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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 be between 48 and 63 mS for TTF and CoCp \begin_inset script subscript \begin_layout Plain Layout 2 \end_layout \end_inset respectively with both having a cutoff frequency of around 20 GHz. Beyond the cutoff frequency the value is around 60 \begin_inset Formula $\mu S$ \end_inset by 5 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. Beyond 100 THz, the imaginary component dips below zero, with a trough of -0.5 mS around 250 THz. \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 \begin_inset Formula $1.3\times10^{13}$ \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 \begin_inset Formula $2.2\times10^{13}$ \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 collapsed \begin_layout Plain Layout \noindent \align center \begin_inset Graphics filename ../Resources/david-recreation-mag.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 two contributions to this complex conductance, intraband and interband, can be seen individually in figure \begin_inset CommandInset ref LatexCommand ref reference "fig:david-simulation-inter-intra" plural "false" caps "false" noprefix "false" \end_inset . Comparing the two, it can be seen that the interactions happen over largely separate frequency ranges. The interband interactions begin after the 10 THz range, initially the imaginary component sharply drops and relaxes with a minima at 187 THz and 248 THz for TTF and CoCp \begin_inset script subscript \begin_layout Plain Layout 2 \end_layout \end_inset . As the imaginary component minimises, the real component begins sharply rising over a 100 THz range to a maximum of 60 \begin_inset Formula $\mu S$ \end_inset . This continues throughout the hundreds of terahertz range and beyond the region of interest. \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-intra-mag.png lyxscale 20 width 50col% \end_inset \begin_inset Graphics filename ../Resources/david-recreation-inter-mag.png lyxscale 20 width 50col% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Intraband and interband 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-inter-intra" \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 . To select a suitable range to visualise, the values from table \begin_inset CommandInset ref LatexCommand ref reference "tab:david-values" plural "false" caps "false" noprefix "false" \end_inset and figure \begin_inset CommandInset ref LatexCommand ref reference "fig:fermi-concentration-func" plural "false" caps "false" noprefix "false" \end_inset were considered. Realistic dopant carrier concentrations can be seen to of the order of \begin_inset Formula $1\times10^{13}$ \end_inset \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none cm \begin_inset script superscript \begin_layout Plain Layout \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none -2 \end_layout \end_inset or \family default \series default \shape default \size default \emph default \bar default \strikeout default \xout default \uuline default \uwave default \noun default \color inherit \begin_inset Formula $1\times10^{17}$ \end_inset m \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none \begin_inset script superscript \begin_layout Plain Layout \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none -2 \end_layout \end_inset . For the simulation, values up to \family default \series default \shape default \size default \emph default \bar default \strikeout default \xout default \uuline default \uwave default \noun default \color inherit \begin_inset Formula $1\times10^{18}$ \end_inset m \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none \begin_inset script superscript \begin_layout Plain Layout \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none -2 \end_layout \end_inset for a Fermi level of 1.13 eV were chosen. \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. Room temperature with a scatter lifetime of \begin_inset Formula $5\times10^{-12}$ \end_inset s and a Fermi velocity energy scale of 2.8 eV \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 Standard The conductance can broadly be seen to follow the same spectral profile over the range of carrier concentrations as can be seen in figure \begin_inset CommandInset ref LatexCommand ref reference "fig:david-simulation-conductivity" plural "false" caps "false" noprefix "false" \end_inset . Variation comes in the magnitude of the various regions. For both the real and imaginary component, the max value (pre-cutoff for the real component and the peak of the imaginary component) can be seen to be constant over net carrier concentrations up until around \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none 10 \begin_inset script superscript \begin_layout Plain Layout \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none 15 \end_layout \end_inset \family default \series default \shape default \size default \emph default \bar default \strikeout default \xout default \uuline default \uwave default \noun default \color inherit m \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none \begin_inset script superscript \begin_layout Plain Layout \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none -2 \end_layout \end_inset , 21 mS for the real component and 11 mS for the imaginary. Beyond \family default \series default \shape default \size default \emph default \bar default \strikeout default \xout default \uuline default \uwave default \noun default \color inherit \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none a net carrier concentration of 10 \begin_inset script superscript \begin_layout Plain Layout \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none 15 \end_layout \end_inset \family default \series default \shape default \size default \emph default \bar default \strikeout default \xout default \uuline default \uwave default \noun default \color inherit m \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none \begin_inset script superscript \begin_layout Plain Layout \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none -2 \end_layout \end_inset the maximum values begin to rapidly increase and by 10 \begin_inset script superscript \begin_layout Plain Layout \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none 17 \end_layout \end_inset \family default \series default \shape default \size default \emph default \bar default \strikeout default \xout default \uuline default \uwave default \noun default \color inherit m \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none \begin_inset script superscript \begin_layout Plain Layout \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none -2 \end_layout \end_inset they have increased by an order of magnitude to hundreds of milli-siemens. \end_layout \begin_layout Standard For the real conductance component, beyond this previously mentioned \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none 10 \begin_inset script superscript \begin_layout Plain Layout \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none 15 \end_layout \end_inset \family default \series default \shape default \size default \emph default \bar default \strikeout default \xout default \uuline default \uwave default \noun default \color inherit m \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none \begin_inset script superscript \begin_layout Plain Layout \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none -2 \end_layout \end_inset threshold, the cutoff frequency begins to increase as can be seen from higher peak smearing the lighter blue across a higher frequency band. This moves the cutoff from around 120 GHz to about 180 GHz. The value that the real conductance takes above the cutoff frequency decreases past the 10 \begin_inset script superscript \begin_layout Plain Layout \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none 15 \end_layout \end_inset \family default \series default \shape default \size default \emph default \bar default \strikeout default \xout default \uuline default \uwave default \noun default \color inherit m \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none \begin_inset script superscript \begin_layout Plain Layout \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none -2 \end_layout \end_inset carrier concentration threshold, from 58 \family default \series default \shape default \size default \emph default \bar default \strikeout default \xout default \uuline default \uwave default \noun default \color inherit \begin_inset Formula $\mu S$ \end_inset to 2 \begin_inset Formula $\mu S$ \end_inset at \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none \begin_inset Formula $1\times10^{17}$ \end_inset \family default \series default \shape default \size default \emph default \bar default \strikeout default \xout default \uuline default \uwave default \noun default \color inherit m \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none \begin_inset script superscript \begin_layout Plain Layout \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none -2 \end_layout \end_inset . For the imaginary component, at low carrier concentrations the peak value decreases to around 1 \family default \series default \shape default \size default \emph default \bar default \strikeout default \xout default \uuline default \uwave default \noun default \color inherit \begin_inset Formula $\mu S$ \end_inset by 500 THz. As the carrier concentration decreases beyond \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none \begin_inset Formula $1\times10^{12}$ \end_inset \family default \series default \shape default \size default \emph default \bar default \strikeout default \xout default \uuline default \uwave default \noun default \color inherit m \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none \begin_inset script superscript \begin_layout Plain Layout \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none -2 \end_layout \end_inset this value decreases into the small negative values that can be seen in figure \begin_inset CommandInset ref LatexCommand ref reference "fig:david-simulation-conductivity" plural "false" caps "false" noprefix "false" \end_inset , the frequency at which the drop occurs lowers and the steeper colour gradient indicates that the change happens faster. The earliest frequency that this occurs at is around 10 THz and \begin_inset Formula $1\times10^{15}$ \end_inset \family default \series default \shape default \size default \emph default \bar default \strikeout default \xout default \uuline default \uwave default \noun default \color inherit m \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none \begin_inset script superscript \begin_layout Plain Layout \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none -2 \end_layout \end_inset . Finally, as the carrier concentration further increases and the 120 GHz peak increases in magnitude, the frequency for this high frequency conductance drop begins to increase again. \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/intraband-lines-mag.png lyxscale 20 width 50col% \end_inset \begin_inset Graphics filename ../Resources/carrier-density/interband-lines-mag.png lyxscale 20 width 50col% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Inter- and intraband conductance for high and low carrier concentration graphene species \begin_inset CommandInset label LatexCommand label name "fig:inter-intra-carrier-conc" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \end_layout \end_inset \end_layout \begin_layout Standard Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:inter-intra-carrier-conc" plural "false" caps "false" noprefix "false" \end_inset presents the conductance for three graphene species of differing carrier concentrations decomposed into the intraband and interband components. From comparing the relative magnitudes from the two, it is clear that the majority contribution for conductance throughout the selected frequency range is from the intraband transitions. This is also apparent from the similarity in spectral profile between the intraband conductivity and both the surfaces of figure \begin_inset CommandInset ref LatexCommand ref reference "fig:surf-carrier-concentration" plural "false" caps "false" noprefix "false" \end_inset and the reproduced results of figure \begin_inset CommandInset ref LatexCommand ref reference "fig:david-simulation-conductivity" plural "false" caps "false" noprefix "false" \end_inset . In general, the intraband conductivity can be seen to exist up to the THz portion of the spectrum while the interband has the majority of it's contributi ons above the THz range. The interband conductance can be seen to be responsible for the previously noted negative imaginary conductance behaviour seen in the surface of figure \begin_inset CommandInset ref LatexCommand ref reference "fig:surf-carrier-concentration" plural "false" caps "false" noprefix "false" \end_inset . Low carrier concentration result in a higher initial imaginary component that does not lower into negative values. As concentration increases, the imaginary component decreases more forming a sharp trough that also bottoms out at a higher frequency. \end_layout \begin_layout Standard Alongside this imaginary decrease, the real component can be seen to increase from a value between 1 \begin_inset Formula $\mu S$ \end_inset and 30 \begin_inset Formula $\mu S$ \end_inset depending on carrier concentration to the limit of 60 \begin_inset Formula $\mu S$ \end_inset . Although the differing species reach this same limit, their approach is different. The lower carrier concentration species begins at the higher 30 \begin_inset Formula $\mu S$ \end_inset value and decreases only slightly to the limit over a wider spectral range. The higher carrier concentration species begins much lower at 1 \begin_inset Formula $\mu S$ \end_inset before increasing to 60 \begin_inset Formula $\mu S$ \end_inset in what is closer to a step action at the higher frequency of 110 THz. \end_layout \begin_layout Subsubsection Temperature \end_layout \begin_layout Standard Values from 0 K to the breakdown temperature of graphene, 2230 K \begin_inset CommandInset citation LatexCommand cite key "graphene-high-temp" literal "false" \end_inset , were simulated in order to investigate the effect on conductance. Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:surf-temperature" plural "false" caps "false" noprefix "false" \end_inset shows a surface of the conductance spectrum over the prescribed temperature range. In general, temperature can be seen to have little effect on conductance, both real and imaginary. \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 \begin_inset CommandInset label LatexCommand label name "fig:surf-temperature" \end_inset \end_layout \end_inset \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/temperature/intraband-lines-mag.png lyxscale 20 width 50col% \end_inset \begin_inset Graphics filename ../Resources/temperature/interband-lines-mag.png lyxscale 20 width 50col% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Inter- and intraband conductance for low, room and high temperature graphene using TTF doping \begin_inset CommandInset label LatexCommand label name "fig:inter-intra-temperature" \end_inset \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 \begin_inset CommandInset label LatexCommand label name "fig:surf-scatter-lifetime" \end_inset \end_layout \end_inset \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/scatter-lifetime/intraband-lines-mag.png lyxscale 20 width 50col% \end_inset \begin_inset Graphics filename ../Resources/scatter-lifetime/interband-lines-mag.png lyxscale 20 width 50col% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Inter- and intraband conductance with 3 different scattering times for graphene using TTF doping \begin_inset CommandInset label LatexCommand label name "fig:inter-intra-scatter-lifetime" \end_inset \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 CommandInset include LatexCommand lstinputlisting filename "../2D-Conductivity/fermi_conc.m" lstparams "caption={Script for plotting net carrier concentrations against Fermi level},label={fermi_concentration_script}" \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