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Analysis of carbon nanotubes using nanoelectromechanical oscillators

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Analysis of carbon nanotubes using nanoelectromechanical oscillators

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Content ANALYSIS OF CARBON NANOTUBES
USING NANOELECTROMECHANICAL OSCILLATORS

by

Mehmet Aykol

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)

August 2012

Copyright 2012 Mehmet Aykol

ii

DEDICATION

To my beloved parents, Nevin and Şükrü, for their constant support.
iii

ACKNOWLEDGEMENTS
Firstly, I have to acknowledge my advisor, Prof. Stephen B. Cronin. I am forever
thankful for his guidance and support from day one. I have studied many diverse subjects in the
exciting field of nanotechnology and his trust in my abilities helped me immensely in
accomplishing significant academic progress. I will always be reminded of this in my future
studies.
Nobody finishes a dissertation without fellow researchers. Cronin Group members, I have
to thank every single one of you for listening to my crazy ideas and even sometimes letting me
tinker with your experiments in the name of science. Not in any particular order, my good friends
Jesse Theiss, David Valley, Prathamesh Pavaskar, Chia-Chi Chang, Rohan Dhall, Shun-Wen
Chang, Wenbo Hou, Zuwei Liu, Adam Bushmaker, Rajay Kumar, Wayne Hung, Chun-Chung
Chen, Moh Amer, and I-Kai Hsu, thank you all for making this journey fun. David, you are
probably the best cubicle–mate ever. Jesse, thanks for helping me, and backing me up in fiery
arguments with Steve about experiments. I-Kai, you are the most grounded man I have seen with
the best work ethic. Rohan and Shun-Wen, I am very grateful to both of you for sharing all those
samples in dire times. Adam and Rajay, you were good mentors in my early days of graduate
school life.
I have mention a very special group of people that made my life very colorful. People of
Normandie House, including the ones who left, thank you very much. Brad, Eliseo, Wendy,
Aysen, Whitney, Ian, Mazin, Benedicte, Amanda, Ulrike, and Thomas, it was a great experience
living with you. I will always remember the food, the drinks, the music, the arguments, the
laughter and the chickens with a stupid grin on my face.
I am very grateful to have worked with Prof. Jack Feinberg early in my PhD life. His
experience, vision and teachings helped me a lot during my Ph.D. studies. I have learned a great
iv

deal on how to be a good experimentalist from him. I also want to thank Dr. Wei Wu for serving
in my dissertation committee.
My girlfriend Tuğba, thank you for being with me during this hectic time of my life. You
have a special place in my heart. Your understanding, your endless support, your joy turned this
year into a great one. I hope one day I can help you the same way.
v

TABLE OF CONTENTS
DEDICATION ........................................................................................................................... ii
ACKNOWLEDGEMENTS ....................................................................................................... iii
LIST OF FIGURES .................................................................................................................. vii
ABSTRACT ............................................................................................................................ xvi
CHAPTER 1: FUNDAMENTAL PROPERTIES OF CARBON NANOTUBES ......................... 1
1.1 Introduction ............................................................................................................... 1
1.2 Structure.................................................................................................................... 2
1.3 Electronic Properties ................................................................................................. 5
1.4 Mechanical Properties ............................................................................................... 9
1.5 Thermal Properties .................................................................................................. 11
1.6 Previous Work on Nanoelectromechanical Resonators ............................................. 14
1.7 Summary and Outline of Thesis ............................................................................... 17
CHAPTER 2: CARBON NANOTUBE ELECTROMECHANICAL RESONATORS ............... 18
2.1 Introduction ............................................................................................................. 18
2.2 Beam Mechanics ..................................................................................................... 20
2.3 Observing The Mechanical Resonances ................................................................... 24
2.4 Tuning The Resonance Frequency ........................................................................... 29
2.5 Losses ..................................................................................................................... 36
2.6 Conclusions ............................................................................................................. 38
CHAPTER 3: DEVICE FABRICATION AND MEASUREMENT SETUP .............................. 39
3.1 Introduction ............................................................................................................. 39
3.2 Device Fabrication .................................................................................................. 39
3.3 Actuation and Detection Methods ............................................................................ 46
3.3.1 Two-Signal Mixing ............................................................................................ 47
3.3.2 Amplitude Modulation ....................................................................................... 49
3.3.3 Frequency Modulation ........................................................................................ 50
3.3.4 Nearby Antenna ................................................................................................. 52
3.4 Low Temperature and Pressure Setup ...................................................................... 54
3.5 Conclusions ............................................................................................................. 57
CHAPTER 4: THERMAL EXPANSION OF CARBON NANOTUBES ................................... 58
4.1 Introduction ............................................................................................................. 58
4.2 Thermal Expansion of Carbon Allotropes ................................................................ 59
4.3 Measuring Thermal Expansion of Individual Carbon Nanotubes .............................. 67
4.4 Extracting Coefficient of Thermal Expansion from Mechanical Resonance .............. 71
4.5 Conclusions ............................................................................................................. 75
vi

CHAPTER 5: CLAMPING IN CARBON NANOTUBE ELECTROMECHANICAL
RESONATORS ........................................................................................................... 77
5.1 Introduction ............................................................................................................. 77
5.2 Van der Waals Forces .............................................................................................. 78
5.3 Carbon Nanotube–Substrate Interaction ................................................................... 80
5.4 Temperature Induced Unclamping in Carbon Nanotube Resonators ......................... 83
5.5 Conclusions ............................................................................................................. 92
CHAPTER 6: OBSERVING OPTICAL PHONONS THROUGH CARBON
NANOTUBE MECHANICAL RESONATORS ........................................................... 93
6.1 Introduction ............................................................................................................. 93
6.2 Electron Transport in Carbon Nanotubes ................................................................. 94
6.3 Phonon Scattering in Carbon Nanotubes .................................................................. 98
6.4 Optical Phonon Scattering in Carbon Nanotube Electromechanical Resonators ...... 101
6.5 Conclusions ........................................................................................................... 108
CHAPTER 7: FUTURE DIRECTIONS .................................................................................. 110
7.1 Optical Excitation of Mechanical Motion in Carbon Nanotube Mechanical
Resonators ............................................................................................................. 110
7.2 Quantifying Doping by Adsorbates in Carbon Nanotubes ...................................... 114
7.3 Summary and Conclusion ...................................................................................... 117
BIBLIOGRAPHY ................................................................................................................... 118
APPENDIX ............................................................................................................................ 126
Labview Program for Measuring Single Source Setup ........................................................ 126

vii

LIST OF FIGURES
Figure 1.1 - Transmission electron microscopy images of (a) a single–walled carbon
nanotube from reference [51] and (b) a multi–walled carbon nanotube with 5
walls. The concentric structure of multi-walled nanotubes are clearly visible as
shown in reference [50]. ................................................................................................. 3
Figure 1.2 - (a) Chiral vector C
h
on the graphene honeycomb lattice, (b) an armchair
nanotube (n, n), (c) a zigzag nanotube (n, 0), and (d) a chiral nanotube (n, m). ................ 4
Figure 1.3 - Band diagram of graphene. The contact points between the conduction and
the valence band can be approximated as a Dirac cone with relevant energy
limits. ............................................................................................................................. 5
Figure 1.4 - (a) Schematic illustration of the 6 Dirac cones in the graphene Brilluoin zone
with cutting lines that correspond to discrete k⊥ values. If the cutting lines (b)
go through a Dirac point, the nanotube has no bandgap (metallic), otherwise (c)
the nanotube has a wide gap (semiconducting). Adapted from Minot [78]. ...................... 6
Figure 1.5 - Illustration of a carbon nanotube field effect transistor device with (a) a
nanotube on substrate and (b) suspended......................................................................... 8
Figure 1.6 - Conductance of (a) semiconducting nanotube and (b) quasi–metallic
nanotube. (c) Suspended nanotubes exhibit negative differential conductance at
high bias voltages. .......................................................................................................... 9
Figure 1.7 - Young's modulus of single-walled nanotubes measured by (a) an atomic
force microscope tip pushing down on a suspended carbon nanotube [112], (b)
the displacement of carbon nanotubes in a magnetic field [118]. Left two panels
in (b) show the Rayleigh spectrum used to obtain the chirality of the measured
nanotubes. Once the nanotubes are characterized, a magnetic field is applied
(lower right panel) on the nanotube and the deflection due to Lorentz forces is
measured. ..................................................................................................................... 11
Figure 1.8 - Specific heat of various carbon allotropes from reference [46]. SWNT refers
to single–walled carbon nanotubes. ............................................................................... 12
Figure 1.9 - Calculated thermal conductivity of a (10, 10) single-walled nanotube as a
function of temperature from reference [13]. ................................................................. 14

viii

Figure 1.10 - Yoctogram resolution mass sensitivity has been achieved using carbon
nanotube electromechanical resonators [25]. Each arrow represents a Xe atom
landing on the carbon nanotube surface (b) Using the mass sensitivity of a carbon
nanotube electromechanical resonator, it is possible to measure the coverage of a
monatomic gas adsorbed on the surface of a carbon nanotube [115]. (c) With a
high–Q resonator, Steele et al. measured the hopping of an electron on to the
nanotube through the change in mechanical resonance frequency [102]. ....................... 15
Figure 1.11 - (a) First graphene nanoelectromechanical resonator with optical actuation
and optical detection [18]. (b) Using mechanically exfoliated graphene Singh et
al. measured the coefficient of thermal expansion of a single layer of graphene
using a nanoelectromechanical resonator [100]. (c) Optical detection of
mechanical resonances can be utilized to image the mode shape of the
mechanical motion [12]. ............................................................................................... 16
Figure 2.1 - False colored scanning electron microscope image and the schematic
drawing of the first carbon nanotube electromechanical resonator [99]. ......................... 19
Figure 2.2 - Motion of a simple harmonic oscillator. A mass m is attached to a spring
with spring constant k. There is no friction or driving force. The displacement
z(t) of the mass can be described by a simple harmonic oscillator. ................................ 20
Figure 2.3 - The amplitude and phase of the simple harmonic oscillator response plotted
as a function of the driving frequency. The amplitude of the response reaches a
maximum at f
0
= ω
0
/2π and flips phase 180°. Adapted from Sazonova [98]. .................. 22
Figure 2.4 - A doubly clamped beam made up of a material with Young's modulus of E,
thickness t, width w, and length L. The beam has a tension T and a uniform
downward force of K. It has a cross-sectional area of A and moment of inertia of
I with respect to the z-axis. ........................................................................................... 22
Figure 2.5 - Schematic of the electrical actuation and detection method. DC and AC
voltages are applied to the gate electrode to excite the mechanical vibration. On
resonance, the capacitance between the nanotube and the gate electrode is
modulated. Current though the nanotube I
NT
is monitored to detect mechanical
motion of the carbon nanotube. ..................................................................................... 26
Figure 2.6 - Current measured at ∆ω plotted as a function of the driving frequency ω near
the mechanical resonance. The upper points (black circles) show the amplitude of
the mixing current, and the lower points (red crosses) show the phase of the
mixing current plotted as a function of the drive frequency. .......................................... 29
Figure 2.7 - Effect of gate voltage on the mechanical resonance frequency. Waterfall plot
of the mixing current amplitude taken at different gate voltages, showing the V
g

dependence of the position of the resonance signal. ...................................................... 30

ix

Figure 2.8 - (a) Raw data show a clear parabolic relation between the mechanical
resonance frequency and the gate voltage under vacuum. When the chamber is
filled with gas at P = 760 Torr, (b) mechanical resonance features disappear. (c) -
(f) Examples of mixing signal plotted as a function of ω and V
g
for 4 different
devices. Adapted from Sazonova [98]. Detected mixing current amplitude (in
colorscale) is plotted with the driving frequency on the y-axis and the DC gate
voltage on the x-axis. .................................................................................................... 31
Figure 2.9 - Scanning electron microscope image of (a) a suspended nanotube device
with source, drain, and gate electrodes. The approximate location of the carbon
nanotube (CNT) is shown with a black line. (b) The close up image of a carbon
nanotube crossing the trench. Note that the nanotube is taut. ......................................... 32
Figure 2.10 - (a) Free body diagram for the bending regime. The loading force F is
applied in the center of the carbon nanotube of length L. The distance between
the clamping points is W. (b) Free body diagram for the elastic regime. The
carbon nanotube is now subject to a larger force, that stretches the carbon
nanotube by ∆L. (c) A diagram of the applicability of different regimes in the
slack-gate voltage space calculated for a typical device [110]. ...................................... 34
Figure 2.11 - Calculated (empty circles) and measured resonance frequencies (solid line)
of the first two vibrational modes for a 2.2 µm long nanotube. ...................................... 36
Figure 3.1 - Post growth processing using electron beam lithography. Method #1 uses
metal electrodes as an etch mask. Method #2 requires a second lithography step
to define the trench. ...................................................................................................... 40
Figure 3.2 - All the fabrication up to step 7 in completed at Nanofabrication Facility of
University of California, Santa Barbara. Catalyst deposition and nanotube growth
is completed in the Cronin Research Laboratory at University of Southern
California. .................................................................................................................... 42
Figure 3.3 - (a) Optical image of a 5 × 5 mm chip with 28 individual pairs of source and
drain electrodes with a shared gate electrode on the bottom of the trench. The
dotted square is the area of the (b) close up optical image of the devices. (c)
Scanning electron microscope image of a device with two individual nanotubes
crossing the trench. (d) A Y-junction formed by a bundle of carbon nanotubes.
Arrows are placed to guide the eye at the ends of the nanotubes. ................................... 44
Figure 3.4 - Low bias sweep of a nanotube device. In this regime, the nanotube acts like a
linear resistor. Each line corresponds to a different gate voltage. If there is a gate
short, at V
sd
= 0 V, the I–V
sd
lines would not meet at I
sd
= 0 A. ..................................... 45

x

Figure 3.5 - Frequency spectrum of the different detection techniques for (a) a sample
two-terminal carbon nanotube field effect transistor. The detection techniques
used are: (b) Two signal mixing (TS), (c) amplitude modulation (AM), (d)
frequency modulation (FM). Red lines are the frequency spectrum of the signal at
the drain, green at the source and blue at the gate terminals. Broken lines
represent the lock-in amplifiers pass band. In all configurations, gate terminals
have a DC component not shown here. ......................................................................... 47
Figure 3.6 - Circuit diagram for two signal mixing technique. ................................................... 48
Figure 3.7 - Amplitude modulation technique. From reference [98]. .......................................... 49
Figure 3.8 - Circuit diagram for the FM technique for nanotube mechanical resonance
detection. ...................................................................................................................... 50
Figure 3.9 - (a) Amplitude and (b) phase of the mixing current detected using FM
technique. Amplitude of the carrier signal at the source electrode is 20 mV,
deviation frequency ω
∆
= 100 kHz, and the modulation frequency ω
LO
= 654 Hz.
The gate voltage is swept between 6 V and 7 V with 0.25 V intervals. .......................... 52
Figure 3.10 - (a) Schematic drawing of the chip geometry, antenna, and measurement
electronics. (b) When the frequency ν of a signal on the antenna is swept with
fixed V
g
and V
sd
, a resonant peak emerges in I(ν). An example of such a
resonance is shown for a driving power of −17.8 dBm at a temperature of 20 mK.
(c) Zoom of the resonance of (b) at low power (-64.5 dBm). The red line is a fit
of a squared damped driven harmonic oscillator response to the resonance peak.
For both (b) and (c) V
g
= −5.16 V and V
sd
= 0.35 mV. From reference [49]. ................. 53
Figure 3.11 - Image of Cryo Industries liquid He and liquid N
2
cryostat with 28 pin
electrical feed through (chamber #1). This chamber can operate between 4K -
475K with liquid He (or between 77K - 475K with liquid N
2
). This chamber is
modified to include gas lines to achieve partial gas pressures between 10
-4
Torr to
760 Torr. ...................................................................................................................... 55
Figure 3.12 - Image of custom vacuum (chamber #2). Left side of the stage has 12-pin
hermetically sealed electrical feed through. The chamber can be pumped down to
10 mTorr with a 1/4" Swagelok connection from the bottom port. The top port
can be used to send gas in, to create partial pressures of any gas with a mass flow
controller. Right side has the connection for a liquid N
2
cooled stage, which
works in the range 77 K - 800 K. .................................................................................. 56
Figure 3.13 - Computer controlled setup with the electronics and the gas line. Top shelf
carriers the electronics: Agilent 4432B Signal Generator, Keithley 2400 Source-
Measure Unit, SR830 DSP Lock-in Amplifier and the temperature controller.
Breakout box in the bottom is used to interface the wires from the chamber with
coaxial cables. It also has grounding connections to protect against electrostatic
discharge during loading and unloading the sample. ..................................................... 57
xi

Figure 4.1 - Potential energy (solid line) for a typical chemical bond between two atoms
as function of interatomic distance ∆r. The dashed line represents the average
interatomic distance. Illustration adapted from Takenaka [103]. .................................... 60
Figure 4.2 - Calculations of thermal expansion of (a) buckyballs [65], (b) graphene and
graphite [80]. (c) Substrate interaction affects the coefficient of thermal
expansion of graphene [55]. Here γ denotes the interaction strength between the
substrate and graphene. Figures are adapted from respective references. ....................... 63
Figure 4.3 - Ripples in suspended graphene due to thermal expansion from reference
[27]. ............................................................................................................................. 64
Figure 4.4 - Calculation of the axial coefficient of thermal expansion of carbon nanotubes
using (a) molecular dynamics technique [65] and (b) non-equilibrium Green's
functions technique [55]. (c) Coefficient of thermal expansion of carbon
nanotubes depend on its diameter [2]. Figures are adapted from respective
references. .................................................................................................................... 65
Figure 4.5 - Measured value for graphene in-plane thermal expansion from reference
[100]. ........................................................................................................................... 66
Figure 4.6 - Thermal expansion coefficient of a carbon nanotube measured by Chaste et
al. using a NEMS resonator [26]. (a) Thermal response of the mechanical
resonance frequency of a resonator is measured. From the change in frequency
the corresponding (b) change in length is calculated. Finally, (c) coefficient of
thermal expansion of the single-walled nanotube is extracted from the change in
length. From reference [26]. ......................................................................................... 67
Figure 4.7 - (a) Scanning electron microscope image of a carbon nanotube
electromechanical resonator device and (b) a close-up of the nanotube crossing
the trench. (c) Schematic diagram of a device with FM mixing circuit to detect
mechanical resonances. The substrate is oxidized p-type silicon with a 100nm
SiN
x
layer on top. ......................................................................................................... 68
Figure 4.8 - (a) Sample resonance peak taken at V
G
= -4.1V and (b) gate dispersion plot
of sample 1. .................................................................................................................. 69
Figure 4.9 - (a) Resonance frequency of a semiconducting nanotube device (sample 1)
and (b) a quasi-metallic nanotube device (sample 2). .................................................... 71
Figure 4.10 - (a) Linear coefficient of thermal expansion α
i
for the overhang (Pt),
substrate (Si) and the trench. Using the expansion coefficient, we can calculate
(b) the ratios of contraction or expansion with respect to the room temperature
(300K) dimensions. ...................................................................................................... 73
Figure 4.11 - Axial coefficient of thermal expansion of (a) sample 1 and (b) sample 2
calculated based on the experimental data shown in Figures 4.7a and 4.7b. ................... 74
xii

Figure 4.12 - Vibrational modes of (a) an unperturbed carbon nanotube that lead to
negative thermal expansion at moderate temperature: (b) transverse acoustic
bending mode, (c) optical pinch mode, and (d) twist mode. ........................................... 75
Figure 5.1 - (a) The Lennard Jones potential curve of (b) a physisorbed molecule M on a
surface S. The minimum potential energy U
bond
, denotes the energy needed for
the molecule to escape the surface. The location of the minima, R
bond
, shows the
bond length between the adsorbed molecule and the surface. ........................................ 78
Figure 5.2 - Calculated and experimental adsorption energies of (a) BDA on Au and (b)
azobenzene on Ag from reference [71]. ........................................................................ 80
Figure 5.3 - Van der Waals–density functional theory (vdw-DF - black diamonds),
revised Perdew–Burke–Ernzerhof (revPBE - green squares), and local–density
approximation (LDA - blue triangles) simulations of graphene adsorbed on Ni at
a distance d. It is clear that long range interactions are important in defining the
contact between graphene and metals. Adapted from Vanin et al. [111]. ....................... 82
Figure 5.4 - Mechanical resonance spectrum of a carbon nanotube electromechanical
resonator with a 2 µm trench at room temperature. The lowest three modes of
operation (n = 0, 1, 2) are clearly visible. The color indicates the amplitude of the
mixing current measured at the lock-in amplifier. ......................................................... 84
Figure 5.5 - (a) Color plot of mixing current for a higher order mechanical resonance
mode (n = 1) of sample 1 as a function of temperature. (b) Frequency of the
fundamental mechanical resonance mode (n = 0) of sample 1 during a high
temperature cycle. (c) Schematic diagram of the nanotube position when T < T
C

(case I) and T > T
C
(case II). When the nanotube unclamps at higher
temperatures, the tension on the nanotube is reduced by the addition of an extra
suspended length of ∆l
th
. (d) Quality factor of the resonator plotted as a function
of temperature. ............................................................................................................. 85
Figure 5.6 - (a) Schematic diagram illustrating the effects of the nanotube–surface
interactions in creating a taut single-walled nanotube bridging the gap between
two pillars. (b) Transmission electron microscope image of a single-walled
nanotube following the contour of a 2µm hole, creating a taut single-walled
nanotube bridging the hole. (c) High resolution scanning electron microscope
image of an extending single-walled nanotube adhering to as much of the pillar
edge as possible, over a scale of almost a micron. The resolution of the high
resolution scanning electron microscope limits the capability of affirming either
the warping or buckling of the nanotube. Figure is adapted from Abrams et al.
[1]. ............................................................................................................................... 87
Figure 5.7 - Repeated thermal cycling shows that the mechanical resonance of the carbon
nanotube on temperature is different in consecutive cycles. ........................................... 89
Figure 5.8 - Carbon nanotubes showing 'peeling' effect at similar temperatures. Six of the
seven tested samples showed this repeatable behavior. .................................................. 90
xiii

Figure 5.9 - Low tension dispersion graph of a carbon nanotube electromechanical
resonator device at two different temperatures. The fundamental resonance
frequency at T = 425 K (blue solid triangles) is significantly lower than at T =
300 K (red empty circles). ............................................................................................ 91
Figure 6.1 - Schematic of a carbon nanotube field effect transistor. Top panel is an on-
substrate device, and the bottom panel shows a suspended device. ................................ 94
Figure 6.2 - Reflection and transmission in multiple barrier (scattering) system. ........................ 96
Figure 6.3 - Particle transmission probability plotted as a function of device length for a
system of weak scatterers, considering the case of no scattering transmission and
multiple reflection transmission. Adapted from reference [21]. ..................................... 97
Figure 6.4 - Suspended nanotube current–voltage (I–V) characteristics demonstrating the
effect of optical-phonon scattering. (a) Scanning electron microscope image,
taken at a 45° angle, of the on-substrate (on nitride) and suspended (over a ∼ 0.5
µm deep trench) segments of the nanotube. (b) A schematic diagram of the
device cross-section. (c) I–V characteristics of the same-length (L ∼ 3 µm)
suspended and on-substrate portions of a single-walled nanotube (d
t
∼ 2.4nm) at
room temperature measured in vacuum. The symbols represent experimental
data; the lines are calculations based on average nanotube temperature (similar
within ∼ 5% to that based on actual tube-temperature profile and resistance
integrated over the ∼ 3 µm tube length). Adapted from Pop et al. [90]. ......................... 99
Figure 6.5 - Schematic drawing of threshold for optical emission in carbon nanotubes at
high bias voltages. (a) Elastic scattering of electron by an optical phonon [119].
(b) Two relevant length scales for high-bias electron-phonon scattering in an
electric field, Ԑ [86]. The population of the decay state N(T
OP
) in (a) is important
in determining the extra energy required for an electron to emit an optical
phonon. ...................................................................................................................... 100
Figure 6.6 – Scanning electron microscope images of (a) a suspended carbon nanotube
sample device. (b) A high magnification scanning electron microscope image of
an individual carbon nanotube crossing the trench. ..................................................... 102
Figure 6.7 – (a) Electric current plotted as a function of gate frequency. The solid arrows
show the resonance due to the nanotube and dashed arrows point to resonances
due to passive circuitry elements. (b) Color plot of the mixing current plotted as a
function of the gate frequency (f
G
) and DC gate voltage. The red box corresponds
to the experimental regime shown in Figure 4b. .......................................................... 103
Figure 6.8 - Bias voltage dependence of the mechanical resonance frequency of two
different suspended nanotube devices. In (a) the different datasets were taken on
different days at a constant gate voltage of V
G
= -3 V, demonstrating the
repeatability of the observed effect. The inset in (a) shows the I-V characteristics
of this device, which exhibits metallic behavior. The red solid lines represent the
I-V
bias
and conductance-V
gate
characteristics of the device. .......................................... 105
xiv

Figure 6.9 - Effective gating of the carbon nanotube due to asymmetric source-drain bias
voltage. ...................................................................................................................... 106
Figure 7.1 - (a) Temperature dependence of the measured Q-factors by Singh et al. using
electrical actuation of a graphene resonator [100]. (b) In a similar device, Barton
et al. induced mechanical vibrations optically to achieve significantly higher Q
values at room temperature [12]. ................................................................................. 111
Figure 7.2 - (a) Top panel shows the polarization dependence of the optical absorption of
a carbon nanotube and the lower panel shows the corresponding optical
absorption cross-sections [82]. (b) Van Hove singularities in the electron density
of states (adapted from reference [97]) results in (c) peaks in optical absorption
spectra of carbon nanotubes. Adapted from reference [113]. ....................................... 112
Figure 7.3 - Schematic of optical actuation, electrical detection of the mechanical motion
of carbon nanotube electromechanical resonators. ....................................................... 113
Figure 7.4 - Electrical response of a semiconducting single-walled nanotube to gas
molecules. (a) Conductance (under V
g
= +4 V, in an initial insulating state)
versus time in a 200-ppm NO
2
flow. (b) Data for a different semiconducting
single-walled nanotube sample in 20- and 2-ppm NO
2
flows. The two curves are
shifted along the time axis for clarity. (c) Conductance (V
g
= 0, in an initial
conducting state) versus time recorded with the same nanotube sample as in (a)
in a flow of Ar containing 1% NH
3
. (d) Data recorded with a different
semiconducting single-walled nanotube sample in a 0.1% NH
3
flow [63]. ................... 114
Figure 7.5 - (a) Plot of low-bias resistance of a quasi-metallic single-walled nanotube
device as a function of V
g
. (b) Experimental setup for scanned gate microscopy.
I, current. (c) Topographic image of the nanotube, and scanned gate image with
V
tip
= 3, 6, 8 V, from left to right. The dark color in indicates increased
resistance. Adapted from Bockrath et al. [16]. (d) Device scheme of a single-
walled nanotube with modulated chemical doping contacted by two Ni/Au
electrodes. (e) Atomic force microscope image of the single-walled nanotube.
Dashed lines are drawn to highlight undoped carbon nanotube region. (c) A band
diagram for the system. E
C
, E
f
, and E
V
represent the conduction band, Fermi
level, and valence band, respectively. Adapted from Zhou et al. [123]......................... 115
Figure 7.6 - Experimental setup for doping efficiency experiment. .......................................... 116
Figure A.1 - Screenshot of the user interface of the Labview program used to measure the
mechanical resonance frequency of carbon nanotube electromechanical oscillator
using single source frequency modulation method. ..................................................... 126
Figure A.2 - Frequency and the gate data points are generated for the set limits in the user
interface with the defined resolution also set in the user interface. ............................... 127
Figure A.3 - The AC generator and gate DC source are initialized and are both set to
initial values of the experiment. .................................................................................. 128
xv

Figure A.4 - The lock-in amplifier is initialized. After initialization there is a two second
wait time for the input of the amplifier to stabilize. ..................................................... 128
Figure A.5 - The file that the data will be stored in is created for both the absolute value
of the current (R) and the phase of the current (T). ...................................................... 129
Figure A.6 - A new data point is collected for each gate voltage and frequency point in
two nested loops. The outer loop varies the gate voltage. The inner loop varies
the frequency of the carrier of the frequency modulated signal generated by the
signal generator. ......................................................................................................... 130
Figure A.7 - Gate voltage is zeroed and the AC generator output is disabled. The GPIB
connections are closed for proper operation. ............................................................... 131
xvi

ABSTRACT
In this thesis the thermo-mechanical properties of single–walled carbon nanotubes are
investigated utilizing carbon nanotube based nanoelectromechanical oscillators. These resonator
devices are highly sensitive to changes in tension on the carbon nanotube. In Chapters 4 the
coefficient of thermal expansion of an individual single–walled carbon nanotube is measured in
the range 4K - 475K. Experimental observation of this parameter has not been reported before
this work and the calculations give different results depending on the models used. The observed
negative thermal expansion is attributed to the free configurational space around the carbon atoms
of the nanotube. When the nanotube is cooled, the entropy of the system is lowered by expanding
the volume of the nanotube through various changes in the structure like pinching twisting or
bending. The minimum of the coefficient of thermal expansion is measured as -4.5 ppm·K
-1
at
100K. The coefficient of thermal expansion remains negative throughout the entire range.
The mechanical response of carbon nanotube electromechanical oscillators at elevated
temperatures is studied in Chapter 5. The weak interaction forces between the carbon nanotube
and underlying platinum electrodes limit the performance of carbon nanotube electromechanical
oscillators, where the devices are built as described in Chapter 3. Van der Waals bond between
the carbon nanotube and the platinum electrode weaken as the temperature increases. At a critical
temperature the nanotube delaminates from the surface completely and a sudden drop is observed
in the mechanical resonance frequency of the oscillator. Using the results obtained, the clamping
force between the carbon nanotube and the underlying platinum electrode is measured to be
around 3 pN. The small value obtained for the clamping force shows that quality factor of carbon
nanotube electromechanical resonators is affected by the clamping efficiency of the nanotube
ends.
xvii

Carbon nanotubes have unique electron ransport properties at high bias voltages. Due to
their one dimensional nature, scattering of electrons by phonon are highly nonlinear. At low bias
voltages (across the nanotube) phonon scattering is suppressed and the electrons exhibit ballistic
transport. At higher bias voltages optical phonon scattering dominates, and subsequently
nanotubes heat suddenly. In Chapter 6, the heating of nanotubes is probed using the mechanical
vibrations of a carbon nanotube based nanoelectromechanical oscillator. Since the substrate
temperature is constant the change in mechanical resonance frequency is attributed to the
contraction of the nanotube due to its negative thermal expansion. The bias voltage, at which the
mechanical resonance shows a sudden drop, corresponds to experimentally observed optical
phonon emission onset voltage for single–walled carbon nanotubes.
1

CHAPTER 1: FUNDAMENTAL PROPERTIES OF CARBON
NANOTUBES

1.1 Introduction
The structural, electronic, and thermal properties of carbon nanotubes have been
extensively studied over the past two decades. Advances in fabrication techniques have led to the
discovery of new and exotic properties of carbon nanotubes. Several aspects of their mechanical
properties, however, remain poorly understood since the temperature dependence of these
properties are often difficult to measure.
Nanoelectromechanical systems based on nanoscale materials like graphene, nanowires,
and carbon nanotubes offer the potential for new applications. These applications, in turn, enable
us to probe the fundamental mechanical properties of these materials in new ways. The very low
mass and high stiffness of carbon nanotubes are particularly advantageous for
nanoelectromechanical resonator devices.
Since the first realization in 2004 by the McEuen group at Cornell University, carbon
nanotube electromechanical resonators have been utilized to observe various properties of carbon
nanotubes [99]. However, the performance of these devices was noticeably lower than the
theoretically predicted behavior due to their low quality factors. To improve the accuracy and
sensitivity of these resonator devices, higher Q–factors are needed. It is, therefore, crucial to
understand the mechanisms limiting the Q–factors in carbon nanotube electromechanical
resonators.
In this thesis, we present several measurements involving carbon nanotube
electromechanical resonators, including measurements of thermal expansion of individual carbon
2

nanotubes. The coefficient of thermal expansion of carbon nanotubes is particularly important,
since carbon nanotubes are used more often as heat dissipation systems in integrated circuits due
to their high thermal conductivity. Therefore, it is important to know how carbon nanotubes
behave at higher temperatures. Another use for carbon nanotube electromechanical resonators is
ultra low mass detection. To achieve the highest possible sensitivity, it is important to identify the
loss mechanisms and imperfections of the system. Later in this thesis, we present an investigation
of the clamping of these mechanical resonators, which reveals the importance of van der Waals
forces in these nanoelectromechanical systems and how it limits the performance of these
devices. Lastly, we will introduce a new method to study electron–phonon coupling mechanisms
in carbon nanotubes.
In this chapter, we give a brief introduction describing the structure of carbon nanotubes
and their electronic properties (Section 1.2), followed by a description of their mechanical and
thermal properties (Sections 1.3 and 1.4). The chapter finishes with a review of relevant work on
carbon nanotube electromechanical resonators.

1.2 Structure
Carbon nanotubes were discovered by Sumio Iijima in 1991 using transmission electron
microscopy [50]. There are two basic types of carbon nanotubes: single–walled carbon nanotubes
(Figure 1.1a)[51] and multi–walled carbon nanotubes (Figure 1.1b), which are made up of
concentric single-walled nanotubes. Single-walled nanotubes are typically 1-2nm in diameter,
whereas multi-walled nanotubes can range between 5-50nm. Various methods exist for
synthesizing carbon nanotubes, including laser ablation [107], arc discharge [57], high pressure
carbon monoxide (HiPco) [19], and chemical vapor deposition [64]. In this thesis, we study
single-walled nanotubes grown by chemical vapor deposition method, which provides high yield

and controllable synthesis of high
this text refer to single-walled nanotube
Figure 1.1 - Transmission electron microscopy images of (a) a single
(b) a multi–walled carbon nanotube with 5 walls. The concentric structure of multi
as shown in reference [50].

Carbon nanotubes can be imagined as a sheet of graphite rolled up into a se
cylindrical tube. Figure 1
In graphite, the distance
up" a graphene sheet into a tubular structure, resulting in many different species or
nanotubes. Figure 1.2 illustrates several different type
and chiral nanotubes, as shown in
The angle ϕ at which the nanotube is rolled up is
nanotube, and the vector defining
has the form na
1
+ ma
2
, as indicated in
nanotube in the form (n,
defines all possible (n, m
be calculated by
(a)
(b)
controllable synthesis of high-quality single-walled nanotubes. All properties discussed in
walled nanotubes unless otherwise noted.
Transmission electron microscopy images of (a) a single–walled carbon nanotube from reference [51] and
walled carbon nanotube with 5 walls. The concentric structure of multi-walled nanotubes are clearly visible
Carbon nanotubes can be imagined as a sheet of graphite rolled up into a se
1.2a shows the honeycomb lattice of a single layer of graphite (graphene).
between these sheets is 3.4 Å. Naturally, there are many ways of "rolling
into a tubular structure, resulting in many different species or
illustrates several different types of nanotubes, including a
, as shown in Figure 1.2b, 1.2c, and 1.2d, respectively.
at which the nanotube is rolled up is referred to
and the vector defining nanotube diameter is called the 'chiral vector'.
, as indicated in Figure 1.2a, and it uniquely specifies
n, m). Due to symmetry of the honeycomb lattice, the range of 0
m) combinations. With its chirality known, the diameter of a nanotube can
3
All properties discussed in

walled carbon nanotube from reference [51] and
walled nanotubes are clearly visible
Carbon nanotubes can be imagined as a sheet of graphite rolled up into a seamless
a single layer of graphite (graphene).
there are many ways of "rolling
into a tubular structure, resulting in many different species or chiralities of
of nanotubes, including armchair, zigzag,
b, 1.2c, and 1.2d, respectively.
referred to the 'chiral angle' of that
is called the 'chiral vector'. The chiral vector
specifies the geometry of a
the honeycomb lattice, the range of 0 ≤ ϕ ≤ 30°
With its chirality known, the diameter of a nanotube can
4

, (1.1)
where a = 0.246 nm.

Figure 1.2 - (a) Chiral vector C
h
on the graphene honeycomb lattice, (b) an armchair nanotube (n, n), (c) a zigzag
nanotube (n, 0), and (d) a chiral nanotube (n, m).

(a)
(b) (c) (d)
ϕ
5

1.3 Electronic Properties
During the discussion in this section, we will closely follow the book "Physical
Properties of Carbon Nanotubes" by Saito, Dresselhaus, and Dresselhaus [96]. Since single-
walled nanotubes are essentially rolled up sheets of graphene, their unique electronic properties
can be derived (to first order) from those of a graphene sheet. The electronic band structure of
graphene, shown in Figure 1.3, has six points of contact between the conduction and the valence
bands in the Brillouin zone. However, for the range of energy values related to our work, this
picture can be simplified by six pairs of Dirac cones representing the conduction and valence
bands due to the six–fold symmetry of the Brillouin zone.

Figure 1.3 - Band diagram of graphene. The contact points between the conduction and the valence band can be
approximated as a Dirac cone with relevant energy limits.

In a carbon nanotube, the periodicity of the wave vector k in the circumferential direction
discretizes the possible values of the perpendicular component of the wave vector

of the
electrons (Figure 1.4a) while the
∥
values remain continuous. This discretization groups
nanotubes into two subsets depending on where the allowed k values intersect the graphene band
diagram. As shown in Figure 1.4a, if one of the

values cuts through the Dirac point, the
resulting nanotubes are metallic (Figure 1.4b), while the remaining nanotubes are considered
semiconducting with wide band gaps of a few 100s of meV (Figure 1.4c). In reality, the curvature

of nanotubes results in a small band gap
this reason, metallic nanotubes are
Figure 1.4 - (a) Schematic illustration of the
correspond to discrete k

values. If the cutting lines (b) go through a Dirac point
(metallic), otherwise (c) the nanotube has a wide gap (semicond

One can show with simple algebra that
define quasi–metallic nanotubes
2/3 are semiconducting in nature
0.7eV/d
t
, where d
t
is the diameter of the nanotube in nm.
The electron transport in nanotubes can be modeled
equation [68]. In this model
nanotubes can reach this value at room temperature and semiconducting
value when their Fermi level is tuned in
However, in an experimental setup, nanotubes are contacted with metals like Pt, Pd or Au
electrical measurements.
(a)
a small band gap in metallic nanotubes, on the order of 10s of meV. For
this reason, metallic nanotubes are often referred to as quasi–metallic.
Schematic illustration of the 6 Dirac cones in the graphene Brilluoin zone with cutting lines that
values. If the cutting lines (b) go through a Dirac point, the nan
(metallic), otherwise (c) the nanotube has a wide gap (semiconducting). Adapted from Minot
One can show with simple algebra that the condition mod[3, m
metallic nanotubes. As a result, 1/3 of all nanotubes are metallic,
in nature. When mod[3, m-n] ≠ 0, the nanotubes have
is the diameter of the nanotube in nm.
The electron transport in nanotubes can be modeled using
. In this model, the nanotube has a maximum conductance of 4
nanotubes can reach this value at room temperature and semiconducting
n their Fermi level is tuned into conduction or the valence band
However, in an experimental setup, nanotubes are contacted with metals like Pt, Pd or Au
measurements. At the metal–nanotube contact, the valence and the conduction band
(c)
(b)
6
in metallic nanotubes, on the order of 10s of meV. For

graphene Brilluoin zone with cutting lines that
the nanotube has no bandgap
dapted from Minot [78].
m-n] = 0 can be used to
all nanotubes are metallic, and the remaining
nanotubes have a band gap of E
g
≈
the Landauer transport
nanotube has a maximum conductance of 4e
2
/h. Quasi–metallic
nanotubes can reach this value at room temperature and semiconducting nanotubes can reach this
conduction or the valence band, as discussed below.
However, in an experimental setup, nanotubes are contacted with metals like Pt, Pd or Au for
nanotube contact, the valence and the conduction bands of
7

the nanotube and the Fermi level of the metal do not match, which results in a Schottky barrier.
This barrier lowers the overall conductance of the nanotube device. In our experiments, the
average device conductance is 25 µS. A more detailed discussion of carbon nanotube transport
characteristics is presented in Chapter 6.
The nanotube conductivity is a function of the Fermi level, which can be tuned between
the conduction band and the valence band or reside in the band gap. The most common method to
change the Fermi level is coupling an electrostatic field to the potential of the nanotube. This is
usually done using a field effect transistor device where the transport channel is the carbon
nanotube. A sample device is shown in Figure 1.5a. When an electrical potential is applied to the
gate electrode, excess charge will be introduced into the nanotube and the Fermi level will shift
with respect to the band edges, as shown in the right panel of Figure 1.4b and Figure 1.4c. A
'positive' gate voltage will attract electrons and shift the Fermi level into the conduction band
turning the device 'n–type'. In the opposite case (negative gate voltage), the nanotube will be 'p–
type' and the charge transport will be dominated by holes. However, for the excess charge to
move along the nanotube, a small source to drain voltage (V
sd
) on the order of 1mV should be
applied.
In this thesis, we use suspended nanotubes as the electron transport channel, where the
underlying substrate is removed, as shown in Figure 1.5b. Fabrication of these devices is
discussed in detail in Section 3.2. Removing the substrate affects the charge transport properties
of a nanotube in several ways. The Fermi level shift due to trapped charges on the substrate
surface is avoided [33]. Also, suspended nanotubes exhibit unique transport properties like
negative differential conductance, as depicted in Figure 1.6c. Figure 1.6a and Figure 1.6b show
typical conductance plots of semiconducting and quasi–metallic nanotubes used in this thesis.
One drawback of using suspended devices is the weak coupling of the gate electrode to the

nanotube. The dielectric constant of air is
between the gate electrode and the nanotube. Lower capacitance means that the number of extra
charges generated on the nanotube is less compared to on
increasing the applied ga
Figure 1.5 - Illustration of a carbon nanotube field effect transistor device with (a) a nanotube on substrate and (b)
suspended.

Carbon nanotube
Therefore, small perturbations to the structure would cause changes in the charge transport
characteristics. Theoretical studies showed that the band gap
semiconducting carbon nanotube
experimentally confirmed
strain on the carbon nanotube

where E
g
is the bandgap of the nanotube,
is the chiral angle of the nanotube.
(a)
(b)
dielectric constant of air is lower than SiO
2
, which reduces the capacitance
between the gate electrode and the nanotube. Lower capacitance means that the number of extra
charges generated on the nanotube is less compared to on–substrate devices
increasing the applied gate voltage.
Illustration of a carbon nanotube field effect transistor device with (a) a nanotube on substrate and (b)
Carbon nanotubes' electronic properties are strongly dependent on
small perturbations to the structure would cause changes in the charge transport
characteristics. Theoretical studies showed that the band gap
carbon nanotubes can be widened with applied strain
confirmed by Minot et al. [79] and the change in nanotube band
carbon nanotube is related by

100
meV
∆ ⁄
cos3
is the bandgap of the nanotube, σ is the strain, ∆L/L is the strain on the nanotube
is the chiral angle of the nanotube.

8
which reduces the capacitance
between the gate electrode and the nanotube. Lower capacitance means that the number of extra
substrate devices. This is addressed by

Illustration of a carbon nanotube field effect transistor device with (a) a nanotube on substrate and (b)
s' electronic properties are strongly dependent on their structure.
small perturbations to the structure would cause changes in the charge transport
characteristics. Theoretical studies showed that the band gap of quasi–metallic and
s can be widened with applied strain [45]. This has been
nanotube band gap with a known
(1.2)
is the strain on the nanotube and ϕ

Figure 1.6 - Conductance of (a) semiconducting nanotube and (b) quasi
exhibit negative differential conductance

1.4 Mechanical Properties
Carbon nanotubes
This makes them an ideal candidate for
mechanical properties of
strain ϵ

Conductance of (a) semiconducting nanotube and (b) quasi–metallic nanotube. (c) Suspended nanotubes
negative differential conductance at high bias voltages.
Mechanical Properties
Carbon nanotubes have the highest stiffness to mass ratio of the
ideal candidate for nanoelectromechanical resonator devices.
of carbon nanotubes are the Young's modulus (E), which relates stress

!

(a)
(c)
(b)
9

metallic nanotube. (c) Suspended nanotubes
ness to mass ratio of the materials known today.
resonator devices. Two important
), which relates stress σ to
(1.3)
10

and the maximum stress a nanotube can endure σ
max
. The Young's modulus of single–walled
carbon nanotubes has been experimentally measured to be ≈ 1TPa [118]. Recently, it is also
shown that single-walled nanotubes can withstand 14% strain in repeated cycles [24].
Measuring the mechanical properties of carbon nanotubes is challenging owing to their
small size. Several measurement techniques utilizing transmission electron microscope [109],
Raman spectroscopy [74], and atomic force microscopy [112] have been used to measure these
parameters, as illustrated in Figure 1.7a. Atomic force microscopy proved to be the most
proficient way to measure the Young's modulus of individual single-walled nanotubes with high
precision. However, the uncertainty in the diameter of the nanotube is reflected in the measured
values of E. Since the tension applied on the nanotube by an external force is carried along the
nanotube cross-section, the determination of the diameter is crucial. Wu et al. characterized their
single-walled nanotubes by Rayleigh scattering spectroscopy and were able to extract the chirality
of their nanotubes. Then the nanotubes are placed in a magnetic field and the displacement of the
nanotubes due to Lorentz force are measured. With the precise value of the diameter of the
nanotube calculated using its chirality (Section 1.1), Wu et al. extracted the Young's modulus of a
nanotube [118], as shown in Figure 1.7b.
For the work in this thesis, characterization of the fundamental mechanical properties of
single-walled nanotubes as a function of temperature is very important. Especially mechanical
properties are of great interest in the discussion of the carbon nanotube electromechanical
resonators in Chapters 4 and 5. Raravikar et al. calculated the temperature dependence of Young's
modulus using molecular dynamics simulations [94]. The downward trend they predicted has not
been observed experimentally in the literature to our best knowledge. Therefore, we assumed this
hardening effect at lower temperatures to be negligible throughout this thesis.
11

Figure 1.7 - Young's modulus of single-walled nanotubes measured by (a) an atomic force microscope tip pushing
down on a suspended carbon nanotube [112], (b) the displacement of carbon nanotubes in a magnetic field [118]. Left
two panels in (b) show the Rayleigh spectrum used to obtain the chirality of the measured nanotubes. Once the
nanotubes are characterized, a magnetic field is applied (lower right panel) on the nanotube and the deflection due to
Lorentz forces is measured.

1.5 Thermal Properties
Besides their electronic and mechanical properties, carbon nanotubes have unique
thermal properties. They have been studied extensively both theoretically and experimentally.
Because of the small size of carbon nanotubes, the quantum effects are important. The low–
(a)
(b)
12

temperature specific heat (C) and thermal conductivity (K) show direct evidence of 1-D
quantization.

Figure 1.8 - Specific heat of various carbon allotropes from reference [46]. SWNT refers to single–walled carbon
nanotubes.

As with many properties of nanotubes, the specific heat of carbon nanotubes are similar
to that of graphene. At high temperatures (T > 100K), graphene, individual and bundled single-
walled nanotubes, and graphite have the same specific heat [46]. The major contribution to the
specific heat in these systems is the phonon specific heat. The electronic contribution of specific
heat is two orders of magnitude smaller than the phonon contribution. Saito et al. calculated the
phonon dispersion of single-walled nanotubes and showed that when a graphene sheet is folded
(to produce a nanotube), its 2-D band structure folds into many 1-D sub-bands [97]. For this
reason, at lower temperatures the single-walled nanotube behavior diverges from that of graphene
and graphite, as shown in Figure 1.8. Measurements on highly purified bulk single-walled
13

nanotube samples agree with theoretical values almost over the entire temperature range, except
below T < 5K where predictions overestimated the inter-tube coupling.
Carbon nanotubes also possess a high thermal conductivity of 37,000 W/m·K around
100K, which is comparable to diamond's 44,000 W/m·K. This high thermal conductivity is due to
both electron and phonon contributions. The electronic contribution (K
el
) to the thermal
conductivity in a 1-D ballistic channel, such as a carbon nanotube, can be calculated using the
Wiedemann-Franz law [60], which has a linear dependence on temperature. The phonon or lattice
contribution (K
ph
) is calculated by summing over all the phonon modes of the nanotube. At lower
temperatures, K
ph
is linear with temperature. At higher temperatures, however, scattering is
significant, which reduces the lifetime of these phonon modes. This causes the K
ph
to decrease
with increasing temperature. Figure 1.9 shows the temperature dependence of the thermal
conductivity of a carbon nanotube, calculated by Berber et al. [13]. K(T) peaks around 100 K,
above which phonon scattering events dominate and the overall value of the thermal conductivity
decreases.
Various methods had been employed to measure the thermal conductivity of single-
walled nanotubes. Pop et al. used the electron transport measurements of an individual single-
walled nanotube to extract the thermal conductivity [90]. In a more direct measurement, Yu et al.
used micro thermometers to measure heat conducted by an individual single-walled nanotube
between two heating pads [121]. Hsu et al. measured changes in Raman spectrum of heated
single-walled nanotube bundles to extract the thermal conductivity [47].

Figure 1.9 - Calculated thermal conductivity of a (10, 10) single
reference [13].

1.6 Previous Work on
The high stiffness and low mass of
candidate for ultra-low mass sensing applications.
masses utilizing the single electron transistor properties of individual
low temperatures [28].
balances to 1.7 yoctogram
1.10a. They achieved this improvement by
carbon nanotube electromechanical
frequencies due to their short length, as discussed in C
carbon nanotube electromechanical
mono atomic molecules (Ar, Kr, etc.) on
temperature and the pressure of the gasses, isotherm curves of these molecules on the
nanotube surface were obtained (
these molecules were obtained.
Calculated thermal conductivity of a (10, 10) single-walled nanotube as a function of temperature from
Previous Work on Nanoelectromechanical Resonators
stiffness and low mass of carbon nanotubes make this
low mass sensing applications. Chiu et al. measured ~50 zeptogram
masses utilizing the single electron transistor properties of individual single
. Recently, Chaste et al. improved the sensitivity of
yoctograms (10
-24
g), which is the mass of a single proton
. They achieved this improvement by using very short carbon nanotubes
electromechanical resonators. These devices had very high resonance
short length, as discussed in Chapter 2. Using the
electromechanical resonators, Wang et al. extracted the adsorption energy of
mono atomic molecules (Ar, Kr, etc.) on carbon nanotube surfaces
temperature and the pressure of the gasses, isotherm curves of these molecules on the
obtained (Figure 1.10b). From these isotherm curves
these molecules were obtained.
14

walled nanotube as a function of temperature from
e this material a prime
Chiu et al. measured ~50 zeptograms (10
-21
g)
single-walled nanotubes at
tivity of carbon nanotube
gle proton [25], as shown in Figure
using very short carbon nanotubes (150 nm) in their
resonators. These devices had very high resonance
e high mass sensitivity of
resonators, Wang et al. extracted the adsorption energy of
[115]. By changing the
temperature and the pressure of the gasses, isotherm curves of these molecules on the carbon
From these isotherm curves, adsorption energy of
15

Nanoelectromechanical resonators were also used for fundamental physics studies. Steele
et al. observed a single electron hopping in the nanotube at very low temperatures where a short
carbon nanotube acts as a quantum dot with discrete energy levels [102], as depicted in Figure
1.10c, by employing a new actuation and detection method [49]. This method is discussed in
detail in Section 3.3.4. The carbon nanotube electromechanical resonators in this study had very
high Q–factors that allowed the observation of the change in the resonance frequency induced by
a single electron.

Figure 1.10 - Yoctogram resolution mass sensitivity has been achieved using carbon nanotube electromechanical
resonators [25]. Each arrow represents a Xe atom landing on the carbon nanotube surface (b) Using the mass sensitivity
of a carbon nanotube electromechanical resonator, it is possible to measure the coverage of a monatomic gas adsorbed
on the surface of a carbon nanotube [115]. (c) With a high–Q resonator, Steele et al. measured the hopping of an
electron on to the nanotube through the change in mechanical resonance frequency [102].

(a) (b)
(c)
16

Shortly after carbon nanotube resonators were demonstrated, graphene resonators were
realized using mechanically exfoliated graphene by Bunch et al. [20]. They used an optical
method to induce vibrations on the graphene by creating a local hot spot with a intensity
modulated laser. The temperature gradient on the surface caused ripples on the graphene. When
the laser modulation frequency matched the mechanical resonance frequency, the ripples formed
a well defined shape on the surface. This mechanical motion is detected by an interferometric
technique using a second laser. This method is later utilized by Barton et al. to image the
mechanical modes of the vibrations on a graphene drum resonator [12], as shown in Figure 1.11c.
Singh et al. measured the temperature dependence of the thermal expansion of graphene using an
electrical actuation and detection technique [100].

Figure 1.11 - (a) First graphene nanoelectromechanical resonator with optical actuation and optical detection [18]. (b)
Using mechanically exfoliated graphene Singh et al. measured the coefficient of thermal expansion of a single layer of
graphene using a nanoelectromechanical resonator [100]. (c) Optical detection of mechanical resonances can be utilized
to image the mode shape of the mechanical motion [12].
(a)
(b) (c)
17

Recently, Teufel et al. demonstrated that mechanical quantum ground states can be
accessed [106]. They used a nanoelectromechanical resonator with an aluminum membrane as the
resonator coupled to a superconducting high–Q RC circuit. By employing a method similar to
laser cooling, they probed the quantum ground states at relatively high temperatures (25 mK).

1.7 Summary and Outline of Thesis
In this thesis, we will describe our measurements on the temperature dependence of the
mechanical properties of carbon nanotubes. Chapter 2 gives an overview of the carbon nanotube
electromechanical resonator platform used. Chapter 3 describes the fabrication of the devices and
measurement techniques used in our research. In Chapter 4, we will discuss the thermal
expansion of carbon allotropes and our findings on the temperature dependence of coefficient of
thermal expansion of the individual single-walled nanotubes. Chapter 5, will discuss the clamping
losses in the studied resonators and our measurements of the clamping force of these resonators.
Van der Waals interactions of carbon nanotubes on Pt electrodes is also discussed in this chapter.
Chapter 6 discusses an alternative method to observe electron-phonon interactions in carbon
nanotubes. Chapter 7 will conclude this thesis with a summary and future work that can be built
on the findings presented here.

18

CHAPTER 2: CARBON NANOTUBE
ELECTROMECHANICAL RESONATORS

2.1 Introduction
Small size, low mass density, and high stiffness make carbon nanotubes an ideal material
for mechanical resonators. Earlier attempts to build these devices focused on multi-walled carbon
nanotubes due to their ease of handling and interfacing [89, 36, 92]. These early experiments
consisted of a carbon nanotube attached to an atomic force microscope tip on one end with the
other end suspended in air. To actuate the mechanical motion, an AC field was created between
the nanotube and a nearby electrode. Poncharal et al. [89] and Gao et al. [36] used transmission
electron microscope or scanning electron microscope imaging to detect the motion of the
nanotube. Purcell et al. engineered a multi-walled nanotube as a field emitter and monitored the
emission current in order to extract information about the mechanical oscillations [92]. With these
methods, it is possible to observe the mechanical vibrations of the nanotube. However, there are
several drawbacks. Using scanning electron microscope or transmission electron microscope to
image the nanotube motion is fairly destructive and the data collection is not very accurate, since
both methods are limited to the resolution of the instrument used. Also for practical sensing
applications, ultra low pressures required for scanning electron microscope, transmission electron
microscope, or a field emitter setup is not feasible. Therefore, direct electrical actuation and
electrical detection of mechanical motion is very desirable.
The first electrically driven nanotube resonator was built in 2004 by Sazonova et al. [99].
Unlike previous work on carbon nanotube resonators, Sazonova used a doubly clamped carbon
19

nanotube, as shown in Figure 2.1. This device also functions as a field effect transistor, which
enables both electrical actuation, and electrical detection of mechanical motion.

Figure 2.1 - False colored scanning electron microscope image and the schematic drawing of the first carbon nanotube
electromechanical resonator [99].

The small mass density of carbon nanotubes makes it possible to measure the weight of
individual molecules directly using a carbon nanotube electromechanical resonator as a very
sensitive balance. The first such application measured Au atoms deposited on the carbon
nanotube [53]. This work demonstrated that carbon nanotube electromechanical resonators can, in
fact, be used as mass sensors. However, the metal deposition causes irreversible changes in the
response of the system, and the measurement sensitivity was quite low. Later, Chiu et al. utilized
the single electron transistor property of short carbon nanotubes at 4 K to achieve a resolution of
50 zeptograms using Xe gas for mass loading [28]. The highest sensitivity reported to date using
these resonators is 1.7 yoctograms (10
-24
g) at 4 K, which is equal to the mass of a single proton
[25].
In addition to mass sensing applications, carbon nanotube electromechanical resonators
were characterized by various groups. Garcia-Sanchez et al. used an atomic force microscope to

image the shapes of the fundamental and higher vibrational modes of these resonators [37].
Eichler et al. studied the effects of nonlinear damping in these resonators, and showed that the
actuation force can change the resonance frequency [32]. Steele et al. studied the coupling of
electrons to the mechanical motion of the resonator [102].
In this chapter, we will develop a model of
general discussion of Sazonova [98]. We will first give a background on simple harmonic
oscillators and doubly clamped beam mechanics. Following this brief overview, we will discus
how to tune the frequency of these oscillators and the model used to fit the observed data. We
will conclude with a discussion of loss mechanisms in these resonators.

2.2 Beam Mechanics
Any resonant system, to first order, can be described as a simple harmo
mass m attached to the end of a spring with spring constant
illustrated in Figure 2.2.
Figure 2.2 - Motion of a simple harmonic oscillator. A mass m is attached to a spring with spring constant k. There is
no friction or driving force. The displacement z(t) of the mass can be described by a simple harmonic oscillator.
image the shapes of the fundamental and higher vibrational modes of these resonators [37].
er et al. studied the effects of nonlinear damping in these resonators, and showed that the
actuation force can change the resonance frequency [32]. Steele et al. studied the coupling of
electrons to the mechanical motion of the resonator [102].
apter, we will develop a model of carbon nanotube resonators, following the
general discussion of Sazonova [98]. We will first give a background on simple harmonic
oscillators and doubly clamped beam mechanics. Following this brief overview, we will discus
how to tune the frequency of these oscillators and the model used to fit the observed data. We
will conclude with a discussion of loss mechanisms in these resonators.
Beam Mechanics
Any resonant system, to first order, can be described as a simple harmo
attached to the end of a spring with spring constant k that is fixed on the other end, as
.
Motion of a simple harmonic oscillator. A mass m is attached to a spring with spring constant k. There is
no friction or driving force. The displacement z(t) of the mass can be described by a simple harmonic oscillator.
m
F
s

k
z < 0
z = 0
z > 0
F
s

20
image the shapes of the fundamental and higher vibrational modes of these resonators [37].
er et al. studied the effects of nonlinear damping in these resonators, and showed that the
actuation force can change the resonance frequency [32]. Steele et al. studied the coupling of
resonators, following the
general discussion of Sazonova [98]. We will first give a background on simple harmonic
oscillators and doubly clamped beam mechanics. Following this brief overview, we will discuss
how to tune the frequency of these oscillators and the model used to fit the observed data. We
Any resonant system, to first order, can be described as a simple harmonic oscillator with
that is fixed on the other end, as

Motion of a simple harmonic oscillator. A mass m is attached to a spring with spring constant k. There is
no friction or driving force. The displacement z(t) of the mass can be described by a simple harmonic oscillator.
21

A more realistic case includes a driving force and a damping term. This damping term typically
depends on the velocity of the mass, "#$%, rather than its position [77]. While there are various
damping mechanisms, gas molecules hitting the nanotube is the main source of damping for a
suspended nanotube. For this reason, all measurements in this thesis were performed in a vacuum
chamber with pressure P < 1 Torr. For simplicity we will assume a sinusoidal driving force. The
equation of motion for the system can be described as
#&%"#$%#%'
(
cos -% (2.1)
where m, b, k, F
0
, and ω are the suspended mass, the damping coefficient, spring constant, driving
force amplitude, and the frequency of the driving force. z denotes the displacement of the mass,
#$% and #&% and are the velocity and the acceleration of the mass, respectively. Solving
Equation (2.1 with a sinusoidal response for the position z(t), we get the solution

#%
'
(
⁄
-
(

−-

4-

0

sin -%−
(2.2)
arctan 6
-
(

−-

2-0
8 (2.3)
where
0 " 2 ⁄ (2.4)

-
(
⁄
(2.5)
Figure 2.3 shows the frequency response of the amplitude and phase of mechanical motion. For
small damping, we can define a quality factor Q, as the ratio of the total energy of the system to
the energy lost in each period, Q = ω
0
/∆ω, where ∆ω is the full width-half max of the amplitude
response at the resonance frequency.

Figure 2.3 - The amplitude and phase of the simple harmonic oscillator response plo
frequency. The amplitude of the response reaches a maximum at f
Sazonova [98].

This simple harmonic oscillator analysis
system of a doubly clamped beam
motion of our beam based on
calculate the resonance frequency of a doubly clamped nanotube resonator. Details of the
continuum model can be found in Witkamp et al. [117].
Figure 2.4 - A doubly clamped beam made up
length L. The beam has a tension
of inertia of I with respect to the z
The amplitude and phase of the simple harmonic oscillator response plotted as a function of the driving
frequency. The amplitude of the response reaches a maximum at f
0
= ω
0
/2π and flips phase 180°. Adapted from
This simple harmonic oscillator analysis can be expanded to describe the behavior of our
a doubly clamped beam illustrated in Figure 2.4. We can calculate the shape and
ased on energy considerations. A continuum model
calculate the resonance frequency of a doubly clamped nanotube resonator. Details of the
continuum model can be found in Witkamp et al. [117].
A doubly clamped beam made up of a material with Young's modulus of E, thickness t, width w, and
length L. The beam has a tension T
9 9 :
and a uniform downward force of K
9 9 :
. It has a cross-sectional area of A and moment
of inertia of I with respect to the z-axis.
22

tted as a function of the driving
π and flips phase 180°. Adapted from
can be expanded to describe the behavior of our
. We can calculate the shape and
A continuum model can also be used to
calculate the resonance frequency of a doubly clamped nanotube resonator. Details of the

of a material with Young's modulus of E, thickness t, width w, and
sectional area of A and moment
23

In order to simplify the calculations, we will assume that the mechanical motion is
restricted to the xz plane and that the shape of the beam is symmetric. If the Young's modulus is E
and the tension on the beam is T, then the elastic potential energy is given by
;
1
2
< 6= #′′

+6?
(
+
@
2
< #′

A
B
(
8#′

8A
B
(
(2.6)
where T
0
is the residual tension in the beam, A is the beam's cross-sectional area, I is the moment
of inertia, and the prime denotes the derivative with respect to x [67]. The first term of the
integrand in Equation (2.6 is the flexural energy, and the second term is the potential energy due
to built-up tension in the beam. Here, the product EI is referred to as the flexural rigidity, and it
describes the force needed to bend a beam. EA is the extensional rigidity, which denotes the stress
necessary to produce a unit strain. If we assume the displacement of the beam is small, we can
replace z'' with κ, and 1/2z'
2
with the strain in the beam ϵ. In the presence of a uniform downward
force C
9 9 9 :
, Equation (2.6 simplifies to
;=
1
2
< (= D

+?
(
+@!

+E#)A
B
(
(2.7)
Minimizing the energy U, gives us the following equilibrium equation
= #
FFFF
−?#
FF
−E=0 (2.8)
where ?=?
(
+
GH
B
I #′

A
B
(
, and T
0
is the residual stress on the nanotube due to fabrication.
In the "bending limit", tension in the beam is much smaller than the flexural rigidity
?≪=

⁄ , so the contribution of the second term in Equation (2.8 can be neglected. In this
limit, the equation reduces to a wave equation after substituting K with K#&, where μ is the linear
mass density.
= #
FFFF
=K#& (2.9)

24

To solve Equation (2.9, we plug in a general solution in the form of ##
(
Acos -% .
Enforcing the boundary conditions for a doubly-clamped beam, where displacement of the
nanotube z and its derivative z' at the clamping points are zero, z(x = 0, L) = 0, z'(x = 0, L) = 0, we
get the resonant frequency ω
n
in the bending regime

-
L
M
0
L

N

O
=
K
,
(2.10)
where cos(β
n
)cosh(β
n
) = 1 and for the fundamental mode β
0
= 4.75.
In the opposite limit ?≫=

⁄ , flexural rigidity becomes negligible. We will call this
range the "tension limit". Repeating the substitution for K, Equation (2.8 reduces to
?#
FF
=K#&. (2.11)
The solution to this equation for a doubly clamped beam is #=#
(
cos(A)cos(-%+ ), and the
resonance frequency is given by

-
L
=

O
?
K
,
(2.12)
where ?=?
(
+
GH
B
I #′

A
B
(
, and T
0
is the residual stress on the nanotube due to fabrication. This
result is not surprising since it describes the mechanical vibrational modes of a "simple string
under tension".

2.3 Observing The Mechanical Resonances
There are various methods to actuate an oscillatory motion and detect the response from a
mechanical resonator. In previous works, actuation techniques such as piezo [72], magnetic-
Lorentz force [29], and optical heating [20] have been employed. Piezo actuation is not suitable
for the operational frequencies of carbon nanotube resonators, and optical actuation is difficult
since the optical absorption of carbon nanotubes are very poor. Actuation through the magnetic-
25

Lorenz force is viable; however, a strong magnetic field is required for each resonator, which
may not be very feasible for practical applications. Electrical actuation, as discussed earlier, is the
simplest method to induce vibrations in these resonators. For the work in this thesis, three
different electrical actuation techniques have been utilized. In this chapter, we will describe a
two-signal mixing technique in detail and the underlying physics of electrical actuation and
detection. Most of the discussion in this section is applicable to the other methods discussed in
Section 3.3.
We actuate the nanotube using the electrostatic field generated between the nanotube and
a highly doped silicon back gate or a metal electrode on the bottom of the trench, as shown in
Figure 2.5. A constant DC voltage V
g
is applied to the gate electrode, which induces equal and
opposite charge q on the nanotube and gate. Here, q = CV
g
, and C is the capacitance between the
nanotube and the gate electrode. Coulomb's law states that two oppositely charged objects have
an attractive force pulling them together. In our system, the Si substrate is fixed in space so the
nanotube will be pulled towards the bottom of the trench. We can calculate this force from the
electrical energy stored in the system

'
ST

#
M
1
2
UV

N=
1
2
U′V

(2.13)
where C' = dC/dz is the derivative of the capacitance with respect to the distance between the
nanotube and the gate electrode.

Figure 2.5 - Schematic of the electrical actuation and detection method. DC and AC voltages are applied to the gate
electrode to excite the mechanical vibration. On resonance, the capacitance between the nanotube and the gate
electrode is modulated. Current though t
nanotube.

When an AC signal at frequency

and the total force on the nanotube is

Here, we have neglected the term proportional to
above, one can separate the DC and AC components;

The DC term is used to control the tension on the nanotube, and the AC component is used to
excite the mechanical vibrations. As the modulation frequency gets closer to the resonance
frequencies (calculated in Section 2.2) the amplitude of the vibrations become large.
As explained in Section 1.3, small gap quasi
semiconducting nanotubes can be used to build field effect transistors similar to the structure
shown in Figure 2.5. We will use the transistor properties of
vibration of our nanoelectromechanical
Schematic of the electrical actuation and detection method. DC and AC voltages are applied to the gate
electrode to excite the mechanical vibration. On resonance, the capacitance between the nanotube and the gate
electrode is modulated. Current though the nanotube I
NT
is monitored to detect mechanical motion of the carbon
When an AC signal at frequency ω with a DC offset is applied to the gate electrode
V

V

WX
V

HX
cos-% V

WX
V
Y

Z
,
and the total force on the nanotube is
'
ST

1
2
U
F
V

WX
[V

WX
2V
Y

Z
\
Here, we have neglected the term proportional to V
Y

Z

, since V

WX
≫ V
above, one can separate the DC and AC components;
'
ST
WX

]

U′V

WX

,
'
ST
HX
U′V

WX
V
Y

Z
.
is used to control the tension on the nanotube, and the AC component is used to
excite the mechanical vibrations. As the modulation frequency gets closer to the resonance
frequencies (calculated in Section 2.2) the amplitude of the vibrations become large.
As explained in Section 1.3, small gap quasi–metallic nanotubes, and wide gap
semiconducting nanotubes can be used to build field effect transistors similar to the structure
. We will use the transistor properties of carbon nanotube
nanoelectromechanical resonators. For both types of nanotubes, conductance
26

Schematic of the electrical actuation and detection method. DC and AC voltages are applied to the gate
electrode to excite the mechanical vibration. On resonance, the capacitance between the nanotube and the gate
is monitored to detect mechanical motion of the carbon
with a DC offset is applied to the gate electrode,
(2.14)
(2.15)
V

HX
. From the equation
(2.16)
(2.17)
is used to control the tension on the nanotube, and the AC component is used to
excite the mechanical vibrations. As the modulation frequency gets closer to the resonance
frequencies (calculated in Section 2.2) the amplitude of the vibrations become large.
metallic nanotubes, and wide gap
semiconducting nanotubes can be used to build field effect transistors similar to the structure
carbon nanotubes to detect the
resonators. For both types of nanotubes, conductance
27

depends on the charge induced on the nanotube, as shown in Figures 1.6a and 1.6b [105, 124, 78].
Changes in the charge on the nanotube will change the conductance of the nanotube ^
Y

_`
_a
b c.
Since q = CV
g
, this modulation can be achieved by modulating either the capacitance or the
voltage applied to the gate electrode
dbUdV

V

dU, (2.18)
where the modulation term is neglected again. The first term arises from the regular field effect
transistor property of the nanotube. The second term reflects the mechanical motion of the
nanotube. The capacitance C is modulated as the distance between the gate electrode and the
nanotube changes. When the excitation frequency approaches the mechanical resonance
frequency, induced charge δq becomes large and the amplitude of the mechanical motion peaks.
We have established a method for observing the mechanical motion electrically through
the modulation of the nanotube conductance. However, in practice, this method has some
difficulties associated with it. These devices are expected to operate at the range from a few MHz
to several GHz [99, 28, 102, 26]. First, one has to use a high frequency lock-in amplifier or
network analyzer to measure the change in nanotube conductance. This has problems, since these
instruments usually have a 50 Ω input impedance, which causes a large impedance mismatch with
the 100 kΩ resistance of a common carbon nanotube device. Second, these devices have large
stray capacitances due to the bonding pads. A simple calculation shows that the carbon nanotube
circuit's cut-off frequency is in the order of 100 kHz. This means any signal coming from the
carbon nanotube would be attenuated and, therefore, difficult to measure. This problem can be
circumvented by implementing a smart design of the circuit of the carbon nanotube device. In
particular, down-mixing the high frequency signals below the cut-off frequency enables us to
measure the conductance modulation [39]. This technique has been used with
28

microelectromechanical resonators [61, 11], and can be easily adapted to carbon nanotube
electromechanical resonators. Specific down-mixing methods are discussed in Chapter 3.
The conductance of the nanotube is
^^
WX
^
HX
cos -%, (2.19)
where G
AC
is the modulation of the conductance of the nanotube, which is proportional to δq.
Keep in mind that δq has contributions from the transistor property of the nanotube and the
capacitance modulation due to its mechanical motion. Applying an AC source-drain voltage V
sd

to the nanotube with frequency ω + ∆ω
V
ef
=V
ef
HX
cos[(ω+∆ω)t\ (2.20)
will result in a current through the nanotube given by
=
hi
=^V
ef
=^
WX
V
ef
HX
cos[(ω+∆ω)t\+
1
2
^
HX
V
ef
HX
cos(2ωt)+
1
2
^
HX
V
ef
HX
cos (∆-%).
(2.21)
In Equation (2.21, last two terms are proportional to G
AC
, which is modulated due to the
mechanical motion of the carbon nanotube. These terms will change significantly when the
mechanical resonance of the resonator is reached. However, the 2ω term will be significantly
attenuated due to the circuit cut-off. If the local oscillator frequency ∆ω is chosen below the cut-
off frequency of the device, this scheme can be used to detect the mechanical motion of the
nanotube. By varying the driving frequency ω, the nanotube can be tuned through its mechanical
resonance. The current through the nanotube at frequency ∆ω can then be written as [98]
= (∆-)=
1
2
^
b
[U
F
#(-)V

WX
+UV

HX
\V
ef
WX
. (2.22)
By measuring the current at ∆ω, the mechanical resonance can be observed. Figure 2.6 shows
some sample data exhibiting resonance behavior.
29

10 15 20 25 30
-10
-5
0
5
10
15

Mixing Current (pA)
Frequency (MHz)
60
75
90
105
120
135
150
Phase (
o
)

Figure 2.6 - Current measured at ∆ω plotted as a function of the driving frequency ω near the mechanical resonance.
The upper points (black circles) show the amplitude of the mixing current, and the lower points (red crosses) show the
phase of the mixing current plotted as a function of the drive frequency.

2.4 Tuning The Resonance Frequency
As described in the previous section, a DC voltage applied to the gate electrode creates an
electrostatic force on the nanotube, shown in Equation (2.16, pulling it downward. This
downward pull creates tension T in the nanotube, and tunes the mechanical frequency given by
the Equation (2.12.
-
L

O
?
K
.
Figure 2.7 shows the upshift of the mechanical resonance frequency due to the increased tension
as the gate voltage is increased. In order to understand the effect of gate voltage on the
mechanical resonance frequency nanotube, however, a more complete parameter sweep is
30

required. Figure 2.8 shows several color plots of the mixing current amplitude plotted with the
drive frequency on the y-axis and DC gate voltage on the x-axis.
10 15 20 25 30
V
g
= -4V
V
g
= -4.5V
V
g
= -5.5V
V
g
= -5V

Mixing Current Amplitude (a.u.)
Frequency (MHz)
V
g
= -6V

Figure 2.7 - Effect of gate voltage on the mechanical resonance frequency. Waterfall plot of the mixing current
amplitude taken at different gate voltages, showing the V
g
dependence of the position of the resonance signal.

The mixing current shown in Figure 2.7 has two sources: one from the transistor property
of the nanotube and the second from the mechanical motion of the nanotube. Since these two
signals are superimposed, it is important to subtract the background due to the transconductance
of the carbon nanotube field effect transistor in order to identify the small mechanical component
of the signal. This background can be collected from a carbon nanotube resonator whose
mechanical vibrations are damped due to air friction or the substrate. As mentioned in Section
2.2, air damps the vibrations of the nanotube so the detected current at the mixing frequency is
only the background. Similarly, on substrate nanotubes cannot vibrate due to their strong
substrate interaction. The background signal is collected the same way the mechanical resonances

of the nanotube is identified. The drive frequency
resonance frequency at different gate voltages.
Figure 2.8 - (a) Raw data show a clear parabolic relation between the mechanical resonance frequency and the gate
voltage under vacuum. When the chamber is filled with gas at
disappear. (c) - (f) Examples of mixing sig
Sazonova [98]. Detected mixing current amplitude (in colorscale) is plotted with the driving frequency on the y
and the DC gate voltage on the x

Figure 2.8a show the raw data from a similar sweep and
collected from the same device at
(a)
(c)
(e)
of the nanotube is identified. The drive frequency ω is tuned in a range that covers the mechanical
resonance frequency at different gate voltages.
(a) Raw data show a clear parabolic relation between the mechanical resonance frequency and the gate
voltage under vacuum. When the chamber is filled with gas at P = 760 Torr, (b) mechanical resonance features
(f) Examples of mixing signal plotted as a function of ω and V
g
for 4 different devices. Adapted from
Sazonova [98]. Detected mixing current amplitude (in colorscale) is plotted with the driving frequency on the y
and the DC gate voltage on the x-axis.
a show the raw data from a similar sweep and Figure 2.8b shows the background
m the same device at P = 760 Torr. It is clear that the parabolic relation between the
(b)
(d)
(f)
31
is tuned in a range that covers the mechanical

(a) Raw data show a clear parabolic relation between the mechanical resonance frequency and the gate
= 760 Torr, (b) mechanical resonance features
for 4 different devices. Adapted from
Sazonova [98]. Detected mixing current amplitude (in colorscale) is plotted with the driving frequency on the y-axis
b shows the background
= 760 Torr. It is clear that the parabolic relation between the

gate voltage and the mechanical resonance frequency disappears. However, broad background
features in the spectra remain unchanged.
Now that we have a method to actuate
tuning the resonance frequencies, we will now develop a model to fit our
resonator data. Figure 2
device used in this study.
Figure 2.9 - Scanning electron microscope image of (a) a suspended nanotube device with source, drain, and gate
electrodes. The approximate location of the carbon nanotube (CNT) is shown with a black line. (b) The close up image
of a carbon nanotube crossing the tre

In Section 2.2, we showed that a resonating beam has two regimes of operation: a
bending regime and a tension regime. However,
is a third regime where the nanotube is stretching with increased tension called the "elastic
regime". This regime can only be achieved under very high tensions on the nanotube, which
cannot be generated with our device geometry.
In the bending regime, the nanotube acts like a stiff beam and its resonance frequency is
given by Equation (2.10

(a)
gate voltage and the mechanical resonance frequency disappears. However, broad background
features in the spectra remain unchanged.
Now that we have a method to actuate and detect the mechanical vibrations as well as
tuning the resonance frequencies, we will now develop a model to fit our
2.9 shows the scanning electron microscope image of a sample nanotube
device used in this study.
Scanning electron microscope image of (a) a suspended nanotube device with source, drain, and gate
electrodes. The approximate location of the carbon nanotube (CNT) is shown with a black line. (b) The close up image
of a carbon nanotube crossing the trench. Note that the nanotube is taut.
, we showed that a resonating beam has two regimes of operation: a
bending regime and a tension regime. However, for an elastic beam, like a
is a third regime where the nanotube is stretching with increased tension called the "elastic
regime". This regime can only be achieved under very high tensions on the nanotube, which
with our device geometry.
In the bending regime, the nanotube acts like a stiff beam and its resonance frequency is

-
L
jSLfkL
M
0
L

N

O
=
K
,?≪=

⁄ .
(b)
32
gate voltage and the mechanical resonance frequency disappears. However, broad background
and detect the mechanical vibrations as well as
tuning the resonance frequencies, we will now develop a model to fit our carbon nanotube
image of a sample nanotube

Scanning electron microscope image of (a) a suspended nanotube device with source, drain, and gate
electrodes. The approximate location of the carbon nanotube (CNT) is shown with a black line. (b) The close up image
, we showed that a resonating beam has two regimes of operation: a
for an elastic beam, like a carbon nanotube, there
is a third regime where the nanotube is stretching with increased tension called the "elastic
regime". This regime can only be achieved under very high tensions on the nanotube, which
In the bending regime, the nanotube acts like a stiff beam and its resonance frequency is

33

Here, E is the Young's modulus of the nanotube, I is the moment of inertia of the nanotube, and µ
is the linear mass density of the nanotube. For a typical nanotube, E ≈ 1 TPa, I = π (R
o
4
- R
i
4
)/4 =
3 × 10
-37
kg·m², µ = ρA = ρπ (R
o
2
- R
i
2
) = 5ag/µm, where R
o
and R
i
represent the outer and inner
radius of a nanotube, the mass density of a carbon nanotube is ρ = 1.35 gr/cm
3
, and β
n
= 4.75,
7.85, 11 for n = 0, 1, 2. For the devices presented in this thesis, the average diameter of the
carbon nanotubes is 2 nm and L = 1 - 3 µm.
In the tension limit, the nanotube's tension depends on the applied DC gate voltage. As
described in Section 2.2, the nanotubes' resonant frequencies are modeled like a "string under
tension" or a "guitar string". Figure 2.10a shows the free body diagram of the nanotube in the
tension limit with a DC force '
ST
WX

m
n
U′V

WX

acting on it. Some of our devices exhibit slack s at
the end of the fabrication that is defined by

o
−p

(2.23)
where W is the distance between the clamping points of the carbon nanotube on the electrodes.
Also, we assume the electrical force is concentrated in the center of the nanotube. From Figure
2.10a, one can easily obtain that F
el
= 2Tcos(θ). Using the Equation (2.23 for slack, we can
express the electrical force induced tension in terms of geometrical parameters.

?
'
ST
WX
2cos q
≅
'
ST
WX
√8o

(2.24)
If we change the problem to a uniformly applied force, then only the slack pre-factor changes
[110]. Substituting Equation (2.16 for the electric force in Equation (2.24 gives the relation
between the tension of the nanotube and the applied gate voltage.

?
'
ST
WX
√24o

U′V

WX

√96o
.
(2.25)
Finally, we can write the mechanical resonance frequencies in the tension regime as

Figure 2.10 - (a) Free body diagram
nanotube of length L. The distanc
carbon nanotube is now subject to a larger force, that stretches the
applicability of different regimes in the slack

In the elastic regime, slack is not important since the applied gate voltage starts changing
the length of the nanotube, as il
comparable to the extensional rigidity
nanotube and the length of the nanotube is related to each other by Hooke's Law
(a)
(c)
-
L
SLekwL

O
U′
K√96o
V

WX
.
Free body diagram for the bending regime. The loading force F is applied in the center of the
. The distance between the clamping points is W. (b) Free body diagram
is now subject to a larger force, that stretches the carbon nanotube
applicability of different regimes in the slack-gate voltage space calculated for a typical device
In the elastic regime, slack is not important since the applied gate voltage starts changing
the length of the nanotube, as illustrated in Figure 2.10b. In this regime, the applied force is
comparable to the extensional rigidity F
el
≈ EA of the carbon nanotube. Here, the tens
nanotube and the length of the nanotube is related to each other by Hooke's Law
(b)
Elastic regime
Tension regime
Bending regime
θ
34
(2.26)

is applied in the center of the carbon
Free body diagram for the elastic regime. The
by ∆L. (c) A diagram of the
gate voltage space calculated for a typical device [110].
In the elastic regime, slack is not important since the applied gate voltage starts changing
. In this regime, the applied force is
. Here, the tension on the
nanotube and the length of the nanotube is related to each other by Hooke's Law
θ
35

∆

=
?
@
.
(2.27)
If we repeat the analysis in the tension regime, we get the resonance frequency

-
L
STxeky
=

M
5
6
U′N
m
{
O
(@)
] | ⁄
6K
V

WX
| ⁄

(2.28)
The elastic regime cannot be accessed in a typical device studied in this thesis. None of
the devices studied for this work, showed the 2/3 power dependence of the gate voltage. The gate
voltages needed to get the nanotube into elastic regime are high when s ≠ 0, as seen in Figure
2.10c. Even with devices showing no slack, weak coupling between the gate electrode and the
nanotube limits the electrostatic force, and the nanotube resonator cannot be tuned into elastic
regime.
For a typical device, one can expect the behavior of a carbon nanotube to start in the
bending regime and transition into tension regime, as the gate voltage in increased. To calculate
the values for the resonance frequency where ?≈=

⁄ , we go back to the equation of motion
for a suspended beam given by Equation (2.8.
= #
FFFF
−?#
FF
−E=0
As mentioned in Section 2.2, while calculating the behavior of the resonator in each regime, we
neglected a part of Equation (2.8 in two different limits of tension. To calculate the complete
solution during the transition, we can express each regime with a spring constant that is defined
by k
n
= mω
n
2
. Since the bending and tension regimes are uncoupled, the two springs representing
each can be modeled as parallel springs connected to the same mass m. Therefore, we can add the
two spring constants for each regime, k
n
= k
bending
+ k
tension
, which gives

-
L
=
}
[-
L
jSLfkL
\

+[-
L
SLekwL
\

.
(2.29)
36

Calculated resonances for a 2 µm long nanotube is plotted in Figure 2.11. We will use this model
throughout this thesis to analyze our carbon nanotube resonator devices. There will be minor
modifications for certain analysis, which will be addressed in the context of the discussion in the
following chapters.
-8 -6 -4 -2 0 2 4 6 8
10
20
30
40
50
60
70

Fit
Data
Resonance Frequency (MHz)
Gate Voltage (V)

Figure 2.11 - Calculated (empty circles) and measured resonance frequencies (solid line) of the first two vibrational
modes for a 2.2 µm long nanotube.

2.5 Losses
Loss in a resonator can be specified by the inverse of the quality factor Q, introduced in
Section 2.2. The inverse value Q
-1
is related to the energy loss in the system, and the total loss can
be calculated by adding up Q
-1
of each process. These losses can be categorized as either intrinsic
or extrinsic. Intrinsic losses are due to imperfections in the structure or substrate interaction
including defects, electron-phonon interactions, lattice vibrations, etc. A thorough theoretical and
n = 1
n = 2
37

experimental discussion of these effects can be found in Braginsky [17] and Nowick [83].
Extrinsic losses include air damping (air friction), imperfections at the clamping points, losses in
electrical coupling into the device, etc. In the scope of this thesis, we will present our work on
clamping mechanisms of carbon nanotube resonators and electron-phonon coupling in nanotubes
in Chapters 5 and 6, respectively.
A resonator can lose energy to the support structure by acoustic coupling at the clamping
points. Huang et al. have shown that by changing the clamping geometries of identical nanoscale
beams, the corresponding resonators' Q-factors can be improved by a factor of 2.5 [48]. There are
several approaches to calculating the clamping loss. For a thin, infinitely wide cantilever attached
to an infinite base, Jimbo et al. calculated the loss as ~
]
% ⁄
|
, where t and L are thickness
and length of the cantilever, respectively [56]. Cross et al. calculated ~
]
% ⁄ for a base with
same thickness as the cantilever [31]. Recently, in a more relevant model, Photiadis et al., for a
narrow cantilever attached to a finite thickness base, calculated ~
]
% ⁄

for the
fundamental vibrational mode [88]. Hao et al. derived an analytical form for the loss due to
clamping in Si beam resonators as

~
2.43
(3−)(1+)
+
1.91
Π

1
(0
L

L
)

"

|
,
(2.30)

where L is the beam length, b is the beam width, ν is the Poisson's ratio, β
n
is the mode constant
derived in Section 2.2 and χ
n
is the mode shape for n
th
mode [40].
Equation (2.30 indicates that there are four important design considerations to improving
losses associated with clamping:
38

(i) Dimensional dependency: The support quality factor is proportional to the cubic
power of the ratio of the beam length to the beam width (L/b)
3
, independent of
the beam thickness.
(ii) Material properties: Clamping losses are independent of the Young's modulus E
of the material of a beam resonator, but dependent on the Poisson's ratio ν of the
resonator material.
(iii) Vibrational amplitude: Clamping losses do not depend on the vibration amplitude
of the resonator as long as it is the linear regime.
(iv) Mode order: Losses from clamping increase as the mode order increases.
In our devices, the nanotube is lying on the surface of the platinum electrodes (or silicon
substrate for Sazonova et al. [99]), which is held down by finite forces that are very small. In this
case, the vibration of the nanotube can be damped by imperfect clamping. A more in depth
discussion is given in Sections 5.2 and 5.3.

2.6 Conclusions
In this chapter, we described the physics of a tunable, doubly clamped cantilever
resonator. We applied this general model to a carbon nanotube electromechanical resonator
device. We have showed how electrical actuation and electrical detection using a down-mixing
technique can be used to measure the mechanical motion of such a resonator. Using this
technique we showed that the mechanical resonance frequencies can be tuned, and we developed
a model that can be used to fit our data. Several aspects of the analysis given in this chapter will
be modified slightly to discuss our observations in the following chapters, particularly with
respect to clamping, which was assumed to be perfect in this chapter.

39

CHAPTER 3: DEVICE FABRICATION AND MEASUREMENT
SETUP

3.1 Introduction
There are various ways of making carbon nanotube electromechanical resonator devices.
Electron beam lithography [95, 99] and deep UV photolithography [22, 34] are the most common
methods. The benefits and drawbacks of these methods are detailed in the following sections.
In our studies, we used electrical actuation and electrical detection to analyze carbon
nanotube electromechanical resonators. As discussed in section 2.3, the down-mixing technique is
essential in observing high frequency resonances due to the relatively low cut-off frequency for
the devices in this study. There are three main methods to achieve this: two-signal mixing [99],
amplitude modulation and demodulation [115], and frequency modulation and demodulation [38].
For the temperature dependent studies, we have used a modified vacuum tight chamber
with a proportional–integral–derivative controlled heating and cooling element that can operate in
vacuum. A chip carrier socket is mounted on the heating element for electrical readout. A
pressure gauge and a gas inlet with a flow controller is installed near the sample for partial
pressure measurements (see Section 3.4 for details).

3.2 Device Fabrication
There are two methods to fabricate suspended nanotube devices. The first method,
illustrated in Figure 3.1, starts with carbon nanotube growth and the electrodes are patterned post-
growth. The catalyst, either 1nm thick patterned Fe film [87] or alumina supported FeO
3
/MoO
2

nanoparticles [64], is deposited on Si/SiO
2
substrate. After the deposition of the catalyst,

nanotubes are grown in a quartz tube furnace at 900°C in methane flow or Argon bubbled
through ethanol. Either method yields individual
nm. The electrodes are patterned using electron beam lithography and Cr/Au (5nm/50nm) is
deposited to form the electrodes. Then the nanotubes are suspended by etching the SiO
underneath the carbon nanotube
the metal electrodes are used as etch masks. In method #2, another step of lithography is done to
define the trench area.
Figure 3.1 - Post growth processing using electron beam lithography. Method #1 uses metal electrodes as an etch mask.
Method #2 requires a second lithography step to define the trench.

nanotubes are grown in a quartz tube furnace at 900°C in methane flow or Argon bubbled
through ethanol. Either method yields individual single-walled nanotube
The electrodes are patterned using electron beam lithography and Cr/Au (5nm/50nm) is
deposited to form the electrodes. Then the nanotubes are suspended by etching the SiO
carbon nanotube using buffered oxide etch. In method #1, (shown
the metal electrodes are used as etch masks. In method #2, another step of lithography is done to
Post growth processing using electron beam lithography. Method #1 uses metal electrodes as an etch mask.
Method #2 requires a second lithography step to define the trench.
40
nanotubes are grown in a quartz tube furnace at 900°C in methane flow or Argon bubbled
walled nanotubes with a diameter of 1-4
The electrodes are patterned using electron beam lithography and Cr/Au (5nm/50nm) is
deposited to form the electrodes. Then the nanotubes are suspended by etching the SiO
2
layer
using buffered oxide etch. In method #1, (shown in Figure 3.1)
the metal electrodes are used as etch masks. In method #2, another step of lithography is done to

Post growth processing using electron beam lithography. Method #1 uses metal electrodes as an etch mask.
41

Methods #1 and #2 described above involves post-growth processing where nanotube
gets in contact with photoresist, bombarded with high energy (10 keV) electrons during electron
beam lithography and an etching step. Each of these steps can introduce defects in the nanotube
structure or leave residue on it. For this reason, we have adapted a third method pioneered by the
Dai group in Stanford University [34]. In this method, nanotubes are grown after all the
electrodes are deposited and the trench is etched. Since there are no processing steps after the
nanotube growth, the resulting devices are defect and residue free.
The fabrication starts with a 4" p–type Si wafer with 1µm SiO
2
and 100 nm Si
3
N
4
on both
sides of the wafer. A second generation device uses un-doped Si wafer to minimize stray
capacitances due to bonding pads. The fabrication of the wafer and dicing into 5 × 5 mm chips
are completed at Nanofabrication Facility of University of California, Santa Barbara. A schematic
of the growth steps is shown in Figure 3.2. The fabrication steps are as follows:
1. Photoresist is spun on the entire wafer and a deep UV (190 nm) lithography defines
the trench area over which the nanotube will be suspended. After development, the areas
exposed to UV light is removed, leaving the Si
3
N
4
layer underneath.
2. The Si
3
N
4
layer is etched by reactive ion etching, exposing the SiO
2
layer underneath.
Reactive ion etching (RIE) is an anisotropic etching process so the Si
3
N
4
walls after this
process is perpendicular. Photoresist from step 1 is removed.
3. Buffered oxide etch is used to etch the SiO
2
layer using Si
3
N
4
layer defined in step 2
as a mask. The time of etching is controlled so a partial etching is achieved to define a known
depth of the trench. Buffered oxide etch (BOE) is an isotropic etchant for SiO
2
so the etching
process removes the oxide from under the Si
3
N
4
layer leaving suspended portions. The width
of this overhang is equal to the etched oxide depth. Regular etching depth for SiO
2
is 200-300
nm.
42

RIE
BOE
Cr/Pt
Si
SiO
2

Si
3
N
4
Catalyst
Cr/Pt
Photoresist
Carbon nanotube
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
Figure 3.2 - All the fabrication up to step 7 in completed at Nanofabrication Facility of University of California,
Santa Barbara. Catalyst deposition and nanotube growth is completed in the Cronin Research Laboratory at
University of Southern California.
43

4. A new layer of photoresist is spun on the wafer and another deep UV lithography
step defines the source, drain, and gate electrodes.
5. 5 nm of Cr and 50-100 nm of Pt is deposited covering the entire wafer. Then the
photoresist is removed (from step 4) taking the Cr/Pt stuck to it as well. Cr is used as an
adhesion layer and Pt is the final electrode metal. Pt is used for the electrode metal since it
can withstand 800-900°C where the carbon nanotube growth takes place. The thickness of the
Pt should be adjusted depending on the final growth temperature. We have observed 100 nm
of Pt would withstand 900°C growth temperatures.
6. To integrate the carbon nanotubes on these structure we need to define a catalyst
region which would maximize our chances of getting a working device. For this reason a new
layer of photoresist is spun and the catalyst islands are defined on the source electrodes.
7. The device fabrication is finished at this point. There are a total of 225 chips with 28
devices on each chip. The 4" wafer is diced in to 5 × 5 mm chips and shipped to our
laboratory. The following steps are done at Cronin Research Group at University of Southern
California. Optical images of these chips and an individual device are shown in Figure 3.3a
and 3.3b, respectively.
8. We follow a modified recipe used by Kong et al. to grow our carbon nanotubes [64].
An aqueous solution of alumina supported Fe(NO
3
)
3
and Mo nanoparticles is deposited in the
catalyst windows completed in step 7. After the catalyst dries, photoresist from step 6 is
removed leaving small areas of catalyst on the electrodes.
9. Nanotubes are grown in a 1" quartz furnace at 825-840°C using ethanol as the carbon
feedstock. The details of the growth process can be found in Bushmaker et al. [23]
Sometimes, a carbon nanotube would grow from these predefined catalyst islands crossing
the trench without touching the gate electrode, making a suspended carbon nanotube

resonator. Scanning electron microscope
and 3.4d.
Figure 3.3 - (a) Optical image of a 5 × 5 mm chip with 28 individual pairs of source and drain electrodes with a shared
gate electrode on the bottom of the trench. The dotted square is the area of the (b) close up optical image of the devices.
(c) Scanning electron microscope image of a device with two individual nanotubes crossing the trench. (d) A Y
junction formed by a bundle of carbon nanotubes. Arrows are placed to guide the eye at the ends of the nanotubes.

10. An optional baking step is performed post
to remove the amorphous carbon deposited on the chip and the nanotube surface. Due to their
stable and crystalline structure,
(a)
(c)
Source
Catalyst
canning electron microscope images of sample devices can be seen in
(a) Optical image of a 5 × 5 mm chip with 28 individual pairs of source and drain electrodes with a shared
gate electrode on the bottom of the trench. The dotted square is the area of the (b) close up optical image of the devices.
microscope image of a device with two individual nanotubes crossing the trench. (d) A Y
junction formed by a bundle of carbon nanotubes. Arrows are placed to guide the eye at the ends of the nanotubes.
An optional baking step is performed post-growth in O
2
environment at 200
to remove the amorphous carbon deposited on the chip and the nanotube surface. Due to their
stable and crystalline structure, carbon nanotubes do not oxidize in this medium.
(d)
(b)
Drain
Gate
44
images of sample devices can be seen in Figure 3.4c

(a) Optical image of a 5 × 5 mm chip with 28 individual pairs of source and drain electrodes with a shared
gate electrode on the bottom of the trench. The dotted square is the area of the (b) close up optical image of the devices.
microscope image of a device with two individual nanotubes crossing the trench. (d) A Y-
junction formed by a bundle of carbon nanotubes. Arrows are placed to guide the eye at the ends of the nanotubes.
environment at 200 - 300°C
to remove the amorphous carbon deposited on the chip and the nanotube surface. Due to their
s do not oxidize in this medium.
45

The resulting devices are tested electrically one by one. If there is a nanotube crossing the
trench, a small DC source-drain (V
sd
) bias sweep will show a resistor behavior shown in Figure
3.4. Then a conductance measurement as a function of gate voltage tests if the device is a
semiconducting nanotube or a quasi-metallic one (see Figures 1.6a and 1.6b). Some devices get
gate shorts by a nanotube or by a large amorphous carbon chunk. Nanotube shorts are due to
carbon nanotubes that fail to cross the trench and fall on to the gate electrode creating an
electrical path between source and the gate electrode. There may be multiple individual nanotubes
or carbon nanotube bundles crossing the trench as well, as shown in Figure 3.3c and 3.3d.
-100 -50 0 50 100
-4
-2
0
2
4

Current (μA)
Bias Voltage (mV)

Figure 3.4 - Low bias sweep of a nanotube device. In this regime, the nanotube acts like a linear resistor. Each line
corresponds to a different gate voltage. If there is a gate short, at V
sd
= 0 V, the I–V
sd
lines would not meet at I
sd
= 0 A.

After the screening process, devices with suspended carbon nanotubes are wire bonded to
chip carriers and placed in the vacuum chamber (described in Section 3.4). A high DC current (3-
15 µA depending on the width of the trench) is applied across source-drain electrodes to remove
the adsorbed molecules from the surface of the carbon nanotube. It is important to note that
46

carbon nanotube devices are not imaged with scanning electron microscope before measurements
to avoid possible damage to carbon nanotubes [98].

3.3 Actuation and Detection Methods
Electrical actuation and detection of carbon nanotube electromechanical resonators is
established in Section 2.3. Low cut-off frequency of the circuit fabricated as described in Section
3.2, makes it impossible to observe the mechanical resonances directly. Therefore, a down–
mixing technique has been described for detecting mechanical resonances of the nanotube. We
used three different down–mixing methods in this thesis: two–signal mixing, amplitude
modulation (AM) [115] and frequency modulation (FM) [38]. The frequency spectrum of each of
these techniques is shown in Figure 3.5. Another recent work showed a different approach which
utilizes a nearby antenna to actuate the mechanical motion on the nanotube [49].

Figure 3.5 - Frequency spectrum of the different detection techniques for (a) a sample two
field effect transistor. The detection techniques used are: (b) Two signal mixing (TS), (c) amplitude modulation (AM),
(d) frequency modulation (FM).
blue at the gate terminals. Broken lines represent the lock
have a DC component not shown here.

3.3.1 Two-Signal Mixing
The circuit diagram for two signal mixing is given in
by Sazonova et al. for the first
technique, the nanotube mixes two high
applied to the gate and the source terminal of the device
at the drain terminal: ω, 2
∆ω/2π = 5-10 kHz is applied to the source electrode
signal. The resulting signal is sent to a lock
(a)
(b)
Frequency spectrum of the different detection techniques for (a) a sample two
field effect transistor. The detection techniques used are: (b) Two signal mixing (TS), (c) amplitude modulation (AM),
(d) frequency modulation (FM). Red lines are the frequency spectrum of the signal at the drain, green at the source and
blue at the gate terminals. Broken lines represent the lock-in amplifiers pass band. In all configurations, gate terminals
have a DC component not shown here.
nal Mixing
The circuit diagram for two signal mixing is given in Figure 3.6. This
by Sazonova et al. for the first carbon nanotube electromechanical resonator device [99]. In this
technique, the nanotube mixes two high–frequency signals (with frequencies
applied to the gate and the source terminal of the device, resulting in three frequency components
, 2ω + ∆ω, ∆ω. Oscillator voltages V
sd
AC
and V
g
AC
is applied to the source electrode as a frequency offset from the gate voltage
. The resulting signal is sent to a lock-in amplifier with its reference f
(a)
(b) (d)
(c)
TS
47

Frequency spectrum of the different detection techniques for (a) a sample two-terminal carbon nanotube
field effect transistor. The detection techniques used are: (b) Two signal mixing (TS), (c) amplitude modulation (AM),
Red lines are the frequency spectrum of the signal at the drain, green at the source and
in amplifiers pass band. In all configurations, gate terminals
. This technique was used
resonator device [99]. In this
frequency signals (with frequencies ω and ω + ∆ω)
resulting in three frequency components
AC
are set to 10-20 mV and
a frequency offset from the gate voltage
in amplifier with its reference frequency set at ∆ω.
AM
FM

Figure
The detected signal is in the form of

=
hi
^
WX
V
ef
HX
cos
where the last two terms proportional to
carbon nanotube. The 2ω
in filter and by the stray capacitances between the electrodes and the substrate. The resulting
current that is recorded will be

Even though it is simple to analyze this circuit, there is a substantial background arising
from the transistor property of the nanotube. Besides the large background, there are
synchronization issues between components as well as the noise from various amplifiers and
circuit elements. Also this method requires two signal sources that could be costly.
The DC gate voltage is controlled by a National Instruments data acquisition card
(NIDaq). It can supply ±10 V maximum at its terminals. The high frequency signal sources are
Agilent E4432B. The locking amplifier is a Stanford Research Systems SR830 DSP lock
V
sd
AC
(ω + ∆ω
V
g
AC
(ω
Figure 3.6 - Circuit diagram for two signal mixing technique.
The detected signal is in the form of
cos-∆-%
1
2
^
HX
V
ef
HX
cos2-%
1
2
^
HX
V
ef
where the last two terms proportional to G
AC
are related to the mechanical resonance of the
ω component will be filtered by the low pass filter at the input of the lock
in filter and by the stray capacitances between the electrodes and the substrate. The resulting
current that is recorded will be
=∆-
1
2
^
b
[U
F
#-V

WX
UV

HX
\V
ef
WX
.
Even though it is simple to analyze this circuit, there is a substantial background arising
from the transistor property of the nanotube. Besides the large background, there are
issues between components as well as the noise from various amplifiers and
circuit elements. Also this method requires two signal sources that could be costly.
The DC gate voltage is controlled by a National Instruments data acquisition card
an supply ±10 V maximum at its terminals. The high frequency signal sources are
Agilent E4432B. The locking amplifier is a Stanford Research Systems SR830 DSP lock
ω)
ω)
V
sd
DC

V
g
DC

48

Circuit diagram for two signal mixing technique.
V
ef
HX
cos∆-%
(3.1)
are related to the mechanical resonance of the
component will be filtered by the low pass filter at the input of the lock-
in filter and by the stray capacitances between the electrodes and the substrate. The resulting
(3.2)
Even though it is simple to analyze this circuit, there is a substantial background arising
from the transistor property of the nanotube. Besides the large background, there are
issues between components as well as the noise from various amplifiers and
circuit elements. Also this method requires two signal sources that could be costly.
The DC gate voltage is controlled by a National Instruments data acquisition card
an supply ±10 V maximum at its terminals. The high frequency signal sources are
Agilent E4432B. The locking amplifier is a Stanford Research Systems SR830 DSP lock-in
49

amplifier. DC source voltage is supplied and measured by a Keithley Model 2400 Source-
Measurement Unit. All these instruments are controlled and synchronized by a custom Labview
code given in Appendix. This technique is used for the work in chapter 6.

3.3.2 Amplitude Modulation
This method is also used by Sazonova [98] and later adapted by Wang et al. [115] to
measure the adsorption kinetics of Argon on the nanotube surface. The circuit schematic is given
in Figure 3.7. The input signal has the form
V
ef
V
ef
HX
1+cos (∆-%))cos(-%)+V
ef
WX
(3.3)
where m denotes the modulation depth and is set to 99% for the highest signal. Carrier amplitude
V
ef
HX
is set to 10-20 mV. The modulation frequency ∆ω is set to 500 Hz-1 kHz. This modulation
scheme is better in terms of detection sensitivity and ease of set up. However, there are a few
drawbacks. Since the amplitude envelope of the driving signal is modulated, the force acting on
the nanotube changes both with respect to ω and ∆ω. This causes a large background and makes
the mechanical oscillation undetectable if contact resistances for the carbon nanotube are high.

Figure 3.7 - Amplitude modulation technique. From reference [98].

50

3.3.3 Frequency Modulation
A frequency modulation technique to detect the mechanical resonance of carbon
nanotube electromechanical resonators has been developed by Gouttenoire et al. for a
"nanoradio" capable of demodulating AM and FM radio signals [38]. The circuit schematic for
this method is given in Figure 3.8.

Figure 3.8 - Circuit diagram for the FM technique for nanotube mechanical resonance detection.
The input of the nanotube is connected to a FM source and a DC source through a 'Bias-
Tee' that combines the DC and AC signal. The signal source is terminated with a 50 Ω resistor for
maximum power transfer and to avoid back reflections from the high impedance of the carbon
nanotube contacts. The input signal has the form
V

V
y
∗
cos[%\, where % -
y
%-
∆
-
B
⁄ sin-
B
%, (3.4)
where V
c
is the applied voltage, ω
c
is the carrier frequency, ω
∆
is the frequency deviation, and
ω
LO
is the modulation frequency. The electromechanical mixing current is given by
=

=
AV
ef
V
y
cos[-
k
∆%%\dA%∆%, (3.5)
V

FM

V
sd
DC

V
g
DC

ω
LO

Bias-Tee
51

where δx is the displacement of the nanotube due to mechanical motion. If we investigate the
demodulated signal from the nanotubes' mechanical motion, we can see that there is no
background associated with the transconductance of the nanotube, unlike the two signal mixing
method and the AM technique. We can write the frequency response of the displacement δx*(ω
i
)
as
dA∗(-)=−U′
V

WX
2
S
V
y
-
(

−-
k

+
Z

Z

, (3.6)
where ω
0
is the resonance frequency, m
eff
is the effective mass, and V
g
DC
is the DC gate voltage
that tunes the resonance frequency. The resulting signal is proportional to Equation (3.7) and it
can be used to fit the observed data shown in Figure 3.9. Here the Q–factor is the zero crossings
of the amplitude.

'(-
y
)=
2-
y
-
y

−-
(

−
Z

n

-
y

−-
(

+
Z

n

(-
y

−-
(

)

+
Z

Z

(3.7)
This method is used to detect the resonance frequency for the work discussed in Chapters
4 and 5. For the detailed analysis of FM detection technique refer to Gouttenoire et al. [38] From
our observations, the FM method brings a significant improvement to the signal to noise ratio of
the measurement of the mechanical resonance.
52

37 38 39 40 41 42 43 44

Mixing Current Amplitude (pA)
Frequency (MHz)
37 38 39 40 41 42 43 44

Phase (
o
)

Frequency (MHz)

Figure 3.9 - (a) Amplitude and (b) phase of the mixing current detected using FM technique. Amplitude of the carrier
signal at the source electrode is 20 mV, deviation frequency ω
∆
= 100 kHz, and the modulation frequency ω
LO
= 654
Hz. The gate voltage is swept between 6 V and 7 V with 0.25 V intervals.

3.3.4 Nearby Antenna
This method is first introduced by Hüttel et al. in 2009 [49]. Figure 3.10a shows the
schematic of the device. They have placed an antenna a few mm away from the nanotube and
used the electromagnetic radiation from the antenna to induce the mechanical motion. The
mechanical resonance is detected by the changes in the DC current through the nanotube, as
shown in Figure 3.10b. The tuning mechanism is the same as described earlier and is controlled
through a DC gate voltage applied to the doped Si substrate. They almost achieved the theoretical
limit with Q ≈ 140,000 for such carbon nanotube resonators at 20mK, shown in Figure 3.10c.
Using this method, Steele et al. measured a single electron hopping on to the nanotube through
the mechanical motion of the carbon nanotube [102].
(a) (b)
53

Figure 3.10 - (a) Schematic drawing of the chip geometry, antenna, and measurement electronics. (b) When the
frequency ν of a signal on the antenna is swept with fixed V
g
and V
sd
, a resonant peak emerges in I(ν). An example of
such a resonance is shown for a driving power of −17.8 dBm at a temperature of 20 mK. (c) Zoom of the resonance of
(b) at low power (-64.5 dBm). The red line is a fit of a squared damped driven harmonic oscillator response to the
resonance peak. For both (b) and (c) V
g
= −5.16 V and V
sd
= 0.35 mV. From reference [49].

Unlike other detection methods described earlier, this method achieves 2 orders of
magnitude higher Q-factors. This is achieved with a contact free actuation method, which
eliminates the losses arising from the circuitry. The signal that drives the mechanical resonance is
emitted from a far field antenna. The detection is done by measuring the DC current through the
(a)
(b)
(c)
54

nanotube. When the nanotube is in resonance the measured current increases. By increasing the
averaging time at each frequency point, signal to noise ratio can be greatly improved.

3.4 Low Temperature and Pressure Setup
Measurements are performed inside two different chambers: The first one (chamber #1)
is a Cryo Industries variable temperature probe station. The chamber is pumped by a turbo-
molecular pump backed by a rough pump to pressures P
min
< 10
-7
Torr. The temperature of the
thermal element can be controlled precisely by a proportional–integral–derivative controller
between T
min
= 4.8 K using liquid He (or T
min
= 77.4K with liquid N
2
) and T
max
= 475 K. Second
chamber (chamber #2) is a custom built vacuum chamber with electrical and gas connections.
This chamber also has a variable temperature stage (Linkam TMS600) that can reach P
min
≈ 10
-3

Torr. This chamber can only be cooled down using liquid N
2
to an ultimate temperature T
min
=
77.1 K but can be heated up to T
max
= 600 K. Pictures of these setups can be seen in Figure 3.11
and Figure 3.12, respectively.
55

Figure 3.11 - Image of Cryo Industries liquid He and liquid N
2
cryostat with 28 pin electrical feed through (chamber
#1). This chamber can operate between 4K - 475K with liquid He (or between 77K - 475K with liquid N
2
). This
chamber is modified to include gas lines to achieve partial gas pressures between 10
-4
Torr to 760 Torr.
56

Figure 3.12 - Image of custom vacuum (chamber #2). Left side of the stage has 12-pin hermetically sealed electrical
feed through. The chamber can be pumped down to 10 mTorr with a 1/4" Swagelok connection from the bottom port.
The top port can be used to send gas in, to create partial pressures of any gas with a mass flow controller. Right side has
the connection for a liquid N
2
cooled stage, which works in the range 77 K - 800 K.

The first chamber is fitted with a custom lid to allow partial pressure measurements to
study gas adsorption effects on the nanotube resonator. Gas is introduced into the chamber
through a MKS mass flow controller with 0.05 sccm resolution. The partial pressures are
achieved by balancing the pumping rate of the pump with the amount of gas let into the chamber
through the mass flow controller, imaged in Figure 3.13.
57

Figure 3.13 - Computer controlled setup with the electronics and the gas line. Top shelf carriers the electronics: Agilent
4432B Signal Generator, Keithley 2400 Source-Measure Unit, SR830 DSP Lock-in Amplifier and the temperature
controller. Breakout box in the bottom is used to interface the wires from the chamber with coaxial cables. It also has
grounding connections to protect against electrostatic discharge during loading and unloading the sample.

3.5 Conclusions
In this chapter, we have described how our samples are fabricated. It is important that
these devices have no defects or residue as the mechanical resonance frequency is sensitive to
such changes. To measure the carbon nanotube properties to a certain accuracy using
nanoelectromechanical resonators, one needs to take extreme caution. We described a capacitive
coupling method to actuate carbon nanotube electromechanical resonators then discussed three
different measurement schemes to detect these mechanical resonances. We should note that all
three techniques described above are using a mixing technique. These measurement schemes are
limiting the Q-factor of the observed signal as the stray capacitances in the fabricated devices are
attenuating the signal.
58

CHAPTER 4: THERMAL EXPANSION OF CARBON
NANOTUBES

4.1 Introduction
Solids that shrink with increasing temperatures (i.e., negative thermal expansion) have
long been a focus of research. The fundamental interest lies in the electronic, magnetic, and
geometrical rarities that are present in their properties. Practically, such systems can be used to
make composites with (almost) zero thermal expansion, especially for optical and electronic
applications. Thermal contraction over a broad temperature range had been observed in selenium
and linear polymer chains such as polyethylene, and polyacetlyne [81, 66, 85]. In inorganic
composites, such as BiNiO
3
and Sm
2.75
C
60
, giant negative thermal expansion has been observed
on the order of -100ppm·K
-1
[6, 9].
Over the last two decades, a new class of solids, namely carbon allotropes, sparked
interest due to their potential use as negative thermal expansion materials. Graphene, buckyballs,
and carbon nanotubes have unique electrical, structural and thermal properties, which indicate a
potential for negative thermal expansion. These materials have already been integrated into
commercial products such as transparent conductors for tactile screens, and gas sensors. Although
still in the research phase, new carbon nanotube applications like ultra-sensitive mass sensors [25]
and microcoolers [114] also require a precise measurement of thermal expansion of carbon
nanotubes.
In this chapter, we will discuss the theoretical approach in determining the coefficient of
thermal expansion of these materials and attempts at measuring coefficient of thermal expansion
59

experimentally. Then, we will present our results on the measurement of the coefficient of
thermal expansion of individual single-walled nanotubes.

4.2 Thermal Expansion of Carbon Allotropes
Before tackling the thermal expansion behavior of carbon allotropes, we will briefly
explain the origin of thermal expansion in solids. We will follow the review by Takenaka on
negative thermal expansion materials for explaining thermal expansion in solids [103]. A
textbook on solid state physics [7, 60] shows that description of many properties of solids can be
derived using two assumptions: displacement of ions (in a solid) from their equilibrium position
is small and these oscillations can be approximated by the first term in the expansion of their
interaction energy (harmonic approximation). However, to understand the origin of thermal
expansion we need to include the anharmonicity of lattice vibrations.
Anharmonicity means that the springs that tie atoms together in a lattice model do not
obey Hooke's law. The atoms are prevented from getting too close to each other due to
electrostatic repulsion of their ion cores. As a result, the atoms prefer to diverge from each other
as the total energy of the system is increased, illustrated in Figure 4.1.
60

Figure 4.1 - Potential energy (solid line) for a typical chemical bond between two atoms as function of interatomic
distance ∆r. The dashed line represents the average interatomic distance. Illustration adapted from Takenaka [103].

The coefficient of linear thermal expansion α is expressed as

U
WSjS
3EV
,
(4.1)
where γ is the Grüneisen's parameter, C
Debye
is the Debye specific heat, K is the bulk modulus, and
V is the volume of the material. For isotropic materials, volumetric expansion coefficient β =
(1/V
0
)(dV/dT) is related to α with β = 3α. Grüneisen's parameter represents how the Debye
temperature θ, which dominates the lattice vibrations, depends on volume:

=−
lnq
lnV
.
(4.2)
The elastic properties can be expressed in terms of the shear modulus G, Young's modulus E, and
the Poisson ratio ν. For isotropic materials, K = E/[3(1 - 2ν)] and G = E/[2(1 + ν)] and for
inorganic materials ν = 0.2-0.3. Usually, γ and K have a weak dependence on temperature and α
shows almost an identical temperature dependence to that of C
Debye
.
61

U
WSjS
9

M
?
q
N
|
< A
A

¡
¢
¡
¢
−1)

£/i
(
(4.3)
Consequently, volume V (V
0
: volume at reference temperature T
0
) depends on temperature T as
V(?)=V
(
+
9

E
?M
?
q
N
|
< A
A

¡
¢
(¡
¢
−1)

£/i
(
. (4.4)
The equation above gives a theoretical description of the thermal expansion of solids.
However, this description fails to explain shrinkage of solids with increasing temperature. For all
materials, thermal expansion or contraction is a result of a competition between internal energy
and entropy of the material, F = U - TS where F is the free energy, U is the internal energy and S
is the entropy. Low dimensional systems (0-D, 1-D, and 2-D) at moderate temperatures gain
structural and vibrational entropy with increased temperature by exploring the voids in the
configurational space at relatively little energy cost [65]. This can be explained by the flexibility
of the network of the atoms that the solid is composed of. In this case, entropy dominates the total
energy and the solid contracts. As the temperature is increased further, the anharmonicities
described earlier play a more important role and the solid starts expanding as expected.
Carbon allotropes are low dimensional solids made of sp
2
bonded carbon atoms. The
strength of the interatomic bond and the plenty of empty volume surrounding the crystal structure
make them ideal candidates as negative thermal expansion materials. As expected, simulations on
buckyballs (0-D) [65], carbon nanotubes (1-D) [65, 55, 2], and graphene (2-D) [80, 55] show that
they all exhibit negative thermal expansion at low to moderate temperatures. Kwon et al. used a
molecular dynamics technique to calculate the coefficient of thermal expansion of buckyballs up
to 500 K, shown in Figure 4.2a [65]. Their finding confirm the flexible network theory, that
atoms move in the configurational space to minimize their energy instead of changing their bond
length, causing thermally induced contraction. Mounet et al. used density functional theory
technique to calculate the coefficient of thermal expansion of graphene, shown in Figure 4.2b
62

[80]. Jiang et al. studied the effects of substrate interaction on the coefficient of thermal
expansion of graphene, illustrated in Figure 4.2c [55]. The flexibility of the carbon network of
graphene has been shown by Chen et al.; when a piece of suspended graphene undergoes a
heating cycle, ripples form only in one direction due to strong substrate interaction in the other
axis, as depicted in Figure 4.3 [27].
Due their 1-D structure, carbon nanotubes can expand or contract in two dimension:
radial (α
r
) and axial (α
a
). However, there is no agreement on the sign or the maximum value for
either parameter. Kwon et al. estimated α
r
to have a value of -12ppm·K
-1
at around 400K (Figure
4.4a). On the contrary, Jiang et al., using non-equilibrium Green's function simulations, got a
positive value for α
a
for the entire range of temperatures, shown in Figure 4.4b.
In another study, Alamusi et al., using molecular dynamics simulations, calculated the
diameter dependence of the axial and radial coefficient of thermal expansion of carbon nanotubes,
shown in Figure 4.4c [2]. Their results show a minima for α
a
around 300K with a linear
dependence to the diameter. This result is not surprising since the free space that allows
contraction of the carbon nanotubes is larger in larger diameter nanotubes.
There are various measurement techniques for the coefficient of thermal expansion of
carbon allotropes. Aleksandrovskii et al. used a dilatometer to measure the thermal expansion of
C
60
buckyballs at liquid He temperatures [3]. Bailey et al. used an interferometric measurement
setup to obtain the in plane (graphene) and inter layer coefficient of thermal expansion of graphite
[10]. Maniwa et al. used X-ray diffraction technique to measure the radial coefficient of thermal
expansion of single-walled nanotube bundles [76]. Their results, however, do not fit the
calculated values presented earlier. They concluded that inter-tube interactions play an important
role in their measurement scheme, therefore, the measured values differ from the calculated
coefficient of thermal expansion of an individual single-walled nanotube. In Sections 4.3 and 4.4,

we will present our work on measuring the
walled nanotubes.
Figure 4.2 - Calculations of thermal expansion of (a) buckyballs [65], (b) graphene and graphite [80]. (c) Substrate
interaction affects the coefficient of thermal expansion of graphene [55]. Here
between the substrate and graphene
(a)
(b)
(c)
we will present our work on measuring the coefficient of thermal expansion
Calculations of thermal expansion of (a) buckyballs [65], (b) graphene and graphite [80]. (c) Substrate
interaction affects the coefficient of thermal expansion of graphene [55]. Here γ denotes the interaction strength
between the substrate and graphene. Figures are adapted from respective references.
(a)
(b)
)
C
60

Graphene
Graphene
Graphite
63
coefficient of thermal expansion of individual single-

Calculations of thermal expansion of (a) buckyballs [65], (b) graphene and graphite [80]. (c) Substrate
γ denotes the interaction strength
64

Figure 4.3 - Ripples in suspended graphene due to thermal expansion from reference [27].
(a)
(b)

Figure 4.4 - Calculation of the axial coefficient of thermal expansion of carbon nanotubes using (a) molecular dynamics
technique [65] and (b) non-equilibrium Green's functions technique [55]. (c) Coefficient of thermal expansion of
carbon nanotubes depend on its diameter [2]. Figures are adapted from respective references.
(a)
(b)
(c)
Calculation of the axial coefficient of thermal expansion of carbon nanotubes using (a) molecular dynamics
equilibrium Green's functions technique [55]. (c) Coefficient of thermal expansion of
rbon nanotubes depend on its diameter [2]. Figures are adapted from respective references.

65

Calculation of the axial coefficient of thermal expansion of carbon nanotubes using (a) molecular dynamics
equilibrium Green's functions technique [55]. (c) Coefficient of thermal expansion of
rbon nanotubes depend on its diameter [2]. Figures are adapted from respective references.
66

Figure 4.5 - Measured value for graphene in-plane thermal expansion from reference [100].
Thermal contraction of graphene has also been measured by Singh et al. using a
nanoelectromechanical resonator similar to one described in Chapter 3 [100]. Their findings in
Figure 4.5a matched the calculations by Mounet et al. in Figure 4.5b. Also using
nanoelectromechanical resonators, Chaste et al. measured the coefficient of thermal expansion of
individual carbon nanotubes, shown in Figure 4.6 [26]. The comparison of this work with our
study is detailed in Sections 4.3 and 4.4.
(a)
(b)
67

Figure 4.6 - Thermal expansion coefficient of a carbon nanotube measured by Chaste et al. using a NEMS resonator
[26]. (a) Thermal response of the mechanical resonance frequency of a resonator is measured. From the change in
frequency the corresponding (b) change in length is calculated. Finally, (c) coefficient of thermal expansion of the
single-walled nanotube is extracted from the change in length. From reference [26].

4.3 Measuring Thermal Expansion of Individual Carbon Nanotubes
In this section, we report our results on the coefficient of thermal expansion of individual
single-walled carbon nanotubes. We have used a carbon nanotube based nanoelectromechanical
resonator and measured the temperature dependence of the mechanical resonance frequency. The
coefficient of thermal expansion is extracted from the measured data using a model for vibrations
on a string under tension.

Figure 4.7 - (a) Scanning electron microscope image of a carbon nanotube electromechanical resonator device and (b) a
close-up of the nanotube crossing the trench. (c) Schematic diagram of a device with FM mixing circuit to detect
mechanical resonances. The substrat

Carbon nanotube
Section 3.2. The fundamental mechanical resonance frequencies for these devices were observed
in the range 10 - 45 MHz
mechanical resonance frequenci
oscillations are the only source of the detected signal, we were able to measure bo
semiconducting and metallic nanotubes. However, semiconducting nanotubes' low conductance at
V
g
> 0 made it difficult to observe the resonance peaks for positive values of the gate voltage.
Figure 4.5Figure 4.8a shows a mechanical resonance peak detected using the FM mixing
(a)
(c)
(a) Scanning electron microscope image of a carbon nanotube electromechanical resonator device and (b) a
up of the nanotube crossing the trench. (c) Schematic diagram of a device with FM mixing circuit to detect
mechanical resonances. The substrate is oxidized p-type silicon with a 100nm SiN
x
layer on top.
arbon nanotube based nanoelectromechanical resonators are fabricated as described in
Section 3.2. The fundamental mechanical resonance frequencies for these devices were observed
45 MHz. A single source FM modulation technique was used to detect the
mechanical resonance frequencies, as illustrated in Figure 4.7c. Since the nanotubes' mechanical
oscillations are the only source of the detected signal, we were able to measure bo
semiconducting and metallic nanotubes. However, semiconducting nanotubes' low conductance at
> 0 made it difficult to observe the resonance peaks for positive values of the gate voltage.
a shows a mechanical resonance peak detected using the FM mixing
(b)
68

(a) Scanning electron microscope image of a carbon nanotube electromechanical resonator device and (b) a
up of the nanotube crossing the trench. (c) Schematic diagram of a device with FM mixing circuit to detect
layer on top.
resonators are fabricated as described in
Section 3.2. The fundamental mechanical resonance frequencies for these devices were observed
ingle source FM modulation technique was used to detect the
c. Since the nanotubes' mechanical
oscillations are the only source of the detected signal, we were able to measure both
semiconducting and metallic nanotubes. However, semiconducting nanotubes' low conductance at
> 0 made it difficult to observe the resonance peaks for positive values of the gate voltage.
a shows a mechanical resonance peak detected using the FM mixing
69

technique, which matches the results reported by Gouttenoire et al. [38]. We confirmed that the
resonances are indeed mechanical in nature by tuning the resonance frequency with a DC gate
voltage (Figure 4.7b), which changes the tension on the carbon nanotube. We also took
measurements in air (760 Torr) to identify the electrical resonances due to the passive elements in
the external circuit. The resonances observed in gas can only originate from the passive elements
of the circuit because gas molecules dampen the mechanical oscillations of carbon nanotubes at
small driving signals (A
FM
= 10 mV
peak
). Details of this measurement method can be found in
Section 3.3.3.
35.0 35.5 36.0 36.5 37.0
0
10
20
30
40

Mixing Current (pA)
Frequency (MHz)
-4 -2 0 2 4
10
20
30
40
Fit
Data

Resonance Frequency (MHz)
Gate Voltage (V)

Figure 4.8 - (a) Sample resonance peak taken at V
G
= -4.1V and (b) gate dispersion plot of sample 1.

The mechanical resonances of these devices were studied as a function of gate voltage
and temperature. An applied gate voltage changes the tension on the nanotube and tunes its
mechanical resonance frequency. The nanotube’s mechanical resonance frequency can be
modeled as

-
L

}
[-
L
jSLfkL
\

[-
L
SLekwL
\

,
(4.5)
where
(a) (b)
70

-
L
jSLfkL
M
0
L

N

O
=
K
and -
L
SLekwL

O
¥
(
¥
¦§
K
.
(4.6)
The tension Γ, is the sum of Γ
0
is the initial axial tension, and ¥
ST

¨©
ª«
√ e

X′¬

ª«
¬

n
√®¯e
is the
electrostatic tension on the carbon nanotube due to applied gate voltage. Details of this
calculation can be found in Section 2.4 (We have changed the symbol of tension to Γ to avoid
confusion with the temperature variable T.). The nanotube resonator is characterized at room
temperature with a large gate voltage sweep shown in Figure 4.8b. From this large sweep, we
extracted the residual tension Γ
0
on the device, and the gate offset voltage V
0
.
The temperature response of the resonance frequency of the device was measured at a
fixed gate voltage (or very small V
g
ranges) instead of large sweeps. The value of this fixed gate
voltage is chosen where the detected signal at the lock-in amplifier was the highest. Then the
temperature of the device was varied from room temperature down to 4K with liquid He (or to
77K with liquid N
2
) using chamber #1. The resistivity of an on-chip platinum resistive
temperature detector was continuously monitored to obtain the temperature of the nanotube
during the measurements. Figure 4.9a shows the mechanical resonance frequency of sample 1
measured from 60K to 300K. The resonance frequency exhibits a minimum at approximately
200K. Figure 4.9b shows the mechanical resonance frequency of another sample (sample 2),
which follows a similar trend. This sample was a metallic carbon nanotube, while sample 1 was
semiconducting, as determined by electron transport measurements.
71

0 75 150 225 300
36
41
46
51
56
61

Resonance Frequency (MHz)
Temperature (K)
0 75 150 225 300
30
35
40
45

Resonance Frequency (MHz)
Temperature (K)

Figure 4.9 - (a) Resonance frequency of a semiconducting nanotube device (sample 1) and (b) a quasi-metallic
nanotube device (sample 2).

4.4 Extracting Coefficient of Thermal Expansion from Mechanical Resonance
From the observed shift in the mechanical resonance frequency, we can determine the
change in tension in the nanotube. Using Hooke's law, the change in tension of the nanotube, [Γ -
Γ (300K)], can be attributed to a change in the nanotube's length

∆=
(¥−¥(300K))
(
@
,
(4.7)
where A, Γ(300K), and L
0
are the cross-sectional area, tension, and length of the suspended
portion of the nanotube at 300K, respectively. We assume the changes in cross-sectional area and
the Young's modulus of the carbon nanotube to be negligible.
The change in nanotube length, ∆L, calculated above is a combination of substrate
expansion and nanotube contraction. To obtain the nanotube's thermal expansion we need to
subtract the expansion of the substrate from ∆L. The effect of the substrate thermal expansion is
two-fold: the expansion of the underlying Si substrate and the overhanging Si
3
N
4
/Pt electrode.
The linear coefficient of thermal expansion α of Si [75, 84] and Pt [59] are given in Figure 4.10a.
(a) (b)
72

From these coefficient of thermal expansion values, we can calculate the ratio of contraction to
the initial length by ∆ ⁄ =I?, shown in Figure 4.10b.
Before calculating the effective trench width as a function of temperature, we need to
calculate the expansion of the overhangs in Figure 4.7c. A bilayer structure with two different
coefficient of thermal expansion values will bend due to the difference in the expansion amount
of each layer. The bending radius κ for such a structure is calculated using Stoney's formula
D=
6
]

(ℎ
]
+ℎ

)ℎ
]
ℎ

(
]
−

)∆?

]

ℎ
]

+4
]

ℎ
]
|
ℎ

+6
]

ℎ
]

ℎ

+4
]

ℎ

|
ℎ
]
+

ℎ

, (4.8)
where E
i
, h
i
, and α
i
are the Young's modulus, the thickness and the coefficient of thermal
expansion of the i
th
layer, respectively [35]. Using the dimensions of our system, κ is calculated to
be 3 orders of magnitude larger than the width of the overhangs, so we can neglect this bending.
Also the effective contraction of Si
3
N
4
/Pt bilayer is almost equal to a single Pt film with the same
dimensions. This is not surprising since the thickness and the stiffness of the Pt film is larger than
the Si
3
N
4
layer.
Now we can combine the expansion of the overhangs and the substrate to calculate the
effective contraction (or expansion) of the trench width. Using values for the Si and Pt coefficient
of thermal expansions from literature, α
Si
and α
Pt
, together with the geometry illustrated in Figure
4.7c, we can predict the change in trench width, ∆L
trench
, as a function of temperature by:
∆
±SLy²
=(p
(
+2p
w²
) <
³k
?
i
|((
−2p
w²
<
´
?
i
|((
, (4.9)
where, W
0
and W
oh
are the room temperature trench width and width of the overhanging Pt
electrodes, respectively. Figure 4.10a shows the effective linear coefficient of thermal expansion
of the trench as a function of temperature.
73

0 50 100 150 200 250 300
-4
-2
0
2
4
6
8
10
Overhang (Pt)
Substrate (Si)
Trench

Linear Expansion Coefficient (ppm K
-1
)
Temperature (K)
0 50 100 150 200 250 300
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0

ΔL/L (x 10
-3
)
Temperature (K)

Figure 4.10 - (a) Linear coefficient of thermal expansion α
i
for the overhang (Pt), substrate (Si) and the trench. Using
the expansion coefficient, we can calculate (b) the ratios of contraction or expansion with respect to the room
temperature (300K) dimensions.

Finally, the net length change of the carbon nanotube due to thermal expansion, ∆L
CNT
,
can be written as
∆=∆
±SLy²
−∆
Xhi
. (4.10)
After substituting Equation (4.9 in Equation (4.10, we obtain the net change in the length of the
carbon nanotube as a function of temperature. Then using the definition of the coefficient of
thermal expansion,
Xhi
=
]
B

f
fi
∆
Xhi
, we calculate the coefficient of thermal expansion for a
nanotube as

Xhi
=
1

(

?
µ∆
±SLy²
−6
4K
|
@
(¶

(?)−¶

(300E))8·. (4.11)
(b) (a)
74

0 75 150 225 300
-5
-4
-3
-2
-1
0
Axial CNT CTE (x10
-6
K
-1
)
Temperature (K)
0 75 150 225 300
-4
-3
-2
-1
0
Axial CNT CTE (x10
-6
K
-1
)
Temperature (K)

Figure 4.11 - Axial coefficient of thermal expansion of (a) sample 1 and (b) sample 2 calculated based on the
experimental data shown in Figures 4.7a and 4.7b.

Figure 4.11a and Figure 4.11b show the coefficient of thermal expansion calculated using
the equation above, based on data shown in Figure 4.9. Both datasets show a minimum
coefficient of thermal expansion of approximately -4 ppm·K
-1
around 100 K. This calculated
coefficient of thermal expansion value agrees with previous simulation results in the literature.
One important observation is the temperature at which the coefficient of thermal expansion is
minimum, which is considerably lower than theoretically predicted values.
The mismatch between the measured and the calculated results for coefficient of thermal
expansion of individual suspended carbon nanotubes, albeit not big, requires further explanation.
As discussed earlier in Section 4.2, the nanotubes are showing negative thermal expansion due to
their flexible network of carbon atoms. As shown in Kwon et al., there are multiple different
modes an unperturbed carbon nanotube (Figure 4.12a) can obtain that cause negative thermal
expansion in carbon nanotubes: transverse acoustic bending mode, optical pinch mode, and twist
mode and shown in Figure 4.12b, Figure 4.12c, Figure 4.12d, respectively [65]. These modes are
uncoupled to each other and their combined effect gives the total coefficient of thermal expansion
(b) (a)
75

of a single-walled nanotube. Various loss mechanisms like clamping and damping from gas
molecules can couple into these modes and nullify their effects on the coefficient of thermal
expansion. Clamping is especially important since the initial assumption in our work is that the
clamping points are fixed and the nanotube has no displacement at these points. We will discuss
the effects of clamping and its strength in Chapter 5.

Figure 4.12 - Vibrational modes of (a) an unperturbed carbon nanotube that lead to negative thermal expansion at
moderate temperature: (b) transverse acoustic bending mode, (c) optical pinch mode, and (d) twist mode.

4.5 Conclusions
In this chapter we have presented our work on individual single-walled carbon nanotube
thermal expansion without the effects of a substrate. We have shown that a
nanoelectromechanical resonator can be used to measure the effects of temperature on the length
of the nanotube through its mechanical resonances. Our results confirmed that carbon nanotubes
have a negative coefficient of thermal expansion in the axial direction over a broad temperature
range (4 - 475 K) with a minimum value of -4.5×10
-6
K
-1
observed at 100 K.
Findings reported here does not reflect absolute values but provide an upper limit for the
coefficient of thermal expansion in the axial direction of carbon nanotubes. As discussed in the
previous section, the nanotube resonators are analyzed assuming the clamping points are fixed.
However, our measurements show that this assumption is not valid for the devices used in this
work. A more detailed explanation of the variation of the suspended nanotube length as a function
(b) (a)
(d) (c)
76

of temperature is given in Chapter 5. Also during the extraction of the coefficient of thermal
expansion from mechanical resonance data, we assumed the radial contraction to be negligible.
However, the linear mass density µ is strongly dependant on the radius of the nanotube. Our setup
does not allow simultaneous measurements of radius, thus, it is best to leave that number as a
constant to avoid further errors.

77

CHAPTER 5: CLAMPING IN CARBON NANOTUBE
ELECTROMECHANICAL RESONATORS

5.1 Introduction
High quality factors are desirable for practical applications of carbon nanotube
electromechanical resonators. For mass sensing, low Q-factors limit the minimum mass
detectable by the device. Cooling the resonator (down to 4K) and increasing the resonance
frequency (to a few GHz) can circumvent this problem; however, high frequency circuits are
difficult to design, and sustaining cold temperatures is not practical for most applications.
Carbon nanotube electromechanical resonators have been predicted to have Q ≈ 10
5
[54].
However, most state of the art devices are far below that limit with Q ≈ 10
3
[69, 28]. Only
recently, Hüttel et al. presented a new method to induce mechanical vibrations in carbon nanotube
electromechanical resonators with Q ≈ 140,000 at millikelvin temperatures [49]. They have
engineered an antenna that couples into the mechanical motion of the carbon nanotube through a
radiated electric field. Details of this method are in Section 3.3.4. However, this method is not
suitable for most practical applications due to the technical difficulties in designing an antenna
coupled to the carbon nanotube electromechanical resonator and the requirement of ultra-low
temperatures.
In this chapter, we will investigate the clamping efficiency of the nanotubes, which is one
of the key loss mechanisms affecting the Q-factor. We will start with a brief introduction of the
van der Waals interaction between microscopic solids. Then, we will focus on carbon nanotube–
substrate interactions, and report our findings on the binding energy between a single-walled
nanotube and platinum surface.

5.2 Van der Waals Forces
Van der Waals forces are used to define nonspecific weak intermolecular forces. They are
electronic interactions with three different origins: dipole
instantaneous dipoles, also called London dispersion force, in the order o
force. These interactions are the physical origin of
[115], solubility of substances in polar and non
molecular biology [41].
Figure 5.1 - (a) The Lennard Jones potential
potential energy U
bond
, denotes the energy needed for the molecule to escape the surface. The location
R
bond
, shows the bond length between the adsorbed molecule and the surface.

Van der Waals forces can also be used to describe the strength of interaction between two
molecules in contact such as graphene on SiO
framework of physisorption where a molecule is weakly bound to a surface through
(a)
an der Waals Forces
Waals forces are used to define nonspecific weak intermolecular forces. They are
electronic interactions with three different origins: dipole–dipole, dipole–
instantaneous dipoles, also called London dispersion force, in the order o
force. These interactions are the physical origin of physisorption of noble gasses on surfaces
[115], solubility of substances in polar and non–polar media [91], and protein folding in

(a) The Lennard Jones potential curve of (b) a physisorbed molecule M on a surface S. The minimum
, denotes the energy needed for the molecule to escape the surface. The location
, shows the bond length between the adsorbed molecule and the surface.
Van der Waals forces can also be used to describe the strength of interaction between two
molecules in contact such as graphene on SiO
2
[62]. Such interactions can
framework of physisorption where a molecule is weakly bound to a surface through
(b)
78
Waals forces are used to define nonspecific weak intermolecular forces. They are
–induced dipole, and two
instantaneous dipoles, also called London dispersion force, in the order of strongest to weakest
of noble gasses on surfaces
polar media [91], and protein folding in

curve of (b) a physisorbed molecule M on a surface S. The minimum
, denotes the energy needed for the molecule to escape the surface. The location of the minima,
Van der Waals forces can also be used to describe the strength of interaction between two
[62]. Such interactions can be discussed in the
framework of physisorption where a molecule is weakly bound to a surface through van der
79

Waals forces, as illustrated in Figure 5.1b. The potential energy, usually called Lennard Jones (L–
J) potential, as a function of the distance between the molecules is illustrated in Figure 5.1a.
From a solid state physics textbook, the general form of the Lennard Jones potential for
van der Waals interaction can be written as

¸−
@
¸
¯

¹
¸
]
,
(5.1)
where A and B are positive constants, and R is the distance between the adsorbed molecule and
the surface. This potential is usually written in a more appealing form

(¸)=4!

¸

]
−

¸

¯
,
(5.2)
where σ = (B/A)
1/6
, ϵ = A
2
/4B. There is no reason in choosing the exponential in the repulsive term
to be 12, other than the resulting analytic simplicity and the requirement that the number be larger
than 6. This equation successfully describes the pair wise interactions; however, to generalize it to
a molecule on a surface, one needs to include the all of the interactions between the molecule and
each relevant atom on the surface. We calculate the total energy U
bond
by

;
jwLf
=º (¸
k
)
k
.
(5.3)
Here R
i
is the distance of the adsorbed molecule from the i
th
atom on the surface when the
molecule is at its minimum energy point, R
bond
, over the surface.
We have defined the energy of the adsorbed molecule over a surface. With the potential
energy curve shown in Figure 5.1a, one can simply calculate the restoring force Γ
vdW
(R) due to
van der Waals interactions at the minimum energy rest point by differentiating Equation (5.2 with
respect to the distance between the molecules

¥
»f¼
(¸)=
4!
¸
−12

¸

]
+6

¸

¯
.
(5.4)
80

This force is usually calculated numerically by van der Waals density functional theory. This
method is developed to include the effect of long range interactions between molecules in widely
used density functional theory calculation method. It has been used to calculate the adsorption
energies of 1,4-benzenediamine (BDA) on Au(111) surface and azobenzene on Ag(111) surface
with good agreement between the calculated and experimental values [71], as shown in Figure
5.2a and 5.2b, respectively.

Figure 5.2 - Calculated and experimental adsorption energies of (a) BDA on Au and (b) azobenzene on Ag from
reference [71].

5.3 Carbon Nanotube–Substrate Interaction
Carbon nanotubes in polymer matrices as mechanical strength enhancing additives have
been studied extensively [108, 18, 42]. Carbon nanotubes can improve the mechanical behavior
of the composite under small strains. However, such improvements disappear at relatively large
strains because the completely debonded nanotubes behave like voids in the matrix and may even
weaken the composite. Increasing the adhesion at the interface between carbon nanotubes and the
polymer matrix may significantly improve the composite behavior under large strains [104].
Carbon nanotubes are commonly supported by an underlying substrate due to the ease of
fabrication [105, 30]. Altough recent methods allow suspended carbon nanotube devices, as
(b) (a)
81

discussed in Chapter 3, the nanotube-electrode contact remains unchanged. For electrical
measurements, the nature of this interaction is only investigated from an electrical point of view.
The emergence of mechanical systems relying on carbon nanotubes [99] require a different
modeling of the carbon nanotube–substrate interaction.
It has been argued that clamping losses [98] are an important reason for the low Q-factors
observed in doubly clamped resonators at room temperature. The strength of the interaction
between the nanotube and the electrodes at the point of contact (or the first 1 µm section to be
more precise) is crucial in determining the clamping strength in the suspended carbon nanotube
resonators. For substrates with permanent dipoles such as SiO
2
, the interaction energy is very
strong [44]. Such devices [98] or devices with sandwiched clamping points [26] show
significantly improved Q-factors. For devices with underlying source and drain electrodes such as
the ones used in Chapter 3, the nanotube is weakly bound to the surface of the Pt electrode. This
weak bond can be investigated by modeling the carbon nanotube as an adsorbate on the Pt
surface. In fact, simulations using van der Waals density functional theory, studying graphene
adsorption on platinum, show that the adsorption energy is only E
a
= 43 meV and the adsorption
distance is d
a
= 3.25 Å, as shown in Figure 5.3 [111]. Adsorption values for graphene on other
metals can be found in Table 5.1. If we convert the adsorption energy to a critical temperature T
C
,
we get T
C
= 500K using E
a
= k
B
T
C
. By increasing the temperature of the system, it is possible to
break this weak van der Waals bond and make the clamping point very lossy. The lossy contacts
will cause friction between the nanotube and the electrodes, which will convert part of the stored
mechanical energy into heat, thereby lowering Q.
82

Figure 5.3 - Van der Waals–density functional theory (vdw-DF - black diamonds), revised Perdew–Burke–Ernzerhof
(revPBE - green squares), and local–density approximation (LDA - blue triangles) simulations of graphene adsorbed on
Ni at a distance d. It is clear that long range interactions are important in defining the contact between graphene and
metals. Adapted from Vanin et al. [111].

Carbon nanotubes are structurally similar to graphene. However, the curvature of the
carbon nanotube would mean only a few atoms are in close proximity of the metal surface. This
would lower the binding energy, as the average distance of atoms form the metal surface is larger.
So we expect the critical temperature T
C
= E
a
/k
B
for a carbon nanotube to be lower than that of
graphene. In the next section, we will present our findings on the binding energy of carbon
nanotubes on Pt electrodes using our nanoelectromechanical resonator device.

83

Table 5.1 - Binding energies (E
b
) per carbon atom and binding distances (d) of graphene on metal (111) surfaces. Fermi
level shift ∆E
F
and charge transfer δQ at the vdW-DF equilibrium separation. Negative (positive) ∆E
F
indicates n (p)-
type doping. Negative (positive) δQ indicates electron transfer to (from) the graphene layer (from reference [111]).

Co Ni Pd Ag Au Cu Pt Al
vdW-DF d (Å) 3.40 3.50 3.50 3.55 3.57 3.58 3.67 3.72

E
b
(meV) 30 37 39 33 38 38 43 35

∆E
F
(eV) -0.20 0.13 0.65 -0.40 0.21 -0.43 0.66 -0.51

δQ (10
-3
e) -5.0 -3.0 5.0 -5.0 0.4 -4.0 5.0 -8.0
Expt. d (Å) 1.5-2.2 2.1

5

3.3

5.4 Temperature Induced Unclamping in Carbon Nanotube Resonators
Carbon nanotubes in nanoelectromechanical resonator devices are clamped down by van
der Waals forces on both ends. The strength of this force is especially small when the carbon
nanotubes are grown on top of the electrodes, as the ones described in Section 3.2. To study the
clamping mechanism in carbon nanotube resonators, we heated up the device in vacuum. By
heating the system, we provide kinetic energy to the nanotube and the substrate, which leads to
the carbon nanotube escaping the Lennard-Jones potential and delaminating from the surface.
Carbon nanotube electromechanical resonators' fundamental mechanical resonance are
characterized at room temperature at different gate voltages, as shown in Figure 5.4. Note that
higher mechanical modes are also visible. After room temperature measurements, the device is
heated up slowly monitoring the mechanical resonance frequency of the nanotube. The gate
voltage is set where the resonance signal is clearly distinguishable.
At higher temperatures (340-375 K), a discrete and sudden drop in the mechanical
resonance frequency (on the order of 5 MHz) is observed due to weakening of the binding forces
between the nanotube and its supporting electrodes, as shown in Figure 5.5b. Resonance
frequency of the higher order modes (n > 0), where L
CNT
= (n + 1) λ/2, are also downshifted at the
same critical temperature (T
C
) as the fundamental mode (n = 0), as depicted in Figure 5.5a.
84

Figure 5.4 - Mechanical resonance spectrum of a carbon nanotube electromechanical resonator with a 2 µm trench at
room temperature. The lowest three modes of operation (n = 0, 1, 2) are clearly visible. The color indicates the
amplitude of the mixing current measured at the lock-in amplifier.

We observe no change in the Q-factor of the mechanical resonance above and below this
transition, as shown in Figure 5.5d. This instability in the resonant frequency, results from a
'peeling off' of the nanotube from the trench sidewall electrode, bound weakly by van der Waals
forces, as illustrated in Figure 5.5c. This drop in the mechanical resonance frequency is a direct
consequence of a sudden elongation in the suspended length of the nanotube. This extra length
(∆l
th
), as depicted in Figure 5.5c, is a consequence of the nanotubes being grown on top of the
trench and electrode structure. The high temperatures sustained during the growth process causes
the trench width to increase by thermal expansion. As the temperature drops, the substrate

contracts and nanotubes that had originally grown over the wider trench now acquire an excess
length, or slack, that is taken up by
structure. Using Equation 4.7,
the suspended carbon nano
suspended length.
Figure 5.5 - (a) Color plot of mixing current for a higher order mechanical resonance mode (n = 1) of sample 1 as a
function of temperature. (b) Frequency of the fundamental mechanical resonance mode (n = 0) of sample 1 during a
high temperature cycle. (c) Schematic
When the nanotube unclamps at higher temperatures, the tension on the nanotube is reduced by the addition of an extra
suspended length of ∆l
th
. (d) Quality factor of the resonator

(a)
(b)
contracts and nanotubes that had originally grown over the wider trench now acquire an excess
length, or slack, that is taken up by van der Waals binding to the sidewalls of the electrode/trench
. Using Equation 4.7, the 5 MHz drop shown in Figure 5.5b corresponds to a change in
carbon nanotube length of ∆l
th
= 50 pm for sample 1, which has a 2
(a) Color plot of mixing current for a higher order mechanical resonance mode (n = 1) of sample 1 as a
function of temperature. (b) Frequency of the fundamental mechanical resonance mode (n = 0) of sample 1 during a
high temperature cycle. (c) Schematic diagram of the nanotube position when T < T
C

When the nanotube unclamps at higher temperatures, the tension on the nanotube is reduced by the addition of an extra
. (d) Quality factor of the resonator plotted as a function of temperature.
(d)
(c)
85
contracts and nanotubes that had originally grown over the wider trench now acquire an excess
ewalls of the electrode/trench
b corresponds to a change in
pm for sample 1, which has a 2µm nominal

(a) Color plot of mixing current for a higher order mechanical resonance mode (n = 1) of sample 1 as a
function of temperature. (b) Frequency of the fundamental mechanical resonance mode (n = 0) of sample 1 during a
(case I) and T > T
C
(case II).
When the nanotube unclamps at higher temperatures, the tension on the nanotube is reduced by the addition of an extra
plotted as a function of temperature.
86

The free body diagram illustrated in Figure 5.5c shows the nanotube 'peeling off' from the
electrode/trench sidewall by thermal excitation, as mentioned earlier. From the Figure, it is clear
that the lateral component of the tension on the nanotube at clamping point P is equal to the van
der Waals force that clamps the nanotube to the trench sidewall, Γcos(θ) = Γ
vdW
. That is, if
Γcos(θ) < Γ
vdW
, then the part of the nanotube's suspended length will stick to the sidewall of the
trench in order to increase the tension on the nanotube. For the opposite case, when Γcos(θ) >
Γ
vdW
, part of the nanotube stuck to the sidewall surface will peel off to reduce the tension. Abrams
et al. studied surface-carbon nanotube interactions using scanning electron microscope and
transmission electron microscope with carbon nanotubes grown between SiN
x
pillars, shown in
Figure 5.6a [1]. Their studies confirmed that nanotubes follow the edge of the substrate until the
built up tension on the nanotube overcomes the van der Waals force. This results in a taut
nanotube crossing the trench.
Here the total tension on the nanotube, Γ, is equal to the initial tension, Γ
0
, plus the
additional tension due to the electrostatic force, at the applied gate voltage Γ
el
. A negative Γ
0

signifies slack, and a positive value corresponds to a nanotube that has tension under zero DC
gate voltage. Therefore at zero DC gate voltage (Γ
el
= 0), all the tension on the device is due to
van der Waals forces, Γ
0
= Γ
vdW
. We can extract this value from Figure 5.4 by fitting the
mechanical resonance of the device with Equation 2.29

-
L

}
[-
L
jSLfkL
\

[-
L
SLekwL
\

,

where
87

-
L
jSLfkL
M
0
L

N

O
=
K
,¥≪=

⁄ , and
-
L
SLekwL
=

O
¥
ST
+¥
(
K
,¥≫=

⁄

given in Equations 2.10 and 2.12, respectively.

Figure 5.6 - (a) Schematic diagram illustrating the effects of the nanotube–surface interactions in creating a taut single-
walled nanotube bridging the gap between two pillars. (b) Transmission electron microscope image of a single-walled
nanotube following the contour of a 2µm hole, creating a taut single-walled nanotube bridging the hole. (c) High
resolution scanning electron microscope image of an extending single-walled nanotube adhering to as much of the
pillar edge as possible, over a scale of almost a micron. The resolution of the high resolution scanning electron
microscope limits the capability of affirming either the warping or buckling of the nanotube. Figure is adapted from
Abrams et al. [1].

From this fit, the initial tension on the nanotube is calculated to be Γ
0
= Γ
vdW
≈ 10 EI/L
2
=
3 pN. Previous measurements of single-walled nanotubes on Si substrates show an in-plane van
(a)
(b) (c)
88

der Waals force of 10 pN/nm [101], as determined from the Raman spectra of nanotubes under
strain due to van der Waals friction. Tension due to van der Waals forces has also been observed
in graphene drum resonators after graphene fabricated by chemical vapor deposition is transferred
onto SiN
x
substrates with circular holes [12]. It should be noted, however, this peeling
phenomenon at elevated temperatures has not been observed in graphene.
As the temperature is increased, the atoms on the nanotube surface gain enough energy to
escape the Lennard-Jones potential formed by the van der Waals interaction between the
nanotube and the trench sidewall. At the critical temperature, the nanotube peels off the sidewall
of the electrode causing a sudden drop in the resonance frequency, since the suspended length of
the nanotube is now increased, as illustrated in Figure 5.5c. For sample 1, we calculate this extra
suspended carbon nanotube length to be ∆l
th
= 50 pm using Hooke's law. This critical temperature
(k
B
T
C
= 30 meV) provides enough energy to break the van der Waals bond between the Pt
electrode and the single-walled nanotube completely. This temperature is not surprising as
mentioned earlier the adsorption energy of graphene on Pt is E
a
= 43 meV. It should be noted that
this peeling (or zipper) effect cannot be observed without a gate voltage that electrostatically pulls
on the nanotube.
89

Figure 5.7 - Repeated thermal cycling shows that the mechanical resonance of the carbon nanotube on temperature is
different in consecutive cycles.

Thermal cycling is performed on these samples to ensure repeatability. Figure 5.7 shows
the dependence of the mechanical resonance frequency of a carbon nanotube on the temperature
in 4 consecutive heating/cooling cycles. The resonance frequency at the end of the first run did
not recover to its original value. This irreversible change is due to the realignment of the both
clamping points of the nanotube on the surface of the electrode to minimize the tension. Repeated
cycles show a downshift of the critical temperature for the first 4 runs. After the fourth run, T
C

does not change.
Run 1 Run 2
Run 3 Run 4
90

Figure 5.8 - Carbon nanotubes showing 'peeling' effect at similar temperatures. Six of the seven tested samples showed
this repeatable behavior.

Interestingly, this 'peeling off' or 'zipper effect' does not affect the quality factor of the
resonance, as shown in Figure 5.5d, since the clamping conditions have not changed. The
clamping at the ends of the nanotube is still limited by the out-of-plane van der Waals forces. The
peeled section is attached to the sidewall of the trench as shown in Figure 5.5c. This sudden drop
in frequency at higher temperatures was observed in six of the seven samples tested, all in the
temperature range 340 - 375 K. Two of these devices are shown in Figure 5.8. The quality factor
still remains much lower than what is expected theoretically. Low tension (i.e. bending regime)
mechanical response of the carbon nanotube electromechanical resonators above and below the
critical temperature also confirms an increased suspended length of the nanotube, as shown in
Figure 5.9. Using Equation 2.10, we estimate the change in length to be ∆L/L
0
= 0.06.
91

-1.0 -0.5 0.0 0.5 1.0
16
17
18
19
20
21
22
23
T = 300K
T = 425K

Resonance Frequency (MHz)
Gate Voltage (V)

Figure 5.9 - Low tension dispersion graph of a carbon nanotube electromechanical resonator device at two different
temperatures. The fundamental resonance frequency at T = 425 K (blue solid triangles) is significantly lower than at T
= 300 K (red empty circles).

We observed a hysteretic behavior in the mechanical resonance frequency of a carbon
nanotube as a function of temperature as seen in Figure 5.5b. We attribute this effect to the
thermal resistance between the heater element and the carbon nanotube. The PID controller that
controls the heating element reads the temperature from a different point than the carbon
nanotube. When the temperature probe reaches a certain value, the temperature of the carbon
nanotube could be slightly different. In this case, the temperature reading would have a positive
offset from the actual carbon nanotube temperature in the heating cycle and a negative offset in
the cooling cycle.

92

5.5 Conclusions
In this chapter, we have shown that clamping in carbon nanotube electromechanical
resonators is quite weak and has an impact on the Q-factor of the device. The measurements
discussed in section 5.4 can shed light on the difference of the calculated and measured
coefficient of thermal expansion of individual carbon nanotubes. If we revisit the discussion in
Chapter 4, we have to include this peeling effect at lower temperatures. However, the peeling
would be reversed and the any excess length due to thermal expansion would stick back on to the
electrode. At lower temperatures, T < T
C
, the substrate contraction is smaller than the carbon
nanotube expansion, ∆L
trench
< ∆L
CNT
, creating slack on the nanotube. As described earlier, this
slack portion sticks back on the trench sidewall. This continuous readjustment of suspended
length of the carbon nanotube results in an offset in the expansion coefficient that cannot be
measured through the carbon nanotube mechanical resonance. This readjustment may explain the
discrepancy between calculated coefficient of thermal expansion values by Kwon et al. [65] and
the measured values shown in Figure 4.10a and 4.10b.

93

CHAPTER 6: OBSERVING OPTICAL PHONONS THROUGH
CARBON NANOTUBE MECHANICAL RESONATORS

*This chapter is similar to Aykol et al. [8], published in Journal of Micromechanics and
Microengineering.
6.1 Introduction
Carbon nanotubes have unique electron transport properties as discussed in Chapter 1.
Their electrical properties are highly dependent on the 1-D crystalline structure and its
deformations. Electrical measurements have proven to be one of the most powerful technique to
probe these properties. A carbon nanotube field effect transistor device has been used, where the
nanotube is directly connected to two metallic contacts (source and drain), and is capacitively
coupled to a third electrode (gate), as illustrated in Figure 6.1. Measurements are performed by
applying AC and DC voltages to various terminals and measuring the resulting currents between
source and drain.

Figure 6.1 - Schematic of a carbon nanotube field effect transistor. Top pane
panel shows a suspended device.

We have briefly discussed the electrical properties of carbon nanotubes in Section 1.3.
We have established two different types (quasi
the chirality of the nanotube. In this chapter, we will give a more detailed description of the
electron transport in 1-D systems (like nanotubes) using a classical (incoherent) theory, following
the work by Bushmaker [21] and Biercuk et al. [15]. Lat
effects of electron-phonon scattering on the mechanical vibrations of a
electromechanical resonator.

6.2 Electron Transport in Carbon Nanotubes
Before investigating the effects of electron
carbon nanotube electromechanical resonator, we need to understand the one dimensional
Schematic of a carbon nanotube field effect transistor. Top panel is an on-substrate device, and the bottom
panel shows a suspended device.
We have briefly discussed the electrical properties of carbon nanotubes in Section 1.3.
We have established two different types (quasi-metallic and semiconducting) of nanotubes du
the chirality of the nanotube. In this chapter, we will give a more detailed description of the
D systems (like nanotubes) using a classical (incoherent) theory, following
the work by Bushmaker [21] and Biercuk et al. [15]. Later, we will present our data showing the
phonon scattering on the mechanical vibrations of a
resonator.
Transport in Carbon Nanotubes
Before investigating the effects of electron-phonon coupling in mechanical motion of a
electromechanical resonator, we need to understand the one dimensional
94

substrate device, and the bottom
We have briefly discussed the electrical properties of carbon nanotubes in Section 1.3.
metallic and semiconducting) of nanotubes due to
the chirality of the nanotube. In this chapter, we will give a more detailed description of the
D systems (like nanotubes) using a classical (incoherent) theory, following
er, we will present our data showing the
phonon scattering on the mechanical vibrations of a carbon nanotube
ng in mechanical motion of a
electromechanical resonator, we need to understand the one dimensional
95

transport characteristics of carbon nanotubes. We start with the Landauer formula, which gives
the conductance of a nanotube by:

^^
(
º<
d¶[−

/

?]
d
¾
¾
k
,
(6.1)
where df/dE is the energy derivative of the Fermi function. We will call R
0
= 1/G
0
= h/4e
2
= 6.5
kΩ, the quantum conductance, and to include the effects of scattering in the nanotube we will
simply add a scattering transmission coefficient T(E) in the conductance equation. We can rewrite
the conduction equation by

^=^
(
º<
d¶[−

/

?]
d
?
¾
¾
k
.
(6.2)
We will use a perfectly metallic nanotube with E
g
= 0 to develop our model. For
simplicity, we will assume a constant scattering transmission coefficient T(E) = T. For a uniform
nanotube, the resistance is

¸
¿jS
¸
(
1
?
=¸
(
M1+
1−?
?
N,
(6.3)
where the effects of scattering and the quantum resistance R
0
are separated. For a two barrier
transmission channel, we have to consider multiple reflections, as shown in Figure 6.2. The total
transmission coefficient in this case is given by
?
w
=?
]
?

+?
]
?

Γ
]
Γ

+?
]
?

Γ
]

Γ

+⋯=
?
]
?

1−Γ
]
Γ

. (6.4)

Figure 6.2
If we manipulate this equation, we can write the total transmission coefficient by

Using Equation (6.5, we can calculate the resistance in a system with multiple scattering sources,
such as a carbon nanotube with contact barriers, dif
devices, the total resistance is
and the nanotube resistance, respectively
R
contacts
= 0 and focus on the phonon scattering mechanisms and its effects on
The relationship
be calculated by using a weak scatterer approximation, where

where the average distance between scatterers
transmission probability of a non
The initial slope of the decay is
2 - Reflection and transmission in multiple barrier (scattering) system.
If we manipulate this equation, we can write the total transmission coefficient by
1−?
w
?
w

1−?
]
?
]

1−?

?

⋯
, we can calculate the resistance in a system with multiple scattering sources,
such as a carbon nanotube with contact barriers, diffusive scattering and defects. For practical
devices, the total resistance is R = R
tube
+ R
contacts
, where R
contacts
, and R
tube
and the nanotube resistance, respectively. For the discussion in this chapter, we will assume
= 0 and focus on the phonon scattering mechanisms and its effects on
The relationship between electron mean free path λ and the transmission coefficient
be calculated by using a weak scatterer approximation, where T
s
~ 1. In this approximation,

e
Â
1−?
e

1−?
e
?
e
,
where the average distance between scatterers L
s
is much smaller than
transmission probability of a non-scattered (ballistic) particle as a function of distance travelled.
The initial slope of the decay is - (1 - T
s
)/L
s
, which sets the decay constant or mean free path
96

Reflection and transmission in multiple barrier (scattering) system.
If we manipulate this equation, we can write the total transmission coefficient by
(6.5)
, we can calculate the resistance in a system with multiple scattering sources,
fusive scattering and defects. For practical
tube
is the contact resistance
For the discussion in this chapter, we will assume
= 0 and focus on the phonon scattering mechanisms and its effects on R
tube.
the transmission coefficient T can
~ 1. In this approximation,
(6.6)
is much smaller than λ. Figure 6.3 shows the
scattered (ballistic) particle as a function of distance travelled.
, which sets the decay constant or mean free path λ.
97

Figure 6.3 - Particle transmission probability plotted as a function of device length for a system of weak scatterers,
considering the case of no scattering transmission and multiple reflection transmission. Adapted from reference [21].

For the case of N
s
number of scatterers with identical transmission coefficients T
s
, the
total scattering coefficient can be calculated by

1−?
w
?
w
=
e
1−?
e
?
e
=
e

e
Â
=

Â
⇒?
w
=
Â
Â+
. (6.7)
Using this result, the resistance of a nanotube with perfect contacts is calculated by

¸=¸
(
Â+
Â
.
(6.8)
Metallic carbon nanotubes, as described in Section 1.3, have a small bandgap in the order
of a few 10s of meV due to their small curvature. The presence of a bandgap means that the
integral ∑ I
Å[(GG
Æ
)/Ç
È
i]
ÅG
?()
¾
¾
k
is no longer equal to unity, because the charge carriers are
depleted in the bandgap region. With simple algebra, we can separate the effects of charge
depletion from the effects of scattering by
98

¸¸
(
1
∑ I
d¶É
−

?
Ê
d
?
¾
¾
k

Ë
Ì
Í ¸
(
∑ I
d¶É
−

?
Ê
d
¾
¾
k

Î
Ï
Ð
Ë
Ì
Í
∑ I
d¶É
−

?
Ê
d
¾
¾
k

∑ I
d¶É
−

?
Ê
d
?
¾
¾
k

Î
Ï
Ð

¸¸
(
∗
1
?
∗
=¸
(
∗
61+
1−?
]
∗
?
]
∗
+
1−?

∗
?

∗
+⋯8, (6.9)
where, R
0
*
is the depleted channel resistance, and T
*
is the effective transmission coefficient of
the channel. Using this model, we calculate the resistance of our devices including two identical
contacts and diffusive electron-phonon scattering by
¸=¸
(
∗
61+
1−?
Ñ²
∗
?
Ñ²
∗
+2
1−?
y
∗
?
y
∗
8 (6.10)
where T
ph
*
and T
c
*
are the transmission coefficients for phonon scattering and the contacts.
Various measurements have been used to identify the relative effects of contacts [43] and phonon
scattering [86] on the overall device conductance.

6.3 Phonon Scattering in Carbon Nanotubes
In clean, defect free single-walled nanotubes at room temperature electron-phonon
scattering is the dominant mechanism that causes the resistance to increase in the nanotube. The
effective mean free path of electrons in carbon nanotubes can be expressed in terms of various
scattering events using Matthiessen's rule by

Â
S
=[Â
xy
]
+Â
´,SÒe
]
+Â
´,xje
]
\
]
.
(6.11)
At low bias voltages, the electron mean free path is dominated by the acoustic phonon scattering.
99

Scattering by acoustic phonons λ
ac
-1
dominates λ
eff
at small source-drain biases, consequently the
resistance of the carbon nanotube [5, 58]. However, the maximum current that can be carried by
the carbon nanotube is limited by optical phonon absorption and emission λ
OP
-1
at higher bias
voltages [86, 119].

Figure 6.4 - Suspended nanotube current–voltage (I–V) characteristics demonstrating the effect of optical-phonon
scattering. (a) Scanning electron microscope image, taken at a 45° angle, of the on-substrate (on nitride) and suspended
(over a ∼ 0.5 µm deep trench) segments of the nanotube. (b) A schematic diagram of the device cross-section. (c) I–V
characteristics of the same-length (L ∼ 3 µm) suspended and on-substrate portions of a single-walled nanotube (d
t
∼
2.4nm) at room temperature measured in vacuum. The symbols represent experimental data; the lines are calculations
based on average nanotube temperature (similar within ∼ 5% to that based on actual tube-temperature profile and
resistance integrated over the ∼ 3 µm tube length). Adapted from Pop et al. [90].

High bias transport in nanotubes is unique because optical phonon scattering becomes
dominant. The slope of current–voltage (I–V) characteristic decreases for increasing bias and, if

the nanotube is suspended, can even become negative. Figure 6.4 shows the
a quasi-metallic nanotube with suspended and on
also exhibit strong optical phonon
dramatic increase in resistance and current saturation is first observed by Yao et al. [119]. The
electron travelling in an electric field (
energy equal to the optical phonon
process limits the maximum current that can be carried in all but the shortest nanotubes to less
than I
max
≈ (4e
2
/h)(ħΩ
0
/e
causes a sudden increase in the nanotubes' temperature, which
Figure 6.5 - Schematic drawing of threshold for optical emission in carbon nanotubes at high bias voltages. (a) Elastic
scattering of electron by an optical phonon [119]. (b) Two relevant length scales for high
scattering in an electric field,
extra energy required for an electron to emit an optical phonon.

Optical phonon emission has a crucial parameter,
electron travels until it can find an empty state to decay into. This excess distance decreases as the
nanotube heats up due to
increases due to increased temperature,
Figure 6.1a, this excess heat is transferred to the substrate and the effects of
(a)
the nanotube is suspended, can even become negative. Figure 6.4 shows the
metallic nanotube with suspended and on-substrate portions. Devices used in this thesis
optical phonon scattering at high bias voltages, as shown in Figure 1.6c. This
dramatic increase in resistance and current saturation is first observed by Yao et al. [119]. The
electron travelling in an electric field (Ԑ) emits an optical phonon after it has gained kinetic
optical phonon energy, E
ph
= ħΩ
0
≈ 160 meV, as shown in
cess limits the maximum current that can be carried in all but the shortest nanotubes to less
e) = 25 µA. The vibrational energy released by the onset of this scattering
causes a sudden increase in the nanotubes' temperature, which can exceed 1000K.
Schematic drawing of threshold for optical emission in carbon nanotubes at high bias voltages. (a) Elastic
scattering of electron by an optical phonon [119]. (b) Two relevant length scales for high
scattering in an electric field, Ԑ [86]. The population of the decay state N(T
OP
) in (a) is important in determining the
extra energy required for an electron to emit an optical phonon.
Optical phonon emission has a crucial parameter, λ
OP
min
, which is the excess distance an
ls until it can find an empty state to decay into. This excess distance decreases as the
heats up due to optical phonon scattering. As the population of the empty states
increases due to increased temperature, λ
OP
min
decreases. For nonsuspended na
excess heat is transferred to the substrate and the effects of
ħΩ
(b)
Â
´,SÒe

Ñ²
¡Ó

1
-eԐx
100
the nanotube is suspended, can even become negative. Figure 6.4 shows the I-V characteristics for
substrate portions. Devices used in this thesis
ttering at high bias voltages, as shown in Figure 1.6c. This
dramatic increase in resistance and current saturation is first observed by Yao et al. [119]. The
) emits an optical phonon after it has gained kinetic
160 meV, as shown in Figure 6.5. This
cess limits the maximum current that can be carried in all but the shortest nanotubes to less
A. The vibrational energy released by the onset of this scattering
can exceed 1000K.

Schematic drawing of threshold for optical emission in carbon nanotubes at high bias voltages. (a) Elastic
scattering of electron by an optical phonon [119]. (b) Two relevant length scales for high-bias electron-phonon
) in (a) is important in determining the
, which is the excess distance an
ls until it can find an empty state to decay into. This excess distance decreases as the
the population of the empty states
decreases. For nonsuspended nanotubes, shown in
excess heat is transferred to the substrate and the effects of optical phonon
ħΩ
Â
´
ÒkL
〈?
´
〉

101

emission are limited. For suspended nanotubes, the temperature increases and a negative
differential resistance is observed at high bias voltages, as shown in Figure 6.4c.

6.4 Optical Phonon Scattering in Carbon Nanotube Electromechanical Resonators
We characterize the nanoelectromechanical response of suspended individual carbon
nanotubes under high voltage biases. An abrupt upshift in the mechanical resonance frequency of
approximately 3 MHz is observed at high bias. While several possible mechanisms are discussed,
this upshift is attributed to the onset of optical phonon emission, which results in a sudden
contraction of the nanotube due to its negative thermal expansion coefficient. This, in turn,
causes an increase in the tension in the suspended nanotube, which upshifts its mechanical
resonance frequency. This upshift is consistent with Raman spectral measurements, which show a
sudden downshift of the optical phonon modes at high bias voltages. Using a simple model for
oscillations on a string, we estimate the effective change in length of the nanotube to be ∆L/L ≈ -2
× 10
-5
, at a bias voltage of 1V.
Figure 6.6a shows a scanning electron microscope image of one of the devices used in
this study. The nanotube devices are fabricated by the method described in Chapter 3. Our
electrical and optical measurements are carried out in an optical vacuum chamber. Mechanical
vibrations are detected using the two-signal mixing technique explained in Section 3.3.1.
102

Figure 6.6 – Scanning electron microscope images of (a) a suspended carbon nanotube sample device. (b) A high
magnification scanning electron microscope image of an individual carbon nanotube crossing the trench.

For the measurements in this chapter, we used the two–signal mixing technique detailed
in Section 3.3.1. Briefly, an AC gate voltage, v
G
(t) = A
G
cosω
G
t + V
G
, is applied to the gate
electrode on the bottom of the trench. The source electrode is connected to another AC source
with v
SD
(t) = A
SD
cos[(ω
G
+ ω
LO
)t]+ V
SD
. Another “Bias-Tee” is connected to the input of the lock-
in amplifier to ensure that the DC current does not saturate the input by creating a path to the
ground terminal. Separating the DC path and AC path on the source and detection side of the
device reduces the noise significantly. Nominal values for the applied voltages are A
SD
= 30mV,
A
G
= 40mV, and f
LO
= ω
LO
/2π = 20 kHz. The current at the input of the lock-in amplifier (I
NT
) is
the sum of all frequency components from mixing in the nanotube device (ω
G
+ ω
LO
, 2ω
G
+ ω
LO
,
ω
G
, ω
LO
). The high frequency components are filtered by the lock-in amplifier’s low pass filter.
By tuning ω
G
through a mechanical resonance, we observed a change in current at the local
oscillator frequency (ω
LO
). The mechanical resonance frequency of the nanotube (ω
R
) is
distinguished from other resonances due to the passive elements in the circuit by varying V
G
and
observing whether or not a shift occurs in the mechanical resonance frequency. Figure 6.7a shows
the current through a suspended carbon nanotube at ω = ω
LO
as the gate frequency (ω
G
) is swept
through the mechanical resonance frequency. The different datasets in the Figure correspond to
(a) (b)
103

different applied gate voltages. The shifting peak (or dip) appearing between 34 - 38 MHz
corresponds to a mechanical resonance, while the stationary peaks correspond to other resonances
due to the passive elements in the circuit. The shift in the resonance peak is caused by increased
tension in the nanotube due to the electrostatic force between the nanotube and the DC gate
electrode. Figure 6.7b shows the color plot of nanotube current plotted as a function of gate
frequency (f
G
= ω
G
/2π) and gate voltage (V
G
) measured at a pressure P = 1×10
-4
Torr. The
fundamental resonance frequencies are consistent with the 2μm length and 1-3 nm diameter of
this nanotube. The diameters of the nanotubes in this study were confirmed by Raman
spectroscopy.
15 20 25 30 35 40 45
0
10
20
30
40
50
V
G
=-3V
V
G
=-5V

Current (pA)
Gate Frequency (MHz)
V
G
=-4V

Figure 6.7 – (a) Electric current plotted as a function of gate frequency. The solid arrows show the resonance due to the
nanotube and dashed arrows point to resonances due to passive circuitry elements. (b) Color plot of the mixing current
plotted as a function of the gate frequency (f
G
) and DC gate voltage. The red box corresponds to the experimental
regime shown in Figure 4b.
Gate Voltage (V)
Gate Frequency (MHz)

-6 -4 -2 0 2 4 6
20
25
30
35
40
45
-100
-50
0
50
100
(a)
(b)
104

Once a nanotube with low contact resistance and a strong mechanical resonance is
identified, the bias voltage is varied from 0 to 1.5V across the source and drain electrodes. Figure
6.8 shows the mechanical resonance frequency (f
R
) plotted as a function of the DC bias voltage
V
SD
at P = 1 × 10
-4
Torr. The small positive slope for the resonance frequency with increasing
bias voltage is caused by the effective gating of the nanotube device due to the increased bias
voltage, as shown schematically in Figure 6.9. Increasing the bias voltage increases the
electrostatic potential at the middle of the nanotube by V
SD
/2. This effective increase in gate
voltage increases the electrostatic force between the nanotube and gate electrode, which in turn
increases the axial tension in the nanotube. Therefore, by increasing the bias voltage, the
nanotube is effectively gated, thus causing the resonance frequency to change.
Figure 6.8a shows an abrupt upshift in the mechanical resonance frequency of
approximately 3 MHz at high bias, which is well within the experimental precision of this
measurement. Figure 6.8b shows a similar discrete change in the slope of the resonance frequency
with bias voltage.
105

0.0 0.2 0.4 0.6 0.8 1.0 1.2
20
24
28
32
36
40
44

Resonance Frequency (MHz)
Bias Voltage (V)
0
1
2
3
4
Current (μA)
5
10
15
20
25
30
-5 -3 -1 1 3 5

Gate Voltage (V)

Conductance (μS)
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
28
32
36
40
44

V
G
= -3V
V
G
= -4V
V
G
= -5V
Resonance Frequency (MHz)
Bias Voltage (V)
0
1
2
3
4
5
6
Current (μA)

Figure 6.8 - Bias voltage dependence of the mechanical resonance frequency of two different suspended nanotube
devices. In (a) the different datasets were taken on different days at a constant gate voltage of V
G
= -3 V, demonstrating
the repeatability of the observed effect. The inset in (a) shows the I-V characteristics of this device, which exhibits
metallic behavior. The red solid lines represent the I-V
bias
and conductance-V
gate
characteristics of the device.

Effective gating by itself is not enough to describe this anomalous behavior at high bias. These
abrupt changes in the mechanical resonance frequency occur when the bias voltage is above the
threshold for emission of optical phonons [119]. The onset of this optical phonon scattering
process corresponds to an abrupt increase in the phonon populations and lattice temperature of the
nanotube with bias voltage. Like graphite and graphene [80], carbon nanotubes have been
predicted to have a negative coefficient of thermal expansion [65]. Therefore, as the suspended
(a)
(b)

nanotubes become hot, they contract, thus increasing the tension in the nanotube and its
mechanical resonance frequency.
The onset voltage of
seen in the I
DS
-V
DS
plot in
measurements, which have shown a sudden downshift of the optical phonon frequencies under
high bias voltages [4]. It should be noted that whil
the resonance frequency of a suspended nanotube is nonlinear, there are no abrupt changes in this
gate-induced tension. In fact,
resonance frequency on gate voltage.
corresponding to the three datasets in
linear dependence on gate voltage (at low bias). Therefore, the abrupt changes observed in
6.8 at high bias must originate from a different underlying mechanism.

Figure 6.9 - Effective gating of the carbon nanotube due to asymmetric source
Using the model
relation between the tension
given by Equation (2.12)

nanotubes become hot, they contract, thus increasing the tension in the nanotube and its
mechanical resonance frequency.
The onset voltage of optical phonon emission by electron-phonon scattering can also be
plot in Figure 6.8a. This upshift is consistent with Raman spectral
measurements, which have shown a sudden downshift of the optical phonon frequencies under
It should be noted that while the effect of gate voltage
the resonance frequency of a suspended nanotube is nonlinear, there are no abrupt changes in this
induced tension. In fact, Figure 6.7b shows the parabolic dependence of the mechanical
ance frequency on gate voltage. The red box in the Figure
corresponding to the three datasets in Figure 6.8b. In this regime, the resonance frequency has a
linear dependence on gate voltage (at low bias). Therefore, the abrupt changes observed in
at high bias must originate from a different underlying mechanism.
Effective gating of the carbon nanotube due to asymmetric source-
odel developed in Chapter 2 for a doubly clamped nanotube
relation between the tension (T) on the nanotube and the fundamental resonance freq
):
¶
(

O
?

Ö
2
,
106
nanotubes become hot, they contract, thus increasing the tension in the nanotube and its
phonon scattering can also be
a. This upshift is consistent with Raman spectral
measurements, which have shown a sudden downshift of the optical phonon frequencies under
e the effect of gate voltage-induced tension on
the resonance frequency of a suspended nanotube is nonlinear, there are no abrupt changes in this
shows the parabolic dependence of the mechanical
the Figure indicates the region
b. In this regime, the resonance frequency has a
linear dependence on gate voltage (at low bias). Therefore, the abrupt changes observed in Figure

-drain bias voltage.
for a doubly clamped nanotube resonator, the
) on the nanotube and the fundamental resonance frequency (f
o
) is

107

where, L and m are the suspended length and total mass of the carbon nanotube, which are 2µm
and 6.35 × 10
-18
g, respectively. Using Equation (4.7), we can relate the change in tension of the
nanotube to an effective change in length of the nanotube ∆L as

×

?
@
Ö
∆

Ö
⇒?=
@∆

,
where A is the cross-sectional area of the nanotube given by A = π/4[(d
t
+ t)
2
- (d
t
- t)
2
], d
t
is the
diameter of the nanotube, and t is the thickness a graphene sheet t = 0.34 nm. Substituting
tension calculated by Hooke's law into our model yields the change in length due to thermal
contraction to be ∆L/L ≈ -2 × 10
-5
at a bias voltage of 1 V. There are currently no measured
values for the coefficient of thermal expansion of individual suspended carbon nanotubes, and
simulated models report values that vary significantly [55, 76, 120]. Recent experimental data on
carbon nanotube coefficient of thermal expansion by Aykol et al. [8] and Chaste et al. [26]
confirm that the thermal expansion of carbon nanotubes is negative. In principle, we should be
able to compare our results with the coefficient of thermal expansion values from literature.
However, these suspended nanotubes are known to be in a state of extreme thermal non-
equilibrium under these high applied bias voltages [22]. We, therefore, cannot make a meaningful
comparison with coefficient of thermal expansion values established under thermal equilibrium.
The desorption of gas molecules from the surface of the nanotube is another possible
explanation for the anomalous upshift in mechanical resonance frequency [115, 116]. For the
gasses present in the atmosphere, the binding energies of gasses present in the atmosphere to the
nanotube surface vary from approximately 82 meV (Ar) to 300 meV (O
2
) [122]. While these
measurements were all carried out under a vacuum of 1 × 10
-4
Torr, some of the gasses present in
the atmosphere (namely H
2
O) have binding energies (130 meV) that are considerably larger than
k
B
T at room temperature [116, 122]. Therefore, it is possible that, as the nanotube temperature
108

increases under high bias voltages, gas molecules desorb from the nanotube surface, thus
reducing the mass density of the nanotube and increasing the mechanical oscillation frequency. It
should be noted, however, that at room temperature and a pressure of 1x10
-4
Torr, as in this
experiment, the coverage ratio of potential binding cites occupied by H
2
O molecules is < 0.0042.
The desorption of these molecules corresponds to a frequency shift of only 0.01 MHz, which is
considerably lower than the 3 MHz that we observe in this study.
In conclusion, we have observed an anomalous shift in the mechanical resonance
frequency of individual carbon nanotubes under high bias voltages. This shift is attributed to the
onset of optical phonon emission by hot electrons under high bias voltages, which in turn causes
lattice contraction and an associated increase in the mechanical resonance frequency. Based on a
simple string under tension model, we estimate a change in length of the nanotube of ∆L/L ≈ -2 ×
10
-5
, at a bias voltage of 1 V. Gas desorption has been ruled out due to the low pressures in the
experimental setup and lower occupation ratio of the gases on the nanotube.

6.5 Conclusions
In this chapter, we have discussed the electrical transport properties of carbon nanotubes
in more detail. The resistance of the nanotube can be expressed as an effective mean free path of
the electrons in the nanotube. Various scattering mechanisms such as local defects, mechanical
deformations and substrate effects can affect the electron mean free path. However, the main
scattering source is electron-phonon scattering in pristine, defect free nanotubes used in this
thesis. At low bias voltages, acoustic phonon scattering dominates the electron mean free path.
Resistance of the nanotube remains constant in this regime. At high bias voltages, optical phonon
scattering dominates the electron mean free path. Optical phonon emission heats up the nanotube
to approximately 1000 K. In a carbon nanotube electromechanical resonator, this heating
109

combined with the negative coefficient of thermal expansion of carbon nanotubes, causes
contraction of the carbon nanotube and increases the mechanical resonance frequency. This data
represents the first experimental evidence of negative thermal expansion in individual single-
walled carbon nanotubes.

110

CHAPTER 7: FUTURE DIRECTIONS

7.1 Optical Excitation of Mechanical Motion in Carbon Nanotube Mechanical Resonators
Impedance matching is an important design constraint in high frequency analog circuits.
With high input and output impedance of carbon nanotube electromechanical resonators, it is
difficult to reach the highest performance of these devices. In particular, measured quality factors
(Q) reflect the impedance mismatch. The theoretically calculated Q ≈ 10
5
[54] is a distant target
from the measured Q ≈ 10
3
[26, 115] hardly reached at cryogenic temperatures, and at room
temperature the measured values are even further below Q ≈ 10
5
. A quick calculation would give
the reflection of the input signal at the source terminal of the nanotube as
Γ
]

Z

−Z
]
Z
]
Z

0.998, (7.1)
where Γ
12
is the reflection coefficient at the boundary of the RF source and carbon nanotube, and
Z
i
is the impedance of the corresponding medium. High reflections at the boundaries will result in
lower measured Q-factors. This does not reflect the quality of the mechanical motion, rather the
quality of the readout circuitry. For mechanical reasons of lower quality factors, refer to Chapter
5.
Electrical actuation and detection is important for any practical application of carbon
nanotube electromechanical resonators. Down-mixing is a useful technique; however, the
complexity of the circuitry and the low measured Q values, necessitate more sophisticated
methods. One such method is, as discussed in Section 3.3.4, inducing the mechanical modes with
a nearby antenna by coupling a radiated electric field to the nanotube [49]. In this method, the
detection of the mechanical motion is accomplished by measuring the DC current through the
nanotube. This technique achieves very high Q-factors with Q ≈ 140,000 at 20 mK. Another
111

method that would also yield high Q-factors is optical actuation of the mechanical motion. Bunch
et al. (2007) used an intensity modulated 408nm laser to create local hot spots on the graphene
membrane and induced vibrations. Using the same method Barton et al. reported Q ≈ 10
3
[12].
However, using an electrical method, Singh et al. reported Q ≈ 10
2
, which is an order of
magnitude lower, as shown in Figure 7.1 [100].

Figure 7.1 - (a) Temperature dependence of the measured Q-factors by Singh et al. using electrical actuation of a
graphene resonator [100]. (b) In a similar device, Barton et al. induced mechanical vibrations optically to achieve
significantly higher Q values at room temperature [12].

In this section, we propose an optical excitation scheme for carbon nanotube
electromechanical resonators. Similar to Bunch et al., we will heat up the nanotube to induce
ripples in the nanotube. There are a few important issues that need to be addressed. Firstly, the
small optical absorption cross-section of carbon nanotubes, α ≈ 10
-17
cm
2
, means most of the
photons incident on the nanotube will not be absorbed [14, 52]. Secondly, the high diameter to
length ratio of carbon nanotubes would require the polarization of the laser to match the direction
of the nanotube to increase the interaction volume of photons with the nanotube [82], as depicted
in Figure 7.2a. Thirdly, due to their 1-D structure, carbon nanotubes' electronic structure have van
Hove singularities [97], which result in peaks in the absorption spectra, as shown in Figure 7.2b
and 7.2c, respectively [73, 113].
(b) (a)

Figure 7.2 - (a) Top panel shows the polarization dependence of the optical absorption of a carbon nanotube and the
lower panel shows the corresponding optical absorption cross
density of states (adapted from
Adapted from reference [113].

We will use a tunable laser source and an electro
the intensity of the incident light, as illustrated in
match the nanotube's highest absorption, we will modulate the laser intensity and detect the
(a)
(c)
(a) Top panel shows the polarization dependence of the optical absorption of a carbon nanotube and the
lower panel shows the corresponding optical absorption cross-sections [82]. (b) Van Hove singularities in the electron
density of states (adapted from reference [97]) results in (c) peaks in optical absorption spectra of carbon nanotubes.
Adapted from reference [113].
We will use a tunable laser source and an electro-optical modulator (EOM) to modulate
the intensity of the incident light, as illustrated in Figure 7.3. After tuning the laser wavelength to
match the nanotube's highest absorption, we will modulate the laser intensity and detect the
(b)

112

(a) Top panel shows the polarization dependence of the optical absorption of a carbon nanotube and the
sections [82]. (b) Van Hove singularities in the electron
reference [97]) results in (c) peaks in optical absorption spectra of carbon nanotubes.
optical modulator (EOM) to modulate
. After tuning the laser wavelength to
match the nanotube's highest absorption, we will modulate the laser intensity and detect the

mechanical vibrations through the changes in the DC current through the nanotube (I
be modulated due to the m
nearby, as described in Section 2.3 by Equation 2.18;

Since there is no modulation of the gate voltage V
equation. The capacitance C between the nanotube and the gate electrode, will be modulated by
the changing nanotube-electrode distance due to thermal contraction.
Figure 7.3 - Schematic of optical actuation, electri
electromechanical resonators.

The amplitude of the oscillations can be controlled by changing the maximum intensity
of the laser or by changing the duty cycle, because the average temperature of the nanotube will
change. Since the temperature difference between the heating and the cool
different, the amount of contraction of the nanotube will be different. Optical excitation paves the
way to combining many other experiments with
mechanical vibrations through the changes in the DC current through the nanotube (I
be modulated due to the modulation of the capacitance between the nanotube and a gate electrode
nearby, as described in Section 2.3 by Equation 2.18;
dbUdV

V

dU.
Since there is no modulation of the gate voltage V
g
, we can ignore the first term in the above
The capacitance C between the nanotube and the gate electrode, will be modulated by
electrode distance due to thermal contraction.
Schematic of optical actuation, electrical detection of the mechanical motion of carbon nanotube

The amplitude of the oscillations can be controlled by changing the maximum intensity
of the laser or by changing the duty cycle, because the average temperature of the nanotube will
change. Since the temperature difference between the heating and the cool
different, the amount of contraction of the nanotube will be different. Optical excitation paves the
way to combining many other experiments with carbon nanotube electromechanical
113
mechanical vibrations through the changes in the DC current through the nanotube (I
NT
). I
NT
will
odulation of the capacitance between the nanotube and a gate electrode

, we can ignore the first term in the above
The capacitance C between the nanotube and the gate electrode, will be modulated by

cal detection of the mechanical motion of carbon nanotube
The amplitude of the oscillations can be controlled by changing the maximum intensity
of the laser or by changing the duty cycle, because the average temperature of the nanotube will
change. Since the temperature difference between the heating and the cooling cycles will be
different, the amount of contraction of the nanotube will be different. Optical excitation paves the
carbon nanotube electromechanical resonators. If
114

the backscattered Raman light from the nanotube is collected and sent to a spectrometer, the
changes in the lattice can be analyzed at the mechanical resonance of the nanotube. Also by
tuning the laser wavelength for highest absorption, we also identify the chirality (n, m) of the
nanotube, which gives us the diameter of the nanotube by Equation 1.1.

7.2 Quantifying Doping by Adsorbates in Carbon Nanotubes
Carbon nanotube gas sensors were first realized in 2000 by Kong et al. [63]. The high
sensitivity of nanotubes is a direct result of their high surface to volume ratio. The entire charge
carrying channel is interacting with the gas environment, which can change the conducting
properties of the channel, as shown in Figure 7.4.

Figure 7.4 - Electrical response of a semiconducting single-walled nanotube to gas molecules. (a) Conductance (under
V
g
= +4 V, in an initial insulating state) versus time in a 200-ppm NO
2
flow. (b) Data for a different semiconducting
single-walled nanotube sample in 20- and 2-ppm NO
2
flows. The two curves are shifted along the time axis for clarity.
(c) Conductance (V
g
= 0, in an initial conducting state) versus time recorded with the same nanotube sample as in (a) in
a flow of Ar containing 1% NH
3
. (d) Data recorded with a different semiconducting single-walled nanotube sample in a
0.1% NH
3
flow [63].

Effects of adsorbates on carbon nanotubes' conductivity are two
doping. As described in Chapter 6, defects increase scattering in the nanotube and the resistance
of the nanotube subsequently increases, as given by Equations 6.3 and 6.5. Adsorbed molecules
on the nanotube change the Fermi surface of the nanotube through charge transfer. The broken
periodicity in the potential along the nanotube will increase the sc
depicted in Figure 7.5a [16]. On the other hand, adsorbed molecules dope the nanotube by either
injecting or capturing a partial electron. Similar to electrical doping through a gate electrode,
chemical doping also changes the conductivity of the nanotube by moving the Fermi le
as shown in Figure 7.5b.
Figure 7.5 - (a) Plot of low-bias resistance of a quasi
Experimental setup for scanned gate microscopy. I, current. (c) Topographic image of the nanotube, and scanned gate
image with V
tip
= 3, 6, 8 V, from left to right. The dark color in indicates increased resistance. Adapted from Bockrath
et al. [16]. (d) Device scheme of a single
electrodes. (e) Atomic force microscope image of the
undoped carbon nanotube region. (c) A band diagram for the system. E
Fermi level, and valence band, respectively. Adapted from Zhou et al. [123].

(a)
(b)
(c)
Effects of adsorbates on carbon nanotubes' conductivity are two
doping. As described in Chapter 6, defects increase scattering in the nanotube and the resistance
of the nanotube subsequently increases, as given by Equations 6.3 and 6.5. Adsorbed molecules
on the nanotube change the Fermi surface of the nanotube through charge transfer. The broken
periodicity in the potential along the nanotube will increase the scattering rate of electrons, as
a [16]. On the other hand, adsorbed molecules dope the nanotube by either
injecting or capturing a partial electron. Similar to electrical doping through a gate electrode,
chemical doping also changes the conductivity of the nanotube by moving the Fermi le
b.
bias resistance of a quasi-metallic single-walled nanotube device as a function of V
Experimental setup for scanned gate microscopy. I, current. (c) Topographic image of the nanotube, and scanned gate
rom left to right. The dark color in indicates increased resistance. Adapted from Bockrath
et al. [16]. (d) Device scheme of a single-walled nanotube with modulated chemical doping contacted by two Ni/Au
electrodes. (e) Atomic force microscope image of the single-walled nanotube. Dashed lines are drawn to highlight
undoped carbon nanotube region. (c) A band diagram for the system. E
C
, E
f
, and E
V
represent the conduction band,
Fermi level, and valence band, respectively. Adapted from Zhou et al. [123].
(d)
(e)
(f)
115
Effects of adsorbates on carbon nanotubes' conductivity are two-fold: scattering, and
doping. As described in Chapter 6, defects increase scattering in the nanotube and the resistance
of the nanotube subsequently increases, as given by Equations 6.3 and 6.5. Adsorbed molecules
on the nanotube change the Fermi surface of the nanotube through charge transfer. The broken
attering rate of electrons, as
a [16]. On the other hand, adsorbed molecules dope the nanotube by either
injecting or capturing a partial electron. Similar to electrical doping through a gate electrode,
chemical doping also changes the conductivity of the nanotube by moving the Fermi level [123],

walled nanotube device as a function of V
g
. (b)
Experimental setup for scanned gate microscopy. I, current. (c) Topographic image of the nanotube, and scanned gate
rom left to right. The dark color in indicates increased resistance. Adapted from Bockrath
walled nanotube with modulated chemical doping contacted by two Ni/Au
walled nanotube. Dashed lines are drawn to highlight
represent the conduction band,

Electrical measurements of chemically doped nanotubes
number of excess charge carriers
Fermi level, then to excess number of carriers. However, it is difficult to quan
adsorbed molecules on a nanotube surface through electrical measurements.
electromechanical resonators, on the other hand, are extremely sensitive to small changes in mass.
The resonance frequency downshift as the total m
studying the resonance frequency shifts in a controlled gas environment, Wang et al. measured
the number of adsorbed molecules on the surface of a nanotube [115], as shown earlier in Figure
1.10b. In their experiment, the adsorbed molecules were monatomic gasses, such as He, Ar, etc.
The chemical doping of these gasses is minimal due to their complete atomic orbitals. Molecules
like O
2
, NO
2
, NH
3
are known to dope a
effects would be very useful in designing
sensitivity of a carbon nanotube electromechanical
carbon nanotube field effect transistor
molecule. The proposed experimental setup is illustrated in
Figure

trical measurements of chemically doped nanotubes can be used to
number of excess charge carriers [70, 93]. The change in conductivity is converted to a change in
Fermi level, then to excess number of carriers. However, it is difficult to quan
adsorbed molecules on a nanotube surface through electrical measurements.
resonators, on the other hand, are extremely sensitive to small changes in mass.
The resonance frequency downshift as the total mass of the suspended region increases
the resonance frequency shifts in a controlled gas environment, Wang et al. measured
the number of adsorbed molecules on the surface of a nanotube [115], as shown earlier in Figure
eriment, the adsorbed molecules were monatomic gasses, such as He, Ar, etc.
The chemical doping of these gasses is minimal due to their complete atomic orbitals. Molecules
are known to dope a carbon nanotube very effectively, and quantify
effects would be very useful in designing carbon nanotube electronic devices. By combining mass
carbon nanotube electromechanical resonator and the doping sensitivity of a
field effect transistor, we can measure the doping efficiency of an adsorbed
molecule. The proposed experimental setup is illustrated in Figure 7.6.
Figure 7.6 - Experimental setup for doping efficiency experiment.
116
can be used to quantify the
. The change in conductivity is converted to a change in
Fermi level, then to excess number of carriers. However, it is difficult to quantify the number of
adsorbed molecules on a nanotube surface through electrical measurements. Carbon nanotube
resonators, on the other hand, are extremely sensitive to small changes in mass.
ass of the suspended region increases [25]. By
the resonance frequency shifts in a controlled gas environment, Wang et al. measured
the number of adsorbed molecules on the surface of a nanotube [115], as shown earlier in Figure
eriment, the adsorbed molecules were monatomic gasses, such as He, Ar, etc.
The chemical doping of these gasses is minimal due to their complete atomic orbitals. Molecules
very effectively, and quantifying their
electronic devices. By combining mass
resonator and the doping sensitivity of a
he doping efficiency of an adsorbed

Experimental setup for doping efficiency experiment.
117

7.3 Summary and Conclusion
Carbon nanotube based nanoelectromechanical resonators present a new platform to
study mechanics at the nanometer scale. As discussed in Chapters 1 and 2, nanotubes possess
many favorable qualities for mass sensing applications, including low mass density and high
stiffness. Even though many electrical and thermal properties of carbon nanotubes have been
investigated, their mechanical properties still remain inadequately studied. In this thesis we
presented our work on the thermo-mechanical properties of carbon nanotubes using carbon
nanotube electromechanical resonators as our experimental platform. In Chapter 4 we showed
that carbon nanotubes exhibit negative thermal expansion in a broad range of temperatures (4 K -
475 K). The presence of a large empty space around the carbon atoms of the carbon nanotube
lattice allows the free movement of thermally excited atoms without requiring any change in the
bond length, which is predicted by the anharmonic theory of solids. During our high temperature
studies of carbon nanotube electromechanical resonators, we examined the strength of the
interaction between the carbon nanotube and underlying platinum surface. Our findings
confirmed that the bond originates from weak van der Waals forces. With adequate thermal
excitation, this bond can be broken, which is consistent with the calculations carried out by
density functional theory and molecular dynamics. Instead of heating the entire sample, we also
heated the carbon nanotube through Joule heating. The results we obtained showed the
nonlinearity of carbon nanotube conductivity and the importance of electron-phonon scattering in
determining the resistance of carbon nanotubes. Even though this outcome was expected, it was
novel in its manifestation in the mechanical motion of the nanotube. We believe the findings
presented in this thesis offer important insight into the thermal response of carbon nanotube
resonators, as well as important design considerations to improve the quality and performance of
these devices.
118

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126

APPENDIX
Labview Program for Measuring Single Source Setup
The program given in this appendix is used to detect the resonances with single source
frequency modulation technique described in Section 3.3.3. Figure A.1 shows the screenshot of
the user interface window of the Labview program used to collect data in single source method.
The program uses the drivers provided on the National Instruments Developer website for the
equipment used. The details and the model numbers of the equipment used can be found in
Chapter 3.

Figure A.1 - Screenshot of the user interface of the Labview program used to measure the mechanical resonance
frequency of carbon nanotube electromechanical oscillator using single source frequency modulation method.

Figures A.2 - A.7 show the block diagram of the used Labview program. In Figure A.2
list of gate data points and frequency data points are created. The arrays are created using the
127

upper and lower limits set in the user interface. The elements in the arrays are equidistant and are
calculated using the resolution set in the user interface window.

Figure A.2 - Frequency and the gate data points are generated for the set limits in the user interface with the defined
resolution also set in the user interface.

Figure A.3 shows the initialization of the AC signal generator and the DC gate voltage source.
The signal generator is set in frequency modulation mode with parameters set in the user interface
window. After initialization both the AC source and the DC source are set to their initial values
and their outputs are activated. In Figure A.4 the lock-in amplifier is initialized. There is a two
second wait after the input of the lock-in is activated. This wait helps the transient current to be
cleared from the registry of the instrument. In Figure A.5 the file that will store the data acquired
is created. The absolute value of the measured current (R) and the phase (T) are stored in separate
files. A parameter file is also created that saves the parameters used for the experiment. The file
128

name includes the date and the time when the experiment is started. Short text extensions can be
added to the filename for brief comments.

Figure A.3 - The AC generator and gate DC source are initialized and are both set to initial values of the experiment.

Figure A.4 - The lock-in amplifier is initialized. After initialization there is a two second wait time for the input of the
amplifier to stabilize.

129

Figure A.5 - The file that the data will be stored in is created for both the absolute value of the current (R) and the
phase of the current (T).

Figure A.6 shows the nested loop for collecting data. The amplitude (R) and the phase (T) of the
mixing current is recorded at each frequency and gate voltage. The outer loop varies the gate
voltage and the inner loop varies the carrier frequency of the frequency modulated signal at the
output of the AC generator. The measured values are displayed on two separate graphs in real
time. The file that stores the measured values is updated after each execution of the loops to avoid
corruption of the data or loss of data in case of an error or unexpected problem with the computer.
In Figure A.7 the gate voltage is zeroed and the AC generator output is disabled. GPIB
connections with the instruments are closed. If there is an error, the corresponding error message
is displayed in a pop-up window.
130

Figure A.6 - A new data point is collected for each gate voltage and frequency point in two nested loops. The outer
loop varies the gate voltage. The inner loop varies the frequency of the carrier of the frequency modulated signal
generated by the signal generator.

131

Figure A.7 - Gate voltage is zeroed and the AC generator output is disabled. The GPIB connections are closed for
proper operation.

Abstract (if available)

Abstract In this thesis the thermo-mechanical properties of single–walled carbon nanotubes are investigated utilizing carbon nanotube based nanoelectromechanical oscillators. These resonator devices are highly sensitive to changes in tension on the carbon nanotube. In Chapters 4 the coefficient of thermal expansion of an individual single–walled carbon nanotube is measured in the range 4K - 475K. Experimental observation of this parameter has not been reported before this work and the calculations give different results depending on the models used. The observed negative thermal expansion is attributed to the free configurational space around the carbon atoms of the nanotube. When the nanotube is cooled, the entropy of the system is lowered by expanding the volume of the nanotube through various changes in the structure like pinching twisting or bending. The minimum of the coefficient of thermal expansion is measured as -4.5 ppm•K-1 at 100K. The coefficient of thermal expansion remains negative throughout the entire range. ❧ The mechanical response of carbon nanotube electromechanical oscillators at elevated temperatures is studied in Chapter 5. The weak interaction forces between the carbon nanotube and underlying platinum electrodes limit the performance of carbon nanotube electromechanical oscillators, where the devices are built as described in Chapter 3. Van der Waals bond between the carbon nanotube and the platinum electrode weaken as the temperature increases. At a critical temperature the nanotube delaminates from the surface completely and a sudden drop is observed in the mechanical resonance frequency of the oscillator. Using the results obtained, the clamping force between the carbon nanotube and the underlying platinum electrode is measured to be around 3 pN. The small value obtained for the clamping force shows that quality factor of carbon nanotube electromechanical resonators is affected by the clamping efficiency of the nanotube ends. ❧ Carbon nanotubes have unique electron ransport properties at high bias voltages. Due to their one dimensional nature, scattering of electrons by phonon are highly nonlinear. At low bias voltages (across the nanotube) phonon scattering is suppressed and the electrons exhibit ballistic transport. At higher bias voltages optical phonon scattering dominates, and subsequently nanotubes heat suddenly. In Chapter 6, the heating of nanotubes is probed using the mechanical vibrations of a carbon nanotube based nanoelectromechanical oscillator. Since the substrate temperature is constant the change in mechanical resonance frequency is attributed to the contraction of the nanotube due to its negative thermal expansion. The bias voltage, at which the mechanical resonance shows a sudden drop, corresponds to experimentally observed optical phonon emission onset voltage for single–walled carbon nanotubes.

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Creator Aykol, Mehmet (author)

Core Title Analysis of carbon nanotubes using nanoelectromechanical oscillators

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 07/31/2012

Defense Date 06/12/2012

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag delamination,electron phonon interaction,graphene,Heating,nanotube,negative differential conductance,negative thermal expansion,OAI-PMH Harvest,platinum,resonator,slack resonator,thermal expansion,van der Waals

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Cronin, Stephen B. (committee chair), Feinberg, Jack (committee member), Wu, Wei (committee member)

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Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-78757

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