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\pdf_title "Graphene Investigations & Conductivity Modelling"
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\begin_body
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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
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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 it's 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.
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
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2
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V
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-1
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s
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at room temperature
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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 it's electrical
and mechanical behaviour at high frequencies.
Section
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presents an investigation into the 2D sheet conductivity of the material.
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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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second application
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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 requiring innovative
3D device structures
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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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cite, reword?
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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
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\emph default
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Existing Silicon CMOS can expect an on/off ratio of at least the order
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4
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, whereas monolayer graphene can only achieve around 5
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.
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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
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reference "fig:dirac-cones"
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.
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
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An example structure for a graphene nanoribbon FET (GNRFET) can be seen
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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
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Graphene nanoribbon FET structure
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Advantages over silicon
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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
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.
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
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, 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
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Sheet Conductivity Modelling
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\begin_layout Standard
This section presents a model for graphene's high frequency conductivity
using the equation below below
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.
\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}
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Taking this equation, the first term accounts for the intraband transitions
while the latter term refers to the interband transitions
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cite
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.
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
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\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}
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Equation
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was implemented in MatLab, see listing
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, 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
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To validate the model, values for TTF and CoCp
\begin_inset script subscript
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2
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n-type doping taken from
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(see table
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) were simulated and can be seen presented in figure
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reference "fig:david-simulation-conductivity"
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.
Similarly to the original, the magnitude (figure
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LatexCommand ref
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caps "false"
noprefix "false"
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) of the function can be seen to be between 48 and 63 mS for TTF and CoCp
\begin_inset script subscript
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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
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LatexCommand ref
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\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
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2
\end_layout
\end_inset
species.
CoCp
\begin_inset script subscript
\begin_layout Plain Layout
2
\end_layout
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also reaches -56 degrees as compared to TTF's -50 degrees at its minima.
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Dopant
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Carrier Concentration (cm
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-2
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)
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Fermi Level (eV)
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TTF
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\begin_inset Formula $1.3\times10^{13}$
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0.41
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CoCp
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2
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\begin_inset Formula $2.2\times10^{13}$
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0.53
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With Fermi velocity energy scale,
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= 3 eV
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Carrier concentration values for dopants from
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and the Fermi levels derived from the model, see figure
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Complex conductivity magnitude (a) and phase (b) for TTF and CoCp
\begin_inset script subscript
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2
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doping at 300 K with a scatter lifetime of 1 ps
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The two contributions to this complex conductivity, intraband and interband,
can be seen individually in figure
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LatexCommand ref
reference "fig:david-simulation-inter-intra"
plural "false"
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.
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
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LatexCommand ref
reference "fig:david-magnitude"
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.
The interband interactions begin after the 10 THz range, as the real component
steps from 1
\begin_inset Formula $\mu S$
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to 60
\begin_inset Formula $\mu S$
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around 200 THz, the imaginary component displays a negative peak over the
same spectral band.
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Static value continues past THz
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Intraband (a) and interband (b) conductivity for TTF and CoCp
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2
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doping at 300 K with a scatter lifetime of 1 ps
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\end_inset
\end_layout
\end_inset
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\begin_layout Standard
The Fermi level used to calculate conductivity (listing
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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
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LatexCommand ref
reference "fermi_from_carrier_density"
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\end_inset
.
The non-linear function can be seen modelled in figure
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reference "fig:fermi-concentration-func"
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.
From this point, net carrier concentration and dopant concentration may
be used somewhat interchangeably with the understanding that they are related
by this function.
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Fermi level associated with different carrier concentrations
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\end_layout
\end_inset
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\begin_layout Subsubsection
Carrier Density
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The general trends for how the dopant-influenced net carrier concentration
influences conductivity can be seen in the surfaces of figure
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LatexCommand ref
reference "fig:surf-carrier-concentration"
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To select a suitable range to visualise, the values from table
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and figure
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reference "fig:fermi-concentration-func"
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were considered.
Realistic dopant carrier concentrations can be seen to of the order of
\begin_inset Formula $1\times10^{13}$
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cm
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or
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.
For the simulation, values up to
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for a Fermi level of 1.13 eV were chosen.
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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
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Conductivity can broadly be seen to follow the same spectral profile as
a function of carrier concentrations up to
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as can be seen in figure
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.
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.
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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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15
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, 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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the maximum values begin to rapidly increase and by 10
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17
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they have increased by an order of magnitude to hundreds of milli-siemens.
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For the real conductivity component, beyond this previously mentioned
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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 $\mu S$
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by 500 THz.
As the carrier concentration increases beyond
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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
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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.
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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
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\end_layout
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Figure
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presents the conductivity for three graphene species of differing carrier
concentrations decomposed into the intraband and interband components.
The blue series,
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a carrier density of
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-2
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, recreates TTF doping from figure
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with two further theoretical species of lower dopant concentration.
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\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
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