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#LyX 2.3 created this file. For more info see http://www.lyx.org/
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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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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 target of significant research
in the last two decades since it's experimental discovery in 2004.
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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 transisitors 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 intergrated 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 limiting
factors 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
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Limitations of silicon
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Terahertz switching
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Electron mobility
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Thermal conductivity
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Sensitivity
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2D channel
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Short channel effects
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Hard to turn off, low on-off I multiplier (bandgap stuff)
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Need to introduce a bandgap which decimates mobility
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Hard to fabricate, delamination and stuff
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Terahertz Radiation
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Summary
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Sheet Conductivity Modelling
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This section presents a model for graphene's high frequency conductivity
using the equation below below
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.
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\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}
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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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.
These two contributions are separated for reference below,
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\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}
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\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}
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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,
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, the Fermi energy level,
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, the temperature,
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, and the scatter lifetime,
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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.
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Results
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To validate the model, values for TTF and CoCp
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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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.
Similarly to the original, the magnitude (figure
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) of the function can be seen to be between 48 and 63 mS for TTF and CoCp
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2
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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
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2
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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.
Beyond 100 THz, the imaginary component dips below zero, with a trough
of -0.5 mS around 250 THz.
Looking to the phase information (figure
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), before 10 GHz the phase can be seen to be 0, however as the imaginary
component begins to peak, the phase increases to a max of 90 degrees, continuin
g until 100 THz where the negative imaginary peak causes the phase to sharply
drop to -90 degrees.
There is little difference between the two dopants, they are equal until
100 THz where the TTF shows a -100 THz offset from the CoCp
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2
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species.
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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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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
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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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.
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 contribution and is responsible
for the conductivity up to the previously mentioned 100 GHz cutoff.
The interband interactions begin after the 10 THz range, initially the
imaginary component sharply drops and relaxes with a minima at 187 THz
and 248 THz for TTF and CoCp
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2
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.
As the imaginary component minimises, the real component begins sharply
rising over a 100 THz range to a maximum of 60
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This continues throughout the hundreds of terahertz range and beyond the
region of interest.
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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_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
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status open
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\noindent
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\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Fermi level associated with different carrier concentrations
\begin_inset CommandInset label
LatexCommand label
name "fig:fermi-concentration-func"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Subsubsection
Carrier Density
\end_layout
\begin_layout Standard
The general trends for how the dopant-influenced net carrier concentration
influences conductivity can be seen in the surfaces of figure
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:surf-carrier-concentration"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
To select a suitable range to visualise, the values from table
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:david-values"
plural "false"
caps "false"
noprefix "false"
\end_inset
and figure
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:fermi-concentration-func"
plural "false"
caps "false"
noprefix "false"
\end_inset
were considered.
Realistic dopant carrier concentrations can be seen to of the order of
\begin_inset Formula $1\times10^{13}$
\end_inset
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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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\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.
Room temperature with a scatter lifetime of
\begin_inset Formula $5\times10^{-12}$
\end_inset
s and a Fermi velocity energy scale of 2.8 eV
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LatexCommand label
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\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
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\begin_inset Flex TODO Note (inline)
status open
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linear surface plot?
\end_layout
\end_inset
\end_layout
\begin_layout Standard
The conductivity can broadly be seen to follow the same spectral profile
over the range of carrier concentrations as can be seen in figure
\begin_inset CommandInset ref
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reference "fig:david-magnitude"
plural "false"
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\end_inset
.
Variation comes in the magnitude of the various regions.
For both the real and imaginary component, the max value (pre-cutoff for
the real component and the peak of the imaginary component) can be seen
to be constant over net carrier concentrations up until around
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10
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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.
\end_layout
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For the real conductivity component, beyond this previously mentioned
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10
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threshold, the cutoff frequency begins to increase as can be seen from
the higher 20 GHz peak smearing the lighter blue across a higher frequency
band.
This moves the cutoff from 120 GHz to around 180 GHz.
The value that the real conductivity takes above the cutoff frequency decreases
past the 10
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carrier concentration threshold, from 58
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\begin_inset Formula $\mu S$
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to 2
\begin_inset Formula $\mu S$
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at
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.
For the imaginary component, at low carrier concentrations the peak value
decreases to around 1
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by 500 THz.
As the carrier concentration decreases beyond
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\begin_inset Formula $1\times10^{12}$
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this value decreases into the small negative values that can be seen in
figure
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reference "fig:david-magnitude"
plural "false"
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\end_inset
, the frequency at which the drop occurs lowers and the steeper colour gradient
indicates that the change happens faster.
The earliest frequency that this occurs at is around 10 THz and
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\end_inset
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.
Finally, as the carrier concentration further increases and the 120 GHz
peak increases in magnitude, the frequency for this high frequency conductivity
drop begins to increase again.
\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
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LatexCommand label
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\end_inset
\end_layout
\end_inset
\end_layout
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\end_layout
\end_inset
\end_layout
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Figure
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LatexCommand ref
reference "fig:inter-intra-carrier-conc"
plural "false"
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\end_inset
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
\begin_inset Formula $1.3\times10^{17}$
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, recreates TTF doping from figure
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reference "fig:david-simulation-inter-intra"
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\end_inset
with two further theoretical species of lower dopant concentration.
\end_layout
\begin_layout Standard
Looking to the intraband interactions, the real and imaginary components
can be seen to have the same profile as seen previously, the differences
lie in magnitude.
Higher net carrier concentrations can be seen to increase the magnitude
in a non-linear fashion, this behaviour can also be seen 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 higher initial imaginary component
that does not descend into negative values.
As concentration increases, the imaginary component decreases more, 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
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\begin_layout Plain Layout
Complex conductivity phase for three net carrier concentrations
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\end_inset
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\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
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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, particularly in the terahertz range.
The
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\begin_inset Formula $1\times10^{8}$
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series does not have a phase of 0 throughout the GHz range while the higher
doped series' do.
Additionally, this lower doped species does not have a negative phase in
the THz range.
The remaining two have significant drops in phase throughout the terahertz
spectrum but at different frequencies, increasing the carrier concentration.
\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,
\begin_inset Quotes bld
\end_inset
natural
\begin_inset Quotes brd
\end_inset
and high temperatures
\begin_inset Note Comment
status open
\begin_layout Plain Layout
the previously mentioned high frequency behaviour can be seen clearer
\end_layout
\end_inset
.
In general, temperature can be seen to have little effect throughout the
inspected temperature range.
As the temperature increases, the negative imaginary peak gets smaller
in value with a smoother gradient.
For the real component, althought 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.
\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
\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
throughout the subject spectral range.
The lower temperatures result in sharper steps around 100 THz, whereas
the higher temperature species has a smoother motion that does not reach
-90 degrees like the other two.
\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
\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 100 ps and 0.01 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
\begin_inset Flex TODO Note (Margin)
status open
\begin_layout Plain Layout
values
\end_layout
\end_inset
.
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
\begin_inset CommandInset label
LatexCommand label
name "fig:surf-scatter-lifetime"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Flex TODO Note (inline)
status open
\begin_layout Plain Layout
linear surface plot?
\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.
\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
The interband conductivity is restricted to the higher–energy terahertz
portion of the spectrum than the lower energy intraband interactions.
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
.
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 freqency,
\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 doped beyond
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 to obtain gigahertz conductivity above 20 mS 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 would severely limit the conductivity
magnitude.
This flexibility in reactive electrical characteristics could prove applicable
to high frequency applications.
\end_layout
\begin_layout Standard
\begin_inset Flex TODO Note (inline)
status open
\begin_layout Plain Layout
linear plot discussion?
\end_layout
\end_inset
\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 an increase of 20
mS.
Looking to the intraband 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 at gigahertz frequencies, but may prove less applicable to high
temperature terahertz applications considering the already lower magnitude.
\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
\begin_inset Flex TODO Note (inline)
status open
\begin_layout Plain Layout
Why?
\end_layout
\end_inset
\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/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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