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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 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 \begin_inset Text \begin_layout Plain Layout Dopant \end_layout \end_inset \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 \begin_inset Text \begin_layout Plain Layout Fermi Level (eV) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout TTF \end_layout \end_inset \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 \begin_inset Text \begin_layout Plain Layout 0.41 \end_layout \end_inset \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 \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 \begin_inset Text \begin_layout Plain Layout 0.53 \end_layout \end_inset \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 \begin_layout Standard \begin_inset Newpage newpage \end_inset \end_layout \begin_layout Standard \begin_inset CommandInset label LatexCommand label name "sec:bibliography" \end_inset \begin_inset CommandInset bibtex LatexCommand bibtex btprint "btPrintCited" bibfiles "references" options "bibtotoc" \end_inset \begin_inset Newpage pagebreak \end_inset \end_layout \begin_layout Section \start_of_appendix Source Code \begin_inset CommandInset label LatexCommand label name "sec:Code" \end_inset \end_layout \begin_layout Standard \begin_inset CommandInset include LatexCommand lstinputlisting filename "../2D-Conductivity/sheet_conductivity.m" lstparams "caption={Calculation function for 2D sheet conductivity},label={calculation_function}" \end_inset \end_layout \begin_layout Standard \begin_inset Newpage pagebreak \end_inset \end_layout \begin_layout Standard \begin_inset CommandInset include LatexCommand lstinputlisting filename "../2D-Conductivity/conductivity_calculations.m" lstparams "caption={Script for calculating conductivity over a range of frequencies},label={sheet_calculation_script}" \end_inset \end_layout \begin_layout Standard \begin_inset CommandInset include LatexCommand lstinputlisting filename "../2D-Conductivity/conductivity_calc_surface.m" lstparams "caption={Script for calculating conductivity over a range of frequencies and presenting as a surface},label={sheet_calculation_script_surface}" \end_inset \end_layout \begin_layout Standard \begin_inset Newpage pagebreak \end_inset \end_layout \begin_layout Standard \begin_inset CommandInset include LatexCommand lstinputlisting filename "../2D-Conductivity/carrier_density_from_fermi.m" lstparams "caption={Derive the carrier density for a given Fermi energy},label={carrier_density_from_fermi}" \end_inset \end_layout \begin_layout Standard \begin_inset CommandInset include LatexCommand lstinputlisting filename "../2D-Conductivity/fermi_from_carrier_density.m" lstparams "caption={Derive the Fermi energy for a given carrier density},label={fermi_from_carrier_density}" \end_inset \end_layout \begin_layout Standard \begin_inset Newpage pagebreak \end_inset \end_layout \begin_layout Standard \begin_inset CommandInset include LatexCommand lstinputlisting filename "../2D-Conductivity/fermi_velocity.m" lstparams "caption={Derive the Fermi velocity for a given energy scale},label={fermi_velocity}" \end_inset \end_layout \begin_layout Standard \begin_inset CommandInset include LatexCommand lstinputlisting filename "../2D-Conductivity/ev_to_j.m" lstparams "caption={Convert electron-volts to joules},label={ev_to_j}" \end_inset \end_layout \begin_layout Standard \begin_inset CommandInset include LatexCommand lstinputlisting filename "../2D-Conductivity/j_to_ev.m" lstparams "caption={Convert joules to electron-volts},label={j_to_ev}" \end_inset \end_layout \end_body \end_document