2021-02-23 10:48:39 +00:00
#LyX 2.3 created this file. For more info see http://www.lyx.org/
\lyxformat 544
\begin_document
\begin_header
\save_transient_properties true
\origin unavailable
\textclass article
\begin_preamble
\def\changemargin#1#2{\list{}{\rightmargin#2\leftmargin#1}\item[]}
\let\endchangemargin=\endlist
\pagenumbering{roman}
\usepackage{color}
\definecolor{commentgreen}{RGB}{0,94,11}
\end_preamble
\use_default_options true
\begin_modules
customHeadersFooters
minimalistic
todonotes
\end_modules
\maintain_unincluded_children false
2021-04-20 21:19:15 +01:00
\language british
2021-02-23 10:48:39 +00:00
\language_package default
\inputencoding auto
\fontencoding global
\font_roman "default" "default"
\font_sans "default" "default"
\font_typewriter "default" "default"
\font_math "auto" "auto"
\font_default_family default
\use_non_tex_fonts false
\font_sc false
\font_osf false
\font_sf_scale 100 100
\font_tt_scale 100 100
\use_microtype true
\use_dash_ligatures true
\graphics default
\default_output_format default
\output_sync 0
\bibtex_command biber
\index_command default
\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
2021-04-20 21:19:15 +01:00
\quotes_style british
2021-02-23 10:48:39 +00:00
\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={audioread, aryule, xcorr, freqz, spectrogram, mfcc, spectro, fft_, autocorr, clip_segment, islocalmax, ms_to_samples, rceps, cceps, ones, audioplayer, play, get_impulse_train, lpc, strcat, num2str, xlim}"
\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
2021-04-20 21:19:15 +01:00
Graphene Applications & Conductivity Modelling At High Frequencies
2021-02-23 10:48:39 +00:00
\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
2021-04-18 12:36:37 +01:00
EEEM022 Coursework
2021-02-23 10:48:39 +00:00
\end_layout
\begin_layout Left Footer
April 2021
\end_layout
\begin_layout Left Header
2021-04-18 12:36:37 +01:00
6420013
2021-02-23 10:48:39 +00:00
\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
2021-04-20 21:19:15 +01:00
\begin_layout Standard
Graphene is a 2D allotrope of carbon with
\end_layout
\begin_layout Standard
This work explores the suitability of graphene for high frequency applications.
Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Applications"
plural "false"
caps "false"
noprefix "false"
\end_inset
presents two applications of graphene that take advantage of it's behaviour
at high frequencies.
Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Sheet-Conductivity-Modelling"
plural "false"
caps "false"
noprefix "false"
\end_inset
presents an investigation into the 2D sheet conductivity of the material.
\end_layout
2021-02-23 10:48:39 +00:00
\begin_layout Section
2021-04-20 21:19:15 +01:00
Applications
\begin_inset CommandInset label
LatexCommand label
name "sec:Applications"
\end_inset
2021-02-23 10:48:39 +00:00
\end_layout
\begin_layout Section
Sheet Conductivity Modelling
2021-04-20 21:19:15 +01:00
\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.
2021-02-23 10:48:39 +00:00
\end_layout
\begin_layout Subsection
Results
\end_layout
2021-04-20 21:19:15 +01:00
\begin_layout Standard
To validate the model, values for TTF and CoCp
\begin_inset script subscript
\begin_layout Plain Layout
2
\end_layout
\end_inset
doping taken from
\begin_inset CommandInset citation
LatexCommand citet
key "david-paper"
literal "false"
\end_inset
(see table
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:david-values"
plural "false"
caps "false"
noprefix "false"
\end_inset
) were simulated and can be seen presented in figure
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:david-simulation-conductivity"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
Similarly to the original, the real component can be seen to begin between
40 and 70 mS before declining to around 0.5
\begin_inset Formula $\mu S$
\end_inset
.
This decline occurs between 20 GHz and 2 THz.
The imaginary component peaks over the same frequency band that the real
component declines and the two intersect at around 150 GHz with a conductance
of 31 mS with CoCp
\begin_inset script subscript
\begin_layout Plain Layout
2
\end_layout
\end_inset
and 24 mS for TTF.
\end_layout
\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status open
\begin_layout Plain Layout
\noindent
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="3" columns="3">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Dopant
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Carrier Concentration (cm
\begin_inset script superscript
\begin_layout Plain Layout
-2
\end_layout
\end_inset
)
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Fermi Level (eV)
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
TTF
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1.3 x 10
\begin_inset script superscript
\begin_layout Plain Layout
13
\end_layout
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.41
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
CoCp
\begin_inset script subscript
\begin_layout Plain Layout
2
\end_layout
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
2.2 x 10
\begin_inset script superscript
\begin_layout Plain Layout
13
\end_layout
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
0.53
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset VSpace defskip
\end_inset
With Fermi velocity energy scale,
\begin_inset Formula $t$
\end_inset
= 3 eV
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Carrier concentration values for dopants from
\begin_inset CommandInset citation
LatexCommand citet
key "david-paper"
literal "false"
\end_inset
and the Fermi levels derived from the model, see figure
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:fermi-concentration-func"
plural "false"
caps "false"
noprefix "false"
\end_inset
\begin_inset CommandInset label
LatexCommand label
name "tab:david-values"
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open
\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
filename ../Resources/david-recreation.png
lyxscale 20
width 60col%
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Complex conductivity for TTF and CoCp
\begin_inset script subscript
\begin_layout Plain Layout
2
\end_layout
\end_inset
doping at 300 K with a scatter lifetime of 1 ps
\begin_inset CommandInset citation
LatexCommand cite
key "david-paper"
literal "false"
\end_inset
\begin_inset CommandInset label
LatexCommand label
name "fig:david-simulation-conductivity"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
The Fermi level used to calculate conductance (listing
\begin_inset CommandInset ref
LatexCommand ref
reference "calculation_function"
plural "false"
caps "false"
noprefix "false"
\end_inset
) was derived from the net carrier concentration as a result of doping,
see listing
\begin_inset CommandInset ref
LatexCommand ref
reference "fermi_from_carrier_density"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
The non-linear function can be seen modelled in figure
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:fermi-concentration-func"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open
\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
filename ../Resources/fermi-conc.png
lyxscale 20
width 60col%
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Fermi level associated with different carrier concentrations
\begin_inset CommandInset label
LatexCommand label
name "fig:fermi-concentration-func"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Subsubsection
Carrier Density
\end_layout
\begin_layout Standard
The general trends for how the dopant-influenced net carrier concentration
influences conductivity can be seen in the surfaces of figure
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:surf-carrier-concentration"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open
\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
filename ../Resources/carrier-density/real-com-carrier-surf-sl5e-12-T300-logCB.png
lyxscale 20
width 80col%
\end_inset
\end_layout
\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
filename ../Resources/carrier-density/im-com-carrier-surf-sl5e-12-T300-logCB.png
lyxscale 20
width 80col%
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Complex conductivity over frequency for different carrier densities
\begin_inset CommandInset label
LatexCommand label
name "fig:surf-carrier-concentration"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Subsubsection
Temperature
\end_layout
\begin_layout Standard
Values from 0 to the breakdown temperature of graphene, 2230 K
\begin_inset CommandInset citation
LatexCommand cite
key "graphene-high-temp"
literal "false"
\end_inset
, were varied in order to investigate the effect on conductance.
\end_layout
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open
\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
filename ../Resources/temperature/real-com-temp-surf-sl5e-12-TTF.png
lyxscale 20
width 80col%
\end_inset
\end_layout
\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
filename ../Resources/temperature/im-com-temp-surf-sl5e-12-TTF.png
lyxscale 20
width 80col%
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Complex conductivity over frequency for different temperatures
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Subsubsection
Scattering Lifetime
\end_layout
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open
\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
filename ../Resources/scatter-lifetime/real-com-SL-surf-300K-TTF10,14.png
lyxscale 20
width 80col%
\end_inset
\end_layout
\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
filename ../Resources/scatter-lifetime/im-com-SL-surf-300K-TTF10,14.png
lyxscale 20
width 80col%
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Complex conductivity over frequency for different scattering lifetimes
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Subsection
Discussion
\end_layout
2021-02-23 10:48:39 +00:00
\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
2021-04-18 12:36:37 +01:00
\end_layout
\begin_layout Standard
\begin_inset Newpage pagebreak
\end_inset
2021-02-23 10:48:39 +00:00
\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
2021-04-18 12:36:37 +01:00
\end_layout
\begin_layout Standard
\begin_inset CommandInset include
LatexCommand lstinputlisting
filename "../2D-Conductivity/conductivity_calc_surface.m"
lstparams "caption={Script for calculating conductivity over a range of frequencies and presenting as a surface},label={sheet_calculation_script_surface}"
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Newpage pagebreak
\end_inset
\end_layout
\begin_layout Standard
\begin_inset CommandInset include
LatexCommand lstinputlisting
filename "../2D-Conductivity/carrier_density_from_fermi.m"
lstparams "caption={Derive the carrier density for a given Fermi energy},label={carrier_density_from_fermi}"
\end_inset
\end_layout
\begin_layout Standard
\begin_inset CommandInset include
LatexCommand lstinputlisting
filename "../2D-Conductivity/fermi_from_carrier_density.m"
lstparams "caption={Derive the Fermi energy for a given carrier density},label={fermi_from_carrier_density}"
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Newpage pagebreak
\end_inset
\end_layout
\begin_layout Standard
\begin_inset CommandInset include
LatexCommand lstinputlisting
filename "../2D-Conductivity/fermi_velocity.m"
lstparams "caption={Derive the Fermi velocity for a given energy scale},label={fermi_velocity}"
\end_inset
\end_layout
\begin_layout Standard
\begin_inset CommandInset include
LatexCommand lstinputlisting
filename "../2D-Conductivity/ev_to_j.m"
lstparams "caption={Convert electron-volts to joules},label={ev_to_j}"
\end_inset
\end_layout
\begin_layout Standard
\begin_inset CommandInset include
LatexCommand lstinputlisting
filename "../2D-Conductivity/j_to_ev.m"
lstparams "caption={Convert joules to electron-volts},label={j_to_ev}"
\end_inset
2021-02-23 10:48:39 +00:00
\end_layout
\end_body
\end_document
