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\pdf_title "Graphene Investigations & Conductivity Modelling"
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\pdf_subject "EEEM022 Nanoelectronics & Devices"
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\begin_body
\begin_layout Title
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Graphene Applications & Conductivity Modelling At High Frequencies
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6420013
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EEEM022
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November 2020
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Department of Electrical and Electronic Engineering
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Faculty of Engineering and Physical Sciences
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University of Surrey
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\begin_layout Abstract
Graphene has many desirable mechanical and electrical characteristics that
make it an exciting material for use in innovative devices.
In particular, its extremely high carrier mobility leads to high operating
frequencies that potentially make the material suitable for the gigahertz
and terahertz spectrum.
The use of graphene for high-frequency electronics applications is considered,
firstly as a channel material for digital logic transistor technology with
a focus on graphene nanoribbon FETs (GNRFET).
Graphene's applicability is then evaluated for flexible antennae with prospecti
ve use in wearable and IoT devices.
Part two models graphene's sheet conductivity, exploring how net carrier
concentration, temperature and scatter lifetime affect both intraband and
interband electrical characteristics.
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EEEM022 Coursework
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April 2021
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6420013
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\begin_layout Section
Introduction
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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 its experimental discovery in 2004
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Resembling a honeycomb structure, each carbon atoms bonds to three surrounding
atoms by overlapping
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-orbitals.
This leaves one remaining
\begin_inset Formula $\pi$
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-electron whose orbital extends perpendicular to the 2D sheet.
This physical and electronic structure can be seen in figure
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.
As a result, graphene has a 0 energy bandgap, the conduction and valence
band meet at a single point known as the Dirac point at the K points in
momentum space.
Both bands linearly extend away from the Dirac point, a highly unusual
result that allows electrons to act as massless Fermions which move with
extremely high mobility, as high as 200,000 cm
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A section of graphene sheet with the orientation of electron orbitals highlighte
d
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With these highly sought after properties, it is unsurprising that graphene
remains a prospective material for many domains including semiconductor
and high frequency applications.
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This work explores the suitability of graphene for high-frequency applications.
Section
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presents two applications of graphene that take advantage of its electrical
and mechanical behaviour at high frequencies.
Section
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presents an investigation into the 2D sheet conductivity of the material.
The net carrier concentration, temperature and scatter lifetime are varied
in order to visualise how this affects the materials electrical properties.
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Applications
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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.
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Following its prospective use in digital logic, graphene's applicability
as a flexible antennae material will be considered.
As the space of wearable and bio-electronics develop, the need for strong,
flexible and efficient antennae at gigahertz frequencies is clear.
Existing flexible antennae technology will be considered before presenting
how graphene could be included.
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\begin_layout Subsection
Digital Logic
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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
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The number of transistors in commercial CPUs between 1970 and 2020
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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, innovative 3D device
structures would be required
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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
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As previously covered, a sheet of graphene or
\emph on
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\emph default
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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 the ratio of on current to off
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\emph on
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Existing Silicon CMOS can expect an on/off ratio of at least the order
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, whereas monolayer graphene can only achieve around 5
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Therefore, it is clear that to use graphene as a channel material a bandgap
must be formed.
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structures can be seen in figure
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High-quality GNRFETs can be seen to demonstrate the lowest sub-threshold
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FET operating characteristics for two silicon-based (16 nm) and two GNR-based
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Some of the limitations of GNR-based FETs include the difficulty in fabricating
high-quality graphene and specifically graphene nanoribbons.
Edge roughness can significantly affect the properties of the channel,
introducing new scattering centres
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, this can be seen in table
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with rough-edge graphene demonstrating a higher subthreshold swing than
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, further emphasising the requirement for high-quality fabrication methods.
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bandgap as seen in figure
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, curves are introduced to the band structure.
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Despite these drawbacks, GNRFETs look to be one of the most promising avenues
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Flexible Antennae
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The miniaturisation and increased efficiency of electronics have pushed
for smaller-scale devices including IoT and wearable devices.
One critical component for these devices is efficient wireless technologies
and antennae in particular.
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Graphene has a number of properties that make it a suitable prospective
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, graphene should be able to withstand much of the repeated physical deformation
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The previously mentioned electrical properties are obviously also applicable
to the domain - high mobility, conductivity and operating frequencies suggest
that the material could be applicable to gigahertz applications and thus
Wi-Fi and Bluetooth usage.
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Existing technologies for flexible antennae include printed metal-ink components
including copper and silver
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that is too expensive to be used extensively in antennae
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Copper also has disadvantages in its tendency to oxidise when exposed to
the environment as could be expected in wearable or IoT applications.
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Printed graphene antennae using water-transfer technology have been demonstrated
that function at the 2.4 GHz WiFi/Bluetooth spectrum
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This room-temperature method demonstrates that as the performance is comparable
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applicable material for wearable antennae.
\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 antennae-power.jpg
width 60col%
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Normalised radiation power for graphene and copper-based dipole antenna,
a) E-plane, b) H-plane
\begin_inset CommandInset citation
LatexCommand cite
key "water-transfer-graphene-antennae"
literal "false"
\end_inset
\begin_inset CommandInset label
LatexCommand label
name "fig:antenna-power"
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
\end_layout
\end_inset
\end_layout
\begin_layout Subsection
Summary
\end_layout
\begin_layout Standard
From these applications, it is clear that graphene has the potential to
revolutionise cutting-edge electronics domains including high-performance,
high-efficiency transistor technologies and the wearable/IoT devices.
Although the material does not yet have significant penetration in consumer
electronics, it is likely that this is not far away.
\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
\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
.
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 simulated
in order both to validate the model.
\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 at 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
<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
\begin_inset Formula $1.3\times10^{13}$
\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
\begin_inset Formula $2.2\times10^{13}$
\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 n-type 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
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\begin_layout Plain Layout
\noindent
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filename ../Resources/david-recreation-phase.png
lyxscale 20
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\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 its
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 Float figure
wide false
sideways false
status open
\begin_layout Plain Layout
\noindent
\align center
\begin_inset Float figure
wide false
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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 net 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
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cm
\begin_inset script superscript
\begin_layout Plain Layout
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-2
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or
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\begin_inset Formula $1\times10^{17}$
\end_inset
m
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-2
\end_layout
\end_inset
.
For the simulation, values up to
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\begin_inset Formula $1\times10^{18}$
\end_inset
m
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-2
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for a Fermi level of 1.13 eV were chosen.
\end_layout
\begin_layout Standard
\begin_inset Float figure
wide false
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status open
\begin_layout Plain Layout
\noindent
\align center
\begin_inset Float figure
wide false
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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
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\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
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wide false
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\begin_layout Plain Layout
\noindent
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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
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\begin_inset Formula $1\times10^{15}$
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-2
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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
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10
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15
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m
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-2
\end_layout
\end_inset
, 21 mS for the real component and 11 mS for the imaginary.
Beyond
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a net carrier concentration of 10
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15
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-2
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the maximum values begin to rapidly increase and by 10
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17
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-2
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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
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10
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15
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-2
\end_layout
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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
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\begin_inset Formula $\mu S$
\end_inset
by 500 THz.
As the carrier concentration increases beyond
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\begin_inset Formula $1\times10^{12}$
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\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
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\end_layout
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.
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
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lyxscale 20
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\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
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\end_layout
\end_inset
\end_layout
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\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
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\end_layout
\end_inset
\end_layout
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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 different carrier
concentrations decomposed into the intraband and interband components.
The blue series,
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a carrier density of
\begin_inset Formula $1.3\times10^{17}$
\end_inset
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, recreates TTF doping from figure
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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 at 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
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wide false
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status open
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\noindent
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filename ../Resources/carrier-density/complex-lines-phase.png
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\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
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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
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Figure
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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
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\noindent
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\begin_inset Graphics
filename ../Resources/temperature/im-com-temp-surf-sl5e-12-TTF.png
lyxscale 20
width 80col%
\end_inset
\end_layout
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\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
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\noindent
\align center
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\noindent
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lyxscale 20
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\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
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\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
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\noindent
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lyxscale 20
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\end_inset
\end_layout
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\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
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\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 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
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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 at 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, 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
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\noindent
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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
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\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
\noindent
\align center
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wide false
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\begin_layout Plain Layout
\noindent
\align center
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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
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\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 at 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
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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
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\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\begin_inset Float figure
wide false
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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 high-frequency 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
The intraband transitions are derived from the Drude model of carrier transport
\begin_inset CommandInset citation
LatexCommand cite
key "graphene-microwave,graphene-modal-prop-drude"
literal "false"
\end_inset
.
This results in the characteristic real cutoff/imaginary peak behaviour
seen in figure
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:david-intraband"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
From the simulated data from the literature, the gigahertz cutoff for intraband
conductivity can be seen to have an associated phase increase.
This suggests an increase in inductive reactance throughout this spectral
range which would have implications for applications employing graphene
at these frequencies and could suggest an increase in delay.
N-type doped material can be seen to show a negative phase peak at terahertz
frequencies indicating capacitive behaviour over this spectrum.
\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 (straight up on a energy-momentum diagram) 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 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
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 at 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 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 the 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 using the Drude model of charge transport.
The scatter lifetime is the average lifetime between collisions, more collision
s reduce the overall mobility of the material.
A longer lifetime suggests that electrons are able to travel for longer
distances at higher speed without being slowed by scatter events, this
explains the higher magnitude throughout the gigahertz spectrum.
\end_layout
\begin_layout Standard
The interband interactions were seen to be unaffected by the scatter lifetime.
This can be seen from the model where, unlike equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:intra-conductivity"
plural "false"
caps "false"
noprefix "false"
\end_inset
, equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:inter-conductivity"
plural "false"
caps "false"
noprefix "false"
\end_inset
has no reference to a lifetime term.
As a higher energy interaction, electrons are less affected by the scattering
of atoms and thus scatter lifetime is not employed for these high-frequency
transitions.
\end_layout
\begin_layout Section
Conclusion
\end_layout
\begin_layout Standard
Two applications of graphene at high frequencies have been presented.
The use of graphene nanoribbons as a channel material for FET technology
was shown to be a suitable prospective material for challenging the current
silicon CMOS/MOSFET paradigm as short-channel effects become harder to
engineer around.
Following this, graphene's applicability to flexible antennae was evaluated.
With a less established consumer domain, graphene's bio-compatibility,
flexibility, stiffness and electrical properties could overtake the more
limited metal-based ink approaches.
\end_layout
\begin_layout Standard
The sheet conductivity of graphene was modelled and the effect of varying
carrier concentration, temperature and scatter lifetime was evaluated.
Carrier concentration was seen to have a significant effect on conductivity
as a result of its relation to the materials Fermi level.
Temperature, however, was shown to have little effect, a promising result
for the stability of graphene over a wide thermal operating range.
Scatter lifetime was also shown to have a significant effect on intraband
conductivity as a result of its importance in the Drude modelling of carrier
transport.
\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