2440 lines
40 KiB
Plaintext
2440 lines
40 KiB
Plaintext
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Graphene Applications & Conductivity Modelling At High Frequencies
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\begin_layout Author
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6420013
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EEEM022
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November 2020
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\size large
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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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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
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Introduction
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\begin_layout Standard
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Graphene is a 2D allotrope of carbon with
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\end_layout
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\begin_layout Standard
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This work explores the suitability of graphene for high frequency applications.
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Section
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reference "sec:Applications"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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presents two applications of graphene that take advantage of it's behaviour
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at high frequencies.
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Section
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reference "sec:Sheet-Conductivity-Modelling"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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presents an investigation into the 2D sheet conductivity of the material.
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\begin_layout Section
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Applications
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name "sec:Applications"
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Sheet Conductivity Modelling
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This section presents a model for graphene's high frequency conductivity
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using the equation below below
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.
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\begin_inset Formula
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\begin{multline}
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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)\\
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+\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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\end{multline}
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\begin_layout Standard
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Taking this equation, the first term accounts for the intraband transitions
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while the latter term refers to the interband transitions
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.
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These two contributions are separated for reference below,
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\begin_layout Standard
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\begin{equation}
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\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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\end{equation}
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\begin_layout Standard
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\begin_inset Formula
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\begin{equation}
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\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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reference "eq:2d-conductivity"
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plural "false"
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was implemented in MatLab, see listing
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\end_inset
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, such that the inter and intraband contributions were returned separately.
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This allowed for displaying both aspects independently or together by summing.
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From the function it can be seen that the variables are AC frequency,
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\begin_inset Formula $\omega$
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\end_inset
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, the Fermi energy level,
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\begin_inset Formula $E_{F}$
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\end_inset
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, the temperature,
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\begin_inset Formula $T$
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\end_inset
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, and the scatter lifetime,
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\begin_inset Formula $\tau$
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\end_inset
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.
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These were varied within reasonable ranges in order to investigate how
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such variations affect the conductivity, both as a whole and individually.
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\end_layout
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\begin_layout Subsection
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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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doping taken from
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LatexCommand citet
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key "david-paper"
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literal "false"
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(see table
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reference "tab:david-values"
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plural "false"
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noprefix "false"
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\end_inset
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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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plural "false"
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\end_inset
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.
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Similarly to the original, the real component can be seen to be between
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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 20 GHz.
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Beyond the cutoff frequency the value is around 60
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\begin_inset Formula $\mu S$
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\end_inset
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by 5 THz.
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The imaginary component peaks over the same frequency band that the real
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component declines and the two intersect at around 150 GHz with a conductance
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of 31 mS with CoCp
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2
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\end_inset
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and 24 mS for TTF.
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Beyond 100 THz, the imaginary component dips below zero, with a trough
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of -0.5 mS around 250 THz.
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\end_layout
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<features tabularvalignment="middle">
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<column alignment="center" valignment="top">
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<column alignment="center" valignment="top">
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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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0.41
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CoCp
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2
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\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Formula $2.2\times10^{13}$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
0.53
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
</lyxtabular>
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset VSpace defskip
|
|
\end_inset
|
|
|
|
With Fermi velocity energy scale,
|
|
\begin_inset Formula $t$
|
|
\end_inset
|
|
|
|
= 3 eV
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption Standard
|
|
|
|
\begin_layout Plain Layout
|
|
Carrier concentration values for dopants from
|
|
\begin_inset CommandInset citation
|
|
LatexCommand citet
|
|
key "david-paper"
|
|
literal "false"
|
|
|
|
\end_inset
|
|
|
|
and the Fermi levels derived from the model, see figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:fermi-concentration-func"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "tab:david-values"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
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|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float figure
|
|
wide false
|
|
sideways false
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
\noindent
|
|
\align center
|
|
\begin_inset Graphics
|
|
filename ../Resources/david-recreation-mag.png
|
|
lyxscale 20
|
|
width 60col%
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption Standard
|
|
|
|
\begin_layout Plain Layout
|
|
Complex conductivity for TTF and CoCp
|
|
\begin_inset script subscript
|
|
|
|
\begin_layout Plain Layout
|
|
2
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
doping at 300 K with a scatter lifetime of 1 ps
|
|
\begin_inset CommandInset citation
|
|
LatexCommand cite
|
|
key "david-paper"
|
|
literal "false"
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "fig:david-simulation-conductivity"
|
|
|
|
\end_inset
|
|
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|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
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|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The two contributions to this complex conductance, intraband and interband,
|
|
can be seen individually in figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:david-simulation-inter-intra"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
Comparing the two, it can be seen that the interactions happen over largely
|
|
separate frequency ranges.
|
|
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
|
|
\begin_inset script subscript
|
|
|
|
\begin_layout Plain Layout
|
|
2
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
.
|
|
As the imaginary component minimises, the real component begins sharply
|
|
rising over a 100 THz range to a maximum of 60
|
|
\begin_inset Formula $\mu S$
|
|
\end_inset
|
|
|
|
.
|
|
This continues throughout the hundreds of terahertz range and beyond the
|
|
region of interest.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float figure
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\noindent
|
|
\align center
|
|
\begin_inset Graphics
|
|
filename ../Resources/david-recreation-intra-mag.png
|
|
lyxscale 20
|
|
width 50col%
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Graphics
|
|
filename ../Resources/david-recreation-inter-mag.png
|
|
lyxscale 20
|
|
width 50col%
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption Standard
|
|
|
|
\begin_layout Plain Layout
|
|
Intraband and interband conductivity for TTF and CoCp
|
|
\begin_inset script subscript
|
|
|
|
\begin_layout Plain Layout
|
|
2
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
doping at 300 K with a scatter lifetime of 1 ps
|
|
\begin_inset CommandInset citation
|
|
LatexCommand cite
|
|
key "david-paper"
|
|
literal "false"
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "fig:david-simulation-inter-intra"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The Fermi level used to calculate conductance (listing
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "calculation_function"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
) was derived from the net carrier concentration as a result of doping,
|
|
see listing
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fermi_from_carrier_density"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
The non-linear function can be seen modelled in figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:fermi-concentration-func"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float figure
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\noindent
|
|
\align center
|
|
\begin_inset Graphics
|
|
filename ../Resources/fermi-conc.png
|
|
lyxscale 20
|
|
width 60col%
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption Standard
|
|
|
|
\begin_layout Plain Layout
|
|
Fermi level associated with different carrier concentrations
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "fig:fermi-concentration-func"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Subsubsection
|
|
Carrier Density
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The general trends for how the dopant-influenced net carrier concentration
|
|
influences conductivity can be seen in the surfaces of figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:surf-carrier-concentration"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
To select a suitable range to visualise, the values from table
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "tab:david-values"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
and figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:fermi-concentration-func"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
were considered.
|
|
Realistic dopant carrier concentrations can be seen to of the order of
|
|
|
|
\begin_inset Formula $1\times10^{13}$
|
|
\end_inset
|
|
|
|
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
|
|
\emph off
|
|
\bar no
|
|
\strikeout off
|
|
\xout off
|
|
\uuline off
|
|
\uwave off
|
|
\noun off
|
|
\color none
|
|
cm
|
|
\begin_inset script superscript
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
|
|
\emph off
|
|
\bar no
|
|
\strikeout off
|
|
\xout off
|
|
\uuline off
|
|
\uwave off
|
|
\noun off
|
|
\color none
|
|
-2
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
or
|
|
\family default
|
|
\series default
|
|
\shape default
|
|
\size default
|
|
\emph default
|
|
\bar default
|
|
\strikeout default
|
|
\xout default
|
|
\uuline default
|
|
\uwave default
|
|
\noun default
|
|
\color inherit
|
|
|
|
\begin_inset Formula $1\times10^{17}$
|
|
\end_inset
|
|
|
|
m
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
|
|
\emph off
|
|
\bar no
|
|
\strikeout off
|
|
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|
|
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|
|
\uwave off
|
|
\noun off
|
|
\color none
|
|
|
|
\begin_inset script superscript
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
|
|
\emph off
|
|
\bar no
|
|
\strikeout off
|
|
\xout off
|
|
\uuline off
|
|
\uwave off
|
|
\noun off
|
|
\color none
|
|
-2
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
.
|
|
For the simulation, values up to
|
|
\family default
|
|
\series default
|
|
\shape default
|
|
\size default
|
|
\emph default
|
|
\bar default
|
|
\strikeout default
|
|
\xout default
|
|
\uuline default
|
|
\uwave default
|
|
\noun default
|
|
\color inherit
|
|
|
|
\begin_inset Formula $1\times10^{18}$
|
|
\end_inset
|
|
|
|
m
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
|
|
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|
|
\bar no
|
|
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|
|
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|
|
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|
|
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|
|
\noun off
|
|
\color none
|
|
|
|
\begin_inset script superscript
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
|
|
\emph off
|
|
\bar no
|
|
\strikeout off
|
|
\xout off
|
|
\uuline off
|
|
\uwave off
|
|
\noun off
|
|
\color none
|
|
-2
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
for a Fermi level of 1.13 eV were chosen.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float figure
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\noindent
|
|
\align center
|
|
\begin_inset Graphics
|
|
filename ../Resources/carrier-density/real-com-carrier-surf-sl5e-12-T300-logCB.png
|
|
lyxscale 20
|
|
width 80col%
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\noindent
|
|
\align center
|
|
\begin_inset Graphics
|
|
filename ../Resources/carrier-density/im-com-carrier-surf-sl5e-12-T300-logCB.png
|
|
lyxscale 20
|
|
width 80col%
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption Standard
|
|
|
|
\begin_layout Plain Layout
|
|
Complex conductivity over frequency for different carrier densities.
|
|
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
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "fig:surf-carrier-concentration"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The conductance 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
|
|
LatexCommand ref
|
|
reference "fig:david-simulation-conductivity"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\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
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
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|
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|
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\color none
|
|
10
|
|
\begin_inset script superscript
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\family roman
|
|
\series medium
|
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\shape up
|
|
\size normal
|
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\emph off
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|
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|
|
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|
|
15
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\family default
|
|
\series default
|
|
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|
|
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|
|
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|
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|
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|
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\color inherit
|
|
m
|
|
\family roman
|
|
\series medium
|
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\shape up
|
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|
|
|
|
\begin_inset script superscript
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\family roman
|
|
\series medium
|
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\shape up
|
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\size normal
|
|
\emph off
|
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|
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|
|
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|
|
\color none
|
|
-2
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
, 21 mS for the real component and 11 mS for the imaginary.
|
|
Beyond
|
|
\family default
|
|
\series default
|
|
\shape default
|
|
\size default
|
|
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|
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|
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|
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|
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|
|
|
|
\family roman
|
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|
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|
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|
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|
|
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|
|
\uuline off
|
|
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|
|
\noun off
|
|
\color none
|
|
a net carrier concentration of 10
|
|
\begin_inset script superscript
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
|
|
\emph off
|
|
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|
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|
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\uuline off
|
|
\uwave off
|
|
\noun off
|
|
\color none
|
|
15
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\family default
|
|
\series default
|
|
\shape default
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|
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|
|
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|
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|
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|
|
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|
|
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|
|
\color inherit
|
|
m
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
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|
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|
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|
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|
|
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|
|
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|
|
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|
|
\noun off
|
|
\color none
|
|
|
|
\begin_inset script superscript
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
|
|
\emph off
|
|
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|
|
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|
|
\xout off
|
|
\uuline off
|
|
\uwave off
|
|
\noun off
|
|
\color none
|
|
-2
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
the maximum values begin to rapidly increase and by 10
|
|
\begin_inset script superscript
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\family roman
|
|
\series medium
|
|
\shape up
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|
\size normal
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|
\emph off
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|
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|
|
\color none
|
|
17
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\family default
|
|
\series default
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|
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|
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|
|
\color inherit
|
|
m
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
|
|
\emph off
|
|
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|
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|
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|
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|
|
\noun off
|
|
\color none
|
|
|
|
\begin_inset script superscript
|
|
|
|
\begin_layout Plain Layout
|
|
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|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
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|
\emph off
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|
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|
|
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|
|
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|
|
\noun off
|
|
\color none
|
|
-2
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
they have increased by an order of magnitude to hundreds of milli-siemens.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
For the real conductance component, beyond this previously mentioned
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
|
|
\emph off
|
|
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|
|
\noun off
|
|
\color none
|
|
10
|
|
\begin_inset script superscript
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\family roman
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|
\series medium
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|
\shape up
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|
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\emph off
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15
|
|
\end_layout
|
|
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|
\end_inset
|
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|
\family default
|
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\series default
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|
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|
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|
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|
|
m
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\family roman
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|
\series medium
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|
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|
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|
|
\color none
|
|
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|
\begin_inset script superscript
|
|
|
|
\begin_layout Plain Layout
|
|
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|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
|
|
\emph off
|
|
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|
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|
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|
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|
|
\noun off
|
|
\color none
|
|
-2
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
threshold, the cutoff frequency begins to increase as can be seen from
|
|
higher peak smearing the lighter blue across a higher frequency band.
|
|
This moves the cutoff from around 120 GHz to about 180 GHz.
|
|
The value that the real conductance takes above the cutoff frequency decreases
|
|
past the 10
|
|
\begin_inset script superscript
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\family roman
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|
\series medium
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|
\shape up
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|
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\emph off
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|
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|
|
\color none
|
|
15
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\family default
|
|
\series default
|
|
\shape default
|
|
\size default
|
|
\emph default
|
|
\bar default
|
|
\strikeout default
|
|
\xout default
|
|
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|
|
\uwave default
|
|
\noun default
|
|
\color inherit
|
|
m
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
|
|
\emph off
|
|
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|
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|
|
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|
|
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|
|
\uwave off
|
|
\noun off
|
|
\color none
|
|
|
|
\begin_inset script superscript
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
|
|
\emph off
|
|
\bar no
|
|
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|
|
\xout off
|
|
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|
|
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|
|
\noun off
|
|
\color none
|
|
-2
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
carrier concentration threshold, from 58
|
|
\family default
|
|
\series default
|
|
\shape default
|
|
\size default
|
|
\emph default
|
|
\bar default
|
|
\strikeout default
|
|
\xout default
|
|
\uuline default
|
|
\uwave default
|
|
\noun default
|
|
\color inherit
|
|
|
|
\begin_inset Formula $\mu S$
|
|
\end_inset
|
|
|
|
to 2
|
|
\begin_inset Formula $\mu S$
|
|
\end_inset
|
|
|
|
at
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
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|
|
\emph off
|
|
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|
|
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|
|
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|
|
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|
|
\uwave off
|
|
\noun off
|
|
\color none
|
|
|
|
\begin_inset Formula $1\times10^{17}$
|
|
\end_inset
|
|
|
|
|
|
\family default
|
|
\series default
|
|
\shape default
|
|
\size default
|
|
\emph default
|
|
\bar default
|
|
\strikeout default
|
|
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|
|
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|
|
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|
|
\noun default
|
|
\color inherit
|
|
m
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
|
|
\emph off
|
|
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|
|
\strikeout off
|
|
\xout off
|
|
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|
|
\uwave off
|
|
\noun off
|
|
\color none
|
|
|
|
\begin_inset script superscript
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
|
|
\emph off
|
|
\bar no
|
|
\strikeout off
|
|
\xout off
|
|
\uuline off
|
|
\uwave off
|
|
\noun off
|
|
\color none
|
|
-2
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
.
|
|
For the imaginary component, at low carrier concentrations the peak value
|
|
decreases to around 1
|
|
\family default
|
|
\series default
|
|
\shape default
|
|
\size default
|
|
\emph default
|
|
\bar default
|
|
\strikeout default
|
|
\xout default
|
|
\uuline default
|
|
\uwave default
|
|
\noun default
|
|
\color inherit
|
|
|
|
\begin_inset Formula $\mu S$
|
|
\end_inset
|
|
|
|
by 500 THz.
|
|
As the carrier concentration decreases beyond
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
|
|
\emph off
|
|
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|
|
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|
|
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|
|
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|
|
\uwave off
|
|
\noun off
|
|
\color none
|
|
|
|
\begin_inset Formula $1\times10^{12}$
|
|
\end_inset
|
|
|
|
|
|
\family default
|
|
\series default
|
|
\shape default
|
|
\size default
|
|
\emph default
|
|
\bar default
|
|
\strikeout default
|
|
\xout default
|
|
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|
|
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|
|
\noun default
|
|
\color inherit
|
|
m
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
|
|
\emph off
|
|
\bar no
|
|
\strikeout off
|
|
\xout off
|
|
\uuline off
|
|
\uwave off
|
|
\noun off
|
|
\color none
|
|
|
|
\begin_inset script superscript
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
|
|
\emph off
|
|
\bar no
|
|
\strikeout off
|
|
\xout off
|
|
\uuline off
|
|
\uwave off
|
|
\noun off
|
|
\color none
|
|
-2
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
this value decreases into the small negative values that can be seen in
|
|
figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:david-simulation-conductivity"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\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
|
|
\begin_inset Formula $1\times10^{15}$
|
|
\end_inset
|
|
|
|
|
|
\family default
|
|
\series default
|
|
\shape default
|
|
\size default
|
|
\emph default
|
|
\bar default
|
|
\strikeout default
|
|
\xout default
|
|
\uuline default
|
|
\uwave default
|
|
\noun default
|
|
\color inherit
|
|
m
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
|
|
\emph off
|
|
\bar no
|
|
\strikeout off
|
|
\xout off
|
|
\uuline off
|
|
\uwave off
|
|
\noun off
|
|
\color none
|
|
|
|
\begin_inset script superscript
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
|
|
\emph off
|
|
\bar no
|
|
\strikeout off
|
|
\xout off
|
|
\uuline off
|
|
\uwave off
|
|
\noun off
|
|
\color none
|
|
-2
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
.
|
|
Finally, as the carrier concentration further increases and the 120 GHz
|
|
peak increases in magnitude, the frequency for this high frequency conductance
|
|
drop begins to increase again.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float figure
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\noindent
|
|
\align center
|
|
\begin_inset Graphics
|
|
filename ../Resources/carrier-density/intraband-lines-mag.png
|
|
lyxscale 20
|
|
width 50col%
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Graphics
|
|
filename ../Resources/carrier-density/interband-lines-mag.png
|
|
lyxscale 20
|
|
width 50col%
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption Standard
|
|
|
|
\begin_layout Plain Layout
|
|
Inter- and intraband conductance for high and low carrier concentration
|
|
graphene species
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "fig:inter-intra-carrier-conc"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:inter-intra-carrier-conc"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
presents the conductance for three graphene species of differing carrier
|
|
concentrations decomposed into the intraband and interband components.
|
|
From comparing the relative magnitudes from the two, it is clear that the
|
|
majority contribution for conductance throughout the selected frequency
|
|
range is from the intraband transitions.
|
|
This is also apparent from the similarity in spectral profile between the
|
|
intraband conductivity and both the surfaces of figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:surf-carrier-concentration"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
and the reproduced results of figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:david-simulation-conductivity"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
In general, the intraband conductivity can be seen to exist up to the THz
|
|
portion of the spectrum while the interband has the majority of it's contributi
|
|
ons above the THz range.
|
|
The interband conductance can be seen to be responsible for the previously
|
|
noted negative imaginary conductance behaviour seen in the surface of figure
|
|
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:surf-carrier-concentration"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
Low carrier concentration result in a higher initial imaginary component
|
|
that does not lower into negative values.
|
|
As concentration increases, the imaginary component decreases more forming
|
|
a sharp trough that also bottoms out 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 decreases 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 110 THz.
|
|
\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 conductance.
|
|
Figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:surf-temperature"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
shows a surface of the conductance spectrum over the prescribed temperature
|
|
range.
|
|
In general, temperature can be seen to have little effect on conductance,
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both real and imaginary.
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\end_layout
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\begin_layout Standard
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\begin_inset Float figure
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sideways false
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status open
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\begin_layout Plain Layout
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\noindent
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\align center
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\begin_inset Graphics
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filename ../Resources/temperature/real-com-temp-surf-sl5e-12-TTF.png
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lyxscale 20
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width 80col%
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\end_inset
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\end_layout
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\begin_layout Plain Layout
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\noindent
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\align center
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\begin_inset Graphics
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filename ../Resources/temperature/im-com-temp-surf-sl5e-12-TTF.png
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lyxscale 20
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width 80col%
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\end_inset
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\end_layout
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\begin_layout Plain Layout
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\begin_inset Caption Standard
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\begin_layout Plain Layout
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Complex conductivity over frequency for different temperatures
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\begin_inset CommandInset label
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LatexCommand label
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name "fig:surf-temperature"
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\end_inset
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\end_layout
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\end_inset
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset Float figure
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wide false
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sideways false
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status open
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\begin_layout Plain Layout
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\noindent
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\align center
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\begin_inset Graphics
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filename ../Resources/temperature/intraband-lines-mag.png
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lyxscale 20
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width 50col%
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\end_inset
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\begin_inset Graphics
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filename ../Resources/temperature/interband-lines-mag.png
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lyxscale 20
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width 50col%
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\end_inset
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\end_layout
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\begin_layout Plain Layout
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\begin_inset Caption Standard
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\begin_layout Plain Layout
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Inter- and intraband conductance for low, room and high temperature graphene
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using TTF doping
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\begin_inset CommandInset label
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LatexCommand label
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name "fig:inter-intra-temperature"
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\end_inset
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\end_layout
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\end_inset
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Subsubsection
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Scattering Lifetime
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\end_layout
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\begin_layout Standard
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\begin_inset Float figure
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wide false
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sideways false
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status open
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\begin_layout Plain Layout
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\noindent
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\align center
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\begin_inset Graphics
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filename ../Resources/scatter-lifetime/real-com-SL-surf-300K-TTF10,14.png
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lyxscale 20
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width 80col%
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\end_inset
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\end_layout
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\begin_layout Plain Layout
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\noindent
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\align center
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\begin_inset Graphics
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filename ../Resources/scatter-lifetime/im-com-SL-surf-300K-TTF10,14.png
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lyxscale 20
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width 80col%
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\end_inset
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\end_layout
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\begin_layout Plain Layout
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\begin_inset Caption Standard
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\begin_layout Plain Layout
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Complex conductivity over frequency for different scattering lifetimes
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\begin_inset CommandInset label
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LatexCommand label
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name "fig:surf-scatter-lifetime"
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\end_inset
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\end_layout
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\end_inset
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset Float figure
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wide false
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sideways false
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status open
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\begin_layout Plain Layout
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\noindent
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\align center
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\begin_inset Graphics
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filename ../Resources/scatter-lifetime/intraband-lines-mag.png
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lyxscale 20
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width 50col%
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\end_inset
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\begin_inset Graphics
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filename ../Resources/scatter-lifetime/interband-lines-mag.png
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lyxscale 20
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width 50col%
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\end_inset
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\end_layout
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\begin_layout Plain Layout
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\begin_inset Caption Standard
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\begin_layout Plain Layout
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|
Inter- and intraband conductance with 3 different scattering times for graphene
|
|
using TTF doping
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
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name "fig:inter-intra-scatter-lifetime"
|
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\end_inset
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\end_layout
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\end_inset
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Subsection
|
|
Discussion
|
|
\end_layout
|
|
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\begin_layout Section
|
|
Conclusion
|
|
\end_layout
|
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\begin_layout Standard
|
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\begin_inset Newpage newpage
|
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\end_inset
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\end_layout
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\begin_layout Standard
|
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\begin_inset CommandInset label
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LatexCommand label
|
|
name "sec:bibliography"
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\end_inset
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\begin_inset CommandInset bibtex
|
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LatexCommand bibtex
|
|
btprint "btPrintCited"
|
|
bibfiles "references"
|
|
options "bibtotoc"
|
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\end_inset
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\begin_inset Newpage pagebreak
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\end_inset
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\end_layout
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\begin_layout Section
|
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\start_of_appendix
|
|
Source Code
|
|
\begin_inset CommandInset label
|
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LatexCommand label
|
|
name "sec:Code"
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\end_inset
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\end_layout
|
|
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\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}"
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset Newpage pagebreak
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\end_inset
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\end_layout
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\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
|
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|
\end_layout
|
|
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\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}"
|
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|
|
\end_inset
|
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\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}"
|
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|
\end_inset
|
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\end_layout
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\begin_layout Standard
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\begin_inset Newpage pagebreak
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\end_inset
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\end_layout
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|
|
\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
|
|
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|
|
|
\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
|
|
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|
|
\end_layout
|
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\begin_layout Standard
|
|
\begin_inset Newpage pagebreak
|
|
\end_inset
|
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\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
|
|
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|
\end_layout
|
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|
\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
|
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|
\end_layout
|
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\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
|
|
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\end_layout
|
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\end_body
|
|
\end_document
|