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\paperfontsize 11 \spacing onehalf \use_hyperref true \pdf_title "Graphene Investigations & Conductivity Modelling" \pdf_author "6420013" \pdf_subject "EEEM022 Nanoelectronics & Devices" \pdf_keywords "EEEM022" \pdf_bookmarks true \pdf_bookmarksnumbered false \pdf_bookmarksopen false \pdf_bookmarksopenlevel 1 \pdf_breaklinks false \pdf_pdfborder true \pdf_colorlinks false \pdf_backref false \pdf_pdfusetitle true \papersize default \use_geometry true \use_package amsmath 1 \use_package amssymb 1 \use_package cancel 1 \use_package esint 1 \use_package mathdots 1 \use_package mathtools 1 \use_package mhchem 1 \use_package stackrel 1 \use_package stmaryrd 1 \use_package undertilde 1 \cite_engine biblatex \cite_engine_type authoryear \biblio_style plain \biblio_options urldate=long \biblatex_bibstyle ieee \biblatex_citestyle ieee \use_bibtopic false \use_indices false \paperorientation portrait \suppress_date true \justification true \use_refstyle 1 \use_minted 0 \index Index \shortcut idx \color #008000 \end_index \leftmargin 1.8cm \topmargin 2cm \rightmargin 1.8cm \bottommargin 2cm \secnumdepth 3 \tocdepth 3 \paragraph_separation skip \defskip medskip \is_math_indent 0 \math_numbering_side default \quotes_style british \dynamic_quotes 0 \papercolumns 1 \papersides 1 \paperpagestyle fancy \listings_params "breaklines=true,frame=tb,language=Matlab,basicstyle={\ttfamily},commentstyle={\color{commentgreen}\itshape},keywordstyle={\color{blue}},emphstyle={\color{red}},stringstyle={\color{red}},identifierstyle={\color{cyan}},morekeywords={carrier_density_from_fermi, fermi_from_carrier_density, ev_to_j, j_to_ev, fermi_velocity, hz_to_omega, sheet_conductivity}" \bullet 1 0 9 -1 \bullet 2 0 24 -1 \tracking_changes false \output_changes false \html_math_output 0 \html_css_as_file 0 \html_be_strict false \end_header \begin_body \begin_layout Title \size giant Graphene Applications & Conductivity Modelling At High Frequencies \end_layout \begin_layout Author 6420013 \end_layout \begin_layout Standard \begin_inset VSpace 15pheight% \end_inset \end_layout \begin_layout Standard \align center \begin_inset Graphics filename surrey.png lyxscale 15 width 40col% \end_inset \end_layout \begin_layout Standard \begin_inset VSpace vfill \end_inset \end_layout \begin_layout Standard \noindent \align center EEEM022 \begin_inset Newline newline \end_inset November 2020 \size large \begin_inset Newline newline \end_inset Department of Electrical and Electronic Engineering \begin_inset Newline newline \end_inset Faculty of Engineering and Physical Sciences \begin_inset Newline newline \end_inset University of Surrey \end_layout \begin_layout Standard \begin_inset Newpage newpage \end_inset \end_layout \begin_layout Abstract abstract \end_layout \begin_layout Standard \begin_inset CommandInset toc LatexCommand tableofcontents \end_inset \end_layout \begin_layout Standard \begin_inset Newpage newpage \end_inset \end_layout \begin_layout Standard \begin_inset FloatList figure \end_inset \end_layout \begin_layout Standard \begin_inset FloatList table \end_inset \end_layout \begin_layout Standard \begin_inset CommandInset toc LatexCommand lstlistoflistings \end_inset \end_layout \begin_layout Standard \begin_inset Newpage newpage \end_inset \end_layout \begin_layout Right Footer EEEM022 Coursework \end_layout \begin_layout Left Footer April 2021 \end_layout \begin_layout Left Header 6420013 \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash pagenumbering{arabic} \end_layout \begin_layout Plain Layout \backslash setcounter{page}{1} \end_layout \end_inset \end_layout \begin_layout Section Introduction \end_layout \begin_layout Standard Graphene is a 2D allotrope of carbon with highly interesting mechanical and electrical properties that have made it a popular target of research over the last two decades since it's experimental discovery in 2004 \begin_inset CommandInset citation LatexCommand cite key "geim-04" literal "false" \end_inset . Resembling a honeycomb structure, each carbon atoms bonds to three surrounding atoms by overlapping \begin_inset Formula $p$ \end_inset orbitals. This leaves one remaining \begin_inset Formula $\pi$ \end_inset -electron whose orbital extends perpendicular to the 2D sheet. This physical and electronic structure can be seen in figure \begin_inset CommandInset ref LatexCommand ref reference "fig:orbital-sketch" plural "false" caps "false" noprefix "false" \end_inset . As a result, graphene has a 0 energy bandgap, the conduction and valence band meet at a single point known as the Dirac point. Both bands linearly extend away from the Dirac point, a highly unusual result that allows electrons to act as massless Fermions and move with extremely high mobility, as high as 200,000 cm \begin_inset script superscript \begin_layout Plain Layout 2 \end_layout \end_inset V \begin_inset script superscript \begin_layout Plain Layout -1 \end_layout \end_inset s \begin_inset script superscript \begin_layout Plain Layout -1 \end_layout \end_inset at room temperature \begin_inset CommandInset citation LatexCommand cite key "graphene-review-2010" literal "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 orbital-sketch.png width 40col% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout A section of graphene sheet with the orientation of electron orbitals highlighte d \begin_inset CommandInset citation LatexCommand cite key "warda-gfet-review" literal "false" \end_inset \begin_inset CommandInset label LatexCommand label name "fig:orbital-sketch" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \end_layout \end_inset \end_layout \begin_layout Standard With these highly sought after properties, it is unsurprising that graphene remains a prospective material for many domains including semiconductor and high frequency applications. \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 electrical and mechanical 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 Standard This section explores two uses of graphene for high frequency applications. First, the applicability of graphene for field effect transistors will be considered as a channel material. Throughout, a particular focus will be paid to use in digital logic and thus as a possible replacement for the current Silicon CMOS/MOSFET paradigm. \end_layout \begin_layout Standard \begin_inset Flex TODO Note (inline) status open \begin_layout Plain Layout second application \end_layout \end_inset \end_layout \begin_layout Subsection Digital Logic \end_layout \begin_layout Standard Silicon-based CMOS/MOSFET digital logic is the basis on which much of the modern electronics landscape has been built. From integrated logic circuits to CPUs, it is hard to overstate how important this technology has proven to be. The need for more powerful devices has increased pressure for smaller and more efficient transistors, such that more can fit into a single device. This progress is typically described by Moore's Law and can be seen graphically in figure \begin_inset CommandInset ref LatexCommand ref reference "fig:cpu-transistor-number" 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 moores-law-owid.png lyxscale 20 width 80col% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout The number of transistors in commercial CPUs between 1970 and 2020 \begin_inset CommandInset citation LatexCommand cite key "owidtechnologicalprogress" literal "false" \end_inset \begin_inset CommandInset label LatexCommand label name "fig:cpu-transistor-number" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \end_layout \end_inset \end_layout \begin_layout Standard However, as transistors are made smaller, theoretical limits for many engineerin g challenges are approached. In 2015, the ITRS predicted that by 2021 the current push for smaller transisto rs would no longer be economically viable, instead requiring innovative 3D device structures \begin_inset CommandInset citation LatexCommand cite key "transistors-21" literal "false" \end_inset . Some of the most important limiting factors in the current Silicon landscape are short-channel effects, a group of undesirable electrical properties that can occur when the channel length of a MOSFET device is of the same order of magnitude as the depletion layer \begin_inset Flex TODO Note (Margin) status open \begin_layout Plain Layout cite, reword? \end_layout \end_inset . \end_layout \begin_layout Standard As previously covered, a sheet of graphene or \emph on large-area graphene \emph default has no bandgap. As such it is unable to turn off, making it unsuitable as a channel material for a digital MOSFET. Quantitatively, this can be measured with by the ratio of on current to off current or the \emph on on/off ratio \emph default . Existing Silicon CMOS can expect an on/off ratio of at least the order of 10 \begin_inset script superscript \begin_layout Plain Layout 4 \end_layout \end_inset , whereas monolayer graphene can only achieve around 5 \begin_inset CommandInset citation LatexCommand cite key "warda-gfet-review" literal "false" \end_inset . \end_layout \begin_layout Standard Therefore, it is clear that in order to use graphene as a channel material a bandgap must be formed. There are a number of ways to do so, a selection of band diagrams for such structures can be seen in figure \begin_inset CommandInset ref LatexCommand ref reference "fig:dirac-cones" plural "false" caps "false" noprefix "false" \end_inset . For digital applications, one of the most promising methods for creating a band gap is by confining graphene in one dimension to create a \emph on graphene nanoribbon \emph default (GNR). In doing so it has been shown that the produced bandgap is inversely proportion al to the width of this ribbon \begin_inset CommandInset citation LatexCommand cite key "gnrfet-applications" literal "false" \end_inset . An example structure for a graphene nanoribbon FET (GNRFET) can be seen in figure \begin_inset CommandInset ref LatexCommand ref reference "fig:gnrget-structure" 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 cones.png width 40col% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Energy band diagram for 4 different structures of graphene, i) large-area, ii) nanoribbons, iii) unbiased bilayer, iv) bilayer with an applied perpendicul ar electric field \begin_inset CommandInset citation LatexCommand cite key "graphene-review-2010" literal "false" \end_inset \begin_inset CommandInset label LatexCommand label name "fig:dirac-cones" \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 gnrfet.png lyxscale 50 width 40col% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Graphene nanoribbon FET structure \begin_inset CommandInset citation LatexCommand cite key "gnrfet-structure-image" literal "false" \end_inset \begin_inset CommandInset label LatexCommand label name "fig:gnrget-structure" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Flex TODO Note (inline) status open \begin_layout Plain Layout Advantages over silicon \end_layout \end_inset \end_layout \begin_layout Standard Some of the limitations of GNR-based FETs include the difficulty in fabricating high-quality graphene and specifically graphene nanoribbons. On top of the typical issues in graphene fabrication including the requirements for tight environmental control. Edge roughness can significantly affect the channels properties, introducing new scattering centres \begin_inset CommandInset citation LatexCommand cite key "gnrfet-structure-image" literal "false" \end_inset . However, the major constraint of GNRFETs is the limited mobility. Although graphene has incredibly high carrier mobility, by opening the bandgap as seen in figure \begin_inset CommandInset ref LatexCommand ref reference "fig:dirac-cones" plural "false" caps "false" noprefix "false" \end_inset , curves are introduced to the band structure. This increases the carrier's effective mass, significantly reducing mobility. \end_layout \begin_layout Standard Despite these drawbacks, GNRFETs look to be one of the most promising avenues for the post-Silicon, high-performance CMOS/MOSFET electronics. \end_layout \begin_layout Subsection Terahertz Radiation \end_layout \begin_layout Subsection Summary \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 interband 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. Prior to these wider investigations, however, experimental data was recreated in order both to validate the model and to allow better identification of later differences from the planned variations. \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 n-type 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 magnitude (figure \begin_inset CommandInset ref LatexCommand ref reference "fig:david-magnitude" plural "false" caps "false" noprefix "false" \end_inset ) of the function 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 100 GHz. Beyond the cutoff frequency the value is around 1 mS by 10 THz. The imaginary component peaks over the same frequency band that the real component declines and the two intersect at around 150 GHz at a conductivity of 31 mS with CoCp \begin_inset script subscript \begin_layout Plain Layout 2 \end_layout \end_inset and 24 mS for TTF. After this intersection, the imaginary component can be seen to be the dominant term of the complex quantity, this can be seen in the graph as the magnitude tends closer to the imaginary component than the real \begin_inset Note Comment status open \begin_layout Plain Layout . Beyond 100 THz, the imaginary component dips below zero, with a trough of -0.5 mS around 250 THz \end_layout \end_inset . \end_layout \begin_layout Standard Looking to the phase information (figure \begin_inset CommandInset ref LatexCommand ref reference "fig:david-phase" plural "false" caps "false" noprefix "false" \end_inset ), before 10 GHz the phase can be seen to be approximately 0, however as the imaginary component begins to peak, the phase increases to a max of 90 degrees, continuing until 100 THz where the phase sharply drops to -90 degrees. There is little difference between the two dopants, particularly prior to 100 THz. Following this, the TTF shows a -100 THz offset from the CoCp \begin_inset script subscript \begin_layout Plain Layout 2 \end_layout \end_inset species. CoCp \begin_inset script subscript \begin_layout Plain Layout 2 \end_layout \end_inset also reaches -56 degrees as compared to TTF's -50 degrees at its minima. \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 open \begin_layout Plain Layout \noindent \align center \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \noindent \align center \begin_inset Graphics filename ../Resources/david-recreation-mag.png lyxscale 20 width 50col% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \begin_inset CommandInset label LatexCommand label name "fig:david-magnitude" \end_inset \end_layout \end_inset \end_layout \end_inset \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \noindent \align center \begin_inset Graphics filename ../Resources/david-recreation-phase.png lyxscale 20 width 50col% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \begin_inset CommandInset label LatexCommand label name "fig:david-phase" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Complex conductivity magnitude (a) and phase (b) 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 conductivity, 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. In general, the intraband conductivity can be seen to exist up to the 10 THz portion of the spectrum while the interband has the majority of it's contributions above the 10 THz range. The intraband can be seen to dominate the total conductivity, it is largely invisible in the previous combined view of figure \begin_inset CommandInset ref LatexCommand ref reference "fig:david-magnitude" plural "false" caps "false" noprefix "false" \end_inset . The interband interactions begin after the 10 THz range, as the real component steps from 1 \begin_inset Formula $\mu S$ \end_inset to 60 \begin_inset Formula $\mu S$ \end_inset around 200 THz, the imaginary component displays a negative peak over the same spectral band. \end_layout \begin_layout Standard \begin_inset Flex TODO Note (inline) status open \begin_layout Plain Layout Static value continues past THz \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 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 \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \begin_inset CommandInset label LatexCommand label name "fig:david-intraband" \end_inset \end_layout \end_inset \end_layout \end_inset \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \noindent \align center \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 \begin_inset CommandInset label LatexCommand label name "fig:david-interband" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Intraband (a) and interband (b) 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 conductivity (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 . From this point, net carrier concentration and dopant concentration may be used somewhat interchangeably with the understanding that they are related by this function. \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 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-sl1e-12-T300-logCB.png lyxscale 20 width 80col% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \begin_inset CommandInset label LatexCommand label name "fig:surf-carrier-conc-real" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \noindent \align center \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \noindent \align center \begin_inset Graphics filename ../Resources/carrier-density/im-com-carrier-surf-sl1e-12-T300-logCB.png lyxscale 20 width 80col% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \begin_inset CommandInset label LatexCommand label name "fig:surf-carrier-conc-im" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Complex conductivity over frequency for different carrier densities at room temperature with a scatter lifetime of 1 ps 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 Conductivity can broadly be seen to follow the same spectral profile as a function of carrier concentrations up to \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 $1\times10^{15}$ \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 \family default \series default \shape default \size default \emph default \bar default \strikeout default \xout default \uuline default \uwave default \noun default \color inherit as can be seen in figure \begin_inset CommandInset ref LatexCommand ref reference "fig:david-magnitude" plural "false" caps "false" noprefix "false" \end_inset . Beyond this threshold, the conductivity below 10 THz begins to increase exponentially. Above 100 THz, an opposite trend can be identified with areas of the surface showing a lower value. From the high frequency, high carrier smear visible in the surface, the spectral behaviour of this lowering conductivity is not static but a function of both frequency and carrier concentration. \begin_inset Note Comment status open \begin_layout Plain Layout Variation in the lower frequency comes largely 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 \end_inset \end_layout \begin_layout Standard For the real conductivity 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 the higher gigahertz value smearing the lighter blue across a higher frequency band. This moves the cutoff from 100 GHz to around 1 THz. 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 increases 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 negative values, 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 50 THz and \begin_inset Formula $6\times10^{15}$ \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 100 GHz intraband value increases in magnitude, the frequency for this high frequency imaginary conductivity 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 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 \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \begin_inset CommandInset label LatexCommand label name "fig:carrier-conc-intra" \end_inset \end_layout \end_inset \end_layout \end_inset \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \noindent \align center \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 \begin_inset CommandInset label LatexCommand label name "fig:carrier-conc-inter" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Intraband (a) and interband (b) conductivity for high and low carrier concentrat ion graphene species at room temperature and a scatter lifetime of 1 ps \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 conductivity for three graphene species of differing carrier concentrations decomposed into the intraband and interband components. The blue series, \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 carrier density of \begin_inset Formula $1.3\times10^{17}$ \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 \family default \series default \shape default \size default \emph default \bar default \strikeout default \xout default \uuline default \uwave default \noun default \color inherit , recreates TTF doping from figure \begin_inset CommandInset ref LatexCommand ref reference "fig:david-simulation-inter-intra" plural "false" caps "false" noprefix "false" \end_inset with two further theoretical species of lower dopant concentration. \end_layout \begin_layout Standard Looking to the intraband interactions, both components can be seen to have the same profiles as seen previously, the differences lie in magnitude. Higher net carrier concentrations can be seen to increase the magnitude exponentially, this behaviour was previously identified in 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 The interband conductivity can be seen to show more variation over the prescribe d carrier concentration range. Low carrier concentrations result in a high initial imaginary component that does not descend into negative values. As concentration increases, the imaginary component decreases more with a sharper gradient, forming a sharp trough that also reaches its lowest value 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 increases 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 200 THz. \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/complex-lines-phase.png lyxscale 20 width 60col% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Complex conductivity phase for three net carrier concentrations at room temperature and a scatter lifetime of 1 ps \begin_inset CommandInset label LatexCommand label name "fig:carrier-conc-phase" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard The complex phase information for these three dopant species can be seen in figure \begin_inset CommandInset ref LatexCommand ref reference "fig:carrier-conc-phase" plural "false" caps "false" noprefix "false" \end_inset . From these it is clear that the dopant concentration has a significant effect on the conductivity's phase in the terahertz spectrum. As the carrier concentration is increased, the phase can be seen to increase to 90 degrees for longer throughout the 10 GHz decade and begin dropping into negative phase. From figure \begin_inset CommandInset ref LatexCommand ref reference "fig:david-phase" plural "false" caps "false" noprefix "false" \end_inset , it can be seen that once the phase decreases below 0, the frequency of the phase minima is also a function of carrier concentration with higher carrier concentration species having a higher minima frequency. \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 conductivity. \begin_inset Note Comment status open \begin_layout Plain Layout Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:surf-temperature" plural "false" caps "false" noprefix "false" \end_inset shows a surface of the conductivity spectrum over the prescribed temperature range. In general, temperature can be seen to have little effect on conductivity, both real and imaginary. \end_layout \begin_layout Plain Layout From the real component, the pre-cutoff GHz peak can be seen to increase from 224 mS to 253 mS when moving from near-room temperature to the breakdown temperature of graphene. \end_layout \begin_layout Plain Layout Looking to the imaginary component, the peak conductivity increases by roughly 15 mS. More variation occurs at the higher frequency, THz conductivity. The sharper colour gradient at lower temperatures become more gradual at higher temperatures, this indicates that the intraband imaginary negative peak takes place over a more gradual spectral range. Looking to the low temperature behaviour, the imaginary component is less stable, rapidly descending to 0 S by 3 K before the the characteristic negative 0.6 mS 200 THz peak returns at 0 K. \end_layout \begin_layout Plain Layout \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 \end_inset \end_layout \begin_layout Standard Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:inter-intra-temperature" plural "false" caps "false" noprefix "false" \end_inset presents the decomposed intraband and interband conductivity contributions for three different temperatures, 10 K, 300 K and 2230 K in order to compare low, room and high temperatures. For intraband conductivity, temperature can be seen to have little effect throughout the inspected thermal range. The low and room temperature series' effectively overlap, while moving to the upper temperature limit increases the conductivity by only 5 mS or 10%. \end_layout \begin_layout Standard For the interband interactions, as the temperature increases, the negative imaginary peak gets smaller in value with a smoother gradient. For the real component, although the final value does not change, the gradient with which it is approached changes. At low temperatures, the increase takes place over a tight spectral range with a sharp step action. As the temperature increases, the spectral band over which the transition occurs broadens with a smoother gradient while maintaining the centre frequency of 200 THz. Overall, considering the complex magnitude for interband interactions, little variation can be seen throughout the prescribed temperature range. \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \noindent \align center \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 \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \begin_inset CommandInset label LatexCommand label name "fig:temp-intra" \end_inset \end_layout \end_inset \end_layout \end_inset \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \noindent \align center \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 \begin_inset CommandInset label LatexCommand label name "fig:temp-inter" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Intraband (a) and interband (b) conductivity for low, room and high temperature graphene using TTF doping and a a scatter lifetime of 1 ps \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 Standard The phase information for these three temperatures can be seen in figure \begin_inset CommandInset ref LatexCommand ref reference "fig:temperature-phase" plural "false" caps "false" noprefix "false" \end_inset . Similar to the magnitude, the phase can be seen to show little variations below 10 THz. Lower temperatures result in sharper drops around 100 THz with a larger negative peak, whereas the higher temperature species has a smoother motion that does not become negative. \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/complex-lines-phase.png lyxscale 20 width 60col% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Complex conductivity phase for three different temperatures using TTF doping at room temperature and a scatter lifetime of 1 ps \begin_inset CommandInset label LatexCommand label name "fig:temperature-phase" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \end_layout \end_inset \end_layout \begin_layout Subsubsection Scattering Lifetime \end_layout \begin_layout Standard This section explores the effect of varying scatter lifetime, \begin_inset Formula $\tau$ \end_inset , on the conductivity. For the range of values to use, existing data was considered. 1 ps is a typical figure in literature \begin_inset CommandInset citation LatexCommand cite key "david-paper" literal "false" \end_inset , with this in mind values between 10 ps and 0.1 ps were simulated. Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:surf-scatter-lifetime" plural "false" caps "false" noprefix "false" \end_inset explores the general trends throughout the prescribed range. \end_layout \begin_layout Standard Looking to the real component, the scatter lifetime can be seen to affect both the cutoff frequency and the magnitude of the pre-cutoff value. As the lifetime increases, the cutoff frequency occurs at a lower value, from around 500 GHz to 50 GHz. The magnitude of the conductivity also increases exponentially as the lifetime is increased. \end_layout \begin_layout Standard Considering the imaginary component, a somewhat similar behaviour can be seen. The same exponential growth in magnitude can be seen in the 100 GHz peak. With regards to the spectral behaviour, increasing scatter lifetime reduces the frequency of the leading edge of the peak, broadening the bandwidth. \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \noindent \align center \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 \begin_inset Caption Standard \begin_layout Plain Layout \begin_inset CommandInset label LatexCommand label name "fig:surf-scatter-intra" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \noindent \align center \begin_inset Float figure wide false sideways false status open \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 \begin_inset CommandInset label LatexCommand label name "fig:surf-scatter-inter" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Complex conductivity over frequency for different scattering lifetimes simulated for TTF n-type doping at room temperature with a scatter lifetime of 1 ps \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 Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:inter-intra-scatter-lifetime" plural "false" caps "false" noprefix "false" \end_inset presents the interband and intraband conductivity contributions for three different scattering lifetimes. The previously identified spectral changes and magnitude growth can be seen in the intraband conductivity. The bandwidth of the imaginary component for the higher lifetime species is broadened while increasing the magnitude. Looking to the interband contributions, the three series show no variation, the scatter lifetime has no effect. This can be seen in the surfaces of figure \begin_inset CommandInset ref LatexCommand ref reference "fig:surf-scatter-lifetime" plural "false" caps "false" noprefix "false" \end_inset as straight lines of sharp colour gradients through the scatter lifetime range at 200 THz. \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \noindent \align center \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 \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \begin_inset CommandInset label LatexCommand label name "fig:scatter-intraband" \end_inset \end_layout \end_inset \end_layout \end_inset \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \noindent \align center \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 \begin_inset CommandInset label LatexCommand label name "fig:scatter-inter" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Intraband (a) and interband (b) conductivity 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 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/complex-lines-phase.png lyxscale 20 width 60col% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Complex conductivity phase for three different temperatures using TTF doping \begin_inset CommandInset label LatexCommand label name "fig:scatter-phase" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:scatter-phase" plural "false" caps "false" noprefix "false" \end_inset presents the complex phase of the conductivity for the three previous lifetimes. Following the observation that the interband conductivity is unaffected by scatter lifetime, it follows that the phase is also unchanged across the selected lifetimes. the phase is affected in the lower gigahertz ranges, however, with a longer lifetime being associated with an earlier rising edge in the spectrum and a longer bandwidth of 90 degree phase. \end_layout \begin_layout Subsection Discussion \end_layout \begin_layout Standard \begin_inset Flex TODO Note (inline) status open \begin_layout Plain Layout Intraband and drude stuff, graphs look the same \end_layout \end_inset \end_layout \begin_layout Standard As discussed, the interband conductivity is restricted to the higher–energy terahertz portion of the spectrum than the lower energy intraband interactions. \end_layout \begin_layout Standard \begin_inset Note Comment status open \begin_layout Plain Layout The reason for this can be seen reflected in equations \begin_inset CommandInset ref LatexCommand ref reference "eq:intra-conductivity" plural "false" caps "false" noprefix "false" \end_inset and \begin_inset CommandInset ref LatexCommand ref reference "eq:inter-conductivity" plural "false" caps "false" noprefix "false" \end_inset . Unlike the intraband transitions where \begin_inset Formula $E_{F}$ \end_inset is singular, the interband transitions instead has only references to \begin_inset Formula $2E_{F}$ \end_inset . \end_layout \end_inset The reason for this can be seen by considering the required energy for each type of transition. \begin_inset Note Comment status open \begin_layout Plain Layout With a non-zero Fermi level offset from the Dirac point, resistance drops as carriers are available for conduction. \end_layout \end_inset When considering n-type doping, the Fermi level is increased from the Dirac point. As it does, the energy states between it and the Dirac point are filled and thus are unavailable for electrons to transition into. As interband conductivity involve transitions from the valence band or lower Dirac cone to the conduction band or upper cone (when considering n-type doping), in order for an electron to make a direct transition without momentum change it must absorb at least two times the Fermi level energy in order for the destination to be an empty state. This restriction, more formally that incident photons of angular frequency, \begin_inset Formula $\omega$ \end_inset , and thus energy \begin_inset Formula $\hbar\omega$ \end_inset will not be sufficient for interband transitions can be expressed as the following, \begin_inset Formula $\hbar\omega<2|E_{F}|$ \end_inset \begin_inset CommandInset citation LatexCommand cite key "david-paper" literal "false" \end_inset . This is referred to as Pauli blocking after the exclusion principle of the same name which defines the population limit of the energy states in question. \end_layout \begin_layout Standard From the simulated data from literature, the gigahertz cutoff for intraband conductivity can be seen to have an associated phase increase. This suggests an increase in dynamic or reactive electrical behaviour throughou t this spectral range which would have implications for applications employing graphene at these frequencies. This phase is reversed at higher frequencies before beginning to relax as the imaginary component approaches zero which will have similar implications for THz applications. \end_layout \begin_layout Paragraph Net Carrier Concentration \end_layout \begin_layout Standard The differences in conductivity between the two dopants from literature presented in figure \begin_inset CommandInset ref LatexCommand ref reference "fig:david-simulation-conductivity" plural "false" caps "false" noprefix "false" \end_inset are more broadly described by the effects of varying net carrier concentration (figures \begin_inset CommandInset ref LatexCommand ref reference "fig:surf-carrier-concentration" plural "false" caps "false" noprefix "false" \end_inset , \begin_inset CommandInset ref LatexCommand ref reference "fig:inter-intra-carrier-conc" plural "false" caps "false" noprefix "false" \end_inset ) as this is the primary difference between dopants. The non-linear relation between net carrier concentration and thus dopant concentration with conductivity results in the largely constant intraband conductivity up to 10 \begin_inset script superscript \begin_layout Plain Layout 14 \end_layout \end_inset m \begin_inset script superscript \begin_layout Plain Layout -2 \end_layout \end_inset carriers. \end_layout \begin_layout Standard \begin_inset Flex TODO Note (inline) status open \begin_layout Plain Layout Why? \end_layout \end_inset \end_layout \begin_layout Standard From the presented trends it is clear both that graphene must be heavily doped in order to obtain significant gigahertz conductivity and that, more generally, varying the dopant concentration provides a highly-tunable method of altering graphene's electrical characteristics. This has particular implications for terahertz applications when considering the phase information. As presented, the conductivity's complex phase can be shifted in frequency and magnitude in the terahertz spectrum by varying the net carrier concentratio n. The phase can also be kept positive at terahertz frequencies with a lower carrier concentration, although this severely limits the conductivity magnitude. This flexibility in reactive electrical characteristics could prove applicable to high frequency applications. \end_layout \begin_layout Paragraph Temperature \end_layout \begin_layout Standard The conductivity spectrum as a function of temperature shows promising results for the electrical stability of graphene over a wide thermal operating range. The intraband conductivity, specifically, showed little variation in behaviour between 10 K and the highest stable temperatures with the previously reported 10% increase. Looking to the interband interactions, these results showed the opposite trend with the magnitude being decreased as the temperature increased. This would suggest that graphene could prove useful in high temperature devices with special consideration being needed for terahertz applications where the phase can be more variable around the critical frequency. \end_layout \begin_layout Standard \begin_inset Flex TODO Note (inline) status open \begin_layout Plain Layout what does this mean \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 fermi-dirac.gif lyxscale 75 width 50col% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Fermi-Dirac distribution function for the occupancy probability of a fermion as a function of temperature \begin_inset CommandInset citation LatexCommand cite key "fermi-dirac-dist" literal "false" \end_inset \begin_inset CommandInset label LatexCommand label name "fig:example-fermi-dirac" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard The real intraband component also shows interesting behaviour, with the differences in gradient about the 200 THz critical frequency resembling the Fermi-Dirac distribution, see figure \begin_inset CommandInset ref LatexCommand ref reference "fig:example-fermi-dirac" plural "false" caps "false" noprefix "false" \end_inset . Similarly to this function, a decreasing temperature increases the gradient of this transition between two quasi-constant values, tending towards a single step action as the temperature approaches 0 K. \end_layout \begin_layout Standard Higher temperatures allow higher interband conductivity at lower frequencies than would otherwise be required to overcome the previously described \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 $\hbar\omega>2|E_{F}|$ \end_inset restriction. The higher energy of the electrons associated with their temperature reduces the extra required energy to make the transition. From figure \begin_inset CommandInset ref LatexCommand ref reference "fig:temp-inter" plural "false" caps "false" noprefix "false" \end_inset this can be seen in two aspects of the real conductivity. Throughout the gigahertz spectrum, the higher temperature blue line has a higher constant value than the others. This higher conductivity is associated with the slightly lower energy required to make an interband transition as a result of the electron's higher energies. Additionally, as the critical temperature is approached, the higher temperature series begins smoothly rising earlier than lower temperature series'. These behaviours are a result of the statistical distribution of both the temperature of the electrons and the energy of the incident photons. \end_layout \begin_layout Paragraph Scatter Lifetime \end_layout \begin_layout Standard From the presented trends for how conductivity is affected by a varied carrier scatter lifetime, it is clear that it has a significant effect on both the magnitude and spectral behaviour of intraband conductivity. A longer scatter lifetime was shown to increase the magnitude of gigahertz conductivity, this can be justified by considering the meaning of the scatter lifetime. \end_layout \begin_layout Standard \begin_inset Flex TODO Note (inline) status open \begin_layout Plain Layout Equation analysis \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Flex TODO Note (inline) status open \begin_layout Plain Layout Why? \end_layout \end_inset \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/conductivity_phase_calculations.m" lstparams "caption={Script for plotting complex conductivity phase over a range of frequencies},label={phase_script}" \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