4705 lines
87 KiB
Plaintext
4705 lines
87 KiB
Plaintext
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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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\end_layout
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\begin_layout Abstract
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Graphene has many desirable mechanical and electrical characteristics that
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make it an exciting material for use in innovative devices.
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In particular, its extremely high carrier mobility leads to high operating
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||
frequencies that potentially make the material suitable for the gigahertz
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and terahertz spectrum.
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||
The use of graphene for high-frequency electronics applications is considered,
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||
firstly as a channel material for digital logic transistor technology with
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a focus on graphene nanoribbon FETs (GNRFET).
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||
Graphene's applicability is then evaluated for flexible antennae with prospecti
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||
ve use in wearable and IoT devices.
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Part two models graphene's sheet conductivity, exploring how net carrier
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concentration, temperature and scatter lifetime affect both intraband and
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interband electrical characteristics.
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EEEM022 Coursework
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April 2021
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6420013
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\begin_layout Section
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Introduction
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Graphene is a 2D allotrope of carbon with highly interesting mechanical
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||
and electrical properties that have made it a popular target of research
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||
over the last two decades since its experimental discovery in 2004
|
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.
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Resembling a honeycomb structure, each carbon atoms bonds to three surrounding
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atoms by overlapping
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\begin_inset Formula $p$
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\end_inset
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-orbitals.
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||
This leaves one remaining
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\begin_inset Formula $\pi$
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-electron whose orbital extends perpendicular to the 2D sheet.
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.
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As a result, graphene has a 0 energy bandgap, the conduction and valence
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||
band meet at a single point known as the Dirac point at the K points in
|
||
momentum space.
|
||
Both bands linearly extend away from the Dirac point, a highly unusual
|
||
result that allows electrons to act as massless Fermions which move with
|
||
extremely high mobility, as high as 200,000 cm
|
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2
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V
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-1
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s
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-1
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key "graphene-review-2010"
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A section of graphene sheet with the orientation of electron orbitals highlighte
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d
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key "warda-gfet-review"
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With these highly sought after properties, it is unsurprising that graphene
|
||
remains a prospective material for many domains including semiconductor
|
||
and high frequency applications.
|
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\end_layout
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This work explores the suitability of graphene for high-frequency applications.
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Section
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plural "false"
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presents two applications of graphene that take advantage of its electrical
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and mechanical behaviour at high frequencies.
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Section
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reference "sec:Sheet-Conductivity-Modelling"
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|
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presents an investigation into the 2D sheet conductivity of the material.
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|
||
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Applications
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This section explores two uses of graphene for high-frequency applications.
|
||
First, the applicability of graphene for field-effect transistors will
|
||
be considered as a channel material.
|
||
Throughout, a particular focus will be paid to use in digital logic and
|
||
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|
||
\end_layout
|
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Following its prospective use in digital logic, graphene's applicability
|
||
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|
||
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|
||
flexible and efficient antennae at gigahertz frequencies is clear.
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||
Existing flexible antennae technology will be considered before presenting
|
||
how graphene could be included.
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\begin_layout Subsection
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Digital Logic
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Silicon-based CMOS/MOSFET digital logic is the basis on which much of the
|
||
modern electronics landscape has been built.
|
||
From integrated logic circuits to CPUs, it is hard to overstate how important
|
||
this technology has proven to be.
|
||
The need for more powerful devices has increased pressure for smaller and
|
||
more efficient transistors, such that more can fit into a single device.
|
||
This progress is typically described by Moore's Law and can be seen graphically
|
||
in figure
|
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|
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\begin_layout Plain Layout
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The number of transistors in commercial CPUs between 1970 and 2020
|
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|
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\begin_layout Standard
|
||
However, as transistors are made smaller, theoretical limits for many engineerin
|
||
g challenges are approached.
|
||
In 2015, the ITRS predicted that by 2021 the current push for smaller transisto
|
||
rs would no longer be economically viable, instead, innovative 3D device
|
||
structures would be required
|
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|
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.
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||
Some of the most important limiting factors in the current Silicon landscape
|
||
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|
||
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|
||
order of magnitude as the depletion layer
|
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.
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|
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\begin_layout Standard
|
||
As previously covered, a sheet of graphene or
|
||
\emph on
|
||
large-area graphene
|
||
\emph default
|
||
has no bandgap.
|
||
As such it is unable to turn off, making it unsuitable as a channel material
|
||
for a digital MOSFET.
|
||
Quantitatively, this can be measured with the ratio of on current to off
|
||
current or the
|
||
\emph on
|
||
on/off ratio
|
||
\emph default
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||
.
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||
Existing Silicon CMOS can expect an on/off ratio of at least the order
|
||
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4
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|
||
, whereas monolayer graphene can only achieve around 5
|
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key "warda-gfet-review"
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literal "false"
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.
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||
\end_layout
|
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|
||
\begin_layout Standard
|
||
Therefore, it is clear that to use graphene as a channel material a bandgap
|
||
must be formed.
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||
There are a number of ways to do so, a selection of band diagrams for such
|
||
structures can be seen in figure
|
||
\begin_inset CommandInset ref
|
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reference "fig:dirac-cones"
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plural "false"
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|
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|
||
\end_inset
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|
||
.
|
||
For digital applications, one of the most promising methods for creating
|
||
a bandgap is by confining graphene in one dimension to create a
|
||
\emph on
|
||
graphene nanoribbon
|
||
\emph default
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|
||
In doing so it has been shown that the produced bandgap is inversely proportion
|
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\begin_inset CommandInset citation
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LatexCommand cite
|
||
key "gnrfet-applications"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
An example structure for a graphene nanoribbon FET (GNRFET) can be seen
|
||
in figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:gnrget-structure"
|
||
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 cones.png
|
||
width 40col%
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Energy band diagram for 4 different structures of graphene, i) large-area,
|
||
ii) nanoribbons, iii) unbiased bilayer, iv) bilayer with an applied perpendicul
|
||
ar electric field
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "graphene-review-2010"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:dirac-cones"
|
||
|
||
\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 gnrfet.png
|
||
lyxscale 50
|
||
width 40col%
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Graphene nanoribbon FET structure
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "gnrfet-structure-image"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:gnrget-structure"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Table
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "tab:gnrfet-low-power-results"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
compares transistor performance characteristics for both silicon and GNR-based
|
||
FET technology
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "gnrfet-low-power"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
High-quality GNRFETs can be seen to demonstrate the lowest sub-threshold
|
||
swing and a reasonably high value for
|
||
\begin_inset Formula $\nicefrac{I_{on}}{I_{off}}$
|
||
\end_inset
|
||
|
||
.
|
||
Further analyses from
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "gnrfet-low-power"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
suggest that GNRFET demonstrates better performance than silicon for low-power
|
||
applications with lower power consumption.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float table
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\noindent
|
||
\align center
|
||
\begin_inset Tabular
|
||
<lyxtabular version="3" rows="5" columns="4">
|
||
<features tabularvalignment="middle">
|
||
<column alignment="center" valignment="top">
|
||
<column alignment="center" valignment="top">
|
||
<column alignment="center" valignment="top">
|
||
<column alignment="center" valignment="top">
|
||
<row>
|
||
<cell alignment="center" valignment="top" bottomline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
S (mV/dec)
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
I
|
||
\begin_inset script subscript
|
||
|
||
\begin_layout Plain Layout
|
||
on
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
/I
|
||
\begin_inset script subscript
|
||
|
||
\begin_layout Plain Layout
|
||
off
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
V
|
||
\begin_inset script subscript
|
||
|
||
\begin_layout Plain Layout
|
||
DD
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
(V)
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Silicon CMOS (High Performance)
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
93.46
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $3.49\times10^{3}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
0.7
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Silicon CMOS (Low Power)
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
86.96
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $5.12\times10^{6}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
0.9
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
GNRFET (Pristine)
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
66.67
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $1.81\times10^{5}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
0.5
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
GNRFET (Rough Edge)
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
140.85
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $9.85\times10^{3}$
|
||
\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
|
||
0.5
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
</lyxtabular>
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
FET operating characteristics for two silicon-based (16 nm) and two GNR-based
|
||
transistor technologies.
|
||
Subthreshold swing, on/off ratio and threshold voltage reported
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "gnrfet-low-power"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "tab:gnrfet-low-power-results"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Some of the limitations of GNR-based FETs include the difficulty in fabricating
|
||
high-quality graphene and specifically graphene nanoribbons.
|
||
Edge roughness can significantly affect the properties of the channel,
|
||
introducing new scattering centres
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "gnrfet-structure-image"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
, this can be seen in table
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "tab:gnrfet-low-power-results"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
with rough-edge graphene demonstrating a higher subthreshold swing than
|
||
both pristine graphene and the existing silicon technology.
|
||
A rough edge also increases delay and leakage power
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "gnrfet-low-power"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
, further emphasising the requirement for high-quality fabrication methods.
|
||
However, one of the major constraints of GNRFET technology is limited mobility.
|
||
Although graphene has incredibly high carrier mobility, by opening the
|
||
bandgap as seen in figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:dirac-cones"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
, curves are introduced to the band structure.
|
||
This increases the carrier's effective mass, significantly reducing mobility.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Despite these drawbacks, GNRFETs look to be one of the most promising avenues
|
||
for post-silicon, high-performance and/or low-power CMOS/MOSFET electronics.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Flexible Antennae
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The miniaturisation and increased efficiency of electronics have pushed
|
||
for smaller-scale devices including IoT and wearable devices.
|
||
One critical component for these devices is efficient wireless technologies
|
||
and antennae in particular.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Graphene has a number of properties that make it a suitable prospective
|
||
material for such purposes.
|
||
Being flexible, bio-compatible and highly stiff
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "warda-gfet-review"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
, graphene should be able to withstand much of the repeated physical deformation
|
||
s that could be expected in the wearable electronics domain.
|
||
The previously mentioned electrical properties are obviously also applicable
|
||
to the domain - high mobility, conductivity and operating frequencies suggest
|
||
that the material could be applicable to gigahertz applications and thus
|
||
Wi-Fi and Bluetooth usage.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Existing technologies for flexible antennae include printed metal-ink components
|
||
including copper and silver
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "flexible-antennae-review"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
An advantage of metal-based inks are the inherently high conductivities
|
||
and some helpful mechanical properties.
|
||
However there are disadvantages, silver in particular is a valuable metal
|
||
that is too expensive to be used extensively in antennae
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "flexible-antennae-review"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
Copper also has disadvantages in its tendency to oxidise when exposed to
|
||
the environment as could be expected in wearable or IoT applications.
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Printed graphene antennae using water-transfer technology have been demonstrated
|
||
that function at the 2.4 GHz WiFi/Bluetooth spectrum
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "water-transfer-graphene-antennae"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
, the power spectrum compared to a copper antenna can be seen in figure
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:antenna-power"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
This room-temperature method demonstrates that as the performance is comparable
|
||
with metal-based inks, graphene's mechanical properties make it a highly
|
||
applicable material for wearable antennae.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\noindent
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename antennae-power.jpg
|
||
width 60col%
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Normalised radiation power for graphene and copper-based dipole antenna,
|
||
a) E-plane, b) H-plane
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "water-transfer-graphene-antennae"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:antenna-power"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Summary
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
From these applications, it is clear that graphene has the potential to
|
||
revolutionise cutting-edge electronics domains including high-performance,
|
||
high-efficiency transistor technologies and the wearable/IoT devices.
|
||
Although the material does not yet have significant penetration in consumer
|
||
electronics, it is likely that this is not far away.
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Sheet Conductivity Modelling
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "sec:Sheet-Conductivity-Modelling"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
This section presents a model for graphene's high-frequency conductivity
|
||
using the equation below
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "yao"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{multline}
|
||
\sigma_{s}\left(\omega\right)=\frac{2ie^{2}k_{B}T}{\pi\hbar^{2}\left(\omega+\nicefrac{i}{\tau}\right)}\ln\left(2\cosh\left(\frac{E_{F}}{2k_{B}T}\right)\right)\\
|
||
+\frac{e^{2}}{4\hbar}\left(\frac{1}{2}+\frac{1}{\pi}\tan^{-1}\left(\frac{\hbar\omega-2E_{F}}{2k_{B}T}\right)-\frac{i}{2\pi}\ln\left(\frac{\left(\hbar\omega+2E_{F}\right)^{2}}{\left(\hbar\omega-2E_{F}\right)^{2}+4\left(k_{B}T\right)^{2}}\right)\right)\label{eq:2d-conductivity}
|
||
\end{multline}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Taking this equation, the first term accounts for the intraband transitions
|
||
while the latter term refers to the interband transitions
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "david-paper"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
These two contributions are separated for reference below,
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\sigma_{s}^{intra}\left(\omega\right)=\frac{2ie^{2}k_{B}T}{\pi\hbar^{2}\left(\omega+\nicefrac{i}{\tau}\right)}\ln\left(2\cosh\left(\frac{E_{F}}{2k_{B}T}\right)\right)\label{eq:intra-conductivity}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\sigma_{s}^{inter}\left(\omega\right)=\frac{e^{2}}{4\hbar}\left(\frac{1}{2}+\frac{1}{\pi}\tan^{-1}\left(\frac{\hbar\omega-2E_{F}}{2k_{B}T}\right)-\frac{i}{2\pi}\ln\left(\frac{\left(\hbar\omega+2E_{F}\right)^{2}}{\left(\hbar\omega-2E_{F}\right)^{2}+4\left(k_{B}T\right)^{2}}\right)\right)\label{eq:inter-conductivity}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Equation
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:2d-conductivity"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
was implemented in MatLab, see listing
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "calculation_function"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
, such that the interband and intraband contributions were returned separately.
|
||
This allowed for displaying both aspects independently or together by summing.
|
||
From the function it can be seen that the variables are AC frequency,
|
||
\begin_inset Formula $\omega$
|
||
\end_inset
|
||
|
||
, the Fermi energy level,
|
||
\begin_inset Formula $E_{F}$
|
||
\end_inset
|
||
|
||
, the temperature,
|
||
\begin_inset Formula $T$
|
||
\end_inset
|
||
|
||
, and the scatter lifetime,
|
||
\begin_inset Formula $\tau$
|
||
\end_inset
|
||
|
||
.
|
||
These were varied within reasonable ranges in order to investigate how
|
||
such variations affect the conductivity, both as a whole and individually.
|
||
Prior to these wider investigations, however, experimental data was simulated
|
||
in order both to validate the model.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Results
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
To validate the model, values for TTF and CoCp
|
||
\begin_inset script subscript
|
||
|
||
\begin_layout Plain Layout
|
||
2
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
n-type doping taken from
|
||
\begin_inset CommandInset citation
|
||
LatexCommand citet
|
||
key "david-paper"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
(see table
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "tab:david-values"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
) were simulated and can be seen presented in figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:david-simulation-conductivity"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
Similarly to the original, the magnitude (figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:david-magnitude"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
) of the function can be seen to be between 48 and 63 mS for TTF and CoCp
|
||
\begin_inset script subscript
|
||
|
||
\begin_layout Plain Layout
|
||
2
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
respectively with both having a cutoff frequency of around 100 GHz.
|
||
Beyond the cutoff frequency, the value is around 1 mS by 10 THz.
|
||
The imaginary component peaks over the same frequency band that the real
|
||
component declines and the two intersect at around 150 GHz at a conductivity
|
||
of 31 mS with CoCp
|
||
\begin_inset script subscript
|
||
|
||
\begin_layout Plain Layout
|
||
2
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
and 24 mS for TTF.
|
||
After this intersection, the imaginary component can be seen to be the
|
||
dominant term of the complex quantity, this can be seen in the graph as
|
||
the magnitude tends closer to the imaginary component than the real
|
||
\begin_inset Note Comment
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
.
|
||
Beyond 100 THz, the imaginary component dips below zero, with a trough
|
||
of -0.5 mS around 250 THz
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
.
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Looking at the phase information (figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:david-phase"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
), before 10 GHz the phase can be seen to be approximately 0, however as
|
||
the imaginary component begins to peak, the phase increases to a max of
|
||
90 degrees, continuing until 100 THz where the phase sharply drops to -90
|
||
degrees.
|
||
There is little difference between the two dopants, particularly prior
|
||
to 100 THz.
|
||
Following this, the TTF shows a -100 THz offset from the CoCp
|
||
\begin_inset script subscript
|
||
|
||
\begin_layout Plain Layout
|
||
2
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
species.
|
||
CoCp
|
||
\begin_inset script subscript
|
||
|
||
\begin_layout Plain Layout
|
||
2
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
also reaches -56 degrees as compared to TTF's -50 degrees at its minima.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float table
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\noindent
|
||
\align center
|
||
\begin_inset Tabular
|
||
<lyxtabular version="3" rows="3" columns="3">
|
||
<features tabularvalignment="middle">
|
||
<column alignment="center" valignment="top">
|
||
<column alignment="center" valignment="top">
|
||
<column alignment="center" valignment="top">
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Dopant
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Carrier Concentration (cm
|
||
\begin_inset script superscript
|
||
|
||
\begin_layout Plain Layout
|
||
-2
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
)
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Fermi Level (eV)
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
TTF
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $1.3\times10^{13}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
0.41
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
CoCp
|
||
\begin_inset script subscript
|
||
|
||
\begin_layout Plain Layout
|
||
2
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $2.2\times10^{13}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
0.53
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
</lyxtabular>
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset VSpace defskip
|
||
\end_inset
|
||
|
||
With Fermi velocity energy scale,
|
||
\begin_inset Formula $t$
|
||
\end_inset
|
||
|
||
= 3 eV
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Carrier concentration values for n-type dopants from
|
||
\begin_inset CommandInset citation
|
||
LatexCommand citet
|
||
key "david-paper"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
and the Fermi levels derived from the model, see figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:fermi-concentration-func"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "tab:david-values"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\noindent
|
||
\align center
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\noindent
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename ../Resources/david-recreation-mag.png
|
||
lyxscale 20
|
||
width 50col%
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:david-magnitude"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\noindent
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename ../Resources/david-recreation-phase.png
|
||
lyxscale 20
|
||
width 50col%
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:david-phase"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Complex conductivity magnitude (a) and phase (b) for TTF and CoCp
|
||
\begin_inset script subscript
|
||
|
||
\begin_layout Plain Layout
|
||
2
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
doping at 300 K with a scatter lifetime of 1 ps
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "david-paper"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:david-simulation-conductivity"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The two contributions to this complex conductivity, intraband and interband,
|
||
can be seen individually in figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:david-simulation-inter-intra"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
Comparing the two, it can be seen that the interactions happen over largely
|
||
separate frequency ranges.
|
||
In general, the intraband conductivity can be seen to exist up to the 10
|
||
THz portion of the spectrum while the interband has the majority of its
|
||
contributions above the 10 THz range.
|
||
The intraband can be seen to dominate the total conductivity, it is largely
|
||
invisible in the previous combined view of figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:david-magnitude"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
The interband interactions begin after the 10 THz range - as the real component
|
||
steps from 1
|
||
\begin_inset Formula $\mu S$
|
||
\end_inset
|
||
|
||
to 60
|
||
\begin_inset Formula $\mu S$
|
||
\end_inset
|
||
|
||
around 200 THz, the imaginary component displays a negative peak over the
|
||
same spectral band.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\noindent
|
||
\align center
|
||
\begin_inset Float figure
|
||
wide false
|
||
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
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:david-intraband"
|
||
|
||
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|
||
|
||
|
||
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|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
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|
||
wide false
|
||
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|
||
status open
|
||
|
||
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|
||
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|
||
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|
||
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|
||
filename ../Resources/david-recreation-inter-mag.png
|
||
lyxscale 20
|
||
width 50col%
|
||
|
||
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|
||
|
||
|
||
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|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
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|
||
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|
||
LatexCommand label
|
||
name "fig:david-interband"
|
||
|
||
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|
||
|
||
|
||
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|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Intraband (a) and interband (b) conductivity for TTF and CoCp
|
||
\begin_inset script subscript
|
||
|
||
\begin_layout Plain Layout
|
||
2
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
doping at 300 K with a scatter lifetime of 1 ps
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "david-paper"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:david-simulation-inter-intra"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The Fermi level used to calculate conductivity (listing
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "calculation_function"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
) was derived from the net carrier concentration as a result of doping,
|
||
see listing
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fermi_from_carrier_density"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
The non-linear function can be seen modelled in figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:fermi-concentration-func"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
From this point, net carrier concentration and dopant concentration may
|
||
be used somewhat interchangeably with the understanding that they are related
|
||
by this function.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
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|
||
\noindent
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename ../Resources/fermi-conc.png
|
||
lyxscale 20
|
||
width 60col%
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Fermi level associated with different net carrier concentrations
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:fermi-concentration-func"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Carrier Density
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The general trends for how the dopant-influenced net carrier concentration
|
||
influences conductivity can be seen in the surfaces of figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:surf-carrier-concentration"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
To select a suitable range to visualise, the values from table
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "tab:david-values"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
and figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:fermi-concentration-func"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
were considered.
|
||
Realistic dopant carrier concentrations can be seen to of the order of
|
||
|
||
\begin_inset Formula $1\times10^{13}$
|
||
\end_inset
|
||
|
||
|
||
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|
||
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|
||
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|
||
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|
||
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|
||
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|
||
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|
||
\xout off
|
||
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|
||
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|
||
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|
||
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|
||
cm
|
||
\begin_inset script superscript
|
||
|
||
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|
||
|
||
\family roman
|
||
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|
||
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|
||
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|
||
\emph off
|
||
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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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|
||
-2
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
or
|
||
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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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|
||
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|
||
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|
||
|
||
\begin_inset Formula $1\times10^{17}$
|
||
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|
||
|
||
m
|
||
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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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|
||
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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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|
||
|
||
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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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|
||
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|
||
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|
||
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|
||
\noun off
|
||
\color none
|
||
-2
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
.
|
||
For the simulation, values up to
|
||
\family 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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|
||
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|
||
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|
||
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|
||
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|
||
|
||
\begin_inset Formula $1\times10^{18}$
|
||
\end_inset
|
||
|
||
m
|
||
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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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|
||
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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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|
||
|
||
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|
||
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|
||
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|
||
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|
||
\emph off
|
||
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|
||
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|
||
\xout off
|
||
\uuline off
|
||
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|
||
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|
||
\color none
|
||
-2
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
for a Fermi level of 1.13 eV were chosen.
|
||
\end_layout
|
||
|
||
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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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|
||
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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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|
||
\align center
|
||
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|
||
filename ../Resources/carrier-density/real-com-carrier-surf-sl1e-12-T300-logCB.png
|
||
lyxscale 20
|
||
width 80col%
|
||
|
||
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|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:surf-carrier-conc-real"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\noindent
|
||
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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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|
||
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|
||
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|
||
filename ../Resources/carrier-density/im-com-carrier-surf-sl1e-12-T300-logCB.png
|
||
lyxscale 20
|
||
width 80col%
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:surf-carrier-conc-im"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Complex conductivity over frequency for different carrier densities at room
|
||
temperature with a scatter lifetime of 1 ps and a Fermi velocity energy
|
||
scale of 2.8 eV
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:surf-carrier-concentration"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Conductivity can broadly be seen to follow the same spectral profile as
|
||
a function of carrier concentrations up to
|
||
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|
||
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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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|
||
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|
||
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|
||
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|
||
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|
||
|
||
\begin_inset Formula $1\times10^{15}$
|
||
\end_inset
|
||
|
||
|
||
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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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|
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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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|
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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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|
||
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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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|
||
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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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|
||
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|
||
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|
||
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|
||
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|
||
-2
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
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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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|
||
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|
||
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|
||
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|
||
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|
||
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|
||
as can be seen in figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:david-magnitude"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
Beyond this threshold, the conductivity below 10 THz begins to increase
|
||
exponentially.
|
||
Above 100 THz, an opposite trend can be identified with areas of the surface
|
||
showing a lower value.
|
||
From the high frequency, high carrier smear visible in the surface, the
|
||
spectral behaviour of this lowering conductivity is not static but a function
|
||
of both frequency and carrier concentration.
|
||
|
||
\begin_inset Note Comment
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Variation in the lower frequency comes largely in the magnitude of the various
|
||
regions.
|
||
For both the real and imaginary component, the max value (pre-cutoff for
|
||
the real component and the peak of the imaginary component) can be seen
|
||
to be constant over net carrier concentrations up until around
|
||
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|
||
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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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|
||
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|
||
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|
||
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|
||
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|
||
10
|
||
\begin_inset script superscript
|
||
|
||
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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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|
||
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|
||
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|
||
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|
||
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|
||
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|
||
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|
||
15
|
||
\end_layout
|
||
|
||
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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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|
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|
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|
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m
|
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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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|
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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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|
||
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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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|
||
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|
||
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|
||
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|
||
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|
||
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|
||
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|
||
-2
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
, 21 mS for the real component and 11 mS for the imaginary.
|
||
Beyond
|
||
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|
||
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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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|
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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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|
||
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|
||
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|
||
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|
||
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|
||
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|
||
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|
||
a net carrier concentration of 10
|
||
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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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|
||
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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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|
||
15
|
||
\end_layout
|
||
|
||
\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
|
||
|
||
the maximum values begin to rapidly increase and by 10
|
||
\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
|
||
17
|
||
\end_layout
|
||
|
||
\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
|
||
|
||
they have increased by an order of magnitude to hundreds of milli-siemens.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For the real conductivity component, beyond this previously mentioned
|
||
\family roman
|
||
\series medium
|
||
\shape up
|
||
\size normal
|
||
\emph off
|
||
\bar no
|
||
\strikeout off
|
||
\xout off
|
||
\uuline off
|
||
\uwave off
|
||
\noun off
|
||
\color none
|
||
10
|
||
\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
|
||
15
|
||
\end_layout
|
||
|
||
\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
|
||
|
||
threshold, the cutoff frequency begins to increase as can be seen from
|
||
the higher gigahertz value smearing the lighter blue across a higher frequency
|
||
band.
|
||
This moves the cutoff from 100 GHz to around 1 THz.
|
||
For the imaginary component, at low carrier concentrations, the peak value
|
||
decreases to around 1
|
||
\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 increases beyond
|
||
\family roman
|
||
\series medium
|
||
\shape up
|
||
\size normal
|
||
\emph off
|
||
\bar no
|
||
\strikeout off
|
||
\xout off
|
||
\uuline off
|
||
\uwave off
|
||
\noun off
|
||
\color none
|
||
|
||
\begin_inset Formula $1\times10^{12}$
|
||
\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
|
||
|
||
this value decreases into negative values, the frequency at which the drop
|
||
occurs lowers and the steeper colour gradient indicates that the change
|
||
happens faster.
|
||
The earliest frequency that this occurs at is around 50 THz and
|
||
\begin_inset Formula $6\times10^{15}$
|
||
\end_inset
|
||
|
||
|
||
\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 100 GHz
|
||
intraband value increases in magnitude, the frequency for this high-frequency
|
||
imaginary conductivity 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 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
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:carrier-conc-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/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
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:carrier-conc-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 high and low carrier concentrat
|
||
ion graphene species at room temperature and a scatter lifetime of 1 ps
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:inter-intra-carrier-conc"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\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 conductivity for three graphene species of different carrier
|
||
concentrations decomposed into the intraband and interband components.
|
||
The blue series,
|
||
\family roman
|
||
\series medium
|
||
\shape up
|
||
\size normal
|
||
\emph off
|
||
\bar no
|
||
\strikeout off
|
||
\xout off
|
||
\uuline off
|
||
\uwave off
|
||
\noun off
|
||
\color none
|
||
a carrier density of
|
||
\begin_inset Formula $1.3\times10^{17}$
|
||
\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
|
||
|
||
|
||
\family default
|
||
\series default
|
||
\shape default
|
||
\size default
|
||
\emph default
|
||
\bar default
|
||
\strikeout default
|
||
\xout default
|
||
\uuline default
|
||
\uwave default
|
||
\noun default
|
||
\color inherit
|
||
, recreates TTF doping from figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:david-simulation-inter-intra"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
with two further theoretical species of lower dopant concentration.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Looking at the intraband interactions, both components can be seen to have
|
||
the same profiles as seen previously, the differences lie in magnitude.
|
||
Higher net carrier concentrations can be seen to increase the magnitude
|
||
exponentially, this behaviour was previously identified in figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:surf-carrier-concentration"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The interband conductivity can be seen to show more variation over the prescribe
|
||
d carrier concentration range.
|
||
Low carrier concentrations result in a high initial imaginary component
|
||
that does not descend into negative values.
|
||
As concentration increases, the imaginary component decreases more with
|
||
a sharper gradient, forming a sharp trough that also reaches its lowest
|
||
value at a higher frequency.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Alongside this imaginary decrease, the real component can be seen to increase
|
||
from a value between 1
|
||
\begin_inset Formula $\mu S$
|
||
\end_inset
|
||
|
||
and 30
|
||
\begin_inset Formula $\mu S$
|
||
\end_inset
|
||
|
||
depending on carrier concentration to the limit of 60
|
||
\begin_inset Formula $\mu S$
|
||
\end_inset
|
||
|
||
.
|
||
Although the differing species reach this same limit, their approach is
|
||
different.
|
||
The lower carrier concentration species begins at the higher 30
|
||
\begin_inset Formula $\mu S$
|
||
\end_inset
|
||
|
||
value and increases only slightly to the limit over a wider spectral range.
|
||
The higher carrier concentration species begins much lower at 1
|
||
\begin_inset Formula $\mu S$
|
||
\end_inset
|
||
|
||
before increasing to 60
|
||
\begin_inset Formula $\mu S$
|
||
\end_inset
|
||
|
||
in what is closer to a step action at the higher frequency of 200 THz.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\noindent
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename ../Resources/carrier-density/complex-lines-phase.png
|
||
lyxscale 20
|
||
width 60col%
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Complex conductivity phase for three net carrier concentrations at room
|
||
temperature and a scatter lifetime of 1 ps
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:carrier-conc-phase"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The complex phase information for these three dopant species can be seen
|
||
in figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:carrier-conc-phase"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
From these, it is clear that the dopant concentration has a significant
|
||
effect on the conductivity's phase in the terahertz spectrum.
|
||
As the carrier concentration is increased, the phase can be seen to increase
|
||
to 90 degrees for longer throughout the 10 GHz decade and begin dropping
|
||
into negative phase.
|
||
From figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:david-phase"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
, it can be seen that once the phase decreases below 0, the frequency of
|
||
the phase minima is also a function of carrier concentration with higher
|
||
carrier concentration species having a higher minima frequency.
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Temperature
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Values from 0 K to the breakdown temperature of graphene, 2230 K
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "graphene-high-temp"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
, were simulated in order to investigate the effect on conductivity.
|
||
|
||
\begin_inset Note Comment
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:surf-temperature"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
shows a surface of the conductivity spectrum over the prescribed temperature
|
||
range.
|
||
In general, temperature can be seen to have little effect on conductivity,
|
||
both real and imaginary.
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
From the real component, the pre-cutoff GHz peak can be seen to increase
|
||
from 224 mS to 253 mS when moving from near-room temperature to the breakdown
|
||
temperature of graphene.
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Looking to the imaginary component, the peak conductivity increases by roughly
|
||
15 mS.
|
||
More variation occurs at the higher frequency, THz conductivity.
|
||
The sharper colour gradient at lower temperatures become more gradual at
|
||
higher temperatures, this indicates that the intraband imaginary negative
|
||
peak takes place over a more gradual spectral range.
|
||
Looking to the low temperature behaviour, the imaginary component is less
|
||
stable, rapidly descending to 0 S by 3 K before the the characteristic
|
||
negative 0.6 mS 200 THz peak returns at 0 K.
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\noindent
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename ../Resources/temperature/real-com-temp-surf-sl5e-12-TTF.png
|
||
lyxscale 20
|
||
width 80col%
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\noindent
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename ../Resources/temperature/im-com-temp-surf-sl5e-12-TTF.png
|
||
lyxscale 20
|
||
width 80col%
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Complex conductivity over frequency for different temperatures
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:surf-temperature"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:inter-intra-temperature"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
presents the decomposed intraband and interband conductivity contributions
|
||
for three different temperatures, 10 K, 300 K and 2230 K in order to compare
|
||
low, room and high temperatures.
|
||
For intraband conductivity, temperature can be seen to have little effect
|
||
throughout the inspected thermal range.
|
||
The low and room temperature series' effectively overlap, while moving
|
||
to the upper-temperature limit increases the conductivity by only 5 mS
|
||
or 10%.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For the interband interactions, as the temperature increases, the negative
|
||
imaginary peak gets smaller in value with a smoother gradient.
|
||
For the real component, although the final value does not change, the gradient
|
||
with which it is approached changes.
|
||
At low temperatures, the increase takes place over a tight spectral range
|
||
with a sharp step action.
|
||
As the temperature increases, the spectral band over which the transition
|
||
occurs broadens with a smoother gradient while maintaining the centre frequency
|
||
of 200 THz.
|
||
Overall, considering the complex magnitude for interband interactions,
|
||
little variation can be seen throughout the prescribed temperature range.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\noindent
|
||
\align center
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\noindent
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename ../Resources/temperature/intraband-lines-mag.png
|
||
lyxscale 20
|
||
width 50col%
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:temp-intra"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\noindent
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename ../Resources/temperature/interband-lines-mag.png
|
||
lyxscale 20
|
||
width 50col%
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:temp-inter"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Intraband (a) and interband (b) conductivity for low, room and high-temperature
|
||
graphene using TTF doping and a scatter lifetime of 1 ps
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:inter-intra-temperature"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The phase information for these three temperatures can be seen in figure
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:temperature-phase"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
Similar to the magnitude, the phase can be seen to show little variations
|
||
below 10 THz.
|
||
Lower temperatures result in sharper drops around 100 THz with a larger
|
||
negative peak, whereas the higher temperature species has a smoother motion
|
||
that does not become negative.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\noindent
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename ../Resources/temperature/complex-lines-phase.png
|
||
lyxscale 20
|
||
width 60col%
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Complex conductivity phase for three different temperatures using TTF doping
|
||
at room temperature and a scatter lifetime of 1 ps
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:temperature-phase"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Scattering Lifetime
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
This section explores the effect of varying scatter lifetime,
|
||
\begin_inset Formula $\tau$
|
||
\end_inset
|
||
|
||
, on the conductivity.
|
||
For the range of values to use, existing data was considered.
|
||
1 ps is a typical figure in literature
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "david-paper"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
, with this in mind values between 10 ps and 0.1 ps were simulated.
|
||
Figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:surf-scatter-lifetime"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
explores the general trends throughout the prescribed range.
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Looking at the real component, the scatter lifetime can be seen to affect
|
||
both the cutoff frequency and the magnitude of the pre-cutoff value.
|
||
As the lifetime increases, the cutoff frequency occurs at a lower value,
|
||
from around 500 GHz to 50 GHz.
|
||
The magnitude of the conductivity also increases exponentially as the lifetime
|
||
is increased.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Considering the imaginary component, somewhat similar behaviour can be seen.
|
||
The same exponential growth in magnitude can be seen in the 100 GHz peak.
|
||
With regards to the spectral behaviour, increasing scatter lifetime reduces
|
||
the frequency of the leading edge of the peak, broadening the bandwidth.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\noindent
|
||
\align center
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\noindent
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename ../Resources/scatter-lifetime/real-com-SL-surf-300K-TTF10,14.png
|
||
lyxscale 20
|
||
width 80col%
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:surf-scatter-intra"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\noindent
|
||
\align center
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\noindent
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename ../Resources/scatter-lifetime/im-com-SL-surf-300K-TTF10,14.png
|
||
lyxscale 20
|
||
width 80col%
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:surf-scatter-inter"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Complex conductivity over frequency for different scattering lifetimes simulated
|
||
for TTF n-type doping at room temperature with a scatter lifetime of 1
|
||
ps
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:surf-scatter-lifetime"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:inter-intra-scatter-lifetime"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
presents the interband and intraband conductivity contributions for three
|
||
different scattering lifetimes.
|
||
The previously identified spectral changes and magnitude growth can be
|
||
seen in the intraband conductivity.
|
||
The bandwidth of the imaginary component for the higher lifetime species
|
||
is broadened while increasing the magnitude.
|
||
Looking at the interband contributions, the three series show no variation,
|
||
the scatter lifetime has no effect.
|
||
This can be seen in the surfaces of figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:surf-scatter-lifetime"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
as straight lines of sharp colour gradients through the scatter lifetime
|
||
range at 200 THz.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\noindent
|
||
\align center
|
||
\begin_inset Float figure
|
||
wide false
|
||
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 high-frequency phase is also unchanged
|
||
across the selected lifetimes.
|
||
the phase is affected in the lower gigahertz ranges, however, with a longer
|
||
lifetime being associated with an earlier rising edge in the spectrum and
|
||
a longer bandwidth of 90-degree phase.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Discussion
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The intraband transitions are derived from the Drude model of carrier transport
|
||
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "graphene-microwave,graphene-modal-prop-drude"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
This results in the characteristic real cutoff/imaginary peak behaviour
|
||
seen in figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:david-intraband"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
From the simulated data from the literature, the gigahertz cutoff for intraband
|
||
conductivity can be seen to have an associated phase increase.
|
||
This suggests an increase in inductive reactance throughout this spectral
|
||
range which would have implications for applications employing graphene
|
||
at these frequencies and could suggest an increase in delay.
|
||
N-type doped material can be seen to show a negative phase peak at terahertz
|
||
frequencies indicating capacitive behaviour over this spectrum.
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
As discussed, the interband conductivity is restricted to the higher–energy
|
||
terahertz portion of the spectrum than the lower energy intraband interactions.
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Comment
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
The reason for this can be seen reflected in equations
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:intra-conductivity"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:inter-conductivity"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
Unlike the intraband transitions where
|
||
\begin_inset Formula $E_{F}$
|
||
\end_inset
|
||
|
||
is singular, the interband transitions instead has only references to
|
||
\begin_inset Formula $2E_{F}$
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
The reason for this can be seen by considering the required energy for
|
||
each type of transition.
|
||
|
||
\begin_inset Note Comment
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
With a non-zero Fermi level offset from the Dirac point, resistance drops
|
||
as carriers are available for conduction.
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
When considering n-type doping, the Fermi level is increased from the Dirac
|
||
point.
|
||
As it does, the energy states between it and the Dirac point are filled
|
||
and thus are unavailable for electrons to transition into.
|
||
As interband conductivity involve transitions from the valence band or
|
||
lower Dirac cone to the conduction band or upper cone (when considering
|
||
n-type doping), in order for an electron to make a direct transition without
|
||
momentum change (straight up on a energy-momentum diagram) it must absorb
|
||
at least two times the Fermi level energy in order for the destination
|
||
to be an empty state.
|
||
This restriction, more formally that incident photons of angular frequency,
|
||
|
||
\begin_inset Formula $\omega$
|
||
\end_inset
|
||
|
||
, and thus energy
|
||
\begin_inset Formula $\hbar\omega$
|
||
\end_inset
|
||
|
||
will not be sufficient for interband transitions can be expressed as the
|
||
following,
|
||
\begin_inset Formula $\hbar\omega<2|E_{F}|$
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "david-paper"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
This is referred to as Pauli blocking after the exclusion principle of
|
||
the same name which defines the population limit of the energy states in
|
||
question.
|
||
\end_layout
|
||
|
||
\begin_layout Paragraph
|
||
Net Carrier Concentration
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The differences in conductivity between the two dopants from literature
|
||
presented in figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:david-simulation-conductivity"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
are more broadly described by the effects of varying net carrier concentration
|
||
(figures
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:surf-carrier-concentration"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:inter-intra-carrier-conc"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
) as this is the primary difference between dopants.
|
||
The non-linear relation between net carrier concentration and thus dopant
|
||
concentration with conductivity results in the largely constant intraband
|
||
conductivity up to 10
|
||
\begin_inset script superscript
|
||
|
||
\begin_layout Plain Layout
|
||
14
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
m
|
||
\begin_inset script superscript
|
||
|
||
\begin_layout Plain Layout
|
||
-2
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
carriers.
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
From the presented trends it is clear both that graphene must be heavily
|
||
doped in order to obtain significant gigahertz conductivity and that, more
|
||
generally, varying the dopant concentration provides a highly-tunable method
|
||
of altering graphene's electrical characteristics.
|
||
This has particular implications for terahertz applications when considering
|
||
the phase information.
|
||
As presented, the conductivity's complex phase can be shifted in frequency
|
||
and magnitude in the terahertz spectrum by varying the net carrier concentratio
|
||
n.
|
||
The phase can also be kept positive at terahertz frequencies with a lower
|
||
carrier concentration, although this severely limits the conductivity magnitude.
|
||
This flexibility in reactive electrical characteristics could prove applicable
|
||
to high-frequency applications.
|
||
\end_layout
|
||
|
||
\begin_layout Paragraph
|
||
Temperature
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The conductivity spectrum as a function of temperature shows promising results
|
||
for the electrical stability of graphene over a wide thermal operating
|
||
range.
|
||
The intraband conductivity, specifically, showed little variation in behaviour
|
||
between 10 K and the highest stable temperatures with the previously reported
|
||
10% increase.
|
||
Looking at the interband interactions, these results showed the opposite
|
||
trend with the magnitude being decreased as the temperature increased.
|
||
This would suggest that graphene could prove useful in high-temperature
|
||
devices with special consideration being needed for terahertz applications
|
||
where the phase can be more variable around the critical frequency.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\noindent
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename fermi-dirac.gif
|
||
lyxscale 75
|
||
width 50col%
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Fermi-Dirac distribution function for the occupancy probability of a fermion
|
||
as a function of temperature
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "fermi-dirac-dist"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:example-fermi-dirac"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The real intraband component also shows interesting behaviour, with the
|
||
differences in gradient about the 200 THz critical frequency resembling
|
||
the Fermi-Dirac distribution, see figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:example-fermi-dirac"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
Similarly to this function, a decreasing temperature increases the gradient
|
||
of this transition between two quasi-constant values, tending towards a
|
||
single step action as the temperature approaches 0 K.
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Higher temperatures allow higher interband conductivity at lower frequencies
|
||
than would otherwise be required to overcome the previously described
|
||
\family roman
|
||
\series medium
|
||
\shape up
|
||
\size normal
|
||
\emph off
|
||
\bar no
|
||
\strikeout off
|
||
\xout off
|
||
\uuline off
|
||
\uwave off
|
||
\noun off
|
||
\color none
|
||
|
||
\begin_inset Formula $\hbar\omega>2|E_{F}|$
|
||
\end_inset
|
||
|
||
restriction.
|
||
The higher energy of the electrons associated with their temperature reduces
|
||
the extra required energy to make the transition.
|
||
From figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:temp-inter"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
this can be seen in two aspects of the real conductivity.
|
||
Throughout the gigahertz spectrum, the higher temperature blue line has
|
||
a higher constant value than the others.
|
||
This higher conductivity is associated with the slightly lower energy required
|
||
to make an interband transition as a result of the electron's higher energies.
|
||
Additionally, as the critical temperature is approached, the higher temperature
|
||
series begins smoothly rising earlier than the lower temperature series'.
|
||
These behaviours are a result of the statistical distribution of both the
|
||
temperature of the electrons and the energy of the incident photons.
|
||
\end_layout
|
||
|
||
\begin_layout Paragraph
|
||
Scatter Lifetime
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
From the presented trends for how conductivity is affected by a varied carrier
|
||
scatter lifetime, it is clear that it has a significant effect on both
|
||
the magnitude and spectral behaviour of intraband conductivity.
|
||
A longer scatter lifetime was shown to increase the magnitude of gigahertz
|
||
conductivity, this can be justified by considering the meaning of the scatter
|
||
lifetime using the Drude model of charge transport.
|
||
The scatter lifetime is the average lifetime between collisions, more collision
|
||
s reduce the overall mobility of the material.
|
||
A longer lifetime suggests that electrons are able to travel for longer
|
||
distances at higher speed without being slowed by scatter events, this
|
||
explains the higher magnitude throughout the gigahertz spectrum.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The interband interactions were seen to be unaffected by the scatter lifetime.
|
||
This can be seen from the model where, unlike equation
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:intra-conductivity"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
, equation
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "eq:inter-conductivity"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
has no reference to a lifetime term.
|
||
As a higher energy interaction, electrons are less affected by the scattering
|
||
of atoms and thus scatter lifetime is not employed for these high-frequency
|
||
transitions.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Conclusion
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Two applications of graphene at high frequencies have been presented.
|
||
The use of graphene nanoribbons as a channel material for FET technology
|
||
was shown to be a suitable prospective material for challenging the current
|
||
silicon CMOS/MOSFET paradigm as short-channel effects become harder to
|
||
engineer around.
|
||
Following this, graphene's applicability to flexible antennae was evaluated.
|
||
With a less established consumer domain, graphene's bio-compatibility,
|
||
flexibility, stiffness and electrical properties could overtake the more
|
||
limited metal-based ink approaches.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The sheet conductivity of graphene was modelled and the effect of varying
|
||
carrier concentration, temperature and scatter lifetime was evaluated.
|
||
Carrier concentration was seen to have a significant effect on conductivity
|
||
as a result of its relation to the materials Fermi level.
|
||
Temperature, however, was shown to have little effect, a promising result
|
||
for the stability of graphene over a wide thermal operating range.
|
||
Scatter lifetime was also shown to have a significant effect on intraband
|
||
conductivity as a result of its importance in the Drude modelling of carrier
|
||
transport.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Newpage newpage
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "sec:bibliography"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset bibtex
|
||
LatexCommand bibtex
|
||
btprint "btPrintCited"
|
||
bibfiles "references"
|
||
options "bibtotoc"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Newpage pagebreak
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
\start_of_appendix
|
||
Source Code
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "sec:Code"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset CommandInset include
|
||
LatexCommand lstinputlisting
|
||
filename "../2D-Conductivity/sheet_conductivity.m"
|
||
lstparams "caption={Calculation function for 2D sheet conductivity},label={calculation_function}"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Newpage pagebreak
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset CommandInset include
|
||
LatexCommand lstinputlisting
|
||
filename "../2D-Conductivity/conductivity_calculations.m"
|
||
lstparams "caption={Script for calculating conductivity over a range of frequencies},label={sheet_calculation_script}"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset CommandInset include
|
||
LatexCommand lstinputlisting
|
||
filename "../2D-Conductivity/conductivity_calc_surface.m"
|
||
lstparams "caption={Script for calculating conductivity over a range of frequencies and presenting as a surface},label={sheet_calculation_script_surface}"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset CommandInset include
|
||
LatexCommand lstinputlisting
|
||
filename "../2D-Conductivity/conductivity_phase_calculations.m"
|
||
lstparams "caption={Script for plotting complex conductivity phase over a range of frequencies},label={phase_script}"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset CommandInset include
|
||
LatexCommand lstinputlisting
|
||
filename "../2D-Conductivity/fermi_conc.m"
|
||
lstparams "caption={Script for plotting net carrier concentrations against Fermi level},label={fermi_concentration_script}"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Newpage pagebreak
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset CommandInset include
|
||
LatexCommand lstinputlisting
|
||
filename "../2D-Conductivity/carrier_density_from_fermi.m"
|
||
lstparams "caption={Derive the carrier density for a given Fermi energy},label={carrier_density_from_fermi}"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset CommandInset include
|
||
LatexCommand lstinputlisting
|
||
filename "../2D-Conductivity/fermi_from_carrier_density.m"
|
||
lstparams "caption={Derive the Fermi energy for a given carrier density},label={fermi_from_carrier_density}"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Newpage pagebreak
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset CommandInset include
|
||
LatexCommand lstinputlisting
|
||
filename "../2D-Conductivity/fermi_velocity.m"
|
||
lstparams "caption={Derive the Fermi velocity for a given energy scale},label={fermi_velocity}"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset CommandInset include
|
||
LatexCommand lstinputlisting
|
||
filename "../2D-Conductivity/ev_to_j.m"
|
||
lstparams "caption={Convert electron-volts to joules},label={ev_to_j}"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset CommandInset include
|
||
LatexCommand lstinputlisting
|
||
filename "../2D-Conductivity/j_to_ev.m"
|
||
lstparams "caption={Convert joules to electron-volts},label={j_to_ev}"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_body
|
||
\end_document
|