Close
About
FAQ
Home
Collections
Login
USC Login
Register
0
Selected
Invert selection
Deselect all
Deselect all
Click here to refresh results
Click here to refresh results
USC
/
Digital Library
/
University of Southern California Dissertations and Theses
/
Raman spectroscopy and electrical transport in suspended carbon nanotube field effect transistors under applied bias and gate voltages
(USC Thesis Other)
Raman spectroscopy and electrical transport in suspended carbon nanotube field effect transistors under applied bias and gate voltages
PDF
Download
Share
Open document
Flip pages
Contact Us
Contact Us
Copy asset link
Request this asset
Transcript (if available)
Content
i
RAMAN SPECTROSCOPY AND ELECTRICAL TRANSPORT IN
SUSPENDED CARBON NANOTUBE FIELD EFFECT TRANSISTORS
UNDER APPLIED BIAS AND GATE VOLTAGES
by
Adam W. Bushmaker
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(ELECTRICAL ENGINEERING)
May 2010
Copyright 2010 Adam W. Bushmaker
ii
Epigraph
“It is a capital mistake to theorize before one has data. Insensibly
one begins to twist facts to suit theories, instead of theories to suit facts.”
Sir Arthur Conan Doyle – Sherlock Holms, A Scandal In Bohemia.
iii
Dedication
To my parents, Wayne and Katy, for their constant support.
iv
I have been extremely lucky during my tenure as a graduate student at USC, and I
would like to thank all those who have worked with me and enabled my success.
First, I would like to thank my advisor, Professor Stephen Cronin, who showed
me how to be a scientist. Throughout my time in his lab, he was patient, could see right to
the core issue of problems I had, and showed me how to communicate results by taking
something that is simple (data), understanding it as the manifestation of something
complicated (nature), and then transforming it back to something that is simple (a paper).
Thanks to all my lab mates, Mohammed Amer, Mehmet Aykol, Vaibhav Bora,
Chia-Chi Chang, Chun-Chung Chen, Mark Harrison, Wenbo Hou, I-Kai Hsu Rajay
Kumar, Charles Le Pere, Ko-Chun Lin, Zuwei Liu, Brian Luscombe, Hari Mahalingam,
Prathamesh Pavaskar, Fernando Souto, Jesse Theiss, David Valley, and Hung "Wayne!"
Wei-Hsuan, for the immeasurably valuable community they have provided in B113, and
for all the ideas and discussions.
I would especially like to thank my Parents, brother Lee and sister Kinzi for
supporting me through my PhD despite being so far from home. My parents always said
we can do whatever we want with our lives, and that kind of support helped me get
through this thesis. Also, thank you for all the legos growing up!
Thanks to Vikram Deshpande, Scott Hseih, and Marc Bockrath for extending
yourselves in collaboration with us. Without the world class samples you provided, my
research would have progressed more slowly for sure. Also thank you David Boyd for
our plasmonic device collaboration. I would like to thank F. Mauri, M. Saitta, and
Acknowledgements
v
Marcelo Kuroda for discussions concerning electron-phonon coupling theory. I would
also like to thank Kan Lee in the machine shop, Brian Thibeault at UCSB, and Richard
Bormett and Zhenhuan Chi at Renishaw for helping me with instrumentation and sample
fabrication, and Kim Reid, Angelique Miller, Mona Gordon and Jaime Zelada for
administrative support.
I have had the great chance to meet good friends in Los Angeles, who have
supported me in one way or another while here. It has been important to me to have a rich
life outside of school, and these people have helped me achieve that. Valentine Matrat,
Brian McCabe, Alex Fink, Matt Marhefka, Cristina Gonzales, Amy Wan, Atilla
Karakurum, Sidd Bikkannavar, Justus Brevik, Bardia Zandian, Payam Pakbin, Kimberley
Vertanen, Anil Bandhakavi, all the people on the USC Ski and Snowboard Team, the
Marathon Team, the Outdoor Club, and the Caltech Alpine Club, Also, thanks to my
close friends from Wisconsin Rapids, Platteville and elsewhere, who helped me develop
as a younger man. Adam Neitzel, Jason Turbin, Aaron Kroll, Jill Mehlbrech, Nathan
Kublank, Adam Freund, Emily Hixon, Laura Tucker, Gina Hilgers, Markys Wagner,
Grant Novey, Chris Donovan, Chad Moder, Jake Holler, Jack Singal, and John Bailey.
Finally, I would like to thank my girlfriend Jane Ferratt, who has stood by me for over
three years while I worked at USC, and linked her life to mine in so many ways.
In addition to Steve, there have also been many other people who have served as
my mentors. I owe a huge debt to your guidance and instruction on all the basic things in
a lab. Dr. Steve Stevenowski at LHS, Prof. Philip Young and Dr. Daniel Dahlquist at
UWP, Profs. Heinrich Jaeger and Sid Nagel at UC, Prof. Eric Black and Dr. Shanti Rao
vi
at Caltech, Prof. Ken Wharton at SJSU, Francois Treussart at ENSC, Dave Rapchun, Dr.
Alan Kogut, Michele Limon, Paul Mirel, Dr. David Chuss, at NASA GSFC, and Prof.
William Steier at USC.
At last, none of this research would have been possible without funding, which
came from DOE Award No. DE-FG02-07ER46376, the James H. Zumberge Fund, the
Powell Foundation, the ARCS foundation, and the National Science Foundation Graduate
Research Fellowship Program. A portion of this work was done in the UCSB
nanofabrication facility, part of the NSF funded NNIN network.
vii
Epigraph .............................................................................................................................. ii
Dedication .......................................................................................................................... iii
Acknowledgements ............................................................................................................ iv
List of Tables ..................................................................................................................... ix
List of Figures ......................................................................................................................x
Abstract ............................................................................................................................ xiv
Chapter 1: Background ........................................................................................................1
Introduction ..............................................................................................................1
Carbon Nanotube Physics ........................................................................................2
Excitonic Effects ......................................................................................................8
Raman Scattering in Carbon Nanotubes ................................................................11
Electronic Conduction in Carbon Nanotubes ........................................................15
Relationship Between the Mean Free Path and T ..................................................18
Relationship to Boltzmann Transport ....................................................................21
Relationship to Mobility Calculation .....................................................................23
Gate Voltage Response ..........................................................................................26
Phonon Scattering ..................................................................................................27
Experimental Setup and Device Fabrication..........................................................30
Chapter 2: Raman Measurements of Non-Equilibrium Electron Transport ......................35
Introduction ............................................................................................................35
Results ....................................................................................................................37
Preferential Heating ...............................................................................................40
Conclusion .............................................................................................................47
Chapter 3: Gate Voltage Controllable Heating Behavior ..................................................48
Introduction ............................................................................................................48
Results ....................................................................................................................51
Non-Equilibrium Behavior ....................................................................................58
Conclusion .............................................................................................................60
Table of Contents
viii
Chapter 4: Breakdown of the Born-Oppenheimer Approximation....................................61
Introduction ............................................................................................................61
Non-Adiabaticity in Carbon Nanotubes.................................................................63
Results ....................................................................................................................64
Conclusion .............................................................................................................71
Chapter 5: Raman Intensity Modulation ............................................................................72
Introduction ............................................................................................................72
Intensity Changes ...................................................................................................74
Potential Mechanisms ............................................................................................76
Mott Insulation .......................................................................................................80
Conclusion .............................................................................................................82
Chapter 6: Memristive Defect State Behavior ...................................................................83
Introduction ............................................................................................................84
Results ....................................................................................................................85
Current Direction Sensitive Annealing ..................................................................87
Landauer Model .....................................................................................................88
Memristive System Equations ...............................................................................91
Conclusion .............................................................................................................93
Chapter 7: Future Directions ..............................................................................................94
Temperature Dependence of the Intensity Effect ..................................................94
Temperature Dependence of Kohn Anomaly Switching .......................................95
Temperature Dependence of Conductance for an Advanced Landauer Model .....96
Further Investigations into Preferential Phonon Heating: .....................................97
Transverse Electric Fields on CNTs ......................................................................98
CNT p-n Diode Experiments .................................................................................98
Rayleigh Scattering Experiments on CNTs Showing the Intensity Effect ............99
Group Theory Analysis of Phonon-Phonon Coupling in CNTs ..........................100
Bibliography ....................................................................................................................101
Appendix: Matlab codes ..................................................................................................109
CNT Landauer Model ..........................................................................................109
CNT Minimum Conductance versus Temperature ..............................................115
Mean Free Path Calculations ...............................................................................116
CNT Phonon Dispersion Relationship .................................................................119
CNT Electronic dispersion relationship ...............................................................123
CNT Optical Phonon Renormalization (Kohn Anomaly) ...................................128
CNT Electronic Excitation spectrum ...................................................................130
Ab Initio Calculations Using Dacapo Package ....................................................134
ix
Table 2.1: Summary of electron and phonon parameters of 5 suspended nanotubes. .......46
Table 5.1: Data summary of qm-CNTs showing intensity modulation. Listed values
include nanotube diameter, G+/G- integrated Raman intensity ratio, maximum
Raman attenuation, attenuation energy gap (
Raman
), and electronic band gap (Egap). ....80
List of Tables
x
Figure 1.1: From carbon to nanotubes. (a-b) Atomic and molecular orbitals of carbon
(c-d) forming sp
2
hybridized bonding and assembly into (e) graphene and
(f) a CNT. ............................................................................................................3
Figure 1.2: Graphene lattice vectors and a (4,2) nanotube unit cell. ...................................4
Figure 1.3: Carbon bonding and the resulting band structure in graphene. .........................5
Figure 1.4: Wavevector quantization in nanotubes, yielding 1D cutting lines on the
2D graphene dispersion. ......................................................................................6
Figure 1.5: Density of states g(E) for a (9,0) nanotube (a) and a (10,0) nanotube (b)
(solid lines), and g(E) from graphene (dashed lines). .........................................7
Figure 1.6: Two photon absorption-emission energy diagram and spectral data plot
from Wang et al. .................................................................................................8
Figure 1.7: Excitonic wavefunction showing e-h localization to ~5nm along the
length of the nanotube. ........................................................................................9
Figure 1.8: The graphene phonon dispersion relationship (b), calculated in Matlab
using the 4
th
nearest neighbor spring constant dynamical matrix method (a). ..10
Figure 1.9: Diagram of the Raman scattering process. ......................................................11
Figure 1.10: Example Raman spectra from individual carbon nanotubes and selected
displacement eigenvectors ................................................................................13
Figure 1.11: The Kohn anomalies in m-CNTs. (a) Raman spectra of a metallic
nanotube, (b) optical phonon dispersion relationship, and (c) electron
phonon coupling processes giving rise to the two Kohn anomalies in (b)........14
Figure 1.12: Ballistic transport in a 1D system..................................................................16
Figure 1.13: Multiple scattering sources in a 1D system. ..................................................18
Figure 1.14: Particle transmission probability for a system of weak scatterers,
considering the case of no-scattering (ballistic) transmission (red solid) and
total (multiple reflection) transmission (green dashed). ...................................19
Figure 1.15: Energy diagram for a nanotube FET with a positive gate voltage applied. ..26
List of Figures
xi
Figure 1.16: Threshold optical phonon emission during high bias transport. ...................28
Figure 1.17: Current-voltage characteristics for nanotubes under high bias, showing
current saturation and negative differential conductance. .................................29
Figure 1.18: (a) Optical and (b) SEM image of device topography, (c) schematic
diagram of the optical setup, (d) the Renishaw InVia spectrometer, and (e)
3D sample geometry. ........................................................................................31
Figure 1.19: SEM images of CNT devices illustrating undercut etch and subsequent
Pt electrode self-alignment. ...............................................................................33
Figure 2.1: G band Raman spectral data versus bias voltage. G band Raman (a) shift,
(b) width, and (c) intensity. The inset shows the Raman spectra at zero bias
voltage. ..............................................................................................................38
Figure 2.2: (a) The G band Raman shift versus bias voltage, with the I-V
bias
inset
exhibiting NDC. (b) Raman spectra taken at V
bias
= 0V and 1.4V. ...................39
Figure 2.3: Optical phonon temperature versus electrical power. Temperature is
measured for the device in Figure 2.2 by anti-Stokes/Stokes Raman
spectroscopy and by G band downshift. ...........................................................42
Figure 2.4: Electrical resistance plotted as function of phonon population. The
phonon population is fit from the measured data in Figure 2.3, for the
device in Figure 2.2. The two models shown are for LO scattering through
emission plus non-equilibrium OP absorption and through OP emission
alone. .................................................................................................................45
Figure 3.1. (a) G band Raman spectra showing downshifts with increasing
temperature. (b) G band Raman shift versus temperature with linear fits. .......51
Figure 3.2: Electrical data and Landauer model results from a suspended quasi-
metallic CNT, with (a) conductance plotted versus V
g
, (b) I
d
plotted versus
V
b
, and (c) R plotted versus V
b
. The doped state (circles) shows NDC,
while the intrinsic state (squares) has a linear I
d
-V
b
relationship. .....................52
Figure 3.3: Electrical data from two additional suspended, metallic single-walled
carbon nanotubes (a) 5µm and (b) 2µm in length exhibiting NDC electrical
behavior with the application of a gate voltage. ...............................................56
Figure 3.4: Schematic of electronic potential versus position along the nanotube
length for the (a) doped (V
g
= -2V) and (b) undoped (V
g
= ½V
b
) cases. The
electric field strength is indicated by the bold arrows in the top of each
Figure. ...............................................................................................................58
xii
Figure 3.5: Electrical current (a, b) and Raman downshift-calculated temperature (c,
d) for the same CNT at two different values of V
g
. The undoped nanotube
shows equal phonon heating and an Ohmic I
d
-V
b
, while the gated (doped)
nanotube exhibits non-equilibrium heating and negative differential
conductance. ......................................................................................................59
Figure 4.1: Sample G band spectra from two suspended m-CNTs showing the Kohn
anomaly deactivation. The spectra in (a) correspond to the data plotted in
Figure 4.2, while those in 1(b) correspond to Figure 4.3 and Figure 4.4.
Note the complete absence of the defect-mediated D band. .............................63
Figure 4.2: Raman spectral data from a single walled carbon nanotube. LO (a) shift
(filled circles), (b) linewidth (FWHM, filled circles) and (c) -1/q (open
squares), as well as the electrical conductivity (filled circles) are plotted
versus gate voltage (V
g
). The Fermi energy (E
F
) is also indicated on the top
x-axis. The lines in (a) and (b) show the results of the adiabatic (dashed)
and non-adiabatic (solid) phonon renormalization models discussed below
(Equations (1.55) and (1.56)), while the solid line in (c) represents the
Boltzmann-Landauer transport model (Equation (1.57)). .................................65
Figure 4.3: Gate voltage evolution of the Raman spectra from a second device. The
gate voltage dependence of the parameters fit to these spectra are presented
in Figure 4.4(a-c). ..............................................................................................67
Figure 4.4: Raman spectral data from a second CNT device. (Spectra in Figure 4.3)
Raman LO (a) shift (filled circles), (b) linewidth (FWHM, filled circles)
and (c) -1/q (open squares), as well as the electrical conductivity (filled
circles) are plotted versus gate voltage (V
g
) and Fermi energy (E
F
). ................68
Figure 5.1: (a) Device geometry and (b) G band Raman spectra at various gate
voltages, with inset showing the G
-
band intensity as a function of V
g
(sample 8B). ......................................................................................................75
Figure 5.2: (a) Normalized Raman intensities of the RBM, G
+
(TO), and G’ bands
taken with a 633nm laser. (b) RBM intensity taken with 633 and 785nm
lasers, together with the RBM AS/S intensity ratio (633nm, normalized for
T=300K), plotted versus V
g
. ..............................................................................76
Figure 5.3: The G band spectra and conductance (G) for suspended (a) qm- and (b)
sc-CNTs, plotted versus V
g
and E
F
. G and E
F
are fit using the Boltzmann-
Landauer transport model. ................................................................................79
Figure 5.4: (a)
Raman
plotted versus E
gap
, determined by fitting the Boltzmann-
Landauer transport equation to electrical data. (b) E
F
plotted versus V
g
for
nanotube 22B2, showing the method for determining
Raman
. ..........................80
xiii
Figure 6.1: Device geometry and CNT defects. The microscope image of a suspended
metallic single-walled CNT device used in the experiment, and possible
defects in the nanotube wall: a vacancy (b), a 5-7 pair (c), and a
substitutional defect (d). ....................................................................................85
Figure 6.2: Conductance of a defected CNT device. (a) Low bias (V
b
= 100mV)
conductance versus gate voltage characteristics (G-V
g
) before laser
exposure, after laser exposure, and after current annealing. (b)
Conductivity versus time during laser exposure and (c) subsequent high
bias current annealing. Inset (b) shows the gate voltage at which the
measurement was taken. Inset (c) shows the bias voltage at which the
measurement was taken.....................................................................................86
Figure 6.3: Repeatable switching behavior. Gate sweep conductivity profiles after
successive ±1.2V bias voltage annealing show reversible switching
behavior. The data in (b) are offset for clarity. .................................................87
Figure 6.4: Conductance versus V
g
data and Landauer model fits from a suspended
nanotube device (a) before and (b,c) after laser exposure and after (b)
forward and (c) reverse bias annealing. The Landauer model fitting
parameters are shown on the right. ...................................................................90
Figure 7.1: Resistivity versus temperature for a Mott insulator
6
(a) and phase diagram
as a function of carrier density and U/t ratio, showing hypothetical Raman
intensity (University of Augsburg) (b). .............................................................95
Figure 7.2: Raman G
-
band frequency as a function of Fermi level a room temperature
and 77K. ............................................................................................................96
Figure 7.3: Conductance plotted versus gate-voltage at a variety of temperatures ...........97
Figure 7.4: Device schematic for applying transverse electric fields to a suspended
CNT FET. ..........................................................................................................98
Figure 7.5: Device schematic of a split-gate FET for creating a suspended CNT p-n
junction. .............................................................................................................99
Figure 7.6: Rayleigh scattering spectra from individual suspended CNTs of known
chirality. ............................................................................................................99
Figure 7.7: Optical phonon decay channels in graphene. ................................................100
xiv
Abstract
One-dimensional materials exhibit striking, unique phenomena that are not found
in two or three dimensions. For the last twenty years, single walled carbon nanotubes
(CNTs) have served as the prototypical experimental one-dimensional system. In this
thesis, I investigate experimental data and theoretical models of spatially and electrically
isolated single-walled CNTs field-effect transistors.
Carbon nanotubes are grown using chemical vapor deposition, which relies on
small percentage of as-grown CNTs landing across pre-defined Pt electrodes that
define the transistor. A Landauer model is developed, which explains the gate and
bias voltage dependence of the electrical transport in these devices, and which serves
as the basis for much of the analysis of the experimental electrical transport and
Raman data.
Raman spectra are collected from CNTs under high applied bias voltages. When
heated with an electrical current, the Raman spectra of CNTs downshift, and this shift can
be used as an in-situ temperature probe. In some CNTs, the various Raman bands are
observed to downshift unequally, indicating non-equilibrium phonon populations caused
by threshold optical phonon emission. The spatial temperature profile of several CNTs is
measured, and shorter CNTs (L < 2µm) are found to exhibit pronounced non-equilibrium
phonon populations, while longer CNTs (L = 5µm) exhibit thermal equilibrium behavior.
xv
In addition to bias voltage effects (heating), gate voltage effects are also
investigated. When metallic CNTs are doped with an applied gate voltage, two primary
effects are observed. First, the one-dimensional Kohn anomaly is shut off, and secondly,
large modulations are observed in the Raman intensity.
The Kohn anomaly is a phonon damping phenomenon unique to one-dimensional
systems, and caused by electron-phonon coupling at the Fermi energy. In metallic CNTs,
a Kohn anomaly can be observed in the Raman G band spectrum. Doping the CNTs
causes this Kohn anomaly to go away, which results in spectral shifts of the G band. A
phonon renormalization model is implemented to fit the results. This effect is used to
experimentally confirm, for the first time, the theoretically predicted breakdown of the
adiabatic Born-Oppenheimer approximation in individual CNTs.
In addition to the spectral shifts associated with the Kohn anomaly, large
variations of up to two orders of magnitude are observed in the Raman intensity of
pristine, suspended quasi-metallic single-walled CNTs in response to applied gate
potentials. No change in the resonance condition is observed, and all Raman bands
exhibit the same changes in intensity, regardless of phonon energy or laser excitation
energy. The electronic energy gaps correlate with the drop in the Raman intensity, and
the recently observed Mott insulating behavior in CNTs presented as an explanation for
the effect.
Finally, the combined effects of heating and doping are investigated. It is found
that doping can change the high bias electrical behavior of CNTs from Ohmic (linear) to
negative differential conductance (NDC). In addition, in some CNTs, there is an
xvi
accompanying change in phonon population distribution, from equilibrium in the Ohmic
regime, to non-equilibrium in the NDC regime. Threshold phonon emission is identified
as the mechanism behind this phenomenon, and a model is presented which utilizes a
new formulation of Matthiessen’s rule for threshold phonon emission processes.
1
Chapter 1:
Background
Introduction
Carbon nanotubes are a very interesting new material with exceptional
fundamental properties such as room temperature intrinsic carrier mobilities larger than
100,000 cm
2
/V∙s
62
, Young’s moduli equal to 1TPa with breaking strengths of up to 50
GPa
158
(maximum for steel is only ~6 GPa), and thermal conductivities in excess of 6000
W/m∙K, which is almost twice that of diamond
10
. The extremely high carrier mobilities in
carbon nanotubes arise from fundamentally low scattering rates. As a result of this, very
clean carbon nanotubes with high crystalline quality have recently been used to verify
fundamental phenomena and interesting one-dimensional physics such as Wigner
crystallization
52
, spin-orbit coupling
85
, exceptionally strong electron-phonon coupling
156
,
Kohn anomalies
120
, ballistic electron transport
82
, and Luttinger liquid behavior
15
. Based
on these exceptional intrinsic properties and recent groundbreaking scientific results, it is
clear that more work needs to be done in order to further our understanding of nanotube
physics and processing.
This thesis will focus on the collection and analysis of Raman and electron
transport data from pristine, suspended, single-walled carbon nanotube (CNT) field effect
transistors. The combined electrical and optical data yield new insight into the
fundamental physical properties of CNTs. The advantages of this configuration for data
collection and analysis are manifold: (1) The absence of noise arising from CNT-
2
substrate interactions and defects allows me to study the intrinsic properties of CNTs. (2)
The elimination of homogeneous broadening effects associated with averaging
measurements over many CNTs enabled me to observe subtle phenomena that affect
different carbon nanotubes differently. (3) The ability to simultaneously take electrical
and Raman data enables correlations to be made between vibrational and electronic
phenomena. Finally, (4) suspended CNTs are known to exhibit an enhancement in their
Raman intensities by a factor of 20x with respect to nanotube-on-substrate devices, due to
an effect that is not yet fully understood
160
. This enhancement allows for clean data to be
taken from isolated, non-resonant CNTs with reasonable integration times.
Carbon Nanotube Physics and Electronic Structure
CNTs are rolled up sheets of single layered graphite, called graphene. A sheet of
graphene is composed of a honeycomb crystal structure of carbon atoms bonded with sp
2
hybridization, as shown in Figure 1.1.
3
Figure 1.1: From carbon to nanotubes. (a-b) Atomic and molecular orbitals of carbon (c-d) forming sp
2
hybridized bonding and assembly into (e) graphene and (f) a CNT.
First, we must start with graphene. The crystal structure of graphene is hexagonal
with two atoms per unit cell. The lattice vectors are shown below in Figure 1.2 (from
Dresselhaus et al.
56
) as
1
a
and
2
a
, with a a a
2 1
. A nanotube must be formed
with a periodic circumference C
h
, specified by
2 1
a m a n C
h
, where n and m are the
nanotube chiral indices, with 0 m n , to eliminate degeneracies. The circumference of
the nanotube is then
2 2
m nm n a C
h
, with the diameter /
h t
C d , and chiral
CNT
(a) (b) (c)
(d) (e) (f)
4
angle
m n
m
2
3
tan
1
. The length of the unit cell is then
3
h
R
TC
D
, where the
degeneracy factor ) 2 , 2 ( m n n m GCD D
R
. For the special case of n = m, nanotubes
are called armchair nanotubes, and those with m=0 are called zigzag nanotubes
130
.
Figure 1.2: Graphene lattice vectors and a (4,2) nanotube unit cell.
The two bond types in graphene give rise to two bands, the -band and the -
band. The mixing between the sp
2
orbitals in the -band is strong, resulting in bonding
and anti-bonding bands that are separated by an energy gap larger than 10eV. Thus, these
bands do not contribute significantly to the optical or electronic properties of carbon
nanotubes or graphene. The p orbitals in the -band mix weakly, however, resulting in
the bonding and anti-bonding (valence and conduction) bands shown in Figure 1.3 (from
Minot
99
). The key feature of the band structure of graphene is the linear dispersion
relationship at the two half-filling points (one for each atom in the graphene unit cell),
sometimes referred to as the Dirac cones or K points
133
.
5
Figure 1.3: Carbon bonding and the resulting band structure in graphene.
In nanotubes, the electronic wavevector around the circumference of the nanotube
(
k ) is quantized because of the periodic boundary conditions, illustrated in Figure
1.4(b). The “cutting lines” represent the allowed values for the electronic wavevector in
-bond
-bond
s
p
2
p
Z
-bands
p
Z
(a)
(b)
(c)
6
the 2-D graphene dispersion relationship. The cutting lines on the Dirac cone form either
hyperbolic bands with a band gap, or linear metallic bands, depending on the chirality of
the nanotube. Nanotubes with mod
3
(n-m) = 0 are metallic, while those with mod
3
(n-m) =
1 or 2 are semiconductors. Mod
3
(n-m) = 0 nanotubes, in fact, always have a small band
gap, due to curvature effects (except in armchair nanotubes) and Mott insulating behavior
(for pristine nanotubes of all chiralities).
Figure 1.4: Wavevector quantization in nanotubes, yielding 1D cutting lines on the 2D graphene dispersion.
(a)
(b)
(c)
7
The hyperbolic sub-bands create van-Hove singularities in the density of states, as
shown in Figure 1.5 for a (9,0) metallic CNT and a (10,0) semiconducting CNT, which
act as quasi-molecular energy levels, and greatly enhance the scattering cross-section of
optical transitions when the light is resonant with two van-Hove singularities.
Figure 1.5: Density of states g(E) for a (9,0) nanotube (a) and a (10,0) nanotube (b) (solid lines), and g(E)
from graphene (dashed lines).
(a)
(b)
8
Excitonic Effects
Due to the one-dimensional confinement of electrons in carbon nanotubes,
electron-hole interaction is strong, and screening by free electrons is weak. This results in
excitonic binding energies on the order of 1eV, which is much larger than excitonic
binding energies typically found in bulk semiconductors, which are on the order of
10meV
59
. For many years, these large predicted excitonic binding energies were not
observed experimentally due to the difficulty in distinguishing between an excitonic
transition and one dominated by free carrier transitions at the van-Hove singularities. The
excitonic nature of optical transitions was finally experimentally confirmed in 2005 using
two-photon photoluminescence spectroscopy
150
. From that paper, Figure 1.6 shows the
exciton states below the single-particle continuum states, and the corresponding two-
photon absorption and emission data. This paper showed that the excitonic transitions
dominate the optical spectra, and continuum transitions are negligible. Therefore, any
realistic calculations of CNT optical properties need to consider excitonic transitions.
Figure 1.6: Two photon absorption-emission energy diagram and spectral data plot from Wang et al.
9
In addition having similar spectral line-shapes, the predicted single particle and
excitonic transition energies are nearly the same, due to the quasi-particle self-energy,
self
, which nearly cancels out the excitonic binding energy
ex
, as shown in
equation (1.1) below.
optical ii self ex
EE
(1.1)
The relatively small difference between the exciton and self energies gave rise to the
so-called ratio problem, which concerned the discrepancy between the predicted and
experimentally observed 1
st
and 2
nd
transition energies
76
.
The excitonic wavefunction is not simply the sum of two single-particle electronic
and hole wavefuctions; it is a localized wavefunction constructed from a superposition of
many low energy single-particle states near the band edge. This localization is illustrated
by Figure 1.7 from Wang et al.
150
, which shows the excitonic wavefunction to be
localized to approximately 5nm along the length of the carbon nanotube.
Figure 1.7: Excitonic wavefunction showing e-h localization to ~5nm along the length of the nanotube.
10
Phonon Dispersion Relations
As with the electronic structure, the phonon structure in CNTs can be derived
from the phonon dispersion relations of graphene using zone folding as well. The phonon
dispersion relationship of graphene can be calculated to surprising accuracy using a
spring constant model, as shown in Figure 1.8. One must consider 4
th
nearest neighbor
interactions to get a reasonable fit of neutron experimental scattering data, however the
spring constants for the 20 nearest neighbor interactions have also been determined from
fits to data
133
. The details of this model are described by Saito et al.
130
. A Matlab program
for calculating the dispersion relations is provided in the Appendix.
Figure 1.8: The graphene phonon dispersion relationship (b), calculated in Matlab using the 4
th
nearest
neighbor spring constant dynamical matrix method (a).
(a) (b)
11
Raman Scattering in Carbon Nanotubes
Raman scattering occurs when light scatters off a lattice vibration in matter.
Energy and wavevector are conserved in the scattering process where a phonon is either
emitted or absorbed as the incoming photon scatters into the outgoing photon, as depicted
in Figure 1.9.
Figure 1.9: Diagram of the Raman scattering process.
For a phonon to be Raman active, its displacement of the lattice must generate a
change in the electric susceptibility of the material, such as that given by
01
cos( )
ph
t (1.2)
where
is the electric susceptibility,
0
is the baseline susceptibility,
1
is the
change caused by the phonon oscillation, and
ph
is the phonon frequency. When this
interacts with an incoming photon, an electric polarization is generated, which can be
expressed as
12
0 0 0 0 1
1
0 0 0 1 2
cos( ) cos( )
cos cos
ph L
L L ph
P E E t t
E t t
.
(1.3)
This acts as a source for light emission at frequencies
ph L Raman
, where
0
is the permittivity,
L
is the frequency of the incoming (laser) light, and
0
E
is the
electric field strength
66
. If a phonon is emitted, then it is called a stokes (S) process, and if
a phonon is absorbed, it is called an anti-Stokes (AS) process. The relative intensities of
these two processes are given by
) / exp(
1
T k E
N
N
I
I
B ph
AS
S
, (1.4)
where N is the occupation number of the phonon. The Raman process is greatly enhanced
when the incoming light is resonant with an electronic transition. The resonant Raman
intensity is given by
63
2
] ) ( ][ ) ( [
) (
i k E i k E
dk M M M
C I
ph L L
op ep op
L
, (1.5)
where C is a constant, E
µ
is the van-Hove singularity transition for the µ
th
electron band,
E
ph
is the phonon energy, γ is the resonance broadening energy, M
op
is the optical matrix
element for the electron-photon interaction, and M
ep
is the electron-phonon coupling
matrix element, and +/-
ph
is for phonon absorption/emission.
There are several Raman bands in CNTs, as shown in Figure 1.10(a). They
include, (1) the radial breathing mode (RBM) (Figure 1.10(b)), (2) the G band, split by
the nanotube curvature into two bands (Longitudinal Optical (LO) and Transverse
13
Optical (TO)) (Figure 1.10(c) and (d)), (3) the disorder-induced D band, and (4) the 2D
(or G’) band. The diameter of the CNT is inversely proportional to and can be determined
from the RBM
98
. The G band arises from the -point optical phonons. This mode is
important because through its behavior, much interesting physics can be observed.
Figure 1.10: Example Raman spectra from individual carbon nanotubes and selected displacement
eigenvectors
(a)
(b) (c) (d)
14
The LO phonon G
-
Raman feature of metallic CNTs (m-CNTs) is fundamentally
different than that of their semiconducting counterparts (sc-CNT)
57
. In m-CNTs, the G
-
band is broadened and downshifted (reduced in frequency), as shown in Figure 1.10(a)
and Figure 1.11(a); an effect arising from coupling to a continuum of electronic states
22,
35, 48, 50, 64, 79, 125, 145, 152
. Figure 1.11(b) shows the effect of this coupling on the OP
dispersion relationship (from Piscanec et al.
120
). In other words, the LO phonon mode is
damped by the free electrons near the Fermi energy
61, 120
. This coupling is a Kohn
anomaly (KA) and has also been referred to as a weakened Peierl’s-like mechanism.
There are two Kohn anomalies in CNTs, one at the -point, which affects the Raman
signal, and one at the 2k
F
-point.
Figure 1.11: The Kohn anomalies in m-CNTs. (a) Raman spectra of a metallic nanotube, (b) optical phonon
dispersion relationship, and (c) electron phonon coupling processes giving rise to the two Kohn anomalies
in (b).
(a) (b)
(c)
15
Electronic Conduction in Carbon Nanotubes
The resistance to electron transport in a one-dimensional system can be described
well by the Landauer transport model, which calculates conduction by treating the
possible scattering sources in the nanotube as barriers with a certain probability of
electron transmission. This approach allows for the possibility of ballistic transport, i.e.,
transport through a system without scattering. Traditional Boltzmann transport
calculations are not capable of describing a material in this regime, however the Landauer
model handles it easily.
The current through a conductor with only one quantum channel (i.e., an
electronic band whose wavevector is single-valued in the transverse dimension but
continuous in the longitudinal dimension) is described by
2
2
( , ) ( , ) ( ) | |
SD
e
I f E eV f E eV g E v dE
, (1.6)
where g(E) is the density of states, v is the group velocity, and f is the Fermi function.
This calculation is graphically depicted in Figure 1.12. In this model, I have assumed
perfect transmission at the contacts.
16
Figure 1.12: Ballistic transport in a 1D system.
For a 1D system, the density of states is simply the inverse of the group velocity,
and so g(E)∙v = 1/ħ. To calculate the conductance, we then simply divide the current by
the bias voltage V
b
= V
S
-V
D
dE
V
eV E f eV E f
h
e
V
I
G
b
D S
b
) , ( ) , ( 4
2
. (1.7)
The value in front of the integral,
h e G
2
0
4
, is the quantum conductance for a perfect
nanotube k G R 5 . 6 1
0 0
, and consists of two parallel degenerate quantum
channels.
We can approximate the integrand with
dE
df
, which is valid for bias voltages not
larger than the thermal energy, or for larger bias voltages under diffusive transport
conditions (i.e., not ballistic). Since the density of states is zero in the band gap, the
17
integral is only taken over the bands. This correctly incorporates the limit of an insulator,
where the conductance is zero, and we have
0
bands
df
G G dE
dE
. (1.8)
This equation assumes that the position of the Fermi level with respect to the bands is
known, and the method for calculating this is outlined below in Equation (1.37). To
incorporate the effects of scattering in the system, such as by a lattice defect or vibration,
we simply add the scattering transmission coefficient T,
0
bands
df
G G dE
dE
. (1.9)
We can describe this equivalently in terms of resistances, as
0
1
bands
RR
df
E dE
dE
. (1.10)
To start, we’ll consider metallic 1D system (i.e., systems without a band gap). In
this case (assuming T(E) constant), the resistance of the system is
00
11
1 R R R
, (1.11)
where we have separated the quantum resistance R
0
from the effects of the scattering
source. To calculate the transmission through multiple barriers, we have to consider
multiple reflections, as in Figure 1.13. The total transmission coefficient in this case is
given by
22
12
1 2 1 2 1 2 1 2 1 2
12
...
1
tot
R R R R
RR
. (1.12)
18
From this, the resistances of an arbitrary number of scattering sources adds like
12
12
1 11
...
tot
tot
. (1.13)
Using this formula, we can calculate the resistance in a system with multiple scattering
sources, such as a carbon nanotube with contact barriers, diffusive scattering, and local
defects.
Figure 1.13: Multiple scattering sources in a 1D system.
Relationship Between the Mean Free Path and T
In order to calculate the relationship between the mean free path λ and the
transmission coefficient T, we will use a weak scatterer approximation, where T
s
~ 1. In
this approximation,
1
1
ss
s
s
L
, (1.14)
where the average distance between scatterers L
s
is much smaller than λ. This can be seen
in Figure 1.14, which shows the no-scattering transmission probability (probability of
ballistic transmission without a single scattering event) for a particle as a function of
...
1 2 1 2
RR
12
19
distance traveled (solid red curve - an exponential decay). The initial slope of the decay is
1
ss
L , which sets the decay constant or mean free path .
Figure 1.14: Particle transmission probability for a system of weak scatterers, considering the case of no-
scattering (ballistic) transmission (red solid) and total (multiple reflection) transmission (green dashed).
The total scattering coefficient for all the scatterers in the system (dashed green
curve in Figure 1.14), including multiple reflections, can then be calculated using
equation (1.13).
11
total s s
s s total
total s
L L
NN
L
(1.15)
Using this result, the resistance for a nanotube with perfect contacts is
L
R R
0
. (1.16)
Now, we can consider a system with a band gap. The integral over energy in Equations
(1.9) and (1.10) has a maximum value of unity, since T ≤ 1 and . So, in the
presence of the band gap, the integral is less than unity, which, physically, means that
1 lim
0
E f
E
20
charge carriers are being depleted. It is useful to separate out the effects of carrier
depletion due to the band gap from those of reduced mobility due to the scattering source.
Otherwise, adding multiple scatterers to the system double counts the effects of depletion,
which is a non-physical result (for instance, adding many scatterers of T(E) = 1 by
blindly applying Equation (1.11) with T =
bands
df
E dE
dE
would greatly change the
calculated resistance, even though it should not). We can separate these effects by
modifying Equation (1.10) as
* 0
00
*
11 bands
bands bands bands
df
dE
R
dE
R R R
df df df
E dE dE E dE
dE dE dE
**
* 12
0 **
12
11
1 R
. (1.17)
Here, is the depleted quantum channel resistance, or the resistance of the gapped
system in the absence of scattering sources, and
is the effective or weighted average
transmission coefficient in the quantum channel. If E is constant, then
.
From a simple example, one can easily see the importance of the separation of
terms in Equation (1.17). Consider a resonant scattering source in the channel, which
only scatters electrons within a narrow energy range. If that energy range happened to lie
inside the band gap, then the scatterer should not affect the conductance of the system,
because none of the conduction electrons are of the proper energy to scatter. This result is
found with the separation above. However, if we neglect to separate, then an additional
*
0
R
21
resistance due to the depletion is added for each such scattering source we add to the
system, which is a non-physical result.
The total resistance in a single walled nanotube, including contacts and diffusive
electron-phonon scattering, can be written as
*
*
*
0 **
1
1
12
ph
c
ph c
RR
(1.18)
where
*
ph
and
*
c
are the transmission coefficients for phonon scattering and the
contacts, respectively, as discussed below.
Relationship to Boltzmann Transport
We can calculate the electronic resistance of a one dimensional system using
Boltzmann transport theory, which, ignoring defect scattering, is given by
Fields Collisions
ff
tt
. (1.19)
Several assumptions are made, the first of which being that the electron distribution
function f can be expressed as
01
f f f (1.20)
where
0
f is the equilibrium Fermi distribution and
1
f is a small perturbation. Secondly,
we assume that the relaxation rate is
1
f divided by a constant relaxation time ,
1
Collisions
f f
t
. (1.21)
22
We can then use these approximations to express the Boltzmann equation again as
0 1
Fields Fields Fields
f f f k f E k E
t t E k t E k
, (1.22)
where k is the electronic wavevector. The correspondence principle, which states that a
classical force is equivalent to a changing wavevector can be expressed as
p k eE
ke
E
t
(1.23)
We can substitute this into Equation (1.22) to obtain
00 1
g
Fields Fields
ff f k E e
Ev
t E k E
0
1 g
f
f e E v
E
(1.24)
To calculate the net current in a system out of equilibrium, we can simply sum the current
carried by all the k-states out of equilibrium, as in
1
1 1 1
2 2 2
()
g
dk
D g g dE
d
dk
e v f
j e v k f dk e v f dE dE
. (1.25)
Since the group velocity
g
v
is just
d
g dk
v
, we can substitute to get the conductance
2
0 1
4
D
g
f j e
v dE
hE E
. (1.26)
Considering the mean free path
g
v
, and the length of the channel L we can
calculate the conductivity as
0
0
f
G G dE
L L E
, (1.27)
23
where
0
G is the CNT quantum conductance. The only difference between this result and
the result of the Landauer model above is that in the Landauer model,
L
L
accounts for ballistic conduction.
Relationship to Mobility Calculation
Traditionally, the electrical conductivity in semiconductors is calculated using the
mobility µ, by the equation
en
,
(1.28)
where
dE E g E f n
band
conduction
(1.29)
is the charge carrier density of the material, ignoring p-type conduction. This formula
breaks down for degenerately doped semiconductors and metals, because not all charge
carriers contribute to the conductivity equally; only carriers near the Fermi energy
contribute to conduction. Electrons with energies far below the Fermi energy do not
contribute because of Pauli exclusion. This is not a problem for most semiconductor
calculations, however, because only the tail of the Fermi distribution (light doping
regime) extends into the conduction band.
The Landauer model can be shown to be equivalent to the mobility equation
en
G
L
in the light doping approximation. First, using the Einstein mobility relation,
24
T k
eD
B
,
(1.30)
where T k
B
is the thermal energy of the system and D is the diffusion coefficient. The
diffusion coefficient is v D . So assuming two quantum channels for the CNT
(spin up and spin down), we have
2
2
B conduction
band
en e v
G f E g E dE
L L k T
. (1.31)
As in Equation (1.6), the density of states cancels the group velocity in 1D, and we get
2
4
B
conduction
band
fE
e
G dE
h L k T
. (1.32)
In the limit
F
B
EE
kT
>> 1,
T k
E f
T k T k dE
df
B B B
1
exp exp 1
exp 1
exp 1
exp
2
. (1.33)
So,
dE
dE
df
L
G G
band
conduction
0
,
(1.34)
which is the same as Equation (1.27), because the mobility equation is based on a
diffusive transport model. Under ballistic conditions ( λ > L), the mobility equation and
Boltzmann transport break down because the conductance must not go below
0
G .
25
Relaxation Time Energy Dependence
The “constant” relaxation time is not actually constant, but a function of energy.
When the electronic density of states near the band edge becomes large at the so-called
van-Hove singularity, many low-energy AC scattering processes are possible. The
electronic relaxation rate in CNTs caused by acoustic phonon scattering can be calculated
using deformation potential theory
77, 112, 148, 159
(for graphene
11, 70, 140
), where the
scattering rate with respect to phonons of wavevector q is given by Fermi’s golden rule as
2
,,
1
,
2
1
i f i f
if
ac kq q q q q q q
k g q E N E N dq
(1.35)
where N
q
is the phonon population, given by Bose-Einstein statistics,
,
i f f
i
q k k q
E E E
is the energy change,
kq
g is the electron phonon coupling matrix
element, and
i
and
f
are the initial and final electronic bands, respectively. Substituting
ff
ii d
k k q k k q dq
dq d E E E E
, making a quasi-elastic approximation for
acoustic scattering processes, only considering intra-band processes, and approximating
ph B
E k T
gives
2
1
kq B
ac
g
g k T
E
vE
, (1.36)
which shows that the scattering rate for acoustic phonons increases proportionately to the
temperature and the density of states (and so is divergent at the band edges in a one-
dimensional system).
26
Gate Voltage Response
In order to calculate the gate voltage (V
g
) response of the nanotube according to
these equations, we must first find the relationship between E
F
and V
g
. This can be done
by examination of the potential on the charged nanotube shown below in Figure 1.15.
The Fermi energy is given by
C
E Q
E eV
F
F g
) (
, (1.37)
where is the gate efficiency factor (close to unity for the CNT devices used here), C is
the geometric capacitance of the nanotube, and Q(E
F
) is the charge on the nanotube,
which can be found using Equation (1.29) (remembering to include both p and n-type
charging). A suspended nanotube can be thought of as a special case of the MOS
capacitor, where the depletion layer width is much wider than the nanotube, and so the
Fermi energy is constant over the width of the nanotube.
Figure 1.15: Energy diagram for a nanotube FET with a positive gate voltage applied.
27
Phonon Scattering
The effective mean free path (MFP) of electrons in the nanotube according to
Matthiessen’s rule is
1
1
,
1
,
1
abs OP ems OP ac eff
. (1.38)
At low bias,
eff
is dominated by acoustic phonon scattering mean free path,
ac
. This
is because electrons may emit OPs only after gaining a threshold energy, given by the
phonon energy, and OP absorption is limited by the negligible occupation of OPs at room
temperature. Defect scattering in pristine devices is negligible. Other studies have
examined the energy dependence of the scattering rate
116
. However, here, we limit
ourselves to the constant relaxation time approximation, by reducing the scattering length
proportionally to the group velocity (
hi
ac F ac
v E v E v E ) ( ) ( ) ( ), which results in
short scattering lengths near the band edges
84
. The temperature dependence of the
electron-acoustic phonon scattering length is given by K T T
RT hi
ac
hi
ac
300 /
,
, where
RT hi
ac
,
is taken to be 1 m in accordance with previous work
84, 111
.
High bias transport in nanotubes is unique because optical phonon (OP) scattering
becomes the dominant source of scattering. The optical phonon scattering length for
emitted phonons is given by
) ( 1
1
min
,
op op
op
ph
ems OP
T N qV
L E
, (1.39)
28
and for absorbed phonons by
) (
1
min
,
op op
op abs OP
T N
. (1.40)
In these equations,
ph
E is the OP energy, and
min
op
is the scattering length for
electron scattering from OP emission in the nanotube after the electron has accelerated to
high energy ≥
ph
E , as shown in Figure 1.16.
Figure 1.16: Threshold optical phonon emission during high bias transport.
This model has one crucial parameter,
min
op
, which governs the details of high bias
transport, as OP scattering becomes the dominant contribution to the MFP. This threshold
emission process has a peculiar consequence for nanotubes at high bias. Because the
resistance goes up linearly with the bias voltage, the current in metallic CNTs thermally
sunk to a substrate saturates to 25 mA at high bias voltage, regardless of nanotube
chirality, length, contact resistance or temperature
156
, as shown in Figure 1.17(a). Figure
(a) (b)
29
1.17(b) shows that when the nanotube is suspended, however, optical phonon absorption
becomes an important factor, and negative differential conductance (NDC) is observed as
the nanotube becomes hot.
Figure 1.17: Current-voltage characteristics for nanotubes under high bias, showing current saturation and
negative differential conductance.
(a)
(b)
30
Experimental Setup and Device Fabrication
Suspended CNTs were grown using CVD on Pt electrodes prepared with islands
of lithographically patterned catalyst. Out of 40 devices per 5mm x 5mm chip, typically
only a few have just one CNT bridging the contacts of the device. The trench width for
the devices range from 0.5µm to 5µm. Figure 1.18(a) and (b) show one such device
fabricated in this study. Raman spectra are measured in a Renishaw InVia spectrometer
(Figure 1.18(d), resolution ~1cm
-1
) with 532nm, 633nm, and 785nm Ti-Sapphire lasers
focused to a diffraction limited spot through a 100X high numerical aperture objective
lens on a Leica DMLM microscope. Figure 1.1 shows an optical microscope image of a
typical suspended nanotube sample measured in this study. The focused laser spot can be
seen positioned in the center of the trench. The nanotube cannot be seen optically but is
detected by its strong Raman signal and by scanning electron microscopy. An Ithaco
current preamplifier was used to measure the current passed through the nanotube. The
devices are tested in a sample chamber containing argon gas in order to 1) prevent burn-
out of the devices when they are heated to high temperatures by bias voltages and to 2)
eliminate water adsorption onto the surface of the CNT devices, which causes hysteresis
in the gate voltage response of the conductance.
31
Figure 1.18: (a) Optical and (b) SEM image of device topography, (c) schematic diagram of the optical
setup, (d) the Renishaw InVia spectrometer, and (e) 3D sample geometry.
Sample fabrication for the first two years of this project was done at Caltech by
our collaborators Vikram Deshpande and Professor Marc Bockrath, and later at USC. A
portion of this work was done in the UCSB nanofabrication facility, part of the NSF
(a)
(c)
(b) (d)
(e)
32
funded NNIN network. The devices are made on Si substrates capped with 500nm-
1000nm of SiO
2
and 50nm-100nm of Si
3
N
4
31, 124
. A trench is patterned in PR, and the
Si
3
N
4
/SiO
2
/Si substrate is dry-etched in CF
4
plasma to form the trench. Using the same
PR mask, the underlying oxide is wet-etched to a depth of 300nm-800nm, which forms
an overhanging Si
3
N
4
cantilever. The overhang is crucial to prevent shorting between the
metallization layers on the Si
3
N
4
and those in the trench. Using this system allows the
design of self-aligning two-layer metallization electrodes.
It is important not to etch all the way through SiO
2
layer, because the trench
metallization layers can form platinum-silicide with the underlying Si substrate in the
high temperature reducing environment of CNT growth. I attributed this as the cause of
large protruding growths observed near the edges of the trench electrodes when I etched
too deep. The Pt contacts are then patterned. I used 5nm W as an adhesion layer for 25nm
of Pt. Thicker layers of Pt might withstand the temperatures of CNT growth a little
better
65
. I found that W worked better than Ti in the high temperature CNT growth
environment. When designing the electrodes, it is important to avoid using a common
gate and drain, since a single nanotube shorting the two together can ruin the entire chip.
Only one electrode should be common to all devices.
Islands of Fe-Mo catalyst salts in an alumina matrix are then deposited on top of
the contacts in lithographically defined areas
81
. Nanotube growth is carried out by
flowing a mixture of hydrogen (0.7SLM) and either methane (0.5SLM) or ethanol vapor
carried by argon over the wafer for 5 minutes at 825
o
C. Devices that show negative
differential conductance at high bias (1-2V) with a maximum current of ~10/L µA (where
33
L is in µm) correspond to individual suspended single-walled nanotubes, and are selected
for further study
122
. No additional processing was performed on devices after the
nanotube growth, except for an oxygen bake to rid the devices of amorphous carbon.
Figure 1.19: SEM images of CNT devices illustrating undercut etch and subsequent Pt electrode self-
alignment.
34
It is well-known that the 1/f noise in CNT FETs
21
is quite substantial. This leads
to experimental difficulties when trying to measure Raman spectra with respect to the
gate voltage because the effective Fermi energy is not constant with time. The 1/f noise
decreases significantly when CNT FETs are removed from air
21, 51
. Also, a reduction in
1/f noise has been observed in suspended devices, due to the absence of substrate
interactions
93
. All of the gate voltage experiments reported here were done in an inert
argon or nitrogen environment, nearly eliminating the effect. In order to further protect
against signal contamination by 1/f noise, an optical lock-in measurement technique was
used, in which the applied gate potential is switched between positive and negative values
and synchronized with Raman scans, collected at each voltage for 8s. The resulting
Raman scans taken at each gate voltage are then summed a posteriori. By integrating
over many cycles, I obtain sufficient Raman intensity from individual, isolated
nanotubes. Additionally, the noise caused by laser positioning errors and sample drift is
eliminated. In order to implement the optical lock-in technique, the data acquisition
software was edited to synchronize the electronic and optical components.
35
Chapter 2:
Raman Measurements of Non-
Equilibrium Electron Transport
This chapter is similar to Bushmaker et al.
25
, published in Nano Letters.
Chapter 2 Abstract: Raman spectra of individual pristine suspended single-
walled carbon nanotubes are observed under high electrical bias. The LO and TO modes
of the G band behave differently with respect to voltage bias, indicating preferential
electron-phonon coupling and non-equilibrium phonon populations, which cause negative
differential conductance in suspended devices. By correlating the electron resistivity to
the optically measured phonon population, the data are fit using a Landauer model to
determine the key scattering parameters.
Introduction
Electron-phonon (e-ph) coupling in carbon nanotubes has been studied by many
research groups
32, 35, 60, 61, 87, 89, 134
. In metallic carbon nanotubes (m-CNTs), conduction
electrons have been predicted to couple strongly to the Г-point longitudinal optical (LO)
phonons and to the 2k
F
-point transverse optical (TO) phonons
35, 60, 61, 87, 89, 134
. The G band
Raman spectra of m-CNTs and semiconducting CNTs (s-CNT) are qualitatively different
because of this strong electron-phonon coupling
57
. In metallic nanotubes, the lower-
frequency component of the G band (G
−
) exhibits a broad Breit-Wigner-Fano (BWF)
36
lineshape, and is significantly downshifted in frequency with respect to its counterpart in
semiconducting nanotubes
58
. Recent experiments have shown that this phonon softening
can be removed by shifting the Fermi energy of m-CNTs with an applied gate voltage or
chemical doping, which modulates this coupling and results in an upshift of the G
−
band
frequency
48, 79, 125
. Density functional theory calculations have shown that the assignment
of the upper and lower frequency components of the G band Raman modes, G
+
and G
-
, to
the LO and TO Г-point phonon modes may be reversed in m-CNTs and sc-CNTs
120
. This
is thought to be caused by the strong downshifting of the LO mode due to the Kohn
anomaly, also referred to as a Peierls-like distortion phenomenon
61, 127
. Other experiments
have shown that this downshift may be caused by inter-nanotube bundling effects
86
.
Negative differential conductance (NDC) has been observed by several research
groups at high voltage bias in suspended nanotubes and is understood on the basis of
electrons coupling strongly to Г-point and 2k
F
-point optical phonons (OPs)
88, 95, 122
. At
high voltage bias, the electrons emit OPs, causing increased scattering from absorption of
those OPs, and an increase in resistance. In the experiment reported here, I
simultaneously observe the OP Temperature and nanotube resistance, and am able to
correlate the electron scattering length to phonon population.
When CNTs are heated, the G band downshifts in frequency
3, 9, 39, 44, 68, 69, 91, 126
,
broadens
75
, and decreases in intensity
3, 39
due to anharmonic phonon decay
7, 18
. In thermal
equilibrium, both the LO and TO optical phonons downshift together. I observe
preferential downshifting of only one of the OPs at high currents, indicating strong
coupling of electrons to only one band and a non-equilibrium phonon population.
37
Preferential e-ph coupling and coherent phonon generation was first reported in
Ruby in 1961
146
, and was followed by other reports describing the phenomenon in GaAs
and other semiconductor crystals
97, 102, 139
. More recent work analyzed selective
amplification and emission of OPs in electron transport experiments
128
, and a full
quantum treatment of THz phonon laser design
30
. This observation of selective e-ph
coupling in carbon nanotubes supports the possibility of using carbon nanotubes as a
source of coherent phonons as suggested elsewhere
88
. The devices in this work were
fabricated as described in Chapter 1.
Results
Figure 2.1 shows the G band Raman modes of a nanotube device under large
voltage biases. The band gap of this nanotube was determined to be ~60meV from the
current-gate voltage dependence. The G
+
band is observed to downshift by more than
26cm
-1
, while the G
-
on average does not change by more than 1cm
-1
. There is a clear
crossing that occurs at ~1.0V, above which the G
-
band becomes higher in frequency than
the G
+
band. The linewidths of the G
+
and G
-
bands in the nanotube of Figure 2 also vary
with the applied bias voltage. Here the G
+
band broadens while the G
-
band remains
largely unchanged. Finally, the intensity of the G
+
band decreases monotonically with
bias voltage, while the G
-
band remains constant. This behavior suggests preferential
heating of the G
+
phonon mode, since the G band Raman spectra are known to downshift,
broaden, and diminish in intensity with increasing temperature
3, 9, 39, 44, 68, 69, 75, 91, 126
. The
38
integrated areas of both the G
+
and the G
-
Raman peaks remain constant, indicating that
there is no change in the resonance condition of this nanotube with applied bias.
Preferential heating of the G
+
phonon was observed in 4 out of 15 devices measured in
this study, including one semiconducting device. Because the unbiased G
-
band exhibits a
broad, downshifted BWF lineshape, I assign it to the LO phonon mode, and I assign the
G
+
to the TO phonon mode.
Figure 2.1: G band Raman spectral data versus bias voltage. G band Raman (a) shift, (b) width, and (c)
intensity. The inset shows the Raman spectra at zero bias voltage.
Figure 2.2 shows the G band Raman modes of another nanotube under large
voltage biases. NDC can be clearly seen above 1.2V in the current-voltage (I-V
bias
)
characteristics of this device, as shown in the inset of Figure 2.2. Here, the voltage
dependence of the G
+
and G
-
bands are reversed from those shown in Figure 2.1. Over the
range of applied bias voltage, the G
-
band is observed to downshift by 15cm
-1
, while the
G
+
band does not change by more than 1cm
-1
. Furthermore, the linewidth of the G
-
band
increases significantly with bias voltage and drops in intensity, while the G
+
band
remains of constant width and intensity. Contrary to Figure 2.1, this data exhibits
(a) (b) (c)
39
preferential heating of the G
-
band, which I again assign to the LO Г-point phonon mode.
This case is rare and was only observed in one out of fifteen nanotubes measured in this
study. Again, the integrated areas of both the G
+
and G
-
peaks remain constant, indicating
that there is no change in the resonance condition. The broadening of the G
+
feature is
consistent with thermal broadening in CNTs as reported by Jorio et al.
75
. Both nanotubes
shown in Figure 2.1 and Figure 2.2 are metallic, and all changes in the Raman spectra are
reversible.
Figure 2.2: (a) The G band Raman shift versus bias voltage, with the I-V
bias
inset exhibiting NDC. (b)
Raman spectra taken at V
bias
= 0V and 1.4V.
A weak radial breathing mode (RBM) was observed in the Raman spectra of this
nanotube at 146.5cm
-1
, which corresponds to a nanotube diameter of 1.70nm by the
relation 27 / 204
t RBM
d .
98
The strongly enhanced Raman intensities from
suspending the carbon nanotubes off the substrate
160
make it possible to observe RBMs
with the incident laser off resonance. The weak RBM observed for this nanotube implies
an off-resonance condition, which creates significant uncertainty in the excitonic
transition energy and hence the chirality assignment of this nanotube. The maximum
(a) (b)
40
current density of this nanotube can be obtained by dividing the peak current (10 A) by
the cross-sectional area of the nanotube (3.81×10
-18
m
2
), resulting in a peak current
density of 5.3×10
8
A/cm
2
.
Preferential Heating
The behaviors shown in Figure 2.1 and Figure 2.2 can be explained by the
previous theoretical work of Piscanec et al.
120
, which describes the strong electron-
phonon coupling of the Kohn anomalies (KA) in metallic carbon nanotubes. One KA
occurs at zero momentum (Γ-point) in the LO phonon band and gives the G
-
band in
metallic nanotubes its downshifted and broadened BWF lineshape
22
. Another KA occurs
at a finite phonon momentum q = 4π/3T (2k
F
-point) in the TO phonon branch, where T is
the length of the unit cell in the nanotube. These two KAs provide the primary source of
electron-phonon scattering in pristine m-CNTs at high bias voltages.
It is surprising that the narrow G
+
band (TO band) in Figure 2.1 is so strongly
coupled to the electrons, while the broad G
-
band (LO band) remains unchanged with
applied bias voltage. This can be explained by considering that the energy of the 2k
F
-
point phonons associated with the TO KA (~0.16 eV) is significantly lower than the
energy of the Γ-point phonons of the LO KA (~0.195 eV). This results in a lower
threshold energy for TO phonon emission in electron transport. Thus the electrons are
scattered by emitting TO phonons before ever attaining enough energy to emit LO
phonons, which results in heating of only the TO phonon band. This, together with the
fact that the electron-phonon coupling for the 2k
F
-point KA is two times stronger than
that of the Γ-point KA
120
, explains why the G
+
band (TO) is observed to be strongly
41
heated for nanotubes of the type shown in Figure 2.1. The Raman downshift with
increasing temperature is caused by increasing intermediate frequency phonon
populations
7, 18
, which are the decay products of the optical phonons, and couple
differently to the LO and TO phonon bands, which are orthogonal to one-another.
The seemingly contradictory results of Figure 2.2, in which only the G
-
band (LO)
shifts with applied bias voltage, can be understood in terms of a rare chirality, where
1
) 2 , 2 (
) , (
n m m n GCD
m n GCD
R
, (1.41)
which only occurs for slightly less than 1/3 of all metallic nanotubes. In fact, this
behavior was only observed 1 out of 15 nanotubes measured in this study, which is
consistent with the rarity of this chirality. In this case, the Raman active TO phonon
branch does not exhibit a KA
120
, and heating by hot electrons is only observed in the LO
phonon band.
The high temperatures reached under large voltage biases were corroborated by
anti-Stokes (AS) Raman spectroscopy. A G band anti-Stokes peak was observed at biases
above 0.4V on the device shown in Figure 2.2. The ratio of the AS (absorbed phonons) to
the Stokes (emitted phonons) Raman intensity is given by the Maxwell-Boltzmann factor
exp(-E
ph
/k
B
T), where E
ph
is the phonon energy (195meV), k
B
is Boltzmann’s constant and
T is the temperature in Kelvin. Figure 2.3 shows the temperature as determined from the
AS/S ratio plotted as a function of electrical power. The temperature shows a linear
dependence on electrical power that reaches ~700
o
C at high bias. At higher voltages, the
nanotube was destroyed. This temperature is consistent with the work of Cataldo, who
measured the burnout threshold of carbon nanotubes in air to be ~800
o
C.
34
The optical
42
phonon temperature was also determined independently from the downshift of the G
-
band by the relation 5 . 1581 10 5 . 6 10 5 . 3 ) (
3 2 5
,
T T T
LO G
, which was
measured on one of my devices in a temperature controlled stage. This data is also
plotted in Figure 2.3 and is in good agreement with the anti-Stokes/Stokes ratio data,
indicating the optical phonon is in thermal equilibrium with its intermediate frequency
phonon decay products. I would like to point out that, while the anti-Stokes spectra of
the LO mode yields a temperature of ~700
o
C, the TO mode was not observed in the anti-
Stokes spectra. This, together with the lack of change in the TO Stokes Raman
frequency, indicates that the population of the TO phonon remains close to room
temperature.
Figure 2.3: Optical phonon temperature versus electrical power. Temperature is measured for the
device in Figure 2.2 by anti-Stokes/Stokes Raman spectroscopy and by G band downshift.
By measuring electrical resistivity and optical phonon population simultaneously,
one gains new information about the phonon scattering mechanism responsible for the
observed NDC
95, 122
as suggested by Lazzeri
88
. Figure 2.4 shows the electrical resistance
43
plotted as a function of LO phonon population N
op
(T
op
), which is fitted from the
experimental data in Figure 2.3. This can be understood using the Landauer model
developed by Pop
122, 123
, Mann
95
, Park
110
, Yao
156
, and others
77, 88
, in which the nanotube
resistance is expressed as
) , (
) , (
4
) , (
2
T V
T V L
q
h
R T V R
eff
eff
c
, (1.42)
where R
c
is the contact resistance, L is the nanotube length, and
eff
=(
ac
-1
+
op,ems
-
1
+
op,abs
-1
)
-1
is the bias and temperature dependent electron mean free path
95, 122
. The
acoustic scattering length is given by
ac
=
1
300
ac
RT
ac
T K . The acoustic phonon
temperature is T
ac
=(T
op
+
sample
)/(1+ ), where the non-equilibrium phonon coefficient
is taken as 2.3 from Mann, et al.
95
and the optical phonon temperature T
op
is measured by
Raman spectroscopy. The optical phonon scattering length for emitted phonons is given
by
op,ems
=
) ( 1
) 300 ( 1
min
op op
op
op
ph
T N
K N
qV
L E
(1.43)
and for absorbed phonons by
op,abs
=
) (
) 300 ( 1
min
op op
op
op
T N
K N
. (1.44)
In these equations,
ph
E is the OP energy, and
min
op
is the scattering length for electron
scattering from OP emission in the nanotube after the electron has accelerated to high
44
energy ≥
ph
E . Low energy electrons may scatter with this length scale from absorption of
thermally populated OPs as well, as described by Equation (1.44). In addition to the
constant contact resistance R
c
, this model has one fitting parameter,
min
op
. An
approximate value of
RT
ac
= 2400nm was used in the fit in accordance with previous
work
110
, and the fitted value for
min
op
was generally found to be insensitive to the value of
RT
ac
.
The solid and dashed lines in Figure 2.4 correspond to fits of my data using this
model with OP emission and absorption and with OP emission alone, respectively, with
min
op
= 26nm. This value is consistent with those reported previously in the literature
72, 110
.
The model including OP emission and absorption is in good agreement with the
experimental results for phonon populations below 0.09. The failure of the model
without OP absorption indicates the important role that the non-equilibrium optical
phonon population plays in the electron transport of suspended CNTs. At larger phonon
populations, corrections to the model are needed to account for the non-uniformity of the
temperature along the length of the nanotube, as shown previously in finite element
thermal analysis calculations
95
.
45
Figure 2.4: Electrical resistance plotted as function of phonon population. The phonon population is fit
from the measured data in Figure 2.3, for the device in Figure 2.2. The two models shown are for LO
scattering through emission plus non-equilibrium OP absorption and through OP emission alone.
We have performed a systematic study measuring the optical and high bias
electronic properties of 5 suspended nanotubes that exhibited preferential downshifting of
the G
+
or G
-
band. This data has been fit to the model described above and their results
are listed in the table below. The table lists the metallic/semiconducting nature and band
gap of the nanotubes, as determined from the electron transport data. The Raman feature
that is preferentially downshifted with bias voltage is also indicated in the table. The
diameter is indicated for nanotubes that exhibited a RBM in their spectra. Despite the
very different results observed in their optical spectra, there is little variation in the
optical phonon scattering parameter
min
op
amongst m-CNTs.
46
Table 2.1: Summary of electron and phonon parameters of 5 suspended nanotubes.
10 out of the 15 nanotubes measured in this study did not exhibit preferential
downshifting of the G
+
or G
-
bands and were not included in Table 2.1. In 5 of these 10
nanotubes, the relative intensity of the G
+
/G
-
bands was so great that a clear resolution of
both peak positions was not possible, and hence it was not possible to observe whether
preferential heating occurred. The G
+
/G
-
intensity ratio has been theoretically predicted
and experimentally shown to be a function of chiral angle
131, 152
. I attribute the behavior
of these 5 nanotubes to the extreme cases of large and small chiral angles. In the
remaining 5 nanotubes not shown in Table 2.1, both the G
+
and the G
-
bands downshifted
when heated with electrical current. This is attributed to anomalous phonon-phonon
anharmonic coupling, and further indicates the high purity of the pristine nanotube
samples that did exhibit strong selective coupling and extreme non-equilibrium phonon
populations.
47
Conclusion
In conclusion, preferential electron-phonon coupling of the G Raman bands is
observed in carbon nanotubes under high voltage bias. This preferential coupling is
caused by the differences between the two Kohn anomalies in the TO and LO Raman
bands. Surprisingly, in most metallic nanotubes, the narrow G
+
band (TO band) is
strongly heated by electron-phonon scattering at high biases. Because of the preferential
electron-phonon coupling, high voltage biases produce a non-equilibrium phonon
population, as observed by anti-Stokes Raman spectroscopy. By correlating the electron
resistivity to the phonon population, measured by Raman spectroscopy, I determine the
high energy electron-OP scattering length
min
op
in m-CNTs to be ~ 30nm.
48
Chapter 3:
Gate Voltage Controllable Heating
Behavior
This chapter is similar to Bushmaker et al.
28
, published in Nano Letters.
Chapter 3 Abstract: In this work, I measure the electrical conductance and
temperature of individual, suspended quasi-metallic single-walled carbon nanotubes
under high voltage biases using Raman spectroscopy, while varying the doping
conditions with an applied gate voltage. By applying a gate voltage, the high-bias
conductance can be switched dramatically between linear (Ohmic) behavior and non-
linear behavior exhibiting negative differential conductance (NDC). Phonon populations
are observed to be in thermal equilibrium under Ohmic conditions, but switch to non-
equilibrium under NDC conditions. A typical Landauer transport model assuming zero
band gap is found to be inadequate to describe the experimental data. A more detailed
model is presented, which incorporates the doping dependence in order to fit this data.
Introduction
The exceptionally strong electron-phonon coupling in quasi-metallic (qm) single-
walled carbon nanotubes (CNTs) leads to exciting low dimensional physics including
Kohn anomalies
27, 35, 61, 119, 134
, charge-density waves
8
, and Peierls distortions
120
. Long
49
mean free paths and high electron velocities make nanotubes an ideal system for studying
these interesting effects through high-field, quasi-ballistic electron transport
72
. Thermal
heating of suspended SNWTs with power densities several hundred times larger than the
surface of the sun (~1MW/cm
2
) gives rise to non-equilibrium phonon populations
25, 87, 88,
95, 106, 122
and negative differential conductance (NDC)
122
. These characteristics are of
interest for high frequency applications, such as mixers, which can operate in the GHz
and THz frequency ranges
129, 161
. Furthermore, CNTs are of great interest for use as high
conductance interconnects in advanced integrated circuits. Initial studies of these
phenomena have already been carried out; however, a next generation of more
sophisticated experiments is needed to further develop our understanding.
High bias electron transport in qm-CNTs has been shown to follow a Landauer
model, adopted to nanotubes by Yao
156
, Park
111
, Pop and Mann
95, 122
, and others
61, 87, 88,
120, 127
. At high bias voltages (V
b
), optical phonon scattering contributes strongly to the
resistance because of the strong electron-phonon coupling originating from the Kohn
anomalies in carbon nanotubes
61, 111, 120, 156
, and results in non-equilibrium phonon
populations
25, 87, 88, 106, 122
. Recently, surface polar phonons
114, 142
in substrate-supported
nanotubes have been identified as a significant electron scattering mechanism, however
this is absent in suspended nanotubes. Previous theoretical models and experiments on
fully doped qm-CNTs
88, 111, 122, 156
ignored the presence of a band gap, focusing instead on
the linear band-structure approximation. However, qm-CNTs do in fact exhibit small
electronic band gaps
13, 14, 31, 108, 162
. In addition, Deshpande et al. recently showed that
Mott insulating behavior in pristine, suspended, qm-CNTs opens larger band gaps of
50
~100meV
53
. Since this energy scale is larger than room temperature thermal energy, the
effects of the gap on electron transport are substantial. While many studies have observed
the effects of this mini-band gap on low bias electron transport in qm-CNTs
13, 14, 31, 108, 162
,
a quantitative model for the gate voltage (V
g
) dependence of low and high-bias electron
transport in qm-CNTs is still much needed.
In the measurements presented here, an electrical drain current (I
d
) is passed
through qm-CNTs by applying a bias voltage, while the nanotubes are doped
electrostatically with a back gate. The electrical data is fit to a Landauer model, which
incorporates the gate voltage dependence of electron transport in qm-CNTs. The
temperatures of the qm-CNTs are measured using Raman spectroscopy, enabling me to
examine the correlation between the vibrational and electronic properties, and to provide
a more detailed understanding of electrical transport behavior. The devices in this work
were fabricated as described in Chapter 1.
The G band vibrational modes are known to downshift with increasing
temperature due to the anharmonicity of the carbon-carbon bond
18, 19
. Typical values for
this rate of downshift lie in the range from -0.022 to -0.044cm
-1
/K
3, 39, 67
. These
downshifts can, in turn, be used to determine the temperature and phonon populations
54,
121, 142
(see Ch. 2). Figure 3.1 shows the calibration data of the G band temperature
dependence, measured in a temperature controlled stage. For the nanotube shown in
Figure 3.1(a), the temperature coefficients of the G
+
and G
-
bands are 0.028 and 0.027
cm
-1
/K, respectively. Using these temperature coefficients, I can convert the downshifts
observed in the G
+
and G
-
band frequencies under high voltage bias to temperatures.
51
Figure 3.1. (a) G band Raman spectra showing downshifts with increasing temperature. (b) G band Raman
shift versus temperature with linear fits.
Results
Figure 3.2 shows the conductance-gate voltage and current-bias voltage data from
a nanotube under two different gate voltages, plotted together with the results from the
model, outlined below. The conductance shows a strong dip near V
g
= 0V, which is
caused by the reduced carrier concentration when the Fermi energy (E
F
) is in the mini-
band gap. In Figure 3.2(b), the current observed at high bias exhibits negative differential
conductance when V
g
= -2V (i.e., p-doped) and Ohmic behavior when V
g
= ½V
b
(so that
E
F
is always in the middle of the gap, i.e., undoped). The significant band gap in this
nanotube causes the conductivity in the doped state to be significantly higher than the
intrinsic state. As can be seen in Figure 3.2(a), the model agrees well with the data for
low V
b
, and qualitatively reproduces the observed Ohmic-to-NDC behavior transition at
high V
b
.
52
Figure 3.2: Electrical data and Landauer model results from a suspended quasi-metallic CNT, with (a)
conductance plotted versus V
g
, (b) I
d
plotted versus V
b
, and (c) R plotted versus V
b
. The doped state (circles)
shows NDC, while the intrinsic state (squares) has a linear I
d
-V
b
relationship.
The data shown in Figure 3.2 are fit nicely with a numerical Landauer transport
model including the effects of a band gap, as described below. In the Landauer model
13,
25, 88, 95, 106, 111, 122, 156
, the nanotube resistance is expressed as
c s
R R R R 2
*
0
, (1.45)
where
s
R is the resistance arising from scattering in the CNT,
c
R is the contact
resistance, and
bands
dE
df
dE
q
h
R
1
4
2
*
0 , (1.46)
is the depleted quantum channel resistance. ƒ(E)
is the Fermi distribution function, q is
the charge of an electron, h is Planck’s constant, and the integral is taken over all
energies, except those in the band gap. The scattering contribution to the resistance in a
carbon nanotube is given by
*
*
*
0
1
s
s
s
R R , (1.47)
53
where
Bands Bands eff
eff
s
dE
dE
df
dE
dE
df
L
*
(1.48)
is the effective scattering coefficient within the depleted quantum channel. This equation
can be found using the Landauer model, or from the Boltzmann transport equation in
1D
137
. The electron mean free path
eff
is given by Matthiessen’s rule:
eff
=(
ac
-1
+
op,ems
-1
+
op,abs
-1
)
-1
. (1.49)
The acoustic scattering length is given by
ac
=
ac
RT
ac
T K 300 . The optical
phonon scattering length for emitted phonons is given by
op,ems
=
) ( 1
) 300 ( 1
min
op op
op
op
t
ph
T N
K N
qV
L E
, (1.50)
where V
t
is the (contact resistance-corrected) voltage drop across the nanotube and
ph
E =
0.16 eV is the optical phonon energy at the zone boundary. For absorbed phonons, it is
given by
op,abs
=
) (
) 300 ( 1
min
op op
op
op
T N
K N
. (1.51)
In these equations,
min
op
is the scattering length for electron scattering from OP
emission in the nanotube after the electron has accelerated to the threshold energy,
ph
E .
Low energy electrons may scatter with this length scale (
min
op
) from the absorption of
thermally populated OPs, as described by Equation (1.51). The scattering lengths used in
the fit were
RT
ac
= 1 m and
min
op
= 35nm, in accordance with previous work
25, 111, 122
. Other
54
studies examined the energy dependence of the scattering rate
116
. However, here, I use
the constant relaxation time approximation, by reducing the scattering length
proportionately to the group velocity ( ) ( ) ( E E ), resulting in short scattering
lengths near the band edges
The quantum contact resistance is given in the Landauer equation
4, 23
by
*
*
*
0
1
c
c
c
R R , (1.52)
where the effective transmission coefficient
*
c
(in the depleted quantum channel) is given
by
Bands
c
Bands
c c c
dE V f dE E V f ) (
*
, (1.53)
where V
c
is the contact voltage drop,
c F c F c
eV E f eV E f V f
2
1
2
1
, and ) (E
c
is the contact transmission coefficient, which approximates the tunneling through
Schottky barriers at the contacts
4, 23, 51, 96
. In order to account for the n-type and p-type
conductance asymmetry, the two transmission coefficients were fit separately, and were
generally found to be between 0.1 to 0.2 for my Pt contacted qm-CNTs. The voltage
drops (2V
c
+ V
t
= V
b
) and resistances must be solved iteratively. As before (Equation
(1.45)), in order to find the total resistance, I sum the individual resistance contributions.
In the limit of zero band gap,
*
c
c
, the V
g
dependence of the depleted quantum
channel resistance vanishes, with
2
0
*
0
4q
h
R R
, and the above equations reduce to the
55
familiar Landauer model. I can relate E
F
of the nanotube to V
g
by numerically solving the
equation
13, 49
g
F
F
eV
C
E Q
E
) (
(1.54)
by integrating over the density of states. Q is the charge induced on the nanotube, C is the
geometric gate capacitance, and a hyperbolic band structure is used to model the density
of states. The band gap creates a non-linear V
g
-E
F
relationship, which is essential to
fitting the data properly. Using these equations and the temperature, measured in situ by
Raman spectroscopy, I
d
can be modeled as a function of both V
b
and V
g
, and fit the model
to the experimental data taken from a nanotube. These equations can be used to model the
transport in semiconducting nanotubes as well; however, larger band gaps and small n-
type transmission coefficients with Pt electrodes prevent accurate band gap estimation.
The electron and hole transmission coefficients, T
c,n
and T
c,p
(respectively), gate
capacitance C (26 pF/ m), and E
gap
(120meV) were used as the only fitting parameters,
while the temperature was measured using Raman spectroscopy (Figure 3.5(c) and Figure
3.5(d)). The model can be seen to fit the data reasonably well with these parameters,
especially at low bias. The transition from Ohmic to NDC behavior can be explained
qualitatively by considering the temperature dependence of the resistance. The resistance
of metals typically increases linearly with T, because of increased phonon scattering,
while that of intrinsic semiconductors decreases exponentially with T, because of
thermally promoted carrier density increase. For qm-CNTs with E
F
in the middle of the
mini-band gap (i.e., intrinsic), the two effects compete, giving rise to a nearly constant
resistance profile despite heating (Figure 3.2(c)). For qm-CNTs with E
F
in the band
56
(doped), the system acts like a metal, exhibiting an increase in resistance due to self-
heating. These effects are captured by the above model and are illustrated in the model
curves of Figure 3.2(c). Two additional data sets from quasi-metallic nanotubes
exhibiting this effect are shown in Figure 3.3. All nanotubes examined in this study
exhibited suppression of NDC while undoped.
Figure 3.3: Electrical data from two additional suspended, metallic single-walled carbon nanotubes (a) 5µm
and (b) 2µm in length exhibiting NDC electrical behavior with the application of a gate voltage.
Equation (1.48) makes the simplifying assumption that E
F
is constant with
respect to the band gap throughout the length of the CNT. However, the voltage drop
along the CNT effectively changes the gate voltage, causing the Fermi energy to change
also, as outlined by Tersoff et al.
144
. This is accounted for with a finite element analysis
(FEA), which includes this spatial dependence of E
F
, as calculated using Equation (1.54).
In the FEA model at low bias, E
F
is found to change by only a few meV (with respect to
the band edge) along the length of the CNT, and is well approximated as constant.
However, at high biases, E
F
varies by as much as 100 meV (at V
b
= 1.4V) along the
length of the CNT.
(a) (b)
57
Figure 3.4 illustrates several effects of high bias transport in the nanotube. First of
all, in the electrostatically doped nanotube (Figure 3.4(a)), small changes in Fermi energy
induce relatively large charging voltages (because the capacitance in Equation (1.54) is
small). Thus, the Fermi energy remains nearly constant along the length of the nanotube.
In an undoped nanotube, however (Figure 3.4(b)), the Fermi energy is in the band gap.
Therefore, changes in the Fermi energy do not induce much charge on the nanotube. As a
result, the Fermi energy varies strongly with the voltage drop across the nanotube.
Secondly, Figure 4 indicates the magnitude and direction of the electric field in the
nanotube. In the doped nanotube (Figure 3.4(a)), the resistance and electric field are fairly
constant along the length of the nanotube. In the undoped nanotube, however, the
resistance increases when the Fermi energy crosses the band gap, due to charge carrier
depletion. Because of this increased resistance, the electric field becomes stronger in this
location, and a larger fraction of the bias voltage drop occurs in the nanotube instead of at
the contacts.
58
Figure 3.4: Schematic of electronic potential versus position along the nanotube length for the (a) doped (V
g
= -2V) and (b) undoped (V
g
= ½V
b
) cases. The electric field strength is indicated by the bold arrows in the
top of each Figure.
Non-Equilibrium Behavior
The Raman data also display interesting behavior with respect to doping
conditions. Figure 3.5 shows the I
d
-V
b
characteristics taken from the same nanotube under
undoped (intrinsic, V
g
= ½V
b
, Figure 3.5(a)) and doped (V
g
= -2V, Figure 3.5(b)) gating
conditions. At high V
b
, the G
+
and G
-
bands are observed to downshift due to Joule
heating in the nanotube. The effective temperature of the CNT measured from these two
bands is determined from their downshifts and the temperature coefficients established in
59
Figure 3.1(b), and are plotted in Figure 3.5(c) and Figure 3.5(d) for the undoped and
doped gating conditions, respectively. In Figure 3.5(c), the temperatures of the G
+
and G
-
bands increase together, in thermal equilibrium, up to about 400
o
C. Whereas, at V
g
= -2V
(Figure 3.5(d)), the two phonon modes give different temperatures over the whole voltage
range, with a sharp increase in the temperature occurring with the onset of NDC. So, in
addition to changing the I
d
-V
b
characteristics, the gate voltage can be used to switch
between thermal equilibrium and non-equilibrium phonon populations.
Figure 3.5: Electrical current (a, b) and Raman downshift-calculated temperature (c, d) for the same CNT at
two different values of V
g
. The undoped nanotube shows equal phonon heating and an Ohmic I
d
-V
b
, while
the gated (doped) nanotube exhibits non-equilibrium heating and negative differential conductance.
60
As pointed out by the large body of literature on this subject
25, 87, 88, 95, 106, 122
, the
cause of the observed negative differential conductance in suspended CNTs is self-
heating via a threshold optical phonon emission process (Equation (1.50)), which causes
hot, non-equilibrium phonon populations. Here, the switch from Ohmic/equilibrium
behavior to NDC/non-equilibrium behavior is observed, supporting this theory. While
current saturation occurs in any suspended metallic filament system, even ordinary
tungsten light bulbs, the criterion for the occurrence of NDC is an additional, non-linear
increase in resistance. In this case, threshold optical phonon emission and the resulting
non-equilibrium phonon populations serve this purpose.
Conclusion
In conclusion, I have shown that the gate voltage conditions drastically influence
the high bias voltage characteristics of carbon nanotubes, and that the phonon distribution
can also change substantially. Ohmic behavior is observed in undoped qm-SNWTs, while
negative differential conductance (NDC) is observed in qm-CNTs doped with a gate
voltage. To explain this behavior, I presented a model for the electric current in a
nanotube as a function of both applied V
b
and V
g
, and compare the results to experimental
data taken from a qm-CNT. The sudden increase in the phonon population of the
nanotube corresponds with the onset of NDC, supporting the theory that NDC is caused
by non-equilibrium phonon populations and selective electron-phonon coupling in the
nanotube.
61
Chapter 4:
Breakdown of the Born-Oppenheimer
Approximation
This chapter is similar to Bushmaker et al.
27
, published in Nano Letters.
Chapter 4 Abstract: Raman spectra and electrical conductance of individual,
pristine, suspended, metallic single-walled carbon nanotubes are measured under applied
gate potentials. The G
-
band is observed to downshift with small applied gate voltages,
with the minima occurring at E
F
= ±½E
phonon
, contrary to adiabatic predictions. A
subsequent upshift in the Raman frequency at higher gate voltages results in a “W”-
shaped Raman shift profile that agrees well with a non-adiabatic phonon renormalization
model. This behavior constitutes the first experimental confirmation of the theoretically
predicted breakdown of the Born-Oppenheimer approximation in individual single walled
carbon nanotubes.
Introduction
The Born-Oppenheimer (BO) or adiabatic approximation is widely used to
simplify the very complex many-body problem of electrons in solids and molecules
20
,
assuming that electrons equilibrate much faster than the atomic motion of the ionic cores.
Without this approximation, most molecular and solid state problems become difficult or
impossible to solve analytically. Although the BO approximation is valid in most
62
materials and molecular systems, there are a few situations in which it does not hold,
including some low atomic weight compounds
33, 37, 149
, intercalated graphite
132
, and
graphene
118
. Clean, defect-free single-walled carbon nanotubes (CNTs) are systems that
can be used to verify fundamental phenomena such as Wigner crystallization
52
and spin-
orbit coupling
85
, and are ideal candidates for testing fundamental physical predictions. In
nanotubes, the BO approximation is expected to break down because of the relatively
short vibrational period of the longitudinal optical (LO) phonon and the relatively long
electronic relaxation time
35, 50
. This breakdown has been observed in semiconducting
nanotube mats
50
, however, inhomogeneities broaden effects in such systems.
The breakdown of the BO approximation can be observed directly in an
individual nanotube by studying the LO phonon G
-
Raman feature of metallic CNTs (m-
CNTs), which is fundamentally different than that of their semiconducting counterparts
57
(sc-CNT). The G
-
band is broadened and downshifted (reduced in frequency), an effect
arising from coupling to a continuum of electronic states
22, 35, 48, 50, 64, 79, 125, 145, 152
. In other
words, the LO phonon mode is damped by the free electrons near the Fermi energy
61, 120
.
This coupling is a Kohn anomaly (KA) and has also been referred to as a weakened
Peierl’s-like mechanism. The G
-
band Raman feature in m-CNTs is particularly
interesting under applied gate voltages (V
g
) because of the ability to effectively turn off
the Kohn anomaly by shifting the Fermi energy (E
F
). As this happens, the LO phonon
frequency upshifts, due to reduced phonon softening of the extinguished Kohn anomaly.
This effect has been observed by many groups
48, 79, 125
, and generally agrees with phonon
renormalization theory quite well
50, 64, 135, 145
. Selected Raman G band spectra from
63
nanotubes in this study are shown in Figure 4.1. Note the near absence of the defect-
related D band, which shows the pristine nature of the nanotubes used in this study.
Figure 4.1: Sample G band spectra from two suspended m-CNTs showing the Kohn anomaly deactivation.
The spectra in (a) correspond to the data plotted in Figure 4.2, while those in 1(b) correspond to Figure 4.3
and Figure 4.4. Note the complete absence of the defect-mediated D band.
Non-Adiabaticity in Carbon Nanotubes
A striking difference between the predictions of the adiabatic and non-adiabatic
models occurs in the gate voltage response of the LO phonon when E
F
is near the Dirac
point. Phonon energy renormalization calculations done within the adiabatic
approximation predict a monotonic upshift of the phonon frequency with increasing |E
F
|,
with the minimum frequency at E
F
= 0 eV (see ref.
35
). In calculations that relax the
adiabatic approximation, there is an initial frequency downshift of the Г-LO phonon with
increasing |E
F
|, followed by an upshift after the Fermi energy has passed ±½E
phonon
,
forming a “W”-shaped gate voltage profile. The minimum frequency for these
calculations occurs at E
F
= ±½E
phonon
35
. This was recently observed in graphene at
64
cryogenic temperatures
153
, however until now there has been no experimental observation
of this clear signature of the influence of the BO approximation in isolated CNTs, most
likely because of sample inhomogeneities and defect-related electron relaxation. I report
the observation of this initial downshift at room temperature in pristine, isolated m-CNTs,
followed by a subsequent upshift, consistent with theoretical predictions reported
previously
35, 145
. These results directly confirm the breakdown of the Born-Oppenheimer
approximation, indicating the intrinsic non-adiabatic nature of the electron-phonon
coupling in this system.
The G
-
band Raman peak is often observed to be asymmetric, consistent with a
Breit-Wigner-Fano (BWF) lineshape, given by
2
0
2
2
1
0
0
) (
q
BWF
I I , where γ is
the linewidth ω
o
is the center frequency, and q is the Fano factor (negative in m-CNTs).
This asymmetry is due to photon coupling to a discrete phonon state and to a continuum
of electronic states
22
. The temperature
147
and gate voltage
64, 105
dependences of this
parameter have been previously reported. Here, I present the gate voltage dependence of -
1/q and electrical conductance measured simultaneously. The devices in this work were
fabricated as described in Chapter 1.
Results
Figure 4.2 shows the Raman frequency, linewidth, -1/q, and electrical
conductance data plotted as a function of applied gate voltage (V
g
) and Fermi energy
(E
F
), as determined from the gate coupling factor. The laser wavelength used was 532nm
65
at a power of 350µW. The frequency of the LO phonon initially downshifts as |E
F
| is
moved away from zero. As |E
F
| is increased beyond E
ph
/2, the Raman frequency (Figure
4.2(a)) begins to upshift. The Raman linewidth (full-width half-max) of the LO G
-
band is
also plotted versus V
g
and E
F
in Figure 4.2(b), and exhibits a strong narrowing as the
Kohn anomaly is shut off with increasing |E
F
|, dropping from over 50 cm
-1
to just over 10
cm
-1
. Plotted along with the Raman data are the adiabatic (dashed) and non-adiabatic
(solid) phonon renormalization models presented by Caudal et al.
35
and described below.
Finally, -1/q exhibits a strong decrease towards zero with increasing |E
F
|, while the
conductance shows a sharp dip near E
F
= 0, typical of the quasi-metallic nanotubes
measured in this study. The solid line in Figure 4.2(c) represents a conductivity model
based on Boltzmann transport and the Landauer model (Equation (1.57))
1, 13, 16, 111, 122, 137,
156
.
Figure 4.2: Raman spectral data from a single walled carbon nanotube. LO (a) shift (filled circles), (b)
linewidth (FWHM, filled circles) and (c) -1/q (open squares), as well as the electrical conductivity (filled
circles) are plotted versus gate voltage (V
g
). The Fermi energy (E
F
) is also indicated on the top x-axis. The
lines in (a) and (b) show the results of the adiabatic (dashed) and non-adiabatic (solid) phonon
renormalization models discussed below (Equations (1.55) and (1.56)), while the solid line in (c) represents
the Boltzmann-Landauer transport model (Equation (1.57)).
66
Figure 4.3 and Figure 4.4 show the gate voltage dependence of the Raman and
conductance data of another suspended nanotube, taken with a 633nm laser at a power of
1mW. The upshift at large |E
F
| is experimentally difficult to observe, because of the
relatively weak gate coupling, and the fact that suspended nanotubes are eventually
destroyed at high gate voltages. A strong initial frequency downshift was observed in all
6 nanotubes of this study. The values reported here for -1/q are smaller than the values
reported elsewhere
22, 64, 105, 147
, where -1/q ~ 0.2-0.4. An interesting trend is that the gate
voltage dependence of -1/q was noticed to change proportionally to (FWHM-
0
)
2
, where
is the intrinsic linewidth of 10-15 cm
-1
.
67
Figure 4.3: Gate voltage evolution of the Raman spectra from a second device. The gate voltage
dependence of the parameters fit to these spectra are presented in Figure 4.4(a-c).
The -1/q value for this nanotube is substantially smaller than that of the previous
nanotube (shown in Figure 3(c)), and was observed to decrease over the course of the
sample’s lifetime (several months in air and at high temperatures during testing) from -
1/q = 0.07 at E
F
= 0 to -1/q = 0.02 at E
F
= 0. The maximum FWHM for this nanotube
also decreased, however, only by 2 cm
-1
from 44.4 cm
-1
to 42.5 cm
-1
. From the absence of
a D band Raman signal throughout the experiment, one can infer that few defects were
68
introduced into the CNT over this time. However the large variation in -1/q indicates that
it is extremely sensitive to aging and environmental conditions; in fact far more sensitive
than the FWHM or the D band intensity. These experimental results highlight the need
for a quantitative model describing the behavior of the Fano parameter -1/q in response to
gate voltages and environment changes.
Figure 4.4: Raman spectral data from a second CNT device. (Spectra in Figure 4.3) Raman LO (a) shift
(filled circles), (b) linewidth (FWHM, filled circles) and (c) -1/q (open squares), as well as the electrical
conductivity (filled circles) are plotted versus gate voltage (V
g
) and Fermi energy (E
F
).
The non-adiabatic phonon renormalization model used in Figures 3 and 5 was
confirmed by density functional theory (DFT) calculations
35
, and is outlined below. The
equation describing the frequency of the -point optical phonon,
, is given by
M
D
M
D
KA
0
, (1.55)
69
where M is the mass of the carbon atom and
0
D is the intrinsic (no Kohn anomaly)
dynamical matrix. The equation for the non-analytical electron-phonon coupling
contribution to the dynamical matrix,
KA
D
, is given by
2
2 0
2 , ,
2 3 [ ( ')] [ ( ')] [ ( ')] [ ( ')]
'
( ') ( ') ( ') ( ')
k
KA v c c v
LO LO
F k
t v c ph c v ph
a f k f k f k f k
D D dk
d k k E i k k E i
, (1.56)
where a
0
is the graphene lattice constant,
, LO
ph
E is the phonon energy, d
t
is the nanotube
diameter,
F
D
2
is the electron-phonon coupling constant, k is a small integration limit, f
is the Fermi function, ) ' (k
and ) ' (k
c
are the hyperbolic valence and conduction band
dispersion relationships, respectively, and is the electronic lifetime broadening
coefficient, found to be 0.9 meV and 0.8 meV for the nanotubes in Figures 3 and 5,
respectively. The adiabatic case is approximated simply by setting
, LO
ph
E = 0. The
FWHM is given by the imaginary part of the non-adiabatic dynamical matrix plus an
intrinsic linewidth of 10-15 cm
-1
. It is possible that this effect of non-adiabatic phonon
hardening near E
F
= 0 has not been observed until now because the substrate interaction
increases the electronic scattering rate by defect scattering. This would make the Born-
Oppenheimer approximation valid, by eliminating the non-adiabatic phonon hardening
near E
F
= 0. Also, the effect would be difficult to observe bulk samples due to
inhomogeneities in the Fermi level
50
.
From the diameter of the nanotube measured (obtained from the RBM frequency),
the electron-LO phonon coupling constant
2
D
F
can be found directly by fitting the data.
The nanotube in Figure 3 exhibited an RBM in its spectra at 115.6 cm
-1
, corresponding to
70
a diameter of 1.97 nm according to the relation
2
RBM t
d / 227 , giving a value of
F
D
2
= 46 (eV/Å)
2
. Another nanotube (not shown) exhibited an RBM at 174.0 cm
-1
,
corresponding to a diameter of 1.31 nm and
F
D
2
= 52 (eV/Å)
2
.
The electrical data in Figures 3(c) and 5(c) are fit to the Landauer model using the
Boltzmann equation in the constant relaxation time approximation
1, 13, 111, 122, 137, 156
. The
resistance of the nanotube can be found by taking a sum of the phonon scattering
resistance and the quantum resistance,
quantum scatt
R R T V R ) , (
, (1.57)
where each contribution is found by summing over the density of states near the Fermi
energy, following Biercuk and McEuen
13
. The Fermi energy is calculated numerically as
a function of gate voltage using a geometric gate capacitance C, the Fermi function, and a
hyperbolic density of states model
16
, according to the equation
g
F
F
eV
C
E Q
E
) (
,
where Q is the charge induced on the nanotube. This equation includes the effect of the
mini band-gap (where the density of states is zero), which creates a non-linear V
g
-E
F
relationship. Inclusion of this non-linear relationship in the model is key to fitting the
data properly. The mean free path of electrons scattering by acoustic phonons,
ac
, was
taken to be 2 m, in accordance with previous publications
111
. The data was fit to the
frequency, width, and conductivity models self-consistently, with the gate capacitance,
2
D
F
, d
t
, , contact transmission coefficients, and mini-band gap as fitting parameters.
The band gaps for the CNTs in Figures 3 and 5 were found to be 42 meV and 120meV,
71
respectively. For these single, pristine CNTs, the model can be seen to fit the data
reasonably well with a gate capacitance of 1.5-1.8 pF/m.
Conclusion
In conclusion, I report Raman spectra of isolated, suspended metallic CNTs (m-
CNTs) observed with applied gate voltages. The LO phonon Raman band (G
-
) is
observed to initially downshift with applied gate voltage, then subsequently upshift for
|E
F
| > ½
, LO
ph
E . This behavior is attributed to the non-adiabaticity of the Г-point Kohn
anomaly in m-CNTs, and constitutes the first experimental confirmation of the predicted
breakdown of the Born-Oppenheimer approximation in individual CNTs. The Raman
data agree quantitatively with a non-adiabatic model using time-dependent perturbation
theory, while the electron transport data are fit using the Landauer model and the
Boltzmann equation within the constant relaxation time approximation. The results
showcase the use of pristine, defect-free nanotubes as model systems for studying
fundamental phenomena.
72
Chapter 5:
Raman Intensity Modulation
This chapter is similar to Bushmaker et al.
29
, published in Physical Review Letters.
Chapter 5 Abstract: Large variations of up to two orders of magnitude are
observed in the Raman intensity of pristine, suspended, quasi-metallic, single-walled
carbon nanotubes in response to applied gate potentials. No change in the resonance
condition is observed, and all Raman bands exhibit the same changes in intensity,
regardless of phonon energy or laser excitation energy. The effect is not observed in
semiconducting nanotubes. The electronic energy gaps correlate with the drop in the
Raman intensity, and the recently observed Mott insulating behavior is suggested as a
possible explanation for this effect.
Introduction
Single-walled carbon nanotubes (CNTs) provide an excellent system for studying
interesting one-dimensional physics, including strong electron-phonon coupling
88, 122
,
ballistic transport
82
, and strongly correlated electrons
15, 52, 55, 71
. Micro-Raman
spectroscopy is a sensitive technique for observing these unique effects
25, 145
. Despite the
great interest in CNTs, new phenomena such as those mentioned above are still being
discovered with the use of clean, nearly defect free, suspended CNTs. Understanding
these effects in pristine systems is crucial for the development of CNT nano-devices.
73
In this study, the Raman spectra of individual, suspended, pristine quasi-metallic
(small band gap or “qm”) CNTs are found to exhibit changes in intensity by up to two
orders of magnitude with an applied electrostatic gate voltage (V
g
), while for
semiconducting nanotubes (sc-CNTs) the intensity remains constant. The effect is so
strong that it renders some qm-CNTs invisible to Raman spectroscopy, and occurs at
room temperature over small voltage ranges, suggesting possible device applications in
the future. Indeed, there are limited device technologies that have demonstrated the
ability to strongly modify a material’s optical properties using static electric fields.
It is well known that the Raman intensity of CNTs is significantly enhanced when
one of the photons involved is resonant with an excitonic transition
57, 74, 115, 150
. There
have been several reports on slight changes in the Raman spectral intensity from CNTs in
response to gate voltages, which were attributed to shifting of the resonance condition
48,
125
, as well as reports on larger intensity changes in CNTs under extreme electrolytic
doping, due to transition bleaching
46, 79
or otherwise anomalous behavior in complex
nanotube mats
50
. Raman studies of electrostatically doped graphene have also been
undertaken, showing moderate decreases in the 2D band Raman intensity with doping
49
.
In contrast to the previous work
46, 48, 50, 79, 125
, I observe an increase in intensity
with doping, as opposed to a decrease. Furthermore, this increase occurs with relatively
small V
g
, in contrast with other studies that used several volts of electrolytic doping or
several tens of volts with electrostatic doping. Changes in the resonance condition are
ruled out based on the constant Stokes/anti-Stokes intensity ratio and insensitivity to
phonon and laser energy. By performing optical and electrical measurements
74
simultaneously, the electrically measured energy gaps (E
gap
) are compared to the Raman
intensity attenuation. Based on these results, the recently observed Mott insulating
behavior
53
in qm-CNTs is suggested as a possible mechanism for the observed behavior.
Recently, there has been a large focus on the Raman G
-
band’s response to V
g
27, 35,
50, 64, 145, 152
. In these studies, the G
-
band frequency and linewidth change drastically due
to the influence of the -point Kohn anomaly (KA) in the LO phonon band
120
. These
effects were also observed in these devices
27
. The intensity modulation, however, affects
all Raman modes universally, not just those associated with the KA. Furthermore, in
bands not affected by the KA, no noteworthy shifts or changes in linewidth are observed.
The devices in this work were fabricated as described in Chapter 1.
Intensity Changes
G band Raman spectra taken with a 785nm laser from an individual, suspended,
qm-CNT are plotted in Figure 5.1(b) at several V
g
. As with all qm-CNTs measured in
this study, the intensity of the Raman signal increases dramatically with increasing |V
g
|,
varying by up to almost two orders of magnitude (>18.8 dB) in this case. Here, the G
+
and G
-
bands exhibited an identical intensity change. The G band lineshape in Figure
5.1(b) is typical of quasi-metallic nanotubes, exhibiting a broad, downshifted G
-
band,
with a sharp G
+
band. Note the near-absence of the defect-related D band. A radial
breathing mode (RBM) for this nanotube was observed at 173.6±0.5 cm
-1
, indicating that
the diameter of this CNT is 1.31 nm
2
.
75
Figure 5.1: (a) Device geometry and (b) G band Raman spectra at various gate voltages, with inset showing
the G
-
band intensity as a function of V
g
(sample 8B).
Figure 5.2 shows the Raman data for another nanotube device, including the
RBM, G
+
band, and G’ band Raman intensities plotted as a function of V
g
and the Fermi
energy (E
F
). The normalized Raman intensity profiles show early identical V
g
dependences, indicating that this effect affects all of the Raman modes universally,
regardless of phonon energy. The G
-
band also exhibited the same dependence. The
RBM, observed at 153±0.5 cm
-1
using both 633nm and 785nm lasers, shows similar
intensity profiles (Figure 5.2(b)), with a Raman signal attenuation of 8.5 dB at V
g
= 0.
Throughout the measurement, the intensity of the background Si 520 cm
-1
band remained
constant. Also shown in the figure is the temperature normalized (300K) RBM anti-
Stokes/Stokes (AS/S) intensity ratio, which is sensitive to changes in the resonance
condition
63
.
76
Figure 5.2: (a) Normalized Raman intensities of the RBM, G
+
(TO), and G’ bands taken with a 633nm
laser. (b) RBM intensity taken with 633 and 785nm lasers, together with the RBM AS/S intensity ratio
(633nm, normalized for T=300K), plotted versus V
g
.
Potential Mechanisms
Normally, any changes in the Raman intensity of carbon nanotubes would be due
to a change in the resonance condition. However, here I find this is not the case. The
resonant Raman intensity of the Stokes process is
2
] ][ [
) (
ph
op ep op
L
E E E
dk M M M
C E I
, (1.58)
where C is a constant,
i E E E
L
is the resonance condition, E
L
is the laser
energy, E
µ
is the excitonic transition energy for the µ
th
subbands, E
ph
is the phonon
energy, γ is the resonance broadening energy, M
op
is the optical matrix element for the
exciton-photon interaction, and M
ep
is the electron-phonon coupling matrix element
63
. A
large change in the Raman intensity can arise from a change in three quantities: 1.) the
resonance condition E (or
ph
E E ), 2.) M
ep
, or 3.) M
op
.
77
We can rule out transition bleaching immediately, as the changes in Fermi energy
(≤200meV) are drastically smaller than the excitonic transition energy (1.5-2.3eV). It is
tempting to attribute the change in Raman intensity to a strain-induced change in
resonance (case 1), caused by the electrostatic gate force
100
. However, this is not the
case, since the RBM has a narrow resonance window, and, therefore, small changes in the
resonance condition (E
µ
– E
L
) result in large changes in the RBM AS/S intensity ratio
63
,
which are not observed (Figure 5.2(b)). Also, the broad G band resonance window would
require an unreasonably large change in E
µ
to account for such a drastic modulation.
Also, one would expect the Raman signal for different phonon modes and laser energies
to respond differently to a change in resonance condition, which is not observed (Figure
5.2). Finally, it is statistically unlikely that one would observe a shift onto resonance with
increasing |V
g
| for all 8 nanotubes showing this effect. One would expect some nanotubes
to show a shift off of resonance with increasing |V
g
|. I can rule out V
g
-induced bending as
a cause for the observed intensity modulation, as most suspended nanotubes have slack
(and thus bending) as fabricated
136
. Furthermore, the Raman intensity is predicted to
decrease with bending
94
, opposite the observed behavior. This unanimous evidence
suggests that a different mechanism is responsible for the observed behavior.
Ruling out case 1 as a possible explanation for, I consider the electron-phonon
coupling strength, M
ep
(case 2), which is known to be quite different for the various
Raman active modes
73
. Therefore, a variation of this quantity is expected to result in
different intensity modulation profiles for the RBM, G and G’ bands, which is not
observed (Figure 5.2(a)). This is especially true with the G
+
and G
-
bands, which have
78
orthogonal TO and LO polarizations in qm-CNTs, respectively
120
. The electron-phonon
coupling of the LO phonon band is influenced by the KA
27, 35, 64, 120, 152
, and is drastically
different from that of the TO phonon band. Despite this difference, the intensity
behaviors of the TO and LO modes are identical. This leaves a change in the optical
matrix element M
op
(case 3) as the only plausible cause of the observed intensity
modulation. This intensity modulation appears to be attenuation at small |V
g
|, rather than
enhancement at high |V
g
|, because the Raman intensity saturates at high |V
g
| to a value
comparable to that of the sc-CNTs.
A Raman intensity map of the G band of a third nanotube is plotted in Figure
5.3(a), together with the measured conductance. Here, the G band peak around 1580cm
-1
vanishes near E
F
= 0. This corresponds to the drop in the conductance observed in the
electrical data. The conductance is modeled using the Boltzmann-Landauer transport
equation
13, 27
, and the Fermi energy is calculated numerically using a geometric gate
capacitance C, the Fermi function, and a hyperbolic density of states
16
, according to the
equation
g
F
F
eV
C
E Q
E
) (
, where Q is the charge induced on the nanotube. This
accounts for the quantum capacitance
24
and the band gap, which create a non-linear V
g
-E
F
relationship (Figure 5.4(b)). Fitting the data in Figure 5.3(a) with this model yielded C
~10pF/m and E
gap
= 120meV. The small offset of the conductance and Raman intensity
minima near V
g
= 0 arises from the gas doping effect
51
. This nanotube exhibits Raman
attenuation for |V
g
| < 2V and, as with the others, saturation at large V
g
. This same effect
is not observed in sc-CNTs (Figure 5.3(b)), which have band gaps on the order of 1eV.
Therefore, the effect is not simply due to a change in the free carrier density.
79
Figure 5.3: The G band spectra and conductance (G) for suspended (a) qm- and (b) sc-CNTs, plotted versus
V
g
and E
F
. G and E
F
are fit using the Boltzmann-Landauer transport model.
Out of 9 qm-CNTs investigated, 8 showed this intensity modulation effect. The
Raman intensity of the remaining qm-CNT was constant. 4 sc-CNTs were also
investigated using this technique, none showing substantial Raman intensity changes with
V
g
. We, therefore, conclude that the effect is specific to qm-CNTs. The data for the 8 qm-
CNTs showing this effect are summarized in the table below. The diameter is given for
nanotubes that exhibited a RBM
2
. The G
+
/G
-
Raman integrated intensity ratios are also
listed for each nanotube, giving an indication of the chiral angle (G
+
/G
-
= 0 zigzag,
G
+
/G
-
= ∞ armchair)
14, 152
. Also given are the maximum observed Raman attenuation
(in dB) and the Fermi energy change corresponding to the FWHM attenuation of the
Raman intensity (
Raman
), found using the V
g
-E
F
relationship (Figure 5.4(b)). Finally, the
energy gaps (E
gap
) obtained by fitting the Boltzmann-Landauer model to the measured
80
conductance are also given. The correlation between
Raman
and E
gap
(Figure 5.4(a))
suggests that the observed Raman intensity attenuation is caused by the same effect that
causes the electronic energy gaps in qm-CNTs.
Table 5.1: Data summary of qm-CNTs showing intensity modulation. Listed values include nanotube
diameter, G
+
/G
-
integrated Raman intensity ratio, maximum Raman attenuation, attenuation energy gap
(
Raman
), and electronic band gap (E
gap
).
Figure 5.4: (a)
Raman
plotted versus E
gap
, determined by fitting the Boltzmann-Landauer transport equation
to electrical data. (b) E
F
plotted versus V
g
for nanotube 22B2, showing the method for determining
Raman
.
Mott Insulation
The secondary gap in qm-CNTs (those with chiral index difference an integer
multiple of 3) has long been thought to arise from the curvature of the nanotube, which
causes mixing of the and orbitals
14, 162
. A Peierls gap transition, one hallmark feature
81
of most one-dimensional metals, was also initially considered. However, density
functional theory (DFT) has found the Peierls gap to be unstable above T~10
-8
K
120
in all
but ultra-small radius carbon nanotubes
45
. Recently, experimental evidence
53
has
confirmed theoretical predictions
6
that, in nearly defect-free qm-CNTs, a Mott insulator
(MI) transition is primarily responsible for creating E
gap
. In the MI state, strongly
correlated electrons localize to their parent atoms, forming gaps of 10-100 meV, even in
armchair CNTs. Raman intensity attenuation has been previously reported for MI
transitions in other materials systems
78
. I believe that this same effect is causing the
Raman attenuation in these nearly defect free nanotubes. The fit values for E
gap
in Table
5.1 lie in the range predicted for MI gaps, and correlate well with the energy gaps over
which the Raman attenuation is observed (Figure 5.4(a)), corroborating the MI state.
The Mott insulator transition explains why all the Raman bands are affected
equally under applied gate potentials. In this phase transition, the electrons in the 2p-
orbital of the carbon atom localize to their parent atom through Coulomb repulsion,
causing all the electrons in the -band to be affected, including those involved in
excitonic transitions. The details of this interaction are left to future theoretical work. The
MI transition also accounts for the qm/sc difference, because in sc-CNTs, the electronic
gap originates from confinement effects. Absorption studies (optical
103
and X-ray
38
) in
other materials systems have also shown dramatic changes as a result of the MI
transition. Finally, the gate voltage-induced MI transition has already been exploited in
cuprate MI field effect transistors (MTFETs)
104
. It is likely that this modulation has not
been observed until now because most gate voltage experiments with qm-CNTs are
82
performed on nanotube-on-substrate devices, rather than pristine, suspended devices. The
MI state requires the presence of a well-defined charge neutrality point
53
, which may not
occur in samples with defects, substrate contact, or post-processing residue. This may be
the case where, as mentioned above, one nanotube out of a total of nine that were
measured failed to show this effect. It is possible that this one metallic nanotube had a
finite defect density or was in a bundle and slipped by the pre-screening process,
indicating that this effect may be used to characterize qm-CNTs.
Conclusion
In conclusion, a large attenuation was observed of the Raman signal from
individual pristine, suspended quasi-metallic CNTs by up to two orders of magnitude
near zero electrostatic gating, while semiconducting CNTs do not exhibit the effect. The
attenuation is so strong as to render some qm-CNTs undetectable by Raman spectroscopy
without a gate voltage. Changes in the resonance condition and transition bleaching are
ruled out on the basis of the constant anti-Stokes/Stokes intensity ratio and behavior with
respect to different phonon modes and laser energies. The recently observed Mott
insulator transition in qm-CNTs is suggested as a possible mechanism for the changes in
the Raman intensity. The Raman attenuation energy gaps for 8 nanotubes are compared
to the electronic energy gaps, estimated from fits to the Boltzmann-Landauer transport
model, and show correlation consistent with the Mott insulator picture.
83
Chapter 6:
Memristive Defect State Behavior
This chapter is similar to Bushmaker et al.
26
, accepted for publication at IEEE
Transactions on Nanotechnology.
Chapter 6 Abstract: Memristive electrical behavior has recently gained attention
because of technological advances in nano-structuring, which has enabled the fabrication
of working devices. However, such investigations have been limited to mobile ionic
systems, and memristive behavior in other types of nano-scale systems has been largely
overlooked. Here, I report direct measurement of memristive behavior of defect states in
quasi-metallic, single-walled carbon nanotube (CNT) field-effect transistors (FETs).
After exposing CNT FETs to laser irradiation, the conductance-gate voltage profile (G-
V
g
) indicates the creation of a gate-tunable, resonant electron scattering defect. Once a
defect is formed, current flowing in the forward and reverse directions reversibly
switches the G-V
g
characteristics of the device. The changes in conductance are attributed
to current direction-sensitive changes in the structure of an isolated defect state in the
nanotube. The defect scattering spectra are extracted from the G-V
g
data using a Landauer
model.
84
Introduction
The study of defects in carbon nanotubes (CNTs) has been driven by the desire to
understand the effect of defects on the ideal conductance of CNTs
36, 40, 83
, as well as
several specific device applications, such as heterojunction devices joined by Stone-
Wales (5-7) defects
41, 47, 157
. Choi et al. used ab initio methods to examine the results of
atomic substitutional defects (doping), 5-7 defects, and vacancy defects on the density of
states and conductivity of CNTs
42
. These defects were found to manifest themselves as
resonant electron scattering centers, only affecting transport strongly at the defect energy.
Experimental investigations have confirmed the resonant nature of these few-atom
defects, by observing gate-voltage tunable resistance
17, 109
. Scanning tunneling
microscopy (STM) studies have shown that defects can be reversibly created and
annihilated in CNTs
12, 107
.
In this work, defects are created with a focused laser spot in suspended single-
walled metallic CNT field effect transistors. A 532nm laser (300µW) was focused to a
diffraction-limited spot through a 100X high numerical aperture objective lens on a Leica
DMLM microscope. This corresponds to a laser intensity of 1.5x10
5
W/cm
2
. A Renishaw
InVia spectrometer, integrated with the microscope, using 633nm excitation wavelength,
collected Raman spectra. A 532nm Spectra Physics solid-state laser was used to create
defects. No D band was observed from the nanotubes throughout the course of the
experiment. The focused laser-spot incident on the center of a 5µm suspended CNT can
be seen in Figure 6.1(a). All electrical and optical measurements were performed in
argon to prevent oxidation.
85
These defects are observed in the room-temperature conductance-gate voltage
characteristics (G-V
g
) taken at low bias voltage (V
b
). When high currents are passed
through the CNT, changes are observed in the G-V
g
relationship that depend on the
direction of the current flow. This direction-sensitive current annealing behavior is
associated with memristive systems
43, 117, 143, 151, 154, 155
, which is a general class of circuit
elements that includes the memristor, the thermistor, and ionic systems in general.
Memristors are considered to be the fourth
circuit element (in addition to resistors,
capacitors and inductors), and have a resistance that depends on the electrical history of
the device and yet are incapable of storing energy
43
. The devices in this work were
fabricated as described in Chapter 1.
Figure 6.1: Device geometry and CNT defects. The microscope image of a suspended metallic single-
walled CNT device used in the experiment, and possible defects in the nanotube wall: a vacancy (b), a 5-7
pair (c), and a substitutional defect (d).
Results
Figure 6.2(a) shows the effect of laser exposure and annealing on the G-V
g
profile. This nanotube initially showed a sharp dip in the conductivity near V
g
= 0V,
caused by the reduced carrier density in the band gap of this quasi-metallic CNT
13, 31, 53,
108
. After laser exposure, the dip in the G-V
g
profile was broadened and shifted down to
86
negative voltages, indicating a change in the local density of states. The laser spot was
localized to the center of the CNT, so changes in G-V
g
profile are not related to any
changes in the contact resistance. When a bias voltage of V
b
= 1.2V was applied to the
nanotube for 15 seconds, the nanotube heated up to approximately 450
o
C, as determined
by temperature-induced downshifts in the Raman spectra
25, 91
. After this high bias
annealing, a partial restoring of the pristine G-V
g
relationship is observed, as shown in
Figure 6.2(a). However, the CNT exhibited a noticeable and permanent decrease in the p-
type conductance.
Figure 6.2 also shows the change in conductivity with time during a 15-second
laser exposure (Figure 6.2(b)) and subsequent high bias annealing (Figure 6.2(c)). A
sharp drop in the conductivity was observed at the onset of laser exposure, proceeding
with a time constant of approximately 10s, whereas for annealing, the change in
conductivity occurred in approximately 5s following a linear time dependence. The high
bias annealing of the CNT was performed in the region of negative differential
conductance (NDC), as reported previously
25, 88, 122
.
Figure 6.2: Conductance of a defected CNT device. (a) Low bias (V
b
= 100mV) conductance versus gate
voltage characteristics (G-V
g
) before laser exposure, after laser exposure, and after current annealing. (b)
Conductivity versus time during laser exposure and (c) subsequent high bias current annealing. Inset (b)
shows the gate voltage at which the measurement was taken. Inset (c) shows the bias voltage at which the
measurement was taken.
87
Current Direction Sensitive Annealing
Once the defect was created by laser exposure, the low bias conductivity was
found to depend on the direction of the annealing current. This can be seen in Figure 6.3,
where the G-V
g
profile is substantially different after reverse bias annealing (-V
b
) as
compared to that taken after forward bias annealing (+V
b
). -V
b
annealing of the device
results in a broad G-V
g
profile with a single dip centered at V
g
= -0.5V. +V
b
annealing,
however, results in a G-V
g
profile with two distinct local minima: one at V
g
= -2.5V and
another at V
g
= 0.1V. The effects of +V
b
and -V
b
annealing were completely reversible, as
can be seen in Figure 6.3(b). During annealing treatments, +V
b
annealing occurred at V
g
=
-3V and -V
b
annealing at V
g
= +3V, to maintain equal carrier density. In order to rule out
the gate voltage as the cause of the change, annealing was also performed with +V
b
and
+V
g
, as well as with -V
b
and -V
g
. The resulting G-V
g
profiles were qualitatively the same
with respect to the direction of the annealing current. The power dissipation in the CNT
was the same for both +V
b
and -V
b
annealing.
Figure 6.3: Repeatable switching behavior. Gate sweep conductivity profiles after successive ±1.2V bias
voltage annealing show reversible switching behavior. The data in (b) are offset for clarity.
88
Landauer Model
The data shown in Figure 6.4 are fit with a Landauer transport model including
the effects of a band gap, as described below. In the Landauer model
25, 28, 88, 95, 111, 122, 156
,
the nanotube resistance is expressed as
c d ph
R R R R R 2
*
0
, (2.1)
where
bands
dE
df
dE
R
R
0 *
0
, is the depleted channels’ quantum resistance. Here, f is the Fermi
function,
2
0
4q
h
R is the full quantum resistance, and the integral is taken only over the
electronic valence and conduction bands. R
ph
is the resistance arising from phonon
scattering in the CNT, R
d
is the resistance arising from the defect, and R
c
is the contact
resistance. The resistance from an individual scatterer in a one-dimensional system in the
Landauer model is expressed as
*
*
0 *
1
RR
, where
*
df
dE
bands
df
dE
bands
E dE
T
dE
is the effective
transmission coefficient in the depleted quantum channels
4, 23
. For phonon scattering, the
transmission coefficient is
eff
ph
eff
E
E
LE
137
.
At low bias, the electron mean free path
eff
is dominated by acoustic phonon
scattering mean free path. The energy dependence of the scattering rate has been
examined in other studies
116
. However, here, the constant relaxation time approximation
is taken, by reducing the scattering length proportionately to the group velocity
89
( ) ( ) ( )
hi
ac ac F
E v E v E v
, which results in short scattering lengths near the band
edges
84
.
hi
ac
was taken to be 1 m in accordance with previous work
84, 111
. The tunneling
through Schottky barriers at the contacts
4, 23, 51, 96
of quasi-metallic CNTs is well
approximated by a single scatterer with constant n-type and p-type transmission
coefficients (T
n
and T
p
, respectively). These were fit separately to account for n-p
asymmetry. Following the work by Choi et al.
42
, the resonant defect state is treated as a
single Lorentzian scatterer, with a transition coefficient given by
2
,0
2
1
d
d
d
d
EE
d
d
EE
, (2.2)
where T
d,0
is the minimum defect transmission, E
d
is the energy of the defect, and Γ
d
is
the HWHM. I relate E
F
in the nanotube to V
g
by numerically solving the equation
g
F
F
eV
C
E Q
E
) (
, (2.3)
by integrating over the density of states
13, 49
. Q is the charge induced on the nanotube, C
is the geometric gate capacitance, and a hyperbolic band structure is used to model the
density of states.
Using this model, the change in conductance from the forward annealed state to
the reverse annealed state is interpreted as the result of a change in the atomic
configuration, theoretically predicted by Choi et al.
42
. Mobile vacancies can reconfigure
under electro-migratory pressure and, thus, change the local density of states. As seen in
Figure 6.4, the model qualitatively reproduces the observed change in G-V
g
from that of
two smaller, isolated dips in conductance (the first originating from the defect state and
90
the second from the band gap), to one wider dip in conductance, originating from the
defect with its energy shifted into the band gap. Remarkably, only small changes in
fitting parameters are required other than the energy of the defect. The gate capacitance
was held constant at 18 pF/m, the band gap at 110 meV, T
p
= 31%, T
n
= 18% for the
pristine CNT and 16% for the defected CNT. For the defected CNT, the two states
(forward and reverse bias annealed) have the same T
d,0
= 2.8%, with Γ
d
= 400 meV for E
d
= -30meV (Figure 6.4(b)), and Γ
d
= 300 meV for E
d
= -180meV (Figure 6.4(c)). The
energies found here are consistent with previously reported theoretical values for vacancy
type defects
42
.
Figure 6.4: Conductance versus V
g
data and Landauer model fits from a suspended nanotube device (a)
before and (b,c) after laser exposure and after (b) forward and (c) reverse bias annealing. The Landauer
model fitting parameters are shown on the right.
91
Defects modify the local potential in CNTs
5
, resulting in a non-uniform work
function along the length of the nanotube. The associated inhomogeneous broadening
could explain the wider G-V
g
relationship observed after -V
b
annealing; however, the
analysis of such an effect is beyond the scope of this paper. In addition to the theory of a
defect in the nanotube, non-uniform doping was also considered as the primary
mechanism for this behavior, perhaps caused by local, heating-assisted desorption of
surface gas molecules. This explanation relies only on thermal activation and, therefore,
fails to account for the current direction-sensitive annealing behavior. Also, it is possible
that the laser could be ionizing charge traps on the surface the CNT, for example, in
surface-bound soot or other contaminants, which then act as local dopants and scattering
sites
141
. This type of contaminant could behave like the defect discussed above, but reside
on the surface of the intact CNT instead of within the wall of carbon atoms.
Memristive System Equations
Since the resistance state of the nanotube at a given Vg is dependent on the
directional history of the bias current, this system falls into the general category of
memristive systems
43, 143
, as a time-invariant current-controlled memristive system. As
such, these defected states in carbon nanotubes could potentially serve as memory
elements. A general memristive system is described by the equations
i w R i V
b
,
(2.4)
i w f dt dw ,
(2.5)
92
where w is the variable specifying the state of the system, , R w i is the resistance, and
, f w i is a function describing how the state variable changes with applied current. For
a true memristor, the equation is
, f w i i . In this CNT system, the energy of the
defect, E
d
, is the state variable, and
, f w i can be described phenomenologically as
1
0
, exp exp
d d H d th L d th
f E i dE dt E E i i E E i i
, (2.6)
where Ed is the defect energy or state variable,
0
is the transition rate constant for
crystallographic reorganization, and EH and EL are the high (Figure 6.4(b)) and low
(Figure 6.4(c)) energy quasi-stable defect configuration states, respectively. The state
variable, Ed, is then bounded by EH and EL. Equation (4) takes into account the self-
heating temperature dependence of
d
dE dt with the exponential term and the threshold
current
th
i . The gate-tunable resistance of the system ( , , )
dg
R i E v is calculated using the
Landauer scattering model. An interesting particularity of this memristive behavior is
that, depending on the gate voltage, the resistance can be either raised or lowered (see
opposing vertical arrows Figure 6.3(a)). This third terminal control over the device
polarity enables this behavior to be utilized to design devices possessing a broader range
of functionalities.
93
Conclusion
In conclusion, I report current-gate voltage characteristics of defect-induced,
suspended, single-walled carbon nanotubes, which display memristive behavior. The
defect is initially created by laser irradiation and is subsequently modified by annealing
the nanotube with large bias currents. Forward and reverse bias current annealing results
in different G-V
g
characteristics, which act as a time-invariant memory element. The data
is interpreted in terms of a defect state that scatters charge carriers near the defect state
energy, which is sensitive to the direction of annealing currents in the nanotube. A
Landauer model is used to treat the defect as an isolated scattering state in the nanotube.
94
Chapter 7:
Future Directions
In the course of this thesis work, several new research directions became
apparent, which my focused research objectives did not allow me to pursue. Below, I
describe these possible future directions that build on this thesis work.
Temperature Dependence of the Intensity Effect
The temperature dependence of the giant Raman intensity modulation will
provide deeper insight into the fundamental origin of this effect. Due to the drastically
varying attenuation magnitudes and threshold voltages between different nanotubes, these
values are expected to smoothly decrease as the temperature is increased, until some
critical temperature, at which point the effect will no longer play a significant role. If the
effect is indeed caused by a Mott-insulator transition with doping, then the system should
transition to a Luttinger liquid above the transition temperature, as shown in Figure
7.1(a). The temperature dependencies of the various nanotubes are expected to give an
understanding of the drastic variance of these parameters between different nanotubes. At
low temperatures, it is possible that the threshold voltages and attenuation magnitudes
would increase to the point where the nanotube is no longer detectable by Raman
spectroscopy. In Figure 7.1(b), the phase diagram is shown for a Mott insulator system as
a function of carrier density (or electrostatic doping in this case), and the U/t ratio, where
U is the Coulomb repulsion energy for the interacting electrons and t is the overlap
95
integral. In this hypothetical scenario, the Raman intensity is low in the Mott insulating
state and high in the metallic state.
Figure 7.1: Resistivity versus temperature for a Mott insulator
6
(a) and phase diagram as a function of
carrier density and U/t ratio, showing hypothetical Raman intensity (University of Augsburg) (b).
Temperature Dependence of Kohn Anomaly Switching
Using a temperature-controlled stage I built during my research, it should be
possible to observe a dramatic exaggeration at low temperature of the gate voltage
controlled G
-
band downshift caused by the Kohn anomaly, as shown in the theoretical
prediction by Caudal
35
in Figure 7.2. I have already shown room temperature downshifts
of over 20 cm
-1
from the intrinsic Raman shift, and the effect is supposed to be
logarithmically divergent at low temperatures. Because of this, at 82K I expect to see
downshifts of up to 50 cm
-1
from the intrinsic Raman shift. Additionally, at low
temperatures, the linewidth of the Raman G
-
band takes on a step-function-like behavior,
dramatically reducing by a factor of 4-10 with small changes in the Fermi energy on the
order of K
B
T.
a b
Filling
(or E
F
)
0
Mott
Insulator
1D
metal
I
Raman
96
Figure 7.2: Raman G
-
band frequency as a function of Fermi level a room temperature and 77K.
Temperature Dependence of Conductance for an Advanced Landauer Model
The Landauer model for conductance of a CNT as a function of gate voltage
makes several assumptions about the nature of electrical transport. By utilizing electrical
gate voltage data from a temperature controlled stage, these assumptions should be tested
to verify their applicability. Such assumptions include the energy-independence of the
contact transmission coefficients and optical phonon mean free paths, and the dependence
of the electron-acoustic phonon mean free path on the group velocity. Shown in Figure
7.3 is one such data set, and one can clearly see the temperature dependence of the
conductance at various gate voltages. Analysis of data such as this with the Landauer
model will shed light onto the nature of electron transport in CNTs.
97
Figure 7.3: Conductance plotted versus gate-voltage at a variety of temperatures
Further Investigations into Preferential Phonon Heating:
Understanding of this preferential phonon heating phenomenon is limited because
of the limited access to the temperatures of a few Raman active phonon modes. The
following experiments will likely provide a more insightful understanding of these
phenomena: (1) A complete systematic study of the length dependence of the phenomena
described in the previous chapters, (2) a careful examination of the preferential phonon
heating observed in semiconductor nanotubes, (3) a measurement of the temperatures of
other Raman active modes as a function of electrical bias, and finally, (4) an examination
of the bias direction dependence of the spatial temperature profile. There is a significant
asymmetry in this spatial observed in Deshpande et al.
54
, however one cannot determine
if this is caused by thermal contact resistance asymmetry or bias voltage direction until
one takes data with both positive and negative bias voltage.
98
Transverse Electric Fields on CNTs
Until now, all experiments done in our lab have used a single back gate to apply
electric fields to suspended CNTs. The primary effect of such a field application is the
injection of charge onto the CNT, which is charged up like a capacitor. This effect
dominates any other effects. However, it would be interesting to study CNTs devices
with transverse electric fields but without charging of the nanotube, to look for changes
on the electronic and optical properties predicted in the literature
80, 92, 113
.
Figure 7.4: Device schematic for applying transverse electric fields to a suspended CNT FET.
CNT p-n Diode Experiments
In addition to transverse electric fields, a split gate can be used to create a nearly
ideal p-n diode using longitudinal electric fields with application of equal but opposite
gate voltages
90, 101
. Such diode junctions are efficient IR light emitters, and as of yet, no-
one has studied the Raman spectra from such CNT p-n junctions. This experiment would
allow for efficient collection and examination of the IR light emitted by the CNT p-n
junction, and also it would be interesting to examine the influence of longitudinal electric
fields on the Raman spectra.
99
Figure 7.5: Device schematic of a split-gate FET for creating a suspended CNT p-n junction.
Rayleigh Scattering Experiments on CNTs Showing the Intensity Effect
The data from the Raman intensity modulation experiments (Chapter 5) seem to
indicate a bulk modulation of the optical matrix element in CNTs with an applied gate
voltage. If this is so, a similar effect should be observable in the Rayleigh scattering
spectra of these CNTs. In addition, Rayleigh spectra can be used to measure the chirality
of CNTs, which can be used to investigate chirality specific effects. Shown below in
Figure 7.6 are sample Rayleigh scattering spectra from several CNTs of know chirality
Sfeir et al.
138
These spectra are expected to be modulated by application of a gate voltage
in pristine suspended metallic CNTs.
Figure 7.6: Rayleigh scattering spectra from individual suspended CNTs of known chirality.
100
Group Theory Analysis of Phonon-Phonon Coupling in CNTs
Hot optical phonon populations and preferential Raman band downshifting have
been a central focus of this thesis, however the exact mechanism that relate the two is
only understood qualitatively. In order to get a better understanding of how the former
causes the later, the allowed phonon decay paths should be examined using group theory.
Figure 7.7 from Bonini et al.
18
shows the primary phonon decay channels of graphene.
Because of the one-dimensionality of CNTs, the phonon decay channels are expected to
be severely restricted, and thus limited in number. Determination of these channels using
group theory in CNTs would shed light onto how heat from hot optical phonons escapes
the CNTs and causes preferential downshifting of various Raman bands.
Figure 7.7: Optical phonon decay channels in graphene.
101
1. Ando, T. Journal of the Physical Society of Japan 2005, 74, (3), 777-817.
2. Araujo, P. T., et al. Phys. Rev. B 2008, 77, (24), 241403(R).
3. Atashbar, M. Z.; Singamaneni, S. Appl. Phys. Lett. 2005, 86, (12), 123112-3.
4. Bachtold, A., et al. Phys. Rev. Lett. 2000, 84, (26), 6082.
5. Balasubramanian, K., et al. Nano Lett 2005, 5, (3), 507-510.
6. Balents, L.; Fisher, M. P. A. Phys. Rev. B 1997, 55, (18), R11973.
7. Balkanski, M., et al. Phys. Rev. B 1983, 28, (4), 1928.
8. Barnett, R., et al. Phys. Rev. B 2005, 71, (3), 22.
9. Bassil, A., et al. Appl. Phys. Lett. 2006, 88, (17), 173113-3.
10. Berber, S., et al. Phys. Rev. Lett. 2000, 84, (20), 4613.
11. Berdebes, B., et al., Low Bias Transport in Graphene: An Introduction. In
Electronics from the Bottom Up, Purdue: West Lafayette, 2009.
12. Berthe, M., et al. Nano Lett. 2007, 7, (12), 3623-3627.
13. Biercuk, M. J., et al. Topics in Applied Physics 2008, 111, 455-93.
14. Blase, X., et al. Phys. Rev. Lett. 1994, 72, (12), 1878.
15. Bockrath, M., et al. Nature 1999, 397, (6720), 598-601.
16. Bockrath, M., et al. Science 1997, 275, (5308), 1922-1925.
17. Bockrath, M., et al. Science 2001, 291, (5502), 283-285.
18. Bonini, N., et al. Phys. Rev. Lett. 2007, 99, (17), 176802-4.
19. Bonini, N., et al. Physica Status Solidi B 2008, 245, (10), 2149-2154.
20. Born, M.; Oppenheimer, R. Ann. Phys. 1927, 84, (20), 0457-0484.
21. Briman, M., et al. J. Appl. Phys. 2006, 100, 013505.
22. Brown, S. D. M., et al. Phys. Rev. B 2001, 63, (15), 155414.
Bibliography
102
23. Buldum, A.; Lu, J. P. Phys. Rev. B 2001, 63, (16), 161403-6.
24. Burke, P. J. Nanotechnology, IEEE Transactions on 2002, 1, (3), 129-144.
25. Bushmaker, A. W., et al. Nano Lett. 2007, 7, (12), 3618-3622.
26. Bushmaker, A. W., et al. Accepted in IEEE Transactions on Nanotechnology
2009.
27. Bushmaker, A. W., et al. Nano Lett. 2009, 9, (2), 607-611.
28. Bushmaker, A. W., et al. Nano Lett. 2009, 9, (8), 2862-2866.
29. Bushmaker, A. W., et al. Phys. Rev. Lett. 2009, 103, (6), 067401-4.
30. Camps, I., et al. Phys. Rev. B 2001, 64, (12), 125311.
31. Cao, J., et al. Small 2005, 1, (1), 138-141.
32. Capaz, R. B., et al. Phys. Rev. Lett. 2005, 94, (3), 36801.
33. Cappelluti, E.; Pietronero, L. SMEC 2005, Study of matter under extreme
conditions 2006, 67, (9-10), 1941-1947.
34. Cataldo, F. Fuller. Nanotub. Carbon Nanostruct. 2002, 10, (4), 293 - 311.
35. Caudal, N., et al. Phys. Rev. B 2007, 75, (11), 115423-11.
36. Charlier, J. C., et al. Phys. Rev. B 1996, 53, (16), 11108-13.
37. Che, L., et al. Science 2007, 317, (5841), 1061-1064.
38. Chen, C. T., et al. Phys. Rev. Lett. 1991, 66, (1), 104.
39. Chiashi, S., et al. Chem. Phys. Lett. 2004, 386, (1-3), 89-94.
40. Chico, L., et al. Phys. Rev. B 1996, 54, (4), 2600-6.
41. Chico, L., et al. Phys. Rev. Lett. 1996, 76, (6), 971-4.
42. Choi, H. J., et al. Phys. Rev. Lett. 2000, 84, (13), 2917-2920.
43. Chua, L. O.; Sung Mo, K. Proceedings of the IEEE 1976, 64, (2), 209-223.
44. Ci, L., et al. Appl. Phys. Lett. 2003, 82, (18), 3098-3100.
45. Connetable, D., et al. Phys. Rev. Lett. 2005, 94, (1), 015503.
103
46. Corio, P., et al. Chem. Phys. Lett. 2003, 370, (5-6), 675-682.
47. Crespi, V. H., et al. Phys. Rev. Lett. 1997, 79, (11), 2093-6.
48. Cronin, S. B., et al. Appl. Phys. Lett. 2004, 84, (12), 2052.
49. Das, A., et al. Nat. Nano. 2008, 3, (4), 210-215.
50. Das, A., et al. Phys. Rev. Lett. 2007, 99, (13), 136803.
51. Derycke, V., et al. Appl. Phys. Lett. 2002, 80, (15), 2773-2775.
52. Deshpande, V. V.; Bockrath, M. Nat Phys 2008, 4, (4), 314-318.
53. Deshpande, V. V., et al. Science 2009, 323, (5910), 106-110.
54. Deshpande, V. V., et al. Phys. Rev. Lett. 2009, 102, (10), 105501-4.
55. Dora, B., et al. Phys. Rev. Lett. 2007, 99, (16), 166402-4.
56. Dresselhaus, M. S., Carbon Nanotubes: Synthesis, Structure, Properties, and
Applications. Springer: 2001.
57. Dresselhaus, M. S., et al. Carbon 2002, 40, (12), 2043-2061.
58. Dresselhaus, M. S., et al. Phys. Rep. 2005, 409, (2), 47-99.
59. Dresselhaus, M. S., et al. Annual Review of Physical Chemistry 2007, 58, 719-
747.
60. Dubay, O.; Kresse, G. Phys. Rev. B 2003, 67, (3), 035401.
61. Dubay, O., et al. Phys. Rev. Lett. 2002, 88, (23), 235506.
62. Durkop, T., et al. Nano Lett. 2004, 4, (1), 35-39.
63. Fantini, C., et al. Phys. Rev. Lett. 2004, 93, (14), 147406.
64. Farhat, H., et al. Phys. Rev. Lett. 2007, 99, (14), 145506-4.
65. Firebaugh, S. L., et al. Microelectromechanical Systems, Journal of 1998, 7, (1),
128-135.
66. Harris, D. C.; Bertolucci, M. D., Symmetry and spectroscopy. Dover Publications:
1978.
67. Hsu, I.-K., et al. Applied Physics Letters 2008, 92, 063119
104
68. Huang, F., et al. J. Appl. Phys. 1998, 84, (7), 4022-4024.
69. Huong, P. V., et al. Phys. Rev. B 1995, 51, (15), 10048.
70. Hwang, E. H.; Das Sarma, S. Phys. Rev. B 2008, 77, (11), 115449-6.
71. Ishii, H., et al. Nature 2003, 426, (6966), 540-544.
72. Javey, A., et al. Phys. Rev. Lett. 2004, 92, (10), 106804-4.
73. Jiang, J., et al. Phys. Rev. B 2005, 72, (23), 235408-11.
74. Jorio, A., et al., Carbon nanotubes: advanced topics in the synthesis, structure,
properties and applications. Springer: 2008.
75. Jorio, A., et al. Phys. Rev. B 2002, 66, (11), 115411.
76. Kane, C. L.; Mele, E. J. Phys. Rev. Lett. 2003, 90, (20), 207401.
77. Kane, C. L., et al. Europhys. Lett. 1998, 41, (6), 683-688.
78. Katsufuji, T.; Tokura, Y. Phys. Rev. B 1994, 50, (4), 2704.
79. Kavan, L., et al. J. Phys. Chem. B 2001, 105, (44), 10764-10771.
80. Kim, Y.-H.; Chang, K. J. Phys. Rev. B 2001, 64, (15), 153404.
81. Kong, J., et al. Nature 1998, 395, (6705), 878-881.
82. Kong, J., et al. Phys. Rev. Lett. 2001, 87, (10), 106801.
83. Kostyrko, T., et al. Phys. Rev. B 1999, 59, (4), 3241-9.
84. Koswatta, S. O., et al. Appl. Phys. Lett. 2006, 89, (2), 023125-3.
85. Kuemmeth, F., et al. Nature 2008, 452, (7186), 448-452.
86. Kumar, R.; Cronin, S. B. Phys. Rev. B 2007, 75, (15), 155421-4.
87. Lazzeri, M.; Mauri, F. Phys. Rev. B 2006, 73, (16), 165419.
88. Lazzeri, M., et al. Phys. Rev. Lett. 2005, 95, 236802-5.
89. Lazzeri, M., et al. Phys. Rev. B 2006, 73, (15), 155426.
90. Lee, J. U., et al. Appl. Phys. Lett. 2004, 85, 145.
91. Li, H. D., et al. Appl. Phys. Lett. 2000, 76, (15), 2053-2055.
105
92. Li, Y., et al. Nano Lett. 2003, 3, (2), 183-187.
93. Lin, Y. M., et al. Nanotechnology 2007, 18, (29).
94. Malola, S., et al. Phys. Rev. B 2008, 78, (15), 153409-4.
95. Mann, D., et al. J. Phys. Chem. B Lett. 2006, 110, (4), 1502-1505.
96. Martel, R., et al. Phys. Rev. Lett. 2001, 87, (25), 256805.
97. Merlin, R. Solid State Commun. 1997, 102, (2-3), 207-220.
98. Meyer, J. C., et al. Phys. Rev. Lett. 2005, 95, (21), 217401-4.
99. Minot, E. D. Tuning the Band Structure of Carbon Nanotubes. Cornell University,
2004.
100. Minot, E. D., et al. Phys. Rev. Lett. 2003, 90, (15), 156401.
101. Mueller, T., et al. Nat Nano 5, (1), 27-31.
102. Nahory, R. E. Phys. Rev. Lett. 1969, 178, (3), 1293.
103. Nakamura, M., et al. Phys. Rev. B 2007, 75, (15), 155103.
104. Newns, D. M., et al. Journal of Electroceramics 2000, 4, (2), 339-344.
105. Nguyen, K. T., et al. Phys. Rev. Lett. 2007, 98, (14), 145504-4.
106. Oron-Carl, M.; Krupke, R. Phys. Rev. Lett. 2008, 100, (12), 127401-4.
107. Osvath, Z., et al. Phys. Rev. B 2005, 72, (4), 045429-34.
108. Ouyang, M., et al. Science 2001, 292, (5517), 702-705.
109. Park, J. W., et al. Appl. Phys. Lett. 2002, 80, (1), 133-135.
110. Park, J. Y., et al. Nano Lett. 2003, 4, (3), 517.
111. Park, J. Y., et al. Nano Lett. 2003, 4, (3), 517.
112. Pennington, G.; Goldsman, N. Phys. Rev. B 2003, 68, (4), 045426.
113. Perebeinos, V.; Avouris, P. Nano Lett. 2007, 7, (3), 609-613.
114. Perebeinos, V., et al. Nano Lett. 2009, 9, (1), 312-316.
115. Perebeinos, V., et al. Phys. Rev. Lett. 2004, 92, (1), 257402-257402.
106
116. Perebeinos, V., et al. Phys. Rev. Lett. 2005, 94, (8), 086802.
117. Pershin, Y. V.; Di Ventra, M. Phys. Rev. B 2008, 78, (11), 113309-4.
118. Pisana, S., et al. Nature Materials 2007, 6, 198–201.
119. Piscanec, S., et al. Phys. Rev. Lett. 2004, 93, (18), 185503.
120. Piscanec, S., et al. Phys. Rev. B 2007, 75, (3), 035427.
121. Pomeroy, J. W., et al. Phys. Status Solidi B 2008, 245, (5), 910-912.
122. Pop, E., et al. Phys. Rev. Lett. 2005, 95, (15), 155505-8.
123. Pop, E., et al. J. Appl. Phys. 2007, 101, (9), 093710-10.
124. Qi, P., et al. Nano Lett. 2003, 3, (3), 347-351.
125. Rafailov, P. M., et al. Phys. Rev. B 2005, 72, (4), 045411.
126. Raravikar, N. R., et al. Phys. Rev. B 2002, 66, (23), 235424.
127. Reich, S., et al., Carbon Nanotubes: Basic Concepts and Physical Properties.
Wiley-VCH: 2004.
128. Rodrigues, C. G., et al. Solid State Commun. 2006, 140, (3-4), 135-140.
129. Rosenblatt, S., et al. Applied Physics Letters 2005, 87, (15), 153111.
130. Saito, R., et al., Physical Properties of Carbon Nanotubes. Imperial College
Press: London, 1998.
131. Saito, R., et al. Phys. Rev. B 2001, 64, (8), 085312.
132. Saitta, A. M., et al. Phys. Rev. Lett. 2008, 100, (22), 226401-4.
133. Samsonidze, G. G. Photophysics of carbon nanotubes. Massachusetts Institute of
Technology, 2007.
134. Samsonidze, G. G., et al. Phys. Rev. B 2007, 75, (15), 155420-8.
135. Sasaki, K.-i., et al. Phys. Rev. B 2008, 77, (24), 245441-8.
136. Sazonova, V., et al. Nature 2004, 431, (7006), 284.
137. Seri, T.; Ando, T. Journal of the Physical Society of Japan 1997, 66, (1), 169-173.
138. Sfeir, M. Y., et al. Science 2006, 312, (5773), 554-556.
107
139. Shaw, R. W. Phys. Rev. B 1971, 3, (10), 3283.
140. Shishir, R., et al. Journal of Computational Electronics 2009, 8, (2), 43-50.
141. Simmons, J. M., et al. Phys. Rev. Lett. 2007, 98, (8), 86802-5.
142. Steiner, M., et al. Nat Nano 2009, advanced online publication.
143. Strukov, D. B., et al. Nature 2008, 453, (7191), 80-83.
144. Tersoff, J., et al. Appl. Phys. Lett. 2005, 86, (26), 263108-3.
145. Tsang, J. C., et al. Nat. Nano. 2007, 2, (11), 725-730.
146. Tucker, E. B. Phys. Rev. Lett. 1961, 6, (10), 547.
147. Uchida, T., et al. Chem. Phys. Lett. 2004, 400, (4-6), 341-346.
148. Verma, A., et al. J. Appl. Phys. 2005, 97, (11), 114319-8.
149. Vidal-Valat, G., et al., Evidence on the breakdown of the Born-Oppenheimer
approximation in the charge density of crystalline 7LiH/D. In Acta
Crystallographica Section A, 1992; Vol. 48, pp 46-60.
150. Wang, F., et al. Science 2005, 308, (5723), 838-841.
151. Wu, J.; McCreery, R. L. Journal of The Electrochemical Society 2009, 156, (1),
29-37.
152. Wu, Y., et al. Phys. Rev. Lett. 2007, 99, (2), 027402-4.
153. Yan, J., et al. Phys. Rev. Lett. 2007, 101, 136804.
154. Yang, J. J., et al. Nanotechnology 2009, 20, 215201-9.
155. Yang, J. J., et al. Nat Nano 2008, 3, (7), 429-433.
156. Yao, Z., et al. Phys. Rev. Lett. 2000, 84, (13), 2941.
157. Yao, Z., et al. Nature 1999, 402, (6759), 273-276.
158. Yu, M.-F., et al. Phys. Rev. Lett. 2000, 84, (24), 5552.
159. Yu, P. Y.; Cardona, M., Fundamentals of semiconductors: physics and materials
properties. Springer Verlag: 2005.
160. Zhang, Y., et al. J. Am. Chem. Soc. 2005, 127, (49), 17156-17157.
108
161. Zhong, Z. H., et al. Nat. Nanotechnol. 2008, 3, (4), 201-205.
162. Zhou, C., et al. Phys. Rev. Lett. 2000, 84, (24), 5604.
109
Appendix: Matlab codes
While working on CNTs in Dr. Cronin’s lab, I wrote a substantial amount of
Matlab code to theoretically model the behavior of CNTs for general understanding and
to fit experimental data. In this appendix, I have included about 1000 lines of code from
the programs I have found most useful.
CNT Landauer Model
function [Vg,EF,G,Vt] = CNT_Conductivity(plotstate,T,pnCond,CgeomL,...
alpha,Rc,Trp,Trn,Eg,offset,Vb,LAC,LOPmin,L,fullname)
global curve
if plotstate ~= 0
sampledata = load(fullname);
end
hbar = 6.58211814e-16; h = hbar*2*pi; kT = 0.0259*T/300; % eVs, eV
vF = 840000; e = 1.602e-19; % m/s (Fermi velocity), C (elem. charge)
eps = 1e-12; % small value epsilon for limiting infinities
Cgeom = CgeomL*L; NOP = 1/(exp(0.16/kT)-1); LAC = LAC*300/T;
% the purpose of this program is to find the conductivity of a small
% bandgap nanotube by summing over the density of states using the
% boltzmann approximation with ballistic transport
% Legacy code by Adam Bushmaker Jan. 2007 - Feb. 2010
nEF = 51; maxEF = 0.3; minEF = -0.3; % calculate array of EF values
EFrange = maxEF-minEF; EF = minEF:EFrange/(nEF-1):maxEF;
nEFg = 101; maxEFg = maxEF+0.2;
% EF array for spatial doping dependence calculations
minEFg = minEF-0.2; EFgrange = maxEFg-minEFg;
EFg = minEFg:EFgrange/(nEFg-1):maxEFg;
maxE = 0.6; minE = -.6; nE = 501;
% eV, initialize the energy variable, to integrate over
delE = (maxE-minE)/(nE-1); E = minE:delE:maxE;
% integrate several kT past the max EF
vcE = ceil(1e-6*E);
% for calculating the valence/conduction band effects, charge density,
% the conduction band has vcE = 1, the valence band has vcE = 0
[min iCb] = min(abs(Eg/2-E)); [min iVb] = min(abs(-Eg/2-E));
band = [1:iVb iCb:nE]; % find band indicies for integration limits
110
k = real(sqrt(E.^2-Eg^2/4))/(hbar*vF);
% nanotube hyperbolic dispersion relationship with a minigap k(E)
DE(1:nE-1) = 4*abs(k(1:nE-1)-k(2:nE))/(pi*delE); DE(nE) = DE(nE-1);
% calculate the density of states, multiply by 4 to included spin
% and isospin degeneracy above I numerically diferentiate to find
% 1/|dE/dk|. The analytical solution is divergent at the van-Hove
% singularities, and is thus very sensitive to the values you
% chose in E to integrate over (below). This solution gets rid of
% the problem because it discounts the van hove singularites.
vg = vF*real(sqrt(1-(Eg./(2*abs(E+delE*eps))).^2)); vr = vg'/vF;
% calculate electronic group velocities, use hyperbolic band structure
for iEFg = 1:nEFg % calculate stuff for each fermi energy
fg = 1./(1+exp((E-EFg(iEFg))/kT)); % calc the fermi funct.
n = e*sum(fg.*DE.*vcE-(1-fg).*DE.*(1-vcE))*delE; % calc the charge
Q(iEFg) = L*n; % calculate the charge on the tube
Vgeom(iEFg) = Q(iEFg)/Cgeom; % calculate the charging voltage
Vgg(iEFg) = (Vgeom(iEFg)+EFg(iEFg))/alpha+offset; % calculate Vg
end
Vgg = Vgg+0.5*Vb;
for iEF = 1:nEF % calculate stuff for each fermi energy
f = 1./(1+exp((E-EF(iEF))/kT)); % calc the fermi funct.
n = e*sum(f.*DE.*vcE-(1-f).*DE.*(1-vcE))*delE; % calc the charge
Q(iEF) = L*n; % calculate the charge on the tube
Vgeom(iEF) = Q(iEF)/Cgeom; % calculate the charging voltage
Vg(iEF) = (Vgeom(iEF)+EF(iEF))/alpha+offset; % calculate Vg
end
Vg = Vg+0.5*Vb;
C(1:nEF-1)=(Q(1:nEF-1)-Q(2:nEF))./(Vg(1:nEF-1)-Vg(2:nEF));
C(nEF) = C(nEF-1); C0 = Cgeom*ones(length(Vg),1);
% calculate the capacitance by numerically diferentiating
Tc(1:(nE-1)/2) = Trp; Tc((nE+1)/2:nE) = Trn;
% transmission coefs, might add an E dependence for schottky barriers
curve = 1
% used to fit a single lorrentzian defect scatterer
if curve == 1
T0 = 0;%.972; % to turn defect off just set T0 to 0
else
T0 = 0.972;
end
if curve == 2 %after W
Wd = .3; %eV 0.3 for "after W"
Ed = -0.18; %eV -0.18 for "after W"
else %after U
Wd = .4; %eV 0.4 for "after U"
Ed = -0.03; %eV -0.03eV for "after U"
end
Td = 1-T0./(1+((E-Ed)/Wd).^2);
for iEF = 1:nEF
111
% dfdE = exp((E-EF(iEF))/kT)./(kT*(1+exp((E-EF(iEF))/kT)).^2);
VcL(iEF) = Vb/3; VcR = VcL; Vt = Vb-2*VcL;
nx = 21; dx = L/(nx-1);
Vx(iEF,:) = Vt(iEF)*(-0.5:1/(nx-1):0.5)+Vg(iEF);
% calculate the voltage drops across the tube and contacts
dVdx(iEF,1:nx-1) = abs(Vx(iEF,1:nx-1)-Vx(iEF,2:nx))/dx;
dVdx(iEF,nx) = dVdx(iEF,nx-1);
% generate dV(x) from Vx(x)
for i = 1:10
% iterate to find the voltage distribution on the nanotube
LOPems = 0.16./(dVdx(iEF,:)+eps)+LOPmin;
LOPabs = LOPmin*(1+NOP)/NOP;
% find the threshold optical phonon emission scattering length
xa = ones(1,nx); Ea = ones(nE,1);
LambdaEFF = 1./(1./(LAC*(vr.^2)*xa+eps)+...
1./(Ea*LOPems)+1./(LOPabs*vr*xa+eps));
% Use Matthiessen's rule
Tx(:,:,iEF) = LambdaEFF./(LambdaEFF+dx);
% T(E,x,EF) transmission coef / unit length
EFx(iEF,:) = interp1(Vgg,EFg,Vx(iEF,:));
% generate EF(x) from Vx(x)
f = 1./(1+exp((E'*xa-Ea*EFx(iEF,:))/kT));
dfdE = (f(1:nE-1,:)-f(2:nE,:))/delE;
dfdE(nE,:) = dfdE(nE-1,:); dfdEA(:,:,iEF) = dfdE;
% generate f(E,x) and dfdE(E,x) from EF(x) (the fermi funct.)
dqc = sum(dfdE(band,:))*delE; Gdqc = 4*e/h*dqc;
% calculate the depleted quantum channel dqc(x)
% generate Gs(x) from f(x) and
Txstar(iEF,:) = sum(Tx(band,:,iEF).*dfdE(band,:))*delE./dqc;
% Txstar(EF,x) calc the effective transmission coeficient per
% unit length by integrating over the bands
Gx(iEF,:) = Gdqc(1,:).*Txstar(iEF,:)./(1-Txstar(iEF,:));
Gt = 1./(sum(1./(Gx')));
% calculate the nanotube conductance
qfrL = 1./(1+exp((E-(EFx(iEF,1) -VcL(iEF)/2))/kT));
qfrR = 1./(1+exp((E-(EFx(iEF,nx)-VcR(iEF)/2))/kT));
% right moving quasi Fermi energy distr. at L and R contacts
qflL = 1./(1+exp((E-(EFx(iEF,1) +VcL(iEF)/2))/kT));
qflR = 1./(1+exp((E-(EFx(iEF,nx)+VcR(iEF)/2))/kT));
% left moving quasi Fermi energy distr. at L and R contacts
dfL = (qflL-qfrL)/VcL(iEF); dfR = (qflR-qfrR)/VcR(iEF);
dqcL = sum(dfL(band))*delE; dqcR = sum(dfR(band))*delE;
% calculate the depleted quantum channel at the contacts
TL = sum(dfL(band).*Tc(band))*delE/dqcL;
TR = sum(dfR(band).*Tc(band))*delE/dqcR;
GcL(iEF) = 4*e/h*dqcL*TL/(1-TL);
GcR(iEF) = 4*e/h*dqcR*TR/(1-TR);
VcL(iEF) = Vb/(GcL(iEF)*(1/Gt(iEF)+1/GcL(iEF)+1/GcR(iEF)));
VcR(iEF) = Vb/(GcR(iEF)*(1/Gt(iEF)+1/GcL(iEF)+1/GcR(iEF)));
% calculate the transmition coefs, conductance,
% and voltage drops at contacts
dVdx = Vb./(Gx*(1/Gt(iEF)+1/GcL(iEF)+1/GcR(iEF)))/dx;
Vx(iEF,:) = dVdx(iEF,:)*triu(ones(nx))*dx+Vg(iEF);
Vt(iEF)=sum(Vx(iEF,:));
112
% triu = upper triangle, used to integrate dVdx to Vx(EF,x)
Tdeff(iEF) = sum(Td(band)'.*dfdE(band,(nx-1)/2))*...
delE/dqc(1,(nx-1)/2);
Gd(iEF) = Gdqc(1,(nx-1)/2).*Tdeff(iEF)./(1-Tdeff(iEF)+eps);
% calc the defect scattering coeficient and conductance
% (Td = 1 for no defect)
end
G(iEF) = abs(1/(1/Gt(iEF)+1/GcL(iEF)+1/GcR(iEF)+...
1/Gdqc(1,(nx-1)/2)+1/Gd(iEF))); % calculate the total cond.
end
%plotting...
if plotstate == 1
% for use when just ploting conductivity data alone
subplot(1,2,1); plot(Vg,G,'Color','red'); hold on
plot(sampledata(:,1),sampledata(:,2))
ylabel('Conductance'); xlabel('Vg (V)')
xlim([-max(sampledata(:,1)) max(sampledata(:,1))])
% subplot(3,1,1); plot(EF,Q,'Color','red'); hold on
% subplot(2,2,2); plot(EF,Q); hold on
% ylabel('Charge (C)'); xlabel('EF (eV)')
% subplot(2,2,3); plot(VgR,EFR); hold on
% subplot(2,2,2); plot(Vg,Q); hold on
subplot(1,2,2); plot(Vg,EF); hold on
% xlim([min(E) max(E)])
ylabel('Vc'); xlabel('Vg');
xlim([-max(sampledata(:,1)) max(sampledata(:,1))])
% xlabel('Vg (V)')
ylabel('EF (eV)'); xlabel('Vg (V)')
% xlim([min(sampledata(:,1)) max(sampledata(:,1))])
% ylim([0 max(C)*1.2])
% subplot(2,2,4); plot(EF,C); hold on
% ylabel('C'); xlabel('E (eV)')
% xlim([-0.5 0.5])
end
if plotstate == 2
% for use when ploting conductivity data with other data
subplot(2,3,1); plot(Vg,G,'Color','red'); hold on
ylabel('Conductance'); xlabel('Gate Voltage (V)')
plot(sampledata(:,1),sampledata(:,2))
xlim([-max(sampledata(:,1)) max(sampledata(:,1))])
% hold off
subplot(2,3,4); plot(Vg,EF); hold on
ylabel('EF'); xlabel('Vg'); % grid
xlim([-6 6]);
ylim([-.2 .2]);
% subplot(2,3,4); plot(Vg,Vc); hold on
% ylabel('Vc'); xlabel('Vg'); % grid
% plot((sampledata(:,1)),sampledata(:,2))
113
xlim([-max(sampledata(:,1)) max(sampledata(:,1))])
% ylim([0 Vb])
% ylim([0 3e-5])
% hold off
end
% This code for use in plotting spatially varying parameters V(x),etc
x = 0:L/(nx-1):L;
iEFprobe = 22;
% This index chooses which gate voltage/doping level to plot data for
EFprobe = EF(iEFprobe)
Vgprobe = Vg(iEFprobe)
dfdE = dfdEA(:,:,iEFprobe);
[nV d] = size(sampledata)
if plotstate == 3
subplot(4,3,1); plot(Vg,G,'Color','red'); hold on
ylabel('Conductance'); xlabel('Gate Voltage (V)');
plot(sampledata(:,1),sampledata(:,2))
subplot(4,3,4); plot(Vg,VcL);
ylabel('Left Vc'); xlabel('Gate Voltage (V)')
ylim([0 Vb]); xlim([-max(sampledata(:,1)) max(sampledata(:,1))])
subplot(4,3,7); plot(Vg,VcL+VcR);
plot(Vg,Vt,'Color','red');
ylabel('V_C_N_T (red), V_C (blue)'); xlabel('Gate Voltage (V)')
ylim([0 Vb]); xlim([-max(sampledata(:,1)) max(sampledata(:,1))])
subplot(4,3,10);plot(Vg,EF);
ylabel('EF'); xlabel('Gate Voltage (V)')
xlim([-max(sampledata(:,1)) max(sampledata(:,1))])
subplot(4,3,2); plot(Vg,G.^-1);
ylabel('Resistance'); xlabel('Gate Voltage (V)')
xlim([-max(sampledata(:,1)) max(sampledata(:,1))])
subplot(4,3,11);plot(x,max(dfdE));
ylabel('max dfdE'); xlabel('x (m)')
subplot(4,3,3); plot(x,Vx(iEFprobe,:));
ylabel('V drop in tube (V)'); xlabel('x (m)')
subplot(4,3,6); plot(x,dVdx(iEFprobe,:));
ylabel('E-field (V/m)'); xlabel('x (m)')
subplot(4,3,9); plot(x,Gx(iEFprobe,:));
ylabel('conductance'); xlabel('x (m)')
subplot(4,3,12);plot(x,EFx(iEFprobe,:));
ylabel('EF'); xlabel('x (m)')
subplot(4,3,5); plot(Vg,C*1e18,Vg,C0*1e18);
ylabel('Capacitance (aF)'); xlabel('Gate Voltage (V)')
xlim([-max(sampledata(:,1)) max(sampledata(:,1))]);
114
ylim([0 1.2*max(C0*1e18)])
hold off
end
if plotstate ~= 0
% write the model curve to a new data file
modelcurve = [Vg' G']; filelength = length(fullname);
EFcurve = [Vg' EF'];
fullnametrunc = fullname(1:filelength-4);
modelcurvename = [fullnametrunc,'_model.txt'];
EFcurvename = [fullnametrunc,'_EFs.txt'];
dlmwrite(modelcurvename,modelcurve, '\t')
dlmwrite(EFcurvename,EFcurve, '\t')
end
Sample CNT Data Fitting File
function CNT_data_fit = CNT_data_fit
% This program is going to be writen to fit the phonon frequency and
% conductance as a function of gate voltage data taken from a specific
% individual CNT. Each CNT has its own CNT_data_fit file, which holds
% fit parameters and data file path info and calls the modeling
% functions which use these fit parameters as input.
%--------------------------------------------------------------------
% file locations
Freqname = 'pwd\5-OH6,13B_wG_vs_Vg.txt';
FWHMname = 'pwd\5-OH6,13B_FWHM_vs_Vg.txt';
Condname = 'pwd\run 42, Vb = 100mV, Vg = -4000mV to 4000mV_COND.txt';
%--------------------------------------------------------------------
% user fitable parameters
% transport
pnCond = 1; % unitless, to account for p-n asymmetry
Trn = 0.16; % n-type conduction contact transmission coefficient
Trp = 0.31; % p-type conduction contact transmission coefficient
Rc = 0e3; % ohms, the contact resistance
Eg = 0.11; % eV, the minigap size or 0.12
Vb = 0.1; % V, bias voltage
LAC = 1e-6; % m, acoustic phonon scattering mean free path
LOPmin = 35e-9; % m, optical phonon minimum mfp
% Raman G-band
w0 = 1582; % cm^-1, the intrinsic phonon freq. (G-band) w/o KA
FWHM0 = 10; % cm^-1, the intrinsic phonon linewidth
gam = 0.0008; % eV, the broadening factor to prevent singularities
Ecutoff = .29; % eV, cutoff energy for phonon renorm. integral
offset2 = .15; % V, offset of Raman data from electrical data
115
% General tube info
alpha = 1.0; % gate efficiency, to account for electrode screening
CgeomL = 18e-12;% Capacitance / unit length for the nanotube
T = 300; % K, the temperature
L = 5e-6; % m, the nanotube length
dt = 2.1e-9; % m, tube diameter
Deph = 40; % (eV/angstrom)^2,
offset = 0.14; % V, the gate voltage offset from gas doping effects
%--------------------------------------------------------------------
[Vg EF Gt Vt] = ...
CNT_Conductivity(3,T,pnCond,CgeomL,alpha,Rc,Trp,Trn,Eg,offset,...
Vb,LAC,LOPmin,L,Condname);
% for fitting conductance vs Vg data
CNT_Raman_vs_Vg(T,EF,Vg,Eg,offset2,L,w0,gam,Ecutoff,FWHM0,dt,...
Deph,Freqname,FWHMname);
% for fitting Raman-Vg data for metallic CNTs (KA G band shift)
CNT Minimum Conductance versus Temperature
function [Vg,EF,G,Vt] = CNTminG
% This function plots the minimum CNT conductance versus temp
global curve
sampledata = load('root\min G vs. T.txt');
% the data set was min conductance vs. T data taken from a nanotube
% on a temperature controlled stage. The first column is the
% Temperature, the second is the thermal energy in eV, the third is
% the min conductance, the fourth is the band gap (included in the
% source file to allow for changes with temperature
[nT ncol] = size(sampledata)
L = 5e-6;
hbar = 6.58211814e-16; h = hbar*2*pi; % eVs
vF = 840000; e = 1.602e-19; % m/s (Fermi velocity), C (elem. charge)
eps = 1e-12; %small value epsilon for limiting infinities
LambdaAC = 1e-6;
kT = sampledata(:,2); Eg = sampledata(:,4);
% the purpose of this program is to find the conductivity of a small
% bandgap nanotube by summing over the density of states using the
% boltzmann approximation with ballistic transport
EF=0; maxE = 0.3; minE = -.3; % eV
nE = 501; % start integrating several kT past the max/min Fermi energy
delE = (maxE-minE)/(nE-1); E = minE:delE:maxE;
for i = 1:nT
[min iCb] = min(abs(Eg(i)/2-E)); [min iVb] = min(abs(-Eg(i)/2-E));
band = [1:iVb iCb:nE];
k = real(sqrt(E.^2-Eg(i)^2/4))/(hbar*vF);
% nanotube hyperbolic dispersion relationship with a minigap
116
DE(1:nE-1) = 4*abs(k(1:nE-1)-k(2:nE))/(pi*delE); DE(nE)=DE(nE-1);
% calculate the density of states, multiply by 4 to included spin
% and isospin degeneracy. Above I numerically diferentiate to find
% 1/|dE/dk|, because the analytical solution is divergent at the
% van-Hove singularities, and is thus very sensitive to the values
% you chose in E to integrate over (below). This solution gets rid
% of the problem because it discounts the van hove singularites.
vg = vF*real(sqrt(1-(Eg(i)./(2*abs(E+eps))).^2));
% calculate the group velocities, use hyperbolic band gap struct.
LambdaEFF = LambdaAC*(0.026/kT(i))*vg.^2/(vF^2);
% TE(E,x), shorten proportional to vg^2
TE = LambdaEFF./(LambdaEFF+L);
f = 1./(1+exp(E/kT(i))); % generate f and dfdE for EF = 0
dfdE = (f(1:nE-1)-f(2:nE))/delE; dfdE(nE) = dfdE(nE-1);
dqc = sum(dfdE(band))*delE;
% calculate the depleted quantum channel dqc
Tstar = sum(TE(band).*dfdE(band))*delE/dqc;
% calc the eff. transmission coeficient by integrating over bands
GT(i) = 4*e/h*dqc*Tstar; % calculate the nanotube conductance
end
plot(sampledata(:,1),sampledata(:,3),sampledata(:,1),GT)
xlabel('T (C)'); ylabel('conductance (S)')
Mean Free Path Calculations
function MFP
% This program calculates the mean free path as a function of bias
% and gate voltage. It is developed mainly as an illustrative plot
% generator, not to fit actual data. CNT_conductivity.m is better
% suited for that.
T = 300; CgeomL = 5e-12; L = 5; %K; F/m; um;
hbar = 6.58211814e-16; h = hbar*2*pi; kT = 0.0259*T/300; % eV*s, eV
vF = 840000; e = 1.602e-19; % m/s (Fermi velocity), C (elem. charge)
eps = 1e-12; % small value epsilon for limiting infinities
Cgeom = CgeomL*L; NOP = 1/(exp(0.16/kT)-1);
V = .0000001:.002:1.2; % V
Eph = .16; Eg = .12; %eV
LOP = L*Eph./V; %threshold OP emission length
LOPmin = 0.03; %OP post-threshold MFP
lacH = 2.4; %um, high energy (doped) MFP of AC scattering
x = 0:0.005:30; dx = (x(2)-x(1)); Lx = length(x); %um
maxE = 0.9;
minE = -0.9;
nE = 501;% start integrating several kT past the max/min Fermi energy
delE = (maxE-minE)/(nE-1); E = minE:delE:maxE;
% initialize the energy variable, to integrate over
vcE = ceil(1e-6*E);
% for calculating the valence band/conduction band effects, charge
117
% density contributions, the conduction band has vcE = 1,
% the valence band has vcE = 0 (step function)
[dummy iCb] = min(abs(Eg/2-E)); [dummy iVb] = min(abs(-Eg/2-E));
band = [1:iVb iCb:nE];
% find indicies of bands for integration limits
k = real(sqrt(E.^2-Eg^2/4))/(hbar*vF);
% nanotube hyperbolic dispersion relationship with a minigap k(E)
DE(1:nE-1) = 4*abs(k(1:nE-1)-k(2:nE))/(pi*delE); DE(nE) = DE(nE-1);
% calculate the density of states, multiply by 4 to included spin and
% isospin degeneracy above I numerically diferentiate to find
% 1/|dE/dk|. The analytical solution is divergent at the van-Hove
% singularities, and is thus very sensitive to the values you chose
% in E to integrate over (below). This solution gets rid of the
% problem because it discounts the van hove singularites
vsbend=hbar*9.43E+03; vstwis=hbar*1.50E+04; vslong=hbar*2.04E+04;
% sound velocity for low energy acoustic phonon modes in CNTs (m/s)
vg = vF*real(sqrt(1-(Eg./(2*abs(E+delE*eps))).^2));
% calc. the electronic group velocities, use hyperbolic band struct.
nEF = 41; maxEF = 0.5; minEF = -0.5;
% calculate the array of EF values
EFrange = maxEF-minEF; EF = minEF:EFrange/(nEF-1):maxEF;
Nph = 1./(exp(vslong*2*k/kT)-1+eps^3); % ac phonon occupation
mfp = (lacH/343)*vg.^2./(k.*(2*Nph+1)+eps^3);
% most people assume k*Nph ~ constant, true for low energy ac phonons
for iEF = 1:nEF % calculate mean group velocity and mfp for each EF
f = 1./(1+exp((E-EF(iEF))/kT)); %calc the fermi funct.
dfdE(iEF,1:nE-1) = (f(1:nE-1)-f(2:nE))/delE;
dfdE(iEF,nE) = dfdE(iEF,nE-1);
% calculate the derivitive of the fermi function dfdE(E)
dqc(iEF) = sum(dfdE(iEF,band))*delE;
% calculate the size of the depleted quantum channel (max=1)
vgmean(iEF) = sum(dfdE(iEF,band).*vg(band))*delE/dqc(iEF);
% calculate the mean group velocity
mfpeff(iEF) = sum(dfdE(iEF,band).*mfp(band))*delE/dqc(iEF);
% calculate the effective mfp = the dfdE weighted average mfp(E)
% n = e*sum(f.*DE.*vcE-(1-f).*DE.*(1-vcE))*delE; %calc the charge
% Q(iEF) = L*n; %calculate the charge on the tube
% Vgeom(iEF) = Q(iEF)/Cgeom; %calculate the charging voltage
% Vg(iEF) = (Vgeom(iEF)+EF(iEF))/alpha+offset2; %calculate the Vg
end
EFdp = .15; [Y, idoped]=min(abs(EF-EFdp));
lacdoped = mfpeff(idoped); lacundop = min(mfpeff);
% find doped and undoped ac mfps
VeffReduction = min(vgmean)/vF; mfpReduction = lacundop/lacdoped
nuundop = exp(-LOP/lacundop); % percentage power dissipated into OPs
nudoped = exp(-LOP/lacdoped);
118
leffmfpundop = (1/lacundop+1./(LOP+LOPmin)).^-1;
leffmfpdoped = (1/lacdoped+1./(LOP+LOPmin)).^-1;
% calc. the doped/undoped mfps according to Matthiessen's rule
leffthrundopan = lacundop*(1-exp(-(LOP)/lacundop));
leffthrdopedan = lacdoped*(1-exp(-(LOP)/lacdoped));
% analytical solution for mfp replacing Matthiessen's rule
for nn = 1:length(V) % undoped calculations
Pac = exp(-x/lacundop);
% probability of scattering from ac phonon between x and dx
POPmfp = exp(-x/LOP(nn));
% faux probability of scattering from an OP between x and dx
U = sign(LOP(nn)-x)/2+0.5;
POPthr = U+(1-U).*exp(-abs(x-LOP(nn))/LOPmin);
% step function for scattering probability off OPs
Peffthrundop = Pac.*POPthr;
dPthrundop(1:Lx-1) = (Peffthrundop(1:Lx-1)-Peffthrundop(2:Lx));
dPthrundop(Lx) = dPthrundop(Lx-1);
leffthrundop(nn) = sum(x.*dPthrundop);
end
for nn = 1:length(V) % doped calculations
Pac = exp(-x/lacdoped);
% probability of scattering from ac phonon between x and dx
POPmfp = exp(-x/LOP(nn));
% faux probability of scattering from an OP between x and dx
U = sign(LOP(nn)-x)/2+0.5;
POPthr = U+(1-U).*exp(-abs(x-LOP(nn))/LOPmin);
% step function for scattering probability off OPs
Peffthrdoped = Pac.*POPthr;
dPthrdoped(1:Lx-1) = (Peffthrdoped(1:Lx-1)-Peffthrdoped(2:Lx));
dPthrdoped(Lx) = dPthrdoped(Lx-1);
leffthrdoped(nn) = sum(x.*dPthrdoped);
end
plotstate = 1;
if plotstate == 1
subplot(2,2,1);
plot(V,leffmfpundop,V,leffthrundop,V,leffmfpdoped,V,leffthrdoped);
ylabel('MFP (um)'); xlabel('Vb (V)')
ylim([0 lacH]); xlim([0 1.2])
subplot(2,2,2); plot(V,nuundop,V,nudoped);
xlim([0 1.2]); ylim([0 1])
ylabel('nu (fraction of)'); xlabel('Vb (V)')
subplot(2,2,3); plot(E,k.*(2*Nph+1));
xlim([minE maxE]);% ylim([0 vF])
ylabel('1/(2N+1)'); xlabel('EF (eV)')
subplot(2,2,4); plot(E ,mfp,...
EF,mfpeff,...
E,4*kT*dfdE(21,:).*mfp,...
E,4*kT*dfdE(21,:)*lacH);
xlim([minEF maxEF]); % ylim([0 lacH])
119
ylabel('MFP (um)'); xlabel('E (eV)')
% write results to file
Mmfpu = [V;leffmfpundop]';
Mthru = [V;leffthrundop]';
Mnuu = [V;nuundop]';
Mmfpd = [V;leffmfpdoped]';
Mthrd = [V;leffthrdoped]';
Mnud = [V;nudoped]';
csvwrite(['MFP_MatthiesenvsV,L=',num2str(L),'um,lacHigh=',...
num2str(lacH),'um,Eph=',num2str(Eph),'eV,EF=',...
num2str(0 ),'eV,mfp=',num2str(lacundop),'um.csv'],Mmfpu)
csvwrite(['MFP_ThresholdvsV,L=', num2str(L),'um,lacHigh=',...
num2str(lacH),'um,Eph=',num2str(Eph),'eV,EF=',...
num2str(0 ),'eV,mfp=',num2str(lacundop),'um.csv'],Mthru)
csvwrite(['OPpercentvsV,L=',num2str(L),'um,lacHigh=',...
num2str(lacH),'um,Eph=',num2str(Eph),'eV,EF=',...
num2str(0 ),'eV,mfp=',num2str(lacundop),'um.csv'],Mnuu )
csvwrite(['MFP_MatthiesenvsV,L=',num2str(L),'um,lacHigh=',...
num2str(lacH),'um,Eph=',num2str(Eph),'eV,EF=',...
num2str(EFdp),'eV,mfp=',num2str(lacdoped),'um.csv'],Mmfpd)
csvwrite(['MFP_ThresholdvsV,L=', num2str(L),'um,lacHigh=',...
num2str(lacH),'um,Eph=',num2str(Eph),'eV,EF=',...
num2str(EFdp),'eV,mfp=',num2str(lacdoped),'um.csv'],Mthrd)
csvwrite(['OPpercentvsV,L=',num2str(L),'um,lacHigh=',...
num2str(lacH),'um,Eph=',num2str(Eph),'eV,EF=',...
num2str(EFdp),'eV,mfp=',num2str(lacdoped),'um.csv'],Mnud )
end
CNT Phonon Dispersion Relationship
function CNT_phonons = CNT_phonons(varargin)
if size(varargin) == [0 0]
plottype = 1;
else
plottype = varargin{1,1};
end
% This program calculates the phonon dispersion for graphene; one can
% use zone folding to get the dispersion relationship for a nanotube
% The program works as follows: (1) input 3x3 K (spring constant)
% matrix for each atom we're considering (2) put together into
% dynamical matrix, with appropriate phase delay terms (3) find k
% vectors of interest (subbands), find characteristic polynomial and
% roots for each point plot roots (frequencies) vs. k to get bands
hbar = 6.58211814e-16; % eV*s
invcmeV = 8065.541154; % 1/cm*eV
a = 2.49; % Angstroms
rt3 = sqrt(3);
Mc = 1.99430717e-26; % kg
120
[phr phti phto eps] = ForceConstants(4); % input the force constants
% from external ForceConstants.m file. The index determines the source
%jb = band index
%jt = neighbor tier index
%jr = jr'th atom in jt'th tier, from the positive x axis rotating CCW
for jt = 1:4
K0(jt,:,:) = [phr(jt) eps(jt) 0;eps(jt) phti(jt) 0;0 0 phto(jt)];
end
% 3x3 unrotated force tensors for nearest neighbor tiers 1-4
dR(1) = a/rt3; dR(2) = a; dR(3) = 2*dR(1); dR(4) = a*sqrt(7/3);
% radius of nearest neighbor tiers 1-4
% angles for 1st (x3), 2nd (x6) and 3rd (x3) tiers of atoms
Th(1,1)=0; Th(1,2)=2*pi/3; Th(1,3)=4*pi/3;
Th(2,1)=1*pi/6; Th(2,2)=3*pi/6; Th(2,3)=5*pi/6;
Th(2,4)=7*pi/6; Th(2,5)=9*pi/6; Th(2,6)=11*pi/6;
Th(3,1)=1*pi/3; Th(3,2)=3*pi/3; Th(3,3)=5*pi/3;
x41 = 2.5; y41 = sqrt(3)/2;
Th(4,1) = atan(y41/x41); % angles for fourth tier x6
Th(4,6) = 2*pi-Th(4,1);
Th(4,2) = -Th(4,1)+2*pi/3;
Th(4,3) = Th(4,1)+2*pi/3;
Th(4,4) = Th(4,2)+2*pi/3;
Th(4,5) = Th(4,3)+2*pi/3;
% Calculate the rotated force constant matricies for each atom
for jt = 1:4
for jr = 1:6
Um = [cos(Th(jt,jr)) -sin(Th(jt,jr)) 0;...
sin(Th(jt,jr)) cos(Th(jt,jr)) 0; 0 0 1];
% The unitary rotation matrix for each atom
Ktemp(:,:) = K0(jt,:,:);
K(:,:,jt,jr) = Um\Ktemp*Um; % Um\ is equivalent to inv(Um),
% this line rotates the force constant matrix
if jt == 1 | jt == 3
if jr == 4 | jr == 5 | jr == 6
K(:,:,jt,jr) = zeros(3,3); % make sure 4th, 5th
% and 6th elements of tier 1 and 3 matrices are =0
% (no atoms there)
end
end
end
end
Nk(1) = 120; Nk(2) = 60; Nk(3) = 120;
kx = zeros(3,max(Nk)+1); ky = kx;
% specify which 1-D swath of K-space to calculate eigenvalues for
% 1 = gamma to M, 2 = M to K, 3 = K to gamma
delkx1 = 2*pi/(rt3*a)*(1/Nk(1)); % Gamma to M
kx(1,:) = 0:delkx1:2*pi/(rt3*a);
ky(1,1:Nk(1)+1) = 0*ones(1,Nk(1)+1);
% M to K
121
delky2 = 2*pi/(3*a)*(1/Nk(2));
kx(2,1:Nk(2)+1) = 2*pi/(rt3*a)*ones(1,Nk(2)+1);
ky(2,1:Nk(2)+1) = (0:delky2:2*pi/(3*a));
% K to gamma
kx3max = 2*pi/(rt3*a); delkx3 = kx3max*(1/Nk(3));
ky3max = 2*pi/(3*a); delky3 = ky3max*(1/Nk(3));
k3max = 4*pi/(3*a); delk3 = k3max*(1/Nk(3));
kx(3,1:Nk(3)+1) = 0:delkx3:kx3max;
ky(3,1:Nk(3)+1) = 0:delky3:ky3max;
k(3,1:Nk(3)+1) = 0:delk3:k3max;
%build the 3x3 sub-matricies of the 6x6 dynamical matrix
for swath = 1:3 % which swath of k-space?
for jb = 1:Nk(swath)+1 % which band?
Dab = zeros(3,3,Nk(swath)+1);
Daa = zeros(3,3,Nk(swath)+1);
D = zeros(6,6,Nk(swath)+1);
% initialize the sub-matricies and the 6x6 dynamical matrix
for jt = 1:4
for jr = 1:6
Ph = exp(i*dR(jt)*(kx(swath,jb)*cos(Th(jt,jr))+...
ky(swath,jb)*sin(Th(jt,jr))));
Kk(:,:,jt,jr) = K(:,:,jt,jr)*Ph;
% Kk is the phase shifted K force constant matrix
if jt ~= 2
Dab(:,:,jb) = Dab(:,:,jb)-Kk(:,:,jt,jr);
end
if jt == 2
Daa(:,:,jb) = Daa(:,:,jb)-Kk(:,:,jt,jr);
end
Daa(:,:,jb) = Daa(:,:,jb)+K(:,:,jt,jr);
end
end
Dba = Dab(:,:,jb)';
Dbb = Daa(:,:,jb); % assuming the matrix is symmetric
D(:,:,jb) = [Daa(:,:,jb) Dab(:,:,jb);Dba Dbb];
% build the 6x6 Dynamical matrix from the sub-matricies
w(:,jb,swath) = real(sqrt(1/Mc*roots(poly(D(:,:,jb)))));
% calculate the eigen values of the dynamical matrix
end
end
% Plot the results
if plottype == 1 % plot high symmetry lines in graphene k-space
plot(0,0); hold on; cl(1) = 'r'; cl(2) = 'b'; cl(3) = 'c';
cl(4) = 'g'; cl(5) = 'm'; cl(6) = 'g';
for jp = 1:6
plot(kx(1,1:Nk(1)),invcmeV*hbar*w(jp,1:Nk(1),1),...
'linestyle','none',...
'marker','.','color',cl(jp),'MarkerSize',5)
plot(ky(2,1:Nk(2))+2*pi/(rt3*a),...
invcmeV*hbar*w(jp,1:Nk(2),2),'linestyle','none',...
'marker','.','color',cl(jp),'MarkerSize',5)
plot(-k(3,1:Nk(3))+2*pi/(rt3*a)+2*pi/(3*a)+4*pi/(3*a),...
122
invcmeV*hbar*w(jp,1:Nk(3),3),'linestyle','none',...
'marker','.','color',cl(jp),'MarkerSize',5)
end
end
if plottype == 2 % plot gamma to M positive and negative
plot(0,0); hold on; cl(1) = 'r'; cl(2) = 'b'; cl(3) = 'c';
cl(4) = 'g'; cl(5) = 'm'; cl(6) = 'g';
for jp = 1:6
plot(1e10*kx(1,1:Nk(1)),hbar*w(jp,1:Nk(1),1),'Color','blue');
plot(1e10*kx(1,1:Nk(1)),-hbar*w(jp,1:Nk(1),1),'Color','blue');
end
end
% get(gcf,'Interpreter')
if plottype == 1
ylim([0 1700])
ylabel('Frequency (cm^-1)')
xlabel('Wavevector "k"')
set(gca,'XTick',[0 2*pi/(rt3*a) 2*pi/(rt3*a)+2*pi/(3*a)...
2*pi/(rt3*a)+2*pi/(3*a)+4*pi/(3*a)])
set(gca,'XTickLabel',{'Gamma','M','K','Gamma'})
set(gca,'XGrid','on')
end
% return the dispersion relationship
CNT_phonons(:,1) = 1e10*kx(1,1:Nk(1))';
CNT_phonons(:,2:7) = real(hbar*w(:,1:Nk(1),1))';
% save the dispersion relationship to file
dlmwrite('kx_cntphonons.txt',1e10*kx(1,1:Nk(1))');
dlmwrite('E_cntphonons.txt',real(hbar*w(:,1:Nk(1),1)'));
dlmwrite('E_cntphonons_neg.txt',-real(hbar*w(:,1:Nk(1),1)'));
Subroutine: Carbon Atom Force Constants
function [phr,phti,phto,eps] = Forceconstants(model)
% Force constants (N/m) phr = x translation (PHi Radial),
% phti = y translation (PHi Transverse Inplane),
% phto = z translation (PHi Transverse Out of plane)
if model == 1
% "Physical properties of carbon nanotubes"
% by Saito and Dresselhaus^2
%-----------------------------------------------------------------
phr(1) = 365.0; phti(1) = 245.0; phto(1) = 98.2; eps(1) = 0;
phr(2) = 88.0; phti(2) = -32.3; phto(2) = -4.0; eps(2) = 0;
phr(3) = 30.0; phti(3) = -52.5; phto(3) = 1.5; eps(3) = 0;
phr(4) = -19.2; phti(4) = 22.9; phto(4) = -5.8; eps(4) = 0;
%-----------------------------------------------------------------
end
123
if model == 2
% A. Grüneis, R. Saito, et al Phys. Rev. B 65 (2002)
%-----------------------------------------------------------------
phr(1) = 403.7; phti(1) = 251.8; phto(1) = 94.0; eps(1) = 0;
phr(2) = 27.6; phti(2) = 22.2 ; phto(2) = -0.8; eps(2) = 0;
phr(3) = 0.5; phti(3) = -89.9; phto(3) = -0.6; eps(3) = 0;
phr(4) = 13.1; phti(4) = 2.2 ; phto(4) = -6.3; eps(4) = 0;
%-----------------------------------------------------------------
end
if model == 3
% "The phonon dispersion of graphite revisited"
% by Wirtz and Rubio, Solid State Communications 131, (2004)
%-----------------------------------------------------------------
phr(1) = 398.7; phti(1) = 172.8; phto(1) = 98.9; eps(1) = 0;
phr(2) = 72.9; phti(2) = -46.1; phto(2) = -8.2; eps(2) = 0;
phr(3) = -26.4; phti(3) = 33.1; phto(3) = 5.8; eps(3) = 0;
phr(4) = 1.0; phti(4) = 7.9; phto(4) = -5.2; eps(4) = 0;
%-----------------------------------------------------------------
end
if model == 4
% "The phonon dispersion of graphite revisited" by Wirtz and
% Rubio, Solid State Communications 131, (2004). same as above
% except with nonzero eps values (rep. off diagonal elements)
%-----------------------------------------------------------------
phr(1) = 409.8; phti(1) = 145.0; phto(1) = 98.9; eps(1) = 0;
phr(2) = 74.2; phti(2) = -40.8; phto(2) = -8.2; eps(2) = -9.1;
phr(3) = -33.2; phti(3) = 50.1; phto(3) = 5.8; eps(3) = 0;
phr(4) = 6.5; phti(4) = 5.5; phto(4) = -5.2; eps(4) = 0;
%-----------------------------------------------------------------
end
CNT Electronic dispersion relationship
function [EiiG,ntr] = sub_bands_CNT(n,m,plot_state)
clear DOStotal
% This program calculates and plots the band structure, density of
% states, and joint DOS of a (n,m) CNT using tight binding and zone
% folding. It also identifies the van-Hove singularity energies.
% Note that optical transitions are dominated by excitonic
% interactions, not van-Hove singularities, so the j-DOS is not valid
% as such.
% parameters from Reich et at. p. 40 of
% "Carbon nanotubes - basic concepts and physical properties'.
% Values obtained by fitting third nearest neighbor tight binding
% model to ab-initio calculations for optical transition energies
% below 4eV. For this range the agreement with ab-initio results is
% better than 5meV
g0 = -2.79; g1 = -0.68; g2 = -0.30; E2p = -2.03; %eV
s0 = 0.30; s1 = 0.046; s2 = 0.039; Eg2 = 0.05; %eV
124
Eb = 0; a = 0.249; %nm
%--------------------------------------------------------------------
%get CNT geometry information from CNT_info.m subroutine
CNT_info(n,m); %iCNT = [Ch,dt,Theta,dR,t1,t2,T,N,delkL]
global iCNT; Ch = iCNT(1); dt = iCNT(2); Theta = iCNT(3);
dR = iCNT(4);
nc = iCNT(5);
R = dR/nc
t1 = iCNT(6); t2 = iCNT(7);
T = iCNT(8)
N = iCNT(9); delkL = iCNT(10); rt3 = sqrt(3); R = 1;
pd = 2*sqrt(n^2+n*m+m^2);
%-------------------------------------------------------------------
%calculate the sub-bands of the nanotube from the 3rd nearest neighbor
%tight binding approximation for graphine
Nkz = 121; delkz = pi/(T*(Nkz-1)); kz = 0:delkz:pi/T; % nm-1
imax = N-1; % number of sub-bands
i = 0:imax;
for j = 1:imax+1;
% for each sub-band, calculate the energies (as a vector)
kx = (kz*(m-n)+delkL*i(j)*sqrt(3)*(m+n))/pd;
ky = (kz*sqrt(3)*(m+n)-delkL*i(j)*(m-n))/pd;
uxy = 2*cos(a*ky)+4*cos(rt3*a*kx/2).*cos(a*ky/2);
u2xy = 2*cos(a*2*ky)+4*cos(rt3*a*kx).*cos(a*ky);
urt3yx = 2*cos(a*rt3*kx)+4*cos(1.5*a*ky).*cos(a*rt3*kx/2);
fxy = 3+uxy; f2xy = 3+u2xy; gxy = 2*uxy+urt3yx;
E0 = (E2p+g1*uxy).*(1+s1*uxy);
E1 = 2*s0*g0*fxy+(s0*g2+s2*g0)*gxy+2*s2*g2*f2xy;
E2 = (E2p+g1*uxy).^2-g0^2*fxy-g0*g2*gxy-g2^2*f2xy;
E3 = (1+s1*uxy).^2-s0^2*fxy-s0*s2*gxy-s2^2*f2xy;
cbands(j,:) = (-(E1-2*E0)+sqrt((E1-2*E0).^2-4*E2.*E3))./(2*E3);
vbands(j,:) = (-(E1-2*E0)-sqrt((E1-2*E0).^2-4*E2.*E3))./(2*E3);
end
Emax = 3.5; Emin = -3.5; N_E = 5000;
deltaE = (Emax-Emin)/(N_E-1); E = Emin:deltaE:Emax;
[DOStotal JDOStotal cVHS vVHS VHS Evhs EiiG ntr] =...
findDOS(cbands,vbands,E,imax,delkz,kz,i);
if plot_state == 1
plot_bands(cbands,vbands,E,Nkz,kz,n,m,N,T,imax)
plot_DOS(DOStotal,JDOStotal,E,VHS,Evhs,Emax,Emin,EiiG,ntr)
end
Subroutine: CNT geometry
function CNT_info(n,m)
if m > n
'n must be greater than m'
end
a = 0.249 ;
% nm, graphene lattice constant
125
Ch = a*sqrt(n^2+n*m+m^2);
% CNT circumference
dt = Ch/pi;
% CNT diameter
Theta = rad2deg(atan(sqrt(3)*m/(2*n+m)));
% CNT chiral angle (0 = zigzag, 30 = armchair)
dR = gcd(2*n+m,2*m+n);
% unit cell degeneracy factor
nc = gcd(n,m);
% rotational symmetry number
T = sqrt(3)*Ch/dR;
% Unit cell length (nm)
t1 = (2*m+n)/dR;
% t1, t2 are the translation vector indices for T
t2 = -(2*n+m)/dR;
% such that T = t1*a1 + t2*a2
N = 2*(n^2+n*m+m^2)/dR;
% number of carbon atoms in the CNT unit cell and number of sub-bands
delkL = 2*pi/Ch;
% circumferential wavevector spacing between bands
global iCNT
iCNT = [Ch,dt,Theta,dR,nc,t1,t2,T,N,delkL];
Subroutine: Density of States Calculator
function [DOStotal,JDOStotal,cVHS,vVHS,VHS,Evhs,EiiG,ntr]...
= findDOS(cbands,vbands,E,imax,delkz,kz,i)
% find the DOS from the subbands, find the van Hove Singularites and
% their strengths, parse them into conduction bands and valence bands,
% find the joint density of states or joing van hove singularity
% transitions
global EiiG;
clear EiiG;
N = length(E);
deltaE = E(2)-E(1);
Emin = E(1);
Emax = E(N);
JE = 0:4/N:4;
delkz;
deltaE;
[Nm,Nkz] = size(cbands); i = transpose(i);
pioverT = kz(Nkz); kzfine = 0:delkz/round(3000*delkz):pioverT;
cbandsfine = interp2(kz,i,cbands,kzfine,i);
vbandsfine = interp2(kz,i,vbands,kzfine,i);
Jbands = cbandsfine-vbandsfine;
DOStotal = transpose(zeros(N,1)); JDOS = zeros(imax+1,N+1);
126
for jj = 1:imax+1
DOStotal = histc(cbandsfine(jj,:),E)+DOStotal;
DOStotal = histc(vbandsfine(jj,:),E)+DOStotal;
JDOS(jj,:) = histc(Jbands(jj,:),JE);
end
JDOStotal(1,:) = sum(JDOS,1)*delkz/(2*pi*4);
JDOStotal(2,:) = JE;
DOStotal = transpose(DOStotal)*delkz/(2*pi*deltaE*N);
%find the van-Hove Singularities
Esize = size(DOStotal,2); numE = size(E,2); midEindex = round(numE/2);
cts=0; VHSindex = 1; % prominence counter, vHs index
gap = 10; % window in which to look for vHs
lmaxold = 0;
for sp = gap:N-2*gap
[lmax, ind] = max(DOStotal(sp:sp+gap));
itrue = ind+sp;
if ((lmax == lmaxold) & (0.7*lmax > (mean(DOStotal(itrue-gap+1:...
(itrue-2)))+mean(DOStotal(itrue+2:itrue+gap-1)))/2) &...
lmax>max(DOStotal)/20);
cts = cts+1;
end
lmaxold = lmax;
if cts == gap
% if cts reaches gap, then local max is prominent over gap
Evhs(VHSindex) = (itrue-2)*deltaE-Emax;
VHS(VHSindex) = lmax; cts=0;
% index vHs, reset prominence counter
VHSindex = VHSindex+1;
end
end
cn = 1; cv = 1;
cVHS = zeros(20,1); vVHS = zeros(20,1);
for sr = 1:length(VHS)
if Evhs(sr)>0
cVHSe(cn) = Evhs(sr); cVHSs(cn) = VHS(sr);
cn = cn+1;
end
if Evhs(length(VHS)-sr+1)<0
vVHSe(cv) = Evhs(length(VHS)-sr+1);
vVHSs(cv) = VHS(length(VHS)-sr+1);
cv = cv+1;
end
end
ntr = min(cn-1,cv-1);
for mm = 1:ntr
EiiG(1,mm) = cVHSe(mm)-vVHSe(mm);
EiiG(2,mm) = cVHSs(mm)*vVHSs(mm);
end
Subroutine: Plot Dispersion Relationship
function plot_bands(cbands,vbands,E,Nkz,kz,n,m,N,T,imax)
127
hbar = 6.58e-16; Nkt = Nkz*2/3; %ev*s, #
N_E = length(E); deltaE = E(2)-E(1); delkz = abs(kz(1)-kz(2));
Emin = E(1); Emax = E(N_E);
nn = 0; mm = 0; qq = 0; f = 1;
fermiband(1,:) = [1 1]; subplot(1,3,1); plot(0,0)
chiralitytxt = ['chirality: n = ',num2str(n),', m = ',num2str(m),...
', N = ',num2str(N),'.'];
title(chiralitytxt); hold on
for j = 1:imax-1
plot(kz,cbands(j,:),kz,vbands(j,:))
% Check for bands that cross the charge neutrality point (CNP)
if cbands(j,1) < 0.1 && cbands(j,1) > -0.1
text(0+nn*1/T,0.2,num2str(j-1)); % label plot sub-band number
text(0+nn*1/T,-0.2,num2str(j-1));
nn = nn+1;
fermiband(f,:) = [j-1 0]; % write location of CNP crossing
f = f+1;
vf = (-cbands(j,1)+cbands(j,2))/(hbar*delkz*1e9); %m/s
kf = 0.02/(vf*1e9*hbar); %nm-^1
end
if cbands(j,round(Nkt)) < 0.1 && cbands(j,round(Nkt)) > -0.1
text(Nkz/3*delkz+qq*1/T,0.2,num2str(j-1));
text(Nkz/3*delkz+qq*1/T,-0.2,num2str(j-1));
qq = qq+1;
fermiband(f,:) = [j-1 2/3];
f = f+1;
vf = (-cbands(j,round(Nkt))+cbands(j,round(Nkt)+1))/...
(hbar*delkz*1e9); %m/s
kf = kz(round(Nkt)); %nm-1
end
end
if mod((n-m),3)==0
kfermi = [num2str(kf) ' (1/nm)']
warning off MATLAB:divideByZero
de_Broglie_wavelenth = [num2str(1/kf,'%3.3f') ' (nm)']
vfermi = [num2str(vf,'%6.3e') ' (m/s)']
fermiband %print the CNP crossings
kz;
end
ylim([Emin Emax]); xlim([0 pi/T])
ylabel('Energy-Ef (Ev)'); xlabel('kz (nm^-1)')
set(gca,'XTick',0:pi/(2*T):pi/T)
set(gca,'XTickLabel',{'G','pi/2*T','pi/T'})
hold off
Subroutine: Plot Density of States
function plot_DOS(DOStotal,JDOStotal,E,VHS,Evhs,Emax,Emin,EiiG,ntr)
hold off
subplot(1,3,2); plot(DOStotal,E); hold on;
plot(VHS,Evhs,'--rs','LineWidth',2,'MarkerEdgeColor','k',...
'MarkerFaceColor','g','MarkerSize',5);
xlabel('DOS'); ylim([Emin Emax]); xlim([0 1.5*mean(VHS)]);
128
hold off
if ntr > 6; ntr = 6; end
subplot(1,3,3); plot(JDOStotal(1,:),JDOStotal(2,:)); hold on
for ii = 1:ntr
St = [0];% EiiG(2,ii)];
Et = [EiiG(1,ii)];% EiiG(1,ii)];
plot(St,Et,'--rs','LineWidth',2,'MarkerEdgeColor','k',...
'MarkerFaceColor','g','MarkerSize',5)
end
ylim([0 4]);
xlim([0 mean(JDOStotal(1,:))*6]);
xlabel('JDOS'); ylabel('Energy (eV)');
hold off
CNT Optical Phonon Renormalization (Kohn Anomaly)
function CNT_Raman_vs_Vg = CNT_Raman_vs_Vg(T,EF,Vg,Eg,offset3,L,...
w0,gam,Ecutoff,width0invcm,dt,Deph,Freqname,FWHMname);
% This program is going to be writen to plot the phonon frequency and
% width vs. gate voltage from the model in Caudal et al, PRB 75, 2007.
% It is designed to work in conjunction with CNT_Conductivity.m,
% which generates the EF-Vg relationship used by this program (input
% parameters)
RamanFreqdata = load(Freqname); RamanWidthdata = load(FWHMname);
kT = 0.0259*T/300; a = 0.249e-9; %eV, m, thermal E, lattice const
hbar = 6.58211814e-16; cmInvPEReV = 8065.541154; %eV, 1/cm*eV
rt3 = sqrt(3); e = 1.602e-19; %C
vF = 840000; Mc = 1.99e-26; %m/s, kg, mass of a carbon atom
beta = vF*hbar; % eV*m fermi felocity in diff. units
Deph = Deph*1e20; %(eV/m)^2, the electron phonon coupling factor
Vg = Vg + offset3;
nEF = length(Vg)
maxEF = max(EF); maxE = 4*maxEF; minE = -4*maxEF;
nE = 200; delE = (maxE-Eg)/nE; E = minE:delE:maxE;
Eph = w0/cmInvPEReV; k0 = Ecutoff/beta; %OP energy, k0 integr. limit
Nk = 2001; delk = (2*k0/(Nk-1)); k = -k0:delk:k0; kp = Eg/beta;
Cw = a^2*sqrt(3)*Deph*e*hbar^2/(pi^2*dt);
% magnitude of frequency renormalization
width0 = width0invcm/cmInvPEReV;
% eV, the intrinsic width of the G-bands, ~10 cm^-1
width1 = 2*a^2*sqrt(3)*Deph*e*hbar^2/(pi*dt*beta*Mc);
% magnitude of broadening effect
Ek = -sign(k).*sqrt((Eg/2)^2+beta^2*k.^2);
% Hyperbolic dispersion relation
doping = 4*EF/(pi*dt*beta*1e4); % cm-2 electrons per area doping
129
c1 = 6.478e-6; c2 = 1.64e-4; c3 = 1e-13;
da = c1*(abs(doping*c3)).^(3/2)+c2*doping*c3;
% from Lazzeri and Mauri; to account for
% lattice expansion da = (a(doping)-a(0))/a(0)
for iEF = 1:nEF
% cycle through fermi energies, create wG- vs. Vg profile
% and phonon width profile
df = 1./(1+exp((Ek-EF(iEF))/kT))-1./(1+exp((-Ek-EF(iEF))/kT));
geom = 1; %(k.^2-kp^2)/(k.^2+kp^2);
% effects of sub-band offset from K-point
Dk = Cw*(1+geom).*df.*(1./(2*Ek+Eph+i*gam)+1./(2*Ek-Eph-i*gam));
Dgamma = sum(Dk)*delk;
% integrate to find the non-adiabatic dynamical matrix
wG(iEF) = sqrt(Eph^2+Dgamma/Mc)/(1+da(iEF)); % eV
fwGp = 1/(1+exp((EF(iEF)-wG(iEF)/2)/kT));
fwGn = 1/(1+exp((EF(iEF)+wG(iEF)/2)/kT));
width(iEF) = width0 + width1/wG(iEF)*(fwGp-fwGn);
% calculate the FWHM in eV
w2(iEF) = width0-imag(Dgamma)/(Mc*Eph);
% the imaginary portion of non-adiabatic dynamical matrix should
% be the same as the explicit calculation (done for width(iEF))
end
CaudalEq29 = width1/Eph*dt*1e9*cmInvPEReV
% used to check the constant in Caudal et al. eq. (29), should be ~79
% plot results
subplot(2,3,2); plot(RamanFreqdata(:,1),RamanFreqdata(:,2));
hold on; plot(Vg,cmInvPEReV*wG,'Color','red');
xlim([min(RamanFreqdata(:,1)) max(RamanFreqdata(:,1))])
xlabel('Gate Voltage (V)');
subplot(2,3,3); plot(RamanWidthdata(:,1),RamanWidthdata(:,2));
hold on; plot(Vg,cmInvPEReV*width,'Color','red');
xlim([min(RamanFreqdata(:,1)) max(RamanFreqdata(:,1))])
xlabel('Gate Voltage (V)');
subplot(2,3,5); plot((RamanFreqdata(:,1)),RamanFreqdata(:,2));
hold on; plot(EF,cmInvPEReV*wG,'Color','red')
xlim([min(EF) max(EF)]); xlabel('Fermi Energy (eV)');
grid; hold off
subplot(2,3,6); plot((RamanWidthdata(:,1)),RamanWidthdata(:,2));
hold on; plot(EF,cmInvPEReV*w2,'Color','red')
xlim([min(EF) max(EF)]); xlabel('Fermi Energy (eV)');
grid; hold off
subplot(2,3,5); plot(EF,cmInvPEReV*wG,'color','red');
subplot(2,3,6); plot(Vg,cmInvPEReV*width,'color','red');
% write data to file
modelFreq = [EF' cmInvPEReV*abs(wG)'];
modelFWHM = [Vg' cmInvPEReV*abs(width)'];
Freqfilelength = length(Freqname);
FWHMfilelength = length(FWHMname);
130
Freqnamet = Freqname(1:Freqfilelength-4);
FWHMnamet = FWHMname(1:FWHMfilelength-4);
modelFreqname = [Freqnamet,'_model.txt'];
modelFWHMname = [FWHMnamet,'_model.txt'];
dlmwrite(modelFreqname,modelFreq, '\t')
dlmwrite(modelFWHMname,modelFWHM, '\t')
CNT Electronic Excitation spectrum
function CNT_Excitations = CNT_Excitations
% this program calculates the Pauli exclusion-weighted scattering rate
% for a carbon nanotube, as a function of doping. It is intended to be
% used to generate instructional an instructional plot showing which
% phonons are scattering electrons most strongly. This should not be
% used to calculate the total scattering rate however; for that one
% needs to average the relaxation time (inverse of the rate)
type = 0; %abs = 0, ems = 1
Emg = .12; %eV
gam = 0.0003; %eV
T=300; %K
maxEF = 0.15; %eV
nEF = 2; dEF = maxEF/(nEF-1);
EF = 0:dEF:maxEF; % eV, constant to start with
hbar = 6.58211814e-16; h = hbar*2*pi; kT = 0.0259*T/300; % eVs, eV
vF = 840000; e = 1.602e-19; % m/s (Fermi velocity), C (elem. charge)
eps = 1.2345e-8; eps2 = 1.2345e-11; %small value limiting infinities
maxE = 0.07; minE = -maxE; nE = 401;
% start integrating several kT past the max/min Fermi energy
dE = (maxE-minE)/(nE-1); E = minE:dE:maxE;
% initialize the energy variable, to integrate over
vcE = ceil(1e-6*E); % for calc. the valence/conduction band effects,
% charge density contributions, the conduction band has vcE = 1,
% the valence band has vcE = 0
nph = ((1-vcE)+1./(1-exp(abs(E)/kT)+eps));
maxk = 3.5*maxE/(vF*hbar); mink = 3.5*minE/(vF*hbar); %m^-1
nk = 201; dk = (maxk-mink)/(nk-1); k = mink:dk:maxk;
% initialize the energy variable, to integrate over
ktrans = dk:dk:(nk-1)*dk;
Ec = ((Emg/2)^2+(hbar*vF*k).^2).^(1/2); Ev = -Ec;
%Calculate the dispersion relationship
vg = vF*hbar*k./sqrt((Emg/(2*hbar*vF)).^2+k.^2);
[kminval iktranslow150] = min(abs(.15-Ec(1:(nk-1)/2)));
[kminval iktranshigh150] = min(abs(.15-Ec((nk-1)/2:nk)));
131
iktranshigh150 = iktranshigh150+(nk-1)/2;
delktrans150 = k(iktranshigh150)-k(iktranslow150);
for iEF = 1:nEF
clear fc fv hnk dEvA dEvE dEvc dEcv dEcA dEcE PvvABS PvvEMS
clear PvcABS PcvEMS PccABS PccEMS dEv dEi dEc Pdosv Pdosi
clear Pdosc Pintv Pinti Pintc Tph
fc = 1./(1+exp((Ec-EF(iEF))/kT)); % calc the fermi funct.
fv = 1./(1+exp((Ev-EF(iEF))/kT));
transitions = iEF
hnk = (nk-1)/2;
for rk = 1:hnk
% calculate transition energy changes pos=absorption, neg=ems
% up to half the FBZ, excluding null transition
% the second index (over dlk) indexes througth delta k
lk = nk-rk+1; dlk = nk-2*rk+1;
% calc the valence intraband excitation energy
dEvA(rk,1:dlk) = +(Ev(rk+1:lk)-Ev(rk));
dEvE(rk,1:dlk) = -(Ev(rk+1:lk)-Ev(rk));
% calc the interband excitation energy
dEvc(rk,1:dlk) = Ec(rk+1:lk)-Ev(rk);
dEcv(rk,1:dlk) = Ev(rk+1:lk)-Ec(rk);
% calc the conduction intraband excitation energy
dEcA(rk,1:dlk) = +(Ec(rk)-Ec(rk+1:lk));
dEcE(rk,1:dlk) = -(Ec(rk)-Ec(rk+1:lk));
if max(dEcA(rk,1:13))>0.05
rk;
end
% calculate the Pauli exclusion factor weighted joint DOS
dvg = abs(vg(rk)-vg(rk+1:lk));
% valence intraband excitation Pauli exclusion factor
PvvABS(rk,1:dlk) = (fv(rk)*(1-fv(rk+1:lk)))./dvg;
PvvEMS(rk,1:dlk) = (fv(rk+1:lk)*(1-fv(rk)))./dvg;
% interband excitation Pauli exclusion factor
PvcABS(rk,1:dlk) = (fv(rk)*(1-fc(rk+1:lk)))./dvg;
PcvEMS(rk,1:dlk) = (fc(rk)*(1-fv(rk+1:lk)))./dvg;
% conduction intraband excitation Pauli exclusion factor
PccABS(rk,1:dlk) = (fc(rk+1:lk)*(1-fc(rk)))./dvg;
PccEMS(rk,1:dlk) = (fc(rk)*(1-fc(rk+1:lk)))./dvg;
end
% Calculate the bounds of selection-rule-forbidden zone, where
% there will be no transitions regardless of pauli exclusion
for kz = 1:nk-1
clear Ez
vk = hbar*vF*ktrans(kz);
Ez = 2*((Emg/2)^2+(vk/2).^2).^(1/2)-dE:-dE:vk+dE;
nz(kz) = length(Ez); nz2 = 2*nz;
Ez2(1:nz2(kz),kz) = [Ez'; -Ez'];
z(1:nz2(kz),kz) = zeros(nz2(kz),1);
end
% concatenate the different bands' transitions
for j = 1:nk-1 % cycle through the delta k values (ktrans)
j;
132
hdk = nk-j+1; hkc = hnk-ceil(j/2)+1;
rj = 1:hck; rj2 = 1:(hck*2+nz2(j));
dEv(rj2,j) = [dEvA(rj,j);(dEvE(rj,j)+eps2);Ez2(1:nz2(j),j)];
dEi(rj2,j) = [dEvc(rj,j);(dEcv(rj,j)) ;Ez2(1:nz2(j),j)];
dEc(rj2,j) = [dEcA(rj,j);(dEcE(rj,j)+eps2);Ez2(1:nz2(j),j)];
% concatenate the JDOS for same delta k values
Pdosv(rj2,j) = [PvvABS(rj,j);PvvEMS(rj,j);z(1:nz2(j),j)];
Pdosi(rj2,j) = [PvcABS(rj,j);PcvEMS(rj,j);z(1:nz2(j),j)];
Pdosc(rj2,j) = [PccABS(rj,j);PccEMS(rj,j);z(1:nz2(j),j)];
% generate interpolated image
Pintv(:,j) = interp1(dEv(rj2,j),Pdosv(rj2,j),E');
Pinti(:,j) = interp1(dEi(rj2,j),Pdosi(rj2,j),E');
Pintc(:,j) = interp1(dEc(rj2,j),Pdosc(rj2,j),E');
Tph = (Pintv+Pinti+Pintc)/(2*pi);
end
%print data to file
global Tphfile;
Tphfile(2:nE+1,2:nk,iEF) = Tph(:,:,1);
Tphfile(1,2:nk,iEF) = ktrans;
Tphfile(2:nE+1,1,iEF) = E';
Tphfile(1,1,iEF) = nE;
Tphfile(1,nk+1,iEF) = Emg;
Tphfile(2,nk+1,iEF) = T;
Tphfile(3,nk+1,iEF) = gam;
Tphfile(nE+1,nk+1,iEF) = EF(iEF);
filename = ['C:\MATLAB6p5\work_m\CNT_excitations\CNTex,Eg=',...
num2str(1000*Emg),'meV,T=',num2str(T),'K,gam=',...
num2str(1e6*gam),'ueV,EF=',num2str(EF(iEF)*1000),'meV.csv']
start_writing = iEF
QEMcelltocsv(filename,Tphfile(:,:,iEF)) % faster than dlmwrite
done_writing = iEF
end
Subroutine: Plot CNT Electronic Excitation Spectrum
function Plot_CNTex = Plot_CNTex
% this program is used to plot the results of CNT_Excitations.m
hbar = 6.58211814e-16; %ev*s
h = hbar*2*pi;
global Tphfile; % must run CNT_Excitations.m in same session for data
phnorm = 0
hardcut = 1
iEF = 1;
[temp nk nEF] = size(Tphfile);
nk = nk-1;
k = Tphfile(1,2:nk,iEF);
n_E = Tphfile(1,1,iEF);
E = Tphfile(2:n_E+1,1,iEF);
133
raw = Tphfile(2:n_E+1,2:nk,iEF);
Elim = 0.07;
Eg = Tphfile(1,nk+1,iEF);
T = Tphfile(2,nk+1,iEF); kT = 0.0259*T/300; % eV (thermal energy)
gam = Tphfile(3,nk+1,iEF); % broadening function
EF = Tphfile(n_E+1,nk+1,iEF);
eps = 5e-3; eps1 = 0; eps3 = 1e-9;
nph = (1./(exp(abs(E)/kT)-1+eps)+(1-ceil(1e-6*E)));
maxE = max(E);
n_E;
nlim = ceil(0.5*n_E*Elim/maxE);
nlimR = (ceil((n_E+1)/2-nlim+1)):(ceil((n_E-1)/2+nlim));
raw1 = raw(nlimR,:);
size(raw1)
if hardcut == 0
if phnorm ==1
minthr = 0.3;
% best at 3 for non phonon-normalized, 0.5 for phonon-norm.
raw1(:,:,1) = raw1(:,:,1).*(nph(nlimR,:)*ones(1,nk));
else
minthr = 0.10
end
topthr = 0.4;
raw2 = log(raw1(:,:,1)/minthr);
% convert the image to a log scale with cuttoff below eps2
maxim = topthr*max(max(raw2(:,:,1))) % find the max
raw3 = raw2/maxim;
% normalize to the max for the plot colorscale
Tabs(:,:,3) = 1-raw3;
% turn off the other colors where scattering occurs
Tabs(:,:,2) = Tabs(:,:,3);
Tabs(:,:,1) = Tabs(:,:,3);
Tabs = truncate(Tabs,0,1);
elseif hardcut == 1
if phnorm ==1
minthr = 0.9;
% best at 3 for non phonon-normalized, 0.5 for phonon-norm.
raw1(:,:,1) = raw1(:,:,1).*(nph(nlimR,:)*ones(1,nk-1));
else
minthr = 0.10
end
topthr = 1;
raw2 = log(raw1(:,:,1)/minthr+eps3);
% convert the image to a log scale with cuttoff below eps2
maxim = topthr*max(max(raw2(:,:,1))) % find the max
raw3 = raw2/maxim;
% normalize to the max for the plot colorscale
Tabs(:,:,1) = raw3;
Tabs(:,:,3) = real(sqrt(-Tabs(:,:,1)))./...
134
(real(sqrt(-Tabs(:,:,1))) + eps3);
% make all values below cuttoff white (value = 1)
Tabs(:,:,2) = Tabs(:,:,3);
Tabs = truncate(Tabs,0,1);
Tabs(:,:,1) = 1-Tabs(:,:,1);
elseif hardcut == 2
if phnorm ==1
minthr = 0.3;
% best at 3 for non phonon-normalized, 0.5 for phonon-norm.
raw1(:,:,1) = raw1(:,:,1).*(nph(nlimR,:)*ones(1,nk));
else
minthr = 0.04
end
topthr = 0.4;
raw2 = log(raw1(:,:,1)/minthr);
% convert the image to a log scale with cuttoff below eps2
maxim = topthr*max(max(raw2(:,:,1))) % find the max
raw3 = raw2/maxim;
% normalize to the max for the plot colorscale
Tabs(:,:,3) = 1-raw3;
% turn off the other colors where scattering occurs
Tabs(:,:,2) = Tabs(:,:,3);
Tabs(:,:,1) = Tabs(:,:,3);
Tabs = truncate(Tabs,0,1);
end
upperlim = minthr*(max(max(raw1(:,:,1)))/minthr)^topthr;
% subplot(1,2,1);
image(k,E(nlimR),Tabs)
set(gca,'YDir','normal')
ylabel('Energy (eV)')
xlabel('k (m^-1)')
text(7e8,0.1,'Absorption','FontSize',18)
text(7e8,-0.1,'Emission','FontSize',18)
hold on
ylim([-Elim Elim])
xlim([0 0.65e9])
CNT_phonons(2)
text(1.3e9,0.312,['EF=',num2str(EF),'eV'])
hold off
Ab Initio Calculations Using Dacapo Package
Dacapo is a free pseudopotential program that can run on Windows. There is a zip
file online that contains all the code and examples at:
http://dcwww.camd.dtu.dk/~jovo/dacapowin/
The program runs in a DOS shell, and I think it has to be unzipped on the root drive. The
directory with the examples is
135
C:\campos\python244\Lib\site-packages\Dacapo\Examples
The protocol for calling a script is to navigate to the example folder in DOS, then type:
“python scriptname.py”,
where python is the program, and scriptname.py is one of the examples in the folder or a
custom script.
The manual for this code can be found here:
https://wiki.fysik.dtu.dk/dacapo/Manual
Below are two Python scripts I wrote to calculate the band structure for graphene
and for a (6,0) CNT. The scripts output text files with the energies, which can be plotted
with another program. Alternatively, someone could add plotting code to these python
scripts themselves.
Graphene Band Structure Calculation
#!/usr/bin/env python
from Dacapo import Dacapo
from ASE import Atom,ListOfAtoms
from math import *
# setup the static graphene sheet
a = 2.46
a0 = a / sqrt( 3 )
c = 20
grsheet = ListOfAtoms([Atom('C',(1 / 3., 1 / 3.,0)),
Atom('C',(2 / 3., 2 / 3.,0))])
# periodic=(1,1,0))
unitcell = [[a0 * 3 / 2, a / 2, 0],
[a0 * 3 / 2,-a / 2, 0],
[ 0, 0, c]]
grsheet.SetUnitCell(unitcell)
kptsym = [[0, 0, 0],
[0.05, 0, 0],
[0.1, 0, 0],
[0.15, 0, 0],
[0.2, 0, 0],
[0.25, 0, 0],
[0.3, 0, 0],
[0.35, 0, 0],
[0.4, 0, 0],
[0.45, 0, 0],
136
[0.5, 0, 0],
[0.45, -0.1, 0],
[0.4, -0.2, 0],
[0.35, -0.3, 0],
[0.33333, -0.33333, 0],
[0.3, -0.3, 0],
[0.25, -0.25, 0],
[0.2, -0.2, 0],
[0.15, -0.15, 0],
[0.1, -0.1, 0],
[0.05, -0.05, 0]]
calc = Dacapo(kpts=kptsym,
# set the k-points along the high symmetry lines G-K-M-G
planewavecutoff = 340,
nbands = 8,
usesymm=False, # use symmetry to reduce the k-points
out='grsheet2c.nc', # define the out netcdf file
txtout='grsheet2c.txt') # define the ascii out file)
calc.DipoleCorrection()
#calc.SetNetCDFFile('graphene_DIP.nc')
grsheet.SetCalculator(calc)
energy = grsheet.GetPotentialEnergy()
(6,0) CNT Band Structure Calculation
#!/usr/bin/env python
from Dacapo import Dacapo
from ASE import Atom,ListOfAtoms
from math import *
# setup the static graphene sheet
a = 2.49
a0 = a / sqrt( 3 )
c = 20 #
T = 4.312807 #6,0 cnt unit celll length in A
grsheet = ListOfAtoms([Atom('C',(0, 0,0)),
Atom('C',(0, 0,0)),
Atom('C',(0, 0,0)),
Atom('C',(0, 0,0)), #4
Atom('C',(0, 0,0)),
Atom('C',(0, 0,0)),
Atom('C',(0, 0,0)),
Atom('C',(0, 0,0)), #8
Atom('C',(0, 0,0)),
Atom('C',(0, 0,0)),
Atom('C',(0, 0,0)),
Atom('C',(0, 0,0)), #12
Atom('C',(0, 0,0)),
137
Atom('C',(0, 0,0)),
Atom('C',(0, 0,0)),
Atom('C',(0, 0,0)), #16
Atom('C',(0, 0,0)),
Atom('C',(0, 0,0)),
Atom('C',(0, 0,0)),
Atom('C',(0, 0,0)), #20
Atom('C',(0, 0,0)),
Atom('C',(0, 0,0)),
Atom('C',(0, 0,0)),
Atom('C',(0, 0,0))]) #24
import csv
#import the position coordinates for the C atoms from external csv file
spamReader = csv.reader(open('6,0cnt_cartpos.csv'),
delimiter=',', quotechar='|')
ii = range(3)
cartpos = []
kk = 0
for row in spamReader:
cartpos.append([0,0,0])
for jj in ii:
cartpos[kk][jj]=(float(row[jj]))
kk=kk+1
unitcell = [[c, 0, 0],
[0, c, 0],
[0, 0, T]]
grsheet.SetUnitCell(unitcell)
grsheet.SetCartesianPositions(cartpos)
kptsym = [1,1,11] #set up the
calc = Dacapo(kpts=kptsym,
# set the k-points along the high symmetry lines G-K-M-G
planewavecutoff = 340,
nbands = 60,
# 96 bands total, must calculate at least N/2 (up to half filling?)
usesymm=False, # use symmetry to reduce the k-points
out='6,0cnt1b.nc', # define the out netcdf file
txtout='6,0cnt1b.txt') # define the ascii out file)
calc.DipoleCorrection()
grsheet.SetCalculator(calc)
energy = grsheet.GetPotentialEnergy()
Abstract (if available)
Abstract
One-dimensional materials exhibit striking, unique phenomena that are not found in two or three dimensions. For the last twenty years, single walled carbon nanotubes (CNTs) have served as the prototypical experimental one-dimensional system. In this thesis, I investigate experimental data and theoretical models of spatially and electrically isolated single-walled CNTs field-effect transistors.
Linked assets
University of Southern California Dissertations and Theses
Conceptually similar
PDF
Raman spectroscopy of carbon nanotubes under axial strain and surface-enhanced Raman spectroscopy of individual carbon nanotubes
PDF
Electronic and optoelectronic devices based on quasi-metallic carbon nanotubes
PDF
Carbon nanotube macroelectronics
PDF
Carbon nanotube nanoelectronics and macroelectronics
PDF
A study of junction effect transistors and their roles in carbon nanotube field emission cathodes in compact pulsed power applications
PDF
GaAs nanowire optoelectronic and carbon nanotube electronic device applications
PDF
Printed and flexible carbon nanotube macroelectronics
PDF
Printed electronics based on carbon nanotubes and two-dimensional transition metal dichalcogenides
PDF
Optoelectronic properties and device physics of individual suspended carbon nanotubes
PDF
Electrical and electrochemical properties of molecular and conducting polymer coated nanoscale field effect transistors
PDF
Single-wall carbon nanotubes separation and their device study
Asset Metadata
Creator
Bushmaker, Adam W.
(author)
Core Title
Raman spectroscopy and electrical transport in suspended carbon nanotube field effect transistors under applied bias and gate voltages
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Electrical Engineering
Publication Date
05/05/2010
Defense Date
02/24/2010
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
ballistic transport,bias voltage,Born-Oppenheimer approximation,carbon nanotube,doping,electrical transport,electron-phonon coupling,field effect transistor,gate voltage,joule heating,Kohn anomaly,Landauer transport,memristor,non-equilibrium,OAI-PMH Harvest,phonon renormalization,phonons,Raman spectroscopy
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Cronin, Stephen B. (
committee chair
), Thompson, Mark E. (
committee member
), Zhou, Chongwu (
committee member
)
Creator Email
abushmaker@gmail.com,adam.bushmaker@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-m3003
Unique identifier
UC1115385
Identifier
etd-Bushmaker-3504 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-333463 (legacy record id),usctheses-m3003 (legacy record id)
Legacy Identifier
etd-Bushmaker-3504.pdf
Dmrecord
333463
Document Type
Dissertation
Rights
Bushmaker, Adam W.
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Repository Name
Libraries, University of Southern California
Repository Location
Los Angeles, California
Repository Email
cisadmin@lib.usc.edu
Tags
ballistic transport
bias voltage
Born-Oppenheimer approximation
carbon nanotube
doping
electrical transport
electron-phonon coupling
field effect transistor
gate voltage
joule heating
Kohn anomaly
Landauer transport
memristor
non-equilibrium
phonon renormalization
phonons
Raman spectroscopy