	number = {3},
+	pages = {1257--1264},
+	publisher = {American Chemical Society},
+	risfield_0_da = {2013/03/13},
+	risfield_1_t2 = {Nano Letters},
+	title = {Broad Electrical Tuning of Graphene-Loaded Plasmonic Antennas},
+	url = {https://pubs.acs.org/doi/10.1021/nl3047943},
+	urldate = {2021-04-19},
+	volume = {13},
+	year = {2013}
+}
+
+@article{david-paper,
+	author = {Samuels, Alexander J. and Carey, J. David},
+	doi = {10.1021/acsami.5b05140},
+	issn = {1944-8244},
+	journal = {ACS Applied Materials \& Interfaces},
+	note = {doi: 10.1021/acsami.5b05140},
+	number = {40},
+	pages = {22246--22255},
+	publisher = {American Chemical Society},
+	risfield_0_da = {2015/10/14},
+	risfield_1_t2 = {ACS Applied Materials \& Interfaces},
+	title = {Engineering Graphene Conductivity for Flexible and High-Frequency Applications},
+	url = {https://pubs.acs.org/doi/pdf/10.1021/acsami.5b05140},
+	urldate = {2021-04-20},
+	volume = {7},
+	year = {2015}
+}
+
+@article{graphene-high-temp,
+	abstract = {Heat has always been a killing matter for traditional semiconductor machines. The underlining physical reason is that the intrinsic carrier density of a device made from a traditional semiconductor material increases very fast with a rising temperature. Once reaching a temperature, the density surpasses the chemical doping or gating effect, any p-n junction or transistor made from the semiconductor will fail to function. Here, we measure the intrinsic Fermi level (|EF| = 2.93 kBT) or intrinsic carrier density (nin = 3.87 {\texttimes} 10(6) cm(-2)K(-2){\textcdot} T(2)), carrier drift velocity, and G mode phonon energy of graphene devices and their temperature dependencies up to 2400 K. Our results show intrinsic carrier density of graphene is an order of magnitude less sensitive to temperature than those of Si or Ge, and reveal the great potentials of graphene as a material for high temperature devices. We also observe a linear decline of saturation drift velocity with increasing temperature, and identify the temperature coefficients of the intrinsic G mode phonon energy. Above knowledge is vital in understanding the physical phenomena of graphene under high power or high temperature.},
+	author = {Yin, Yan and Cheng, Zengguang and Wang, Li and Jin, Kuijuan and Wang, Wenzhong},
+	doi = {10.1038/srep05758},
+	issn = {2045-2322},
+	journal = {Scientific reports},
+	month = jul,
+	pages = {5758--5758},
+	publisher = {Nature Publishing Group},
+	risfield_0_db = {PubMed},
+	risfield_1_la = {eng},
+	risfield_2_an = {25044003},
+	risfield_3_u1 = {25044003[pmid]},
+	risfield_4_u2 = {PMC4104577[pmcid]},
+	risfield_5_u4 = {srep05758[PII]},
+	title = {Graphene, a material for high temperature devices--intrinsic carrier density, carrier drift velocity, and lattice energy},
+	url = {https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4104577},
+	urldate = {2021-04-20},
+	volume = {4},
+	year = {2014}
+}
+

Report/report.lyx

@@ -21,7 +21,7 @@ minimalistic
 todonotes
 \end_modules
 \maintain_unincluded_children false
-\language english
+\language british
 \language_package default
 \inputencoding auto
 \fontencoding global
@@ -97,7 +97,7 @@ todonotes
 \defskip medskip
 \is_math_indent 0
 \math_numbering_side default
-\quotes_style english
+\quotes_style british
 \dynamic_quotes 0
 \papercolumns 1
 \papersides 1
@@ -117,7 +117,7 @@ todonotes
 \begin_layout Title
 
 \size giant
-Graphene Investigations & Conductivity Modelling
+Graphene Applications & Conductivity Modelling At High Frequencies
 \end_layout
 
 \begin_layout Author
@@ -272,18 +272,778 @@ setcounter{page}{1}
 Introduction
 \end_layout
 
+\begin_layout Standard
+Graphene is a 2D allotrope of carbon with 
+\end_layout
+
+\begin_layout Standard
+This work explores the suitability of graphene for high frequency applications.
+ Section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:Applications"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+ presents two applications of graphene that take advantage of it's behaviour
+ at high frequencies.
+ Section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:Sheet-Conductivity-Modelling"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+ presents an investigation into the 2D sheet conductivity of the material.
+\end_layout
+
 \begin_layout Section
-Graphene Applications
+Applications
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:Applications"
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Section
 Sheet Conductivity Modelling
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:Sheet-Conductivity-Modelling"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+This section presents a model for graphene's high frequency conductivity
+ using the equation below below 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "yao"
+literal "false"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{multline}
+\sigma_{s}\left(\omega\right)=\frac{2ie^{2}k_{B}T}{\pi\hbar^{2}\left(\omega+\nicefrac{i}{\tau}\right)}\ln\left(2\cosh\left(\frac{E_{F}}{2k_{B}T}\right)\right)\\
++\frac{e^{2}}{4\hbar}\left(\frac{1}{2}+\frac{1}{\pi}\tan^{-1}\left(\frac{\hbar\omega-2E_{F}}{2k_{B}T}\right)-\frac{i}{2\pi}\ln\left(\frac{\left(\hbar\omega+2E_{F}\right)^{2}}{\left(\hbar\omega-2E_{F}\right)^{2}+4\left(k_{B}T\right)^{2}}\right)\right)\label{eq:2d-conductivity}
+\end{multline}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Taking this equation, the first term accounts for the intraband transitions
+ while the latter term refers to the interband transitions
+\begin_inset CommandInset citation
+LatexCommand cite
+key "david-paper"
+literal "false"
+
+\end_inset
+
+
+\begin_inset Flex TODO Note (Margin)
+status open
+
+\begin_layout Plain Layout
+cite
+\end_layout
+
+\end_inset
+
+.
+ These two contributions are separated for reference below,
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+\sigma_{s}^{intra}\left(\omega\right)=\frac{2ie^{2}k_{B}T}{\pi\hbar^{2}\left(\omega+\nicefrac{i}{\tau}\right)}\ln\left(2\cosh\left(\frac{E_{F}}{2k_{B}T}\right)\right)\label{eq:intra-conductivity}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+\sigma_{s}^{inter}\left(\omega\right)=\frac{e^{2}}{4\hbar}\left(\frac{1}{2}+\frac{1}{\pi}\tan^{-1}\left(\frac{\hbar\omega-2E_{F}}{2k_{B}T}\right)-\frac{i}{2\pi}\ln\left(\frac{\left(\hbar\omega+2E_{F}\right)^{2}}{\left(\hbar\omega-2E_{F}\right)^{2}+4\left(k_{B}T\right)^{2}}\right)\right)\label{eq:inter-conductivity}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:2d-conductivity"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+ was implemented in MatLab, see listing 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "calculation_function"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+, such that the inter and intraband contributions were returned separately.
+ This allowed for displaying both aspects independently or together by summing.
+ From the function it can be seen that the variables are AC frequency, 
+\begin_inset Formula $\omega$
+\end_inset
+
+, the Fermi energy level, 
+\begin_inset Formula $E_{F}$
+\end_inset
+
+, the temperature, 
+\begin_inset Formula $T$
+\end_inset
+
+, and the scatter lifetime, 
+\begin_inset Formula $\tau$
+\end_inset
+
+.
+ These were varied within reasonable ranges in order to investigate how
+ such variations affect the conductivity, both as a whole and individually.
+ 
 \end_layout
 
 \begin_layout Subsection
 Results
 \end_layout
 
+\begin_layout Standard
+To validate the model, values for TTF and CoCp
+\begin_inset script subscript
+
+\begin_layout Plain Layout
+2
+\end_layout
+
+\end_inset
+
+ doping taken from 
+\begin_inset CommandInset citation
+LatexCommand citet
+key "david-paper"
+literal "false"
+
+\end_inset
+
+ (see table 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:david-values"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+) were simulated and can be seen presented in figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:david-simulation-conductivity"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+.
+ Similarly to the original, the real component can be seen to begin between
+ 40 and 70 mS before declining to around 0.5 
+\begin_inset Formula $\mu S$
+\end_inset
+
+.
+ This decline occurs between 20 GHz and 2 THz.
+ The imaginary component peaks over the same frequency band that the real
+ component declines and the two intersect at around 150 GHz with a conductance
+ of 31 mS with CoCp
+\begin_inset script subscript
+
+\begin_layout Plain Layout
+2
+\end_layout
+
+\end_inset
+
+ and 24 mS for TTF.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float table
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Tabular
+<lyxtabular version="3" rows="3" columns="3">
+<features tabularvalignment="middle">
+<column alignment="center" valignment="top">
+<column alignment="center" valignment="top">
+<column alignment="center" valignment="top">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Dopant
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Carrier Concentration (cm
+\begin_inset script superscript
+
+\begin_layout Plain Layout
+-2
+\end_layout
+
+\end_inset
+
+)
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Fermi Level (eV)
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+TTF
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+1.3 x 10
+\begin_inset script superscript
+
+\begin_layout Plain Layout
+13
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+0.41
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+CoCp
+\begin_inset script subscript
+
+\begin_layout Plain Layout
+2
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+2.2 x 10
+\begin_inset script superscript
+
+\begin_layout Plain Layout
+13
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+0.53
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset VSpace defskip
+\end_inset
+
+With Fermi velocity energy scale, 
+\begin_inset Formula $t$
+\end_inset
+
+ = 3 eV 
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Carrier concentration values for dopants from 
+\begin_inset CommandInset citation
+LatexCommand citet
+key "david-paper"
+literal "false"
+
+\end_inset
+
+ and the Fermi levels derived from the model, see figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:fermi-concentration-func"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:david-values"
+
+\end_inset
+
+ 
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Graphics
+	filename ../Resources/david-recreation.png
+	lyxscale 20
+	width 60col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Complex conductivity for TTF and CoCp
+\begin_inset script subscript
+
+\begin_layout Plain Layout
+2
+\end_layout
+
+\end_inset
+
+ doping at 300 K with a scatter lifetime of 1 ps 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "david-paper"
+literal "false"
+
+\end_inset
+
+
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:david-simulation-conductivity"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+The Fermi level used to calculate conductance (listing 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "calculation_function"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+) was derived from the net carrier concentration as a result of doping,
+ see listing 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fermi_from_carrier_density"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+.
+ The non-linear function can be seen modelled in figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:fermi-concentration-func"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Graphics
+	filename ../Resources/fermi-conc.png
+	lyxscale 20
+	width 60col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Fermi level associated with different carrier concentrations
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:fermi-concentration-func"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Carrier Density
+\end_layout
+
+\begin_layout Standard
+The general trends for how the dopant-influenced net carrier concentration
+ influences conductivity can be seen in the surfaces of figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:surf-carrier-concentration"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Graphics
+	filename ../Resources/carrier-density/real-com-carrier-surf-sl5e-12-T300-logCB.png
+	lyxscale 20
+	width 80col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Graphics
+	filename ../Resources/carrier-density/im-com-carrier-surf-sl5e-12-T300-logCB.png
+	lyxscale 20
+	width 80col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Complex conductivity over frequency for different carrier densities
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:surf-carrier-concentration"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Temperature
+\end_layout
+
+\begin_layout Standard
+Values from 0 to the breakdown temperature of graphene, 2230 K
+\begin_inset CommandInset citation
+LatexCommand cite
+key "graphene-high-temp"
+literal "false"
+
+\end_inset
+
+, were varied in order to investigate the effect on conductance.
+ 
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Graphics
+	filename ../Resources/temperature/real-com-temp-surf-sl5e-12-TTF.png
+	lyxscale 20
+	width 80col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Graphics
+	filename ../Resources/temperature/im-com-temp-surf-sl5e-12-TTF.png
+	lyxscale 20
+	width 80col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Complex conductivity over frequency for different temperatures
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Scattering Lifetime
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Graphics
+	filename ../Resources/scatter-lifetime/real-com-SL-surf-300K-TTF10,14.png
+	lyxscale 20
+	width 80col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Graphics
+	filename ../Resources/scatter-lifetime/im-com-SL-surf-300K-TTF10,14.png
+	lyxscale 20
+	width 80col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Complex conductivity over frequency for different scattering lifetimes
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Discussion
+\end_layout
+
 \begin_layout Section
 Conclusion
 \end_layout

Resources/README.md

@@ -0,0 +1,7 @@
+Surfaces:
+
+35 x 25 x 300 dpi
+
+Lines:
+
+20 x 15 x 300 dpi