© 2014 zuanyi li
TRANSCRIPT
© 2014 Zuanyi Li
THERMAL TRANSPORT IN GRAPHENE-BASED NANOSTRUCTURES AND OTHER
TWO-DIMENSIONAL MATERIALS
BY
ZUANYI LI
DISSERTATION
Submitted in partial fulfillment of the requirements
for the degree of Doctor of Philosophy in Physics
in the Graduate College of the
University of Illinois at Urbana-Champaign, 2014
Urbana, Illinois
Doctoral Committee:
Associate Professor Nadya Mason, Chair
Adjunct Associate Professor Eric Pop, Director of Research
Professor David Cahill
Associate Professor Aleksei Aksimentiev
ii
ABSTRACT
Heat conduction in nanomaterials is an important area of study, with both fundamental and
technological implications. However, little is known about heat flow in two-dimensional (2D)
materials or devices with dimensions comparable to the phonon mean free path (MFP). Here, we
investigated thermal transport in graphene-based nanostructures and several other 2D materials.
First, we measured heat conduction in nanoscale graphene by a substrate-supported
thermometry platform. Short, quarter-micron graphene samples reach ~35% of the ballistic
thermal conductance limit (Gball) up to room temperature, enabled by the relatively large phonon
MFP (~100 nm) in SiO2 substrate-supported graphene. In contrast, patterning similar samples
into nanoribbons leads to a diffusive heat flow regime that is controlled by ribbon width and
edge disorder. These results show how manipulation of device dimensions on the scale of the
phonon MFP can be used to achieve full control of their heat-carrying properties, approaching
fundamentally limited upper or lower bounds.
We also examined the possibility of using this supported platform to measure other materials
through finite element simulations and uncertainty analysis. The smallest thermal sheet
conductance that can be sensed by this method within a 50% error is ~25 nWK-1
at room
temperature, indicating this platform can be applied to most thin films like polymer and nanotube
networks, as well as nanomaterials like nanotube or nanowire arrays, even a single Si nanowire.
Moreover, the platform can be extended to plastic substrates, not limited to the SiO2/Si substrate.
Last, we calculated in-plane (for monolayer and bulk) and cross-plane (for bulk) ballistic
thermal conductances Gball of graphene/graphite, h-BN, MoS2, and WS2, based on full phonon
dispersions from first-principles approach. Then, a rigorous and proper average of phonon mean
free path, λ was simply obtained in terms of Gball and the diffusive thermal conductivity.
Moreover, length-dependent thermal conductivity (k) was estimated using a ballistic-diffusive
model, which agrees with available experimental data and shows increasing k with length until
~100λ before convergence. This indicates that, to observe theoretically predicted k divergence in
low-dimensional systems, simulations and experiments should extend beyond length ~100λ.
Our work provides a comprehensive study of thermal conduction in 2D layered materials in
micro- and nanoscale with an emphasis on ballistic conduction and size effects. The findings
extend our understanding of thermal conduction and how to tune it to reach the requirements for
potential applications like thermal management and thermoelectric conversion.
iii
To my parents
iv
ACKNOWLEDGEMENTS
I would like to thank my advisor, Professor Eric Pop, for his invaluable guidance and
support. Throughout my graduate study, he has been a great mentor and inspired me in various
aspects. His advice and support on research and career development has played an invaluable
role in my graduate life and will be precious to my future life. I would also like to thank the
committee members of my final and preliminary exams: Prof. Nadya Mason, Prof. David Cahill,
Prof. Aleksei Aksimentiev, and Prof. Karin Dahmen, whose valuable feedback is very helpful to
my research projects.
I want to thank all my colleagues in the Pop group, who create a friendly and cooperative
atmosphere within the group. In particular, I would like to thank Dr. Myung-Ho Bae, Dr. Vincent
Dorgan, Dr. David Estrada, and Dr. Feng Xiong, who taught me experimental skills and gave a
lot of valuable advice. I also thank Dr. Albert Liao, Dr. Ashkan Behnam, Dr. Zhun-Yong Ong,
Sharnali Islam, Kyle Grosse, Enrique Carrion, Andrey Serov, Feifei Lian, Chris English, Ning
Wang, and Michal Mleczko for all their help and support. Meanwhile, I would like to thank my
collaborators out of the Pop group: Prof. Wenhui Duan, Prof. Zlatan Aksamija, Prof. Irena
Knezevic, Dr. Lucas Lindsay, Dr. Pierre Martin, Dr. Bing Huang, Dr. Yong Xu, and Yizhou Liu.
I appreciate the resources available to me at the University of Illinois at Urbana-Champaign
(UIUC), especially those at the Micro and Nanotechnology Lab (MNTL) and Materials Research
Lab (MRL). I acknowledge funding support from National Science Foundation (NSF),
Nanoelectronics Research Initiative (NRI), and all scholarships and awards I received from
UIUC.
Finally, I would like to extend special thanks to my parents for their unconditional love and
unwavering support. They have sacrificed so much to make me have better life and education,
and they always believed in me. They are the most important part of my life. Moreover, I thank
all my friends for their help and the wonderful time that we had together, which is an
unforgettable part in my life.
v
TABLE OF CONTENTS
CHAPTER 1: INTRODUCTION ........................................................................................1
1.1 Power Dissipation and Thermal Management ...................................................1
1.2 Two-Dimensional Materials ..............................................................................2
1.3 Organization of Dissertation ..............................................................................4
CHAPTER 2: REVIEW OF THERMAL AND THERMOELECTRIC PROPERTIES OF
GRAPHENE .......................................................................................................................6
2.1 Methods to Measure Thermal Conductivity ......................................................7
2.2 Intrinsic Thermal Conductivity of Graphene ...................................................12
2.3 Extrinsic Thermal Conductivity of Graphene ..................................................14
2.4 Thermoelectric Properties of Graphene ...........................................................22
CHAPTER 3: BALLISTIC TO DIFFUSIVE CROSSOVER OF HEAT FLOW IN
GRAPHENE RIBBONS ...................................................................................................26
3.1 Introduction ......................................................................................................26
3.2 Experimental Method.......................................................................................27
3.3 Model to Analyze Experimental Data ..............................................................36
3.4 Results ..............................................................................................................43
3.5 Discussion and BTE Calculations ....................................................................53
3.6 Summary ..........................................................................................................62
CHAPTER 4: SUBSTRATE-SUPPORTED THERMOMETRY PLATFORM FOR
NANOMATERIALS LIKE GRAPHENE, NANOTUBES, AND NANOWIRES ...........64
4.1 Introduction ......................................................................................................64
4.2 Thermometry Platform Demonstration ............................................................65
4.3 Uncertainty Analysis and Optimization ...........................................................70
4.4 Applicability of the Platform ...........................................................................74
4.5 Summary ..........................................................................................................78
CHAPTER 5: BALLISTIC THERMAL CONDUCTANCE AND LENGTH-DEPENDENT
SIZE EFFECTS IN LAYERED TWO-DIMENSIONAL MATERIALS .........................79
5.1 Introduction ......................................................................................................79
5.2 Phonon Dispersions by First-Principles Calculations ......................................81
5.3 Formalism of Thermal Physical Quantities .....................................................86
5.4 Results and Discussion ....................................................................................93
5.5 Summary ........................................................................................................106
CHAPTER 6: CONCLUSIONS AND OUTLOOK ........................................................108
6.1 Conclusions ....................................................................................................108
6.2 Future Work ...................................................................................................110
REFERENCES ................................................................................................................112
1
CHAPTER 1
INTRODUCTION
1.1 Power Dissipation and Thermal Management
In today’s information technology (IT) era, demands for power usage in electronics are
growing exponentially, and power dissipation has been a big issue in energy consumption.
Taking the United States as an example, the total IT power use in 2007 was estimated to be
~20.6 gigawatts (GW), where the biggest usage is from data centers (~7 GW) [1], as shown in
Fig. 1.1a. The power usage in US data centers continuously increases year-by-year, and has been
tripled from 2000 to 2011 (Fig. 1.1b) [1]. Figure 1.1a also shows that the power usage by work
and home personal computers (PCs) is significant high as well. These issues of power dissipation
are due to increased power density in microprocessor (CPUs) when the transistor size is scaled
down following the Moore’s law. Figure 1.2 shows that the typical CPU power density increased
exponentially over time and flattened out after 2004 when it reached ~100 W/cm2 due to the
introduction of multi-core CPUs [1]. This power density is approximately an order of magnitude
higher than that of a hot plate, implying incredible heat generated in devices. This is clearly
FIG. 1.1: (a) Estimate of US total IT power use in 2007. (b) Estimate of US data center power
use from 2000 to 2011. Adapted from Ref. [1].
a bData centers
(7 GW)
Home PCs
(2.6 GW)Home displays (1.3 GW)
Work displays
(3.2 GW)
Work PCs
(6.5 GW)
Total power
use (2007)
2
reflected in the power usage of US data centers, where about half of it is used for cooling.
To accommodate the increasing power dissipation, measures must be taken to ensure that
the generated waste heat is properly managed. One strategy of managing this waste heat is to
modify the design of the transistor and its surrounding packaging, and to use thermally conduc-
tive materials which can help to transport heat away from regions of localized Joule heating.
Another strategy is to use thermoelectric materials with low thermal conductivity to convert
extra heat to electrical power [2, 3]. Both require effective control of thermal conduction ability
in materials, which relies on comprehensive and deep understanding of thermal transport [4, 5].
1.2 Two-Dimensional Materials
As an ultimately two-dimensional (2D) crystal, graphene, an atomic single-layer of graphite
(Fig. 1.3a), has attracted huge research attention since its discovery in 2004 [6]. It has unique
band structure with a linear dispersion relation near the Brillouin corner (Dirac point), leading to
tunable electron and hole doping [7, 8]. Graphene thus exhibits extraordinary electronic and
FIG. 1.2: Power density vs. time for computer processors manufactured by AMD, Intel, and
Power PC over the past two decades. The exponential trend in power density, although flattened
by the introduction of multi-core CPUs, is a limiting factor for the future scaling of semiconduc-
tor technology. Adapted from Ref. [1].
1
10
100
1990 1994 1998 2002 2006 2010
AMD
Intel
Power PC
CP
U P
ow
er D
ensi
ty (
W/c
m2)
Year
Pentium 4(2005)
Core 2 Duo(2006) Atom
(2008)
3
optical properties like high carrier mobility ~104 cm
2V
-1s
-1 and optical transparency, and presents
promising applications in electronics, optoelectronics, and photonics [9-15]. A drawback of
graphene is the absence of a band gap, which leads to low on/off ratios in transistor performance.
Stimulated by the extensive studies of graphene, other 2D materials start to be fabricated and
investigated. For example, hexagonal boron nitride (h-BN, see Fig. 1.3b) is a 2D insulator (band
gap ~5 eV) and serves as the best dielectric substrate for graphene, because it reduces surface
roughness compared to other substrates like SiO2 [16]. In addition, transition metal dichalcogen-
ides (TMDs, with the form of MX2: M = transition metal, and X = S, Se, or Te) can be exfoliated
into 2D layers (Fig. 1.3c), and they are mostly semiconductors. Among them, 2D molybdenum
disulfide MoS2 was first achieved in experiments and shown a high on/off ratio of ~108 when
used as transistors [17], due to the presence of a band gap ~1.8 eV [18, 19].
These 2D materials can be used independently or further assembled layer-by-layer into van
der Waals (vdW) heterostructures [20] (Fig. 1.3d), which widens their potential applications like
hetero-junctions and tunneling devices. However, compared to extensive studies of the electrical
FIG. 1.3: Schematic structures of (a) graphene [8], (b) single layer of hexagonal boron nitride,
and (c) single layer of transition-metal dichalcogenides MX2 [17], where M is transition metal,
and X is chalcogen atoms. (d) Schematic of vdW heterostructure assembled by graphene, h-BN,
MoS2, and WSe2 [20].
B N
X
M
a
c
bgraphene h-BN
TMD
d vdW heterostructure
graphene
h-BN
graphene
WSe2
MoS2
4
and optical properties of these 2D materials, our understanding of their thermal properties is
relatively lacking, especially for micro and nanoscale in which these materials are used for
devices [21-24]. Their thermal properties inevitably affect heat dissipation in devices, and hence
should be systematically and carefully investigated to understand new physics in micro and
nanoscale as well as explore possible applications in thermal management and thermoelectric
energy conversion.
1.3 Organization of Dissertation
In this dissertation, we investigated thermal transport in graphene and several other 2D lay-
ered materials, with a focus on nanoscale ballistic conduction and size effects. This dissertation
is a combination of experiments, finite element simulations, and theoretical calculations.
In Chapter 2, we briefly reviewed experimental studies on thermal and thermoelectric prop-
erties of graphene, including up-to-date experimental methods, intrinsic and extrinsic thermal
conductivity, interface thermal conductance, and thermoelectric power.
In Chapter 3, we measured thermal conduction in SiO2-supported graphene and graphene
nanoribbons by the substrate-supported thermometry platform. Quasi-ballistic thermal conduc-
tion was observed in quarter-micron long graphene up to room temperature. Graphene nanorib-
bons show clear width-dependent thermal conductivity due to phonon scattering with edge
disorder.
In Chapter 4, we discussed the applicability of the substrate-supported thermometry plat-
form to other materials, based on finite element simulations. We estimated the lower limit of the
measurable thermal sheet conductance by this method, and demonstrated it can be used to
measure most thin films and some nanomaterials like nanotube/nanowire arrays, even a single
nanowire.
5
In Chapter 5, we calculated the ballistic thermal conductances of graphene, h-BN, MoS2,
and WS2 (including their monolayers and bulk) based on full phonon dispersions. From obtained
ballistic thermal conductance, we estimated their average phonon mean free paths and length-
dependent thermal conductivity due to ballistic-diffusive transition.
In Chapter 6, we conclude the dissertation with a summary of key findings and discussions
of possible future directions.
6
CHAPTER 2
REVIEW OF THERMAL AND THERMOELECTRIC PROPERTIES OF GRA-
PHENE*
Since graphene is a single atomic layer of graphite, its thermal properties can be understood
from those of graphite. Figure 2.1a shows the atomic structure of the typical AB (i.e., Bernal)
stacking of graphene layers in graphite. The adjacent carbon atoms within one layer are connect-
ed by strong covalent sp2 bonds with a bonding energy of approximately 5.9 eV [27]. In contrast,
the adjacent graphene layers are stacked via weak van der Waals interactions (~50 meV) [27].
The lattice vibrational modes (phonons) of graphene, directly related to its thermal properties,
are described by the phonon dispersion in the Brillouin zone, as shown in Fig. 2.1b. Due to N = 2
carbon atoms in graphene unit cell, the dispersion contains 3 acoustic (A) and 3N-3 = 3 optical
(O) phonon modes. Besides the usual longitudinal (L) and transverse (T) modes, the unique 2D
nature of graphene leads to flexural (Z) phonon modes, corresponding to out-of-plane atomic
* This chapter is reprinted from Y. Xu, Z. Li, and W. Duan, “Thermal and Thermoelectric Properties of Graphene,”
Small, vol. 10, pp. 2182-2199 (2014). Copyright 2014, Wiley-VCH Verlag GmbH & Co.
FIG. 2.1: (a) Schematic of atomic arrangement in graphene sheets [23]. Dashed lines in the
bottom sheet represent the outline of the unit cell. (b) Calculated phonon dispersions along high-
symmetry lines for monolayer graphene, where red circles [25] and blue triangles [26] are
experimental data plotted for comparison. Inset is the Brillouin zone of graphene.
0
50
100
150
200
Phonon E
(m
eV
)
LO
LA
TA
TO
ZA
ZO
Γ M
K
A plane
B plane
h = 3.35 Å
A planea b
7
displacements. At low phonon wave vector q, the TA and LA modes have linear dispersions of
ωTA ≈ vTAq and ωLA ≈ vLAq, where vTA ≈ 14 km/s and vLA ≈ 21 km/s [28]. These group velocities
are 4-6 times higher than those in Si or Ge due to the strong in-plane sp2 bonds and small mass of
carbon atoms [23]. Whereas, ZA modes show an approximately quadratic dispersion, ωZA ≈ αq2
with α ≈ 6.2 × 10-7
m2/s [29].
2.1 Methods to Measure Thermal Conductivity
Thermal conduction through a solid material is usually modeled on the basis of Fourier's law
that relates the local heat flux J to the local temperature gradient T: J = −k T, where k is the
thermal conductivity and has the units of [Wm-1
K-1
]. The minus sign in front of k implies that
heat flows in the direction opposite to that of the temperature gradient i.e., heat flows from the
hotter to the colder regions of the solid.
Measuring nanoscale thermal transport is quite challenging due to high requirements of
sample fabrication and temperature sensing [30, 31]. So far, methods used to probe thermal
conduction in graphene include optothermal Raman thermometry [32-43], thermoreflectance
technique [44-47], 3ω method [48], micro-resistance thermometry [49-57], electrical self-heating
method [58, 59], and scanning thermal microscopy (SThM) [60-62]. Here we mainly discuss and
compare two techniques widely used to measure in-plane thermal conductivity of graphene, i.e.,
optothermal Raman thermometry and micro-resistance thermometry.
2.1.1 Optothermal Raman Thermometry
The optothermal Raman thermometry technique was developed to measure suspended, mi-
crometer scale (> 2 µm) graphene. A laser light was focused at the center of the suspended
graphene flake to generate a heating power PH and raise temperature locally (Figs. 2.2a-b).
Meanwhile, the Raman spectrum of graphene was recorded and the temperature rise ΔT could be
8
monitored by calibrating it with Raman G peak position [32-36], 2D peak position [37-40] (only
for monolayer), or Stokes/anti-Stokes ratio [41, 42]. By knowing the correlation between PH and
ΔT, as well as the geometry size of suspended graphene, its in-plane thermal conductivity k can
be extracted through the solution of the heat diffusion equation. The early experiments [32-35]
were carried out on graphene strips suspended over a trench (Fig. 2.2a). This was modified in
subsequent experiments [36-41], by adopting circular holes with graphene over them (Fig. 2.2b),
which matches with the radial symmetry of the laser spot and allowed for an analytic solution of
the temperature distribution [22].
FIG. 2.2: (a) Schematic of the optothermal Raman thermometry set-up, where a graphene strip is
suspended over a trench and heated up by a focused laser light [34]. (b) Schematic of the Raman
thermometry set-up with addition of a laser power meter to measure optical transmittance. Inset
is the Raman G peak map of graphene suspended over a circular hole [36]. (c) Scanning electron
microscopy (SEM) image of micro-resistance thermometry device with SLG supported on a
suspended SiO2 membrane between thermometers [49]. Scale bar is 3 µm. (d) Optical image of
bilayer graphene (BLG) suspended over two thermometer pads [53]. Scale bar is 10 µm. (e)
SEM image of a SiO2/Si-supported micro-resistance thermometry device to measure encased
few-layer graphene (FLG) [54]. (f) False-colored SEM image of a GNR array on SiO2/Si with
micro-resistance thermometers. Inset is a zoom-in atomic force microscopy (AFM) image of
GNRs [55].
1 μm
ca
b d
e
f
suspended
BLG
suspended
SLG/SiO2
thermometers
encased FLG
thermometerheater
thermometersheater
9
A major source of uncertainty in different works using the Raman thermometry technique is
determining the laser power absorbed by graphene, that is, determining optical absorbance of
graphene. Balandin et al. [33] and Ghosh et al. [32, 34, 35] evaluated this number by comparing
the integrated Raman G peak intensity of graphene with that of highly oriented pyrolytic graphite
(HOPG), leading to ~9% if converted to optical absorbance for exfoliated single-layer graphene
(SLG) [22]. The measured value of ~9% considered two passes of light (down and reflected back)
and resonance absorption effects due to close proximity of graphene to the substrate, and corre-
sponded to 488-nm wavelength where absorption is higher than the 2.3% long-wavelength limit
[63]. Faugeras et al. [41], Lee et al. [37], and Vlassiouk et al. [42] did not measure the optical
absorbance under the conditions of their experiments and assumed a value of 2.3% for exfoliated
SLG based on a separate optical transmission measurement [63]. Cai et al. [36] and Chen et al.
[38] obtained values of 3.3±1.1% and 3.4±0.7% for CVD SLG by directly measuring the optical
transmittance via addition of a power meter under the suspended portion of graphene (Fig. 2.2b).
The used optical absorbance is very important because it would proportionally change extracted
k. We note that both theory [64] and experiments [65-67] showed an increase of optical absorb-
ance in graphene with decreasing laser wavelength due to many-body effect, and the values of
Cai et al. [36] and Chen et al. [38] are consistent with those experimental results. To obtain
reliable k, it is thus necessary to measure optical absorbance under used laser wavelength and
specific experimental conditions.
Another uncertainty source is the calibration of temperature with features of Raman spec-
trum. It is known that strains and impurities in graphene can affect the Raman peak positions and
their temperature dependence [68], which greatly limits the temperature sensitivity of the Raman
thermometry technique. In addition, heat loss from graphene to the surrounding air was neglect-
10
ed in most experiments, but Chen et al. [38] found that for a large diameter (9.7 µm) graphene
flake, the k obtained in air could be overestimated by 14−40% compared with the value obtained
in vacuum. This implies measurable errors in previous experiments, even though the influence
might be weaker due to smaller sizes of measured graphene. Furthermore, extra uncertainty
could come from the difference in k between suspended and supported portions of graphene, as
well as thermal boundary resistance between graphene and supporting substrates [36]. Overall,
the Raman thermometry technique provides an efficient way to measure k of suspended graphene
with benefits of relatively easy sample fabrication, reduced graphene contamination, and simple
data analysis, but it inevitably has limitations: (i) relatively large uncertainty (up to 40%) [21]; (ii)
difficulty to probe the low temperature regime due to significant heating in graphene by laser; (iii)
inability to be applied to nanometer scale or supported graphene, where edge and interface will
take effect.
2.1.2 Micro-Resistance Thermometry
The micro-resistance thermometry technique is a steady-state method to directly probe heat
flows in materials [30]. It is able to measure both suspended and supported graphene, as well as
at the nanometer scale and in low temperature range, with high resolution of temperature by
employing electrical resistance as thermometers. This technique can be further divided into two
kinds: suspended bridge platform and fully substrate-supported platform. The former was
developed by Shi et al. [69, 70] to measure thermal conductivity of 1D nanostructures, and it has
been widely used for nanotubes [71-76] and nanowires [77-81]. By using this platform, graphene
can be either supported by a suspended SiO2/SiNx membrane connecting two thermometers [49-
52] (Fig. 2.2c) or fully suspended over two thermometer pads [51, 53, 57] (Fig. 2.2d), enabling
measurements of both suspended and supported graphene. The fully substrate-supported plat-
11
form was developed by Jang et al. [54] and Bae et al. [55], where at least two thermometers were
patterned on Si/SiO2-supported graphene (Fig. 2.2e) or GNRs (Fig. 2.2f). In both platforms, one
thermometer serves as the heater to generate heating power PH and a temperature gradient across
graphene by electrical heating, meanwhile all thermometers (including the heater) monitor
temperature changes ΔT in terms of their electrical resistance changes. Then, the thermal con-
ductance/conductivity of measured materials can be extracted in a simple analytic way by
solving its equivalent thermal resistance circuit for the suspended bridge platform [69, 70], while
a complicated 3D numerical (finite element) simulation has to be performed for the substrate-
supported platform due to significant heat leakage into the substrate [54, 55].
Although the data extraction of the suspended bridge platform is easier than that of the sub-
strate-supported platform, as a trade-off the sample fabrication is more complicated for the
former than the latter. Thus, for materials which are hard to be suspended, such as GNRs, the
latter is an advantageous method to be employed. However, the measurable length of interested
materials cannot be longer than a few micrometer for the substrate-supported platform, because
the temperature drops nearly exponentially away from the heater, leading to undetectable ΔT if
other thermometers are far away. It is worth noting that for the platform of graphene supported
by a suspended membrane (Fig. 2.2c) and the substrate-supported platform (Figs. 2.2e and 2.2f),
a control experiment has to be carried out by etching off graphene/GNRs and repeating meas-
urements to calibrate the background heat flow and thermal contact resistance between graphene
and thermometers [49, 50, 54, 55]. This improves the measurement accuracy. Whereas, for the
platform of graphene fully suspended (Fig. 2.2d) such a control experiment cannot be performed,
so the thermal contact resistance could be a main source of uncertainty in results. Overall, the
micro-resistance thermometry technique has very high resolution of temperature (<50 mK) [21,
12
22] and can cover a wide temperature range. It can probe both suspended and supported gra-
phene, as well as in nanometer scale. However, attention should be paid to micro/nanofabrication,
which could introduce contaminations (like residues and defects) and increase the uncertainty
[53].
2.2 Intrinsic Thermal Conductivity of Graphene
In this section, we mainly discuss the “intrinsic” thermal conductivity of SLG based on ex-
perimental results and theoretical analysis. Here, by “intrinsic” we mean isolated, large scale,
pristine graphene without suffering impurity, defect, interface, and edge scatterings, so its
thermal conductivity is only limited by intrinsic phonon-phonon scattering due to crystal anhar-
monicity [21] and electron-phonon scattering. In experiments, suspended, micrometer scale
graphene samples have properties close to intrinsic ones. We thus first summarize current
experimental observations of k in suspended SLG.
Using the Raman thermometry technique described above, suspended micro-scale graphene
flakes obtained by both exfoliation from graphite [32-35, 37, 41] and CVD growth [36, 38-40, 42]
have been measured at room temperature and above. Some representative data versus tempera-
ture from these studies are shown in Fig. 2.3a. The obtained in-plane thermal conductivity values
of suspended SLG generally fall in the range of ~2000-4000 Wm-1
K-1
at room temperature, and
decrease with increasing temperature, reaching about 700-1500 Wm-1
K-1
at ~500 K. The varia-
tion of obtained values could be attributed to different choices of graphene optical absorbance
(see Section 2.1), thermal contact resistance, different sample geometries, sizes, and qualities.
For comparison, we also plot the experimental thermal conductivity of diamond [82], graphite
[82], and carbon nanotubes (CNTs) [71, 83] in Fig. 2.3a.
13
It is clear that suspended graphene has thermal conductivity as high as these carbon allo-
tropes near room temperature, even higher than its 3D counterpart, graphite, whose highest
record of observed in-plane k in HOPG is ~2000 Wm-1
K-1
at 300 K. The presently available data
of graphene based on the Raman thermometry technique only cover the temperature range of
~300-600 K, except one at ~660 K reported by Faugeras et al. [41] showing k ≈ 630 Wm-1
K-1
.
For higher temperature, Dorgan et al. [58] used the electrical breakdown method for thermal
conductivity measurements and found k ≈ 310 Wm-1
K-1
at 1000 K for suspended SLG. The
FIG. 2.3: (a) Experimental thermal conductivity k as a function of temperature T: representative
data for suspended CVD SLG by Chen et al. [38] (solid red square), suspended exfoliated SLG
by Lee et al. [37] (solid purple asterisk) and Faugeras et al. [41] (solid brown pentagon), sus-
pended SLG by Dorgan et al. [58] (solid grey hexagon), suspended exfoliated BLG by Pettes et
al. [53] (solid orange diamond), supported exfoliated SLG by Seol et al. [49] (solid black circle),
supported CVD SLG by Cai et al. [36] (solid blue right-triangle), encased exfoliated 3-layer
graphene (3LG) by Jang et al. [54] (solid cyan left-triangle), supported exfoliated GNR of W ≈
65 nm by Bae et al. [55] (solid magenta square), type IIa diamond [82] (open gold diamond),
graphite in-plane [82] (open blue up-triangle), graphite cross-plane (open blue down-triangle),
suspended single-walled CNT (SWCNT) by Pop et al. [83] (open dark-green circle), and multi-
walled CNT (MWCNT) by Kim et al. [71] (solid light-green circle). (b) Thermal conductance
per unit cross-sectional area, G/A=k/L, converted from thermal conductivity data in (a), com-
pared with the theoretical ballistic limit of graphene (solid line), which can be approximated
analytically as Gball/A ≈ [1/(4.4×105T
1.68)+1/(1.2×10
10)]
-1 Wm
-2K
-1 over the temperature range
1−1000 K [55]. Data in (a) whose sample L is unknown or not applicable are not shown in (b).
100 1000
101
102
103
104
50
k (
Wm
-1K
-1)
T (K)300
encased
3LG
suspended
SLG
Graphite (||)
supported
GNR, W ≈ 65 nm
Graphite (┴)
suspended
BLG
supported
SLG
100 100010
7
108
109
1010
G/A
(W
m-2
K-1
)
T (K)40 300
a b
suspended
SLG
supported
SLG
G/A = k/L
T1.5
T 2
T -1
14
overall trend of present graphene data from 300 K to 1000 K shows a steeper temperature
dependence than graphite (see Fig. 2.3a), consistent with the extrapolation of thermal conductivi-
ty by Dorgan et al. [58]. This behavior could be attributed to stronger second-order three-phonon
scattering (τ ~ T -2
) in graphene than graphite enabled by the flexural (ZA) phonons of suspended
graphene [84], similar to the observations in CNTs [83, 85]. For temperature below 300 K, the
micro-resistance thermometry technique needs to be employed for k measurements. Unfortunate-
ly, there is no reliable data for suspended single-layer graphene until now. However, data do
exist for suspended few-layer graphene (FLG) [51, 53, 56], which will be discussed in Section
2.3.4. It is instructive to compare experimental results with the ballistic limit of graphene as a
check, so we convert measured k in Fig. 2.3a to thermal conductance per unit cross-sectional area,
G/A = k/L, which are re-plotted in Fig. 2.3b with graphene Gball/A (discussed in the next section).
Above room temperature, measured G/A of suspended SLG are more than one order of magni-
tude lower than Gball/A, indicating the diffusive regime. The reason that the value of Faugeras et
al. [41] is much lower than others is because of a much larger L = 22 µm (radius) of their sus-
pended graphene.
2.3 Extrinsic Thermal Conductivity of Graphene
The long phonon MFP in pristine graphene would suggest that it is possible to tune thermal
conductivity more effectively by introducing extrinsic scattering mechanisms which dominate
over intrinsic scattering mechanisms in graphene. For example, isotope scattering, normally
unimportant with respect to other scattering processes, could become significant in graphene
thermal conduction. It could be easier to observe size effects on thermal transport in graphene
because samples do not have to be extremely shrunk. In the following, we discuss various
scattering mechanisms and their influences on thermal conduction separately, giving rise to
15
tunable extrinsic thermal conductivity of graphene.
2.3.1 Isotope Effects
The knowledge of isotope effects on thermal transport properties is valuable for tuning heat
conduction in graphene. Natural abundance carbon materials are made up of two stable isotopes
of 12
C (98.9%) and 13
C (1.1%). Changing isotope composition can modify dynamic properties of
crystal lattices and affect their thermal conductivity [89, 90]. For instance, it has been found that
at room temperature isotopically purified diamond has a thermal conductivity of ~3300 Wm-1
K-1
[91, 92], about 50% higher than that of natural diamond, ~2200 Wm-1
K-1
[82]. Similar effects
have also been observed in 1D nanostructures, boron nitride nanotubes [73]. Very recently, the
first experimental work to show the isotope effect on graphene thermal conduction was reported
by Chen et al. [39]. By using the CVD technique, they synthesized isotopically modified gra-
phene containing various percentages of 13
C. The graphene flakes were subsequently suspended
over 2.8-µm-diameter holes and thermal conductivity was measured by the Raman thermometry
FIG. 2.4: (a) Thermal conductivity of suspended CVD graphene as a function of temperature for
different 13
C concentrations, showing isotope effect [39]. b) Thermal conductivity of suspended
CVD graphene with (red down-triangle) and without (blue up-triangle) wrinkles as a function of
temperature. Also shown in comparison are the literature thermal conductivity data of pyrolytic
graphite samples [86-88]. Inset shows the SEM image of CVD graphene on the Au-coated SiNx
holey membrane. The red arrow indicates a wrinkle. Scale bar is 10 µm [40].
a b
without wrinkle
with wrinkle
6000
300 350 400 450 500 550 600 650
5000
4000
3000
2000
1000
T (K)
k(W
m-1
K-1
)
0 100 200 300 400 500 600
4000
3000
2000
1000
T (K)
k(W
m-1
K-1
)
0
16
technique. As shown in Fig. 2.4a, compared with natural abundance graphene (1.1% 13
C), the k
values were enhanced in isotopically purified samples (0.01% 13
C), and reduced in isotopically
mixed ones (50% 13
C).
2.3.2 Structural Defect Effects
Structural defects are common in fabricated graphene, especially in CVD grown graphene
[93, 94]. The effects of wrinkles [40] and grain size [42] on the thermal conduction of suspended
single-layer CVD graphene have been examined in experiments by using the Raman thermome-
try technique. Chen et al. [40] found that the thermal conductivity of graphene with obvious
wrinkles (indicated by arrows in the inset of Fig. 2.4b) is about 15-30% lower than that of
wrinkle-free graphene over their measured temperature range, ~330-520 K (Fig. 2.4b). Vlassiouk
et al. [42] measured suspended graphene with different grain sizes obtained by changing the
temperature of CVD growth. The grain sizes ℓG were estimated to be 150 nm, 38 nm, and 1.3 nm
in different samples in terms of the intensity ratio of the G peak to D peak in Raman spectra [95].
Since grain boundaries in graphene serve as extended defects and scatter phonons, graphene with
smaller grain sizes are expected to suffer more frequent phonon scattering. Their measured
thermal conductivity does show a decrease for smaller grain sizes, indicating the observation of
the grain boundary effect on thermal conduction. Whereas, the dependence on the grain size
shows a weak power law, k ~ ℓG1/3
, for which there is no theoretical explanation yet [42]. How-
ever, for SiO2-supported graphene, recent theoretical work based on NEGF method showed a
similar but stronger dependence of k on the grain size ℓG in the range of ℓG < 1 µm [96]. Further
experimental studies are required to reveal the grain size effects on the thermal transport of both
suspended and supported graphene.
2.3.3 Substrate Effects in Supported Graphene
17
For practical applications, graphene is usually attached to a substrate in electronic and op-
toelectronic devices, so it is important to understand substrate effects on thermal properties of
supported graphene [97-100]. Seol et al. [49, 50] measured exfoliated SLG on a 300 nm thick
SiO2 membrane by using the micro-resistance thermometry technique with the suspended bridge
platform (Fig. 2.2c). The observed thermal conductivity is k ~ 600 Wm-1
K-1
near room tempera-
ture (solid black circles in Fig. 2.3a). This value is much lower than those reported for suspended
SLG via the Raman thermometry technique, but is still relatively high compared with those of
bulk silicon (~150 Wm-1
K-1
) and copper (~400 Wm-1
K-1
). Another study by Cai et al. [36]
showed CVD SLG supported on Au also has a decreased thermal conductivity, ~370 Wm-1
K-1
(this lower value compared to ~600 Wm-1
K-1
could be caused by grain boundary scattering in
CVD graphene [96]). The thermal conductivity reduction in supported graphene is attributed to
substrate scattering, which strongly affects the out-of-plane flexural (ZA) mode of graphene [49,
97, 101]. This effect becomes stronger in encased graphene, where graphene is sandwiched
between bottom and top SiO2. The thermal conductivity of SiO2-encased exfoliated SLG was
measured to be below 160 Wm-1
K-1
, reported by Jang et al. [54] using the micro-resistance
thermometry technique with the substrate-supported platform (Fig. 2.2e). For encased graphene,
besides the phonon scattering by bottom and top oxides, the evaporation of top oxide could cause
defects in graphene, which can further lower thermal conductivity. Knowing encased graphene k
is useful for analyzing heat dissipation in top-gated graphene devices.
2.3.4 Interlayer Effects in Few-Layer Graphene
Interlayer scattering as well as top and bottom boundary scattering could take place in few-
layer graphene, which could be another mechanism to modulate graphene thermal conductivity.
It is interesting to investigate the evolution of the thermal conductivity of FLG with increasing
18
thickness, denoted by the number of atomic layers (N), and the critical thickness needed to
recover the thermal conductivity of graphite.
Several experimental studies on this topic have been conducted for encased [54], supported
[52], and suspended FLG [35, 56], and their results are summarized in Fig. 2.5. Jang et al. [54]
measured the thermal transport of SiO2-encased FLG by using the substrate-supported, micro-
resistance thermometry platform (Fig. 2.2e). They found that the room-temperature thermal
conductivity increases from ~50 to ~1000 Wm-1
K-1
as the FLG thickness increases from 2 to 21
layers, showing a trend to recover natural graphite k. This strong thickness dependence was
explained by the top and bottom boundary scattering and disorder penetration into FLG induced
by the evaporated top oxide [54]. Very recently, another similar yet less pronounced trend was
observed in SiO2-supported FLG by Sadeghi et al. [52] using the suspended micro-resistance
FIG. 2.5: Experimental in-plane thermal conductivity near room temperature as a function of the
number of layers N for suspended graphene by Ghosh et al. [35] (open blue diamond) and by
Jang et al. [56] (open green square), SiO2-supported graphene by Seol et al. [49] (open red circle)
and Sadeghi et al. [52] (solid red circle), and SiO2-encased graphene by Jang et al. [54] (solid
black triangle). The data show a trend to recover the value (dashed line) measured by Sadeghi et
al. [52] for natural graphite source used to exfoliate graphene. The gray shaded area shows the
highest reported k values of pyrolytic graphite [86-88]. Adapted from Ref. [52].
1 10
100
1000
k (
Wm
-1K
-1)
number of layers
Suspended
Ghosh et al.
Encased
Jang et al.
Suspended
Jang et al. Supported
Sadeghi et al.
Natural graphite
T ~ 300 K
19
thermometry platform (similar to Fig. 2.2c). As shown by red dots in Fig. 2.5, the measured
room-temperature k increases slowly as increasing thickness, and the recovery to natural graphite
would occur even more than 34 layers. The difference between the results by Jang et al. [54] and
Sadeghi et al. [52] is not unexpected, because encased FLG k could be suppressed much more in
thin layers than thick layers due to the effect of top oxide, and hence shows a stronger thickness
dependence.
For suspended FLG, there are two contradictory observations in the thickness dependence.
At first, based on the Raman thermometry technique (Fig. 2.2a) Ghosh et al. [35] showed a
decrease of suspended FLG k from the SLG high value to regular graphite value as thickness
increases from 2 to 8 layers (open diamonds in Fig. 2.5). The k reduction was explained by the
interlayer coupling and increased phase-space states available for the phonon Umklapp scattering
in thicker FLG [35]. However, a very recent study by Jang et al. [56] seems to show a different
thickness trend for suspended FLG. They measured thermal conductivity of suspended graphene
of 2-4 and 8 layers by using a modified T-bridge micro-resistance thermometry technique. The
obtained room-temperature k for 2-4 layers is about 300-400 Wm-1
K-1
with no apparent thickness
dependence, while k for 8-layer shows an increase to ~600 Wm-1
K-1
(open squares in Fig. 2.5).
Surprisingly, this trend is qualitatively in agreement with that of Sadeghi et al. [52] for supported
FLG; both show similar increasing amounts of k from 2 to 8 layers (Fig. 2.5), despite a small
decrease from 2 to 4 layers in the former, which could arise from different sample qualities and
measurement uncertainty. Given opposite thickness trends of Ghosh et al. [35] and Jang et al.
[56], further experimental works are required to clarify the real thickness-dependent k in sus-
pended FLG. Moreover, we want to point out that suspended FLG k values of Jang et al. [56] are
close to those reported by Pettes et al. [53] for suspended bilayer graphene (BLG), ~600 Wm-1
K-
20
1 at room temperature (see Fig. 2.3a). Both are much lower than suspended SLG k. The latter
attributed this to phonon scattering by a residual polymeric layer on graphene [53], even though
the former claimed that electrical current annealing was used to remove polymer residues [56].
2.3.5 Cross-Plane Thermal Conduction
A remarkable feature of graphite and graphene is that their thermal properties are highly ani-
sotropic. Despite high thermal conductivity along the in-plane direction, heat flow along the
cross-plane direction (c-axis) is hundreds of times weaker, limited by weak van der Waals
interactions between layers (for graphite) or with adjacent materials (for graphene). For example,
the thermal conductivity along the c-axis of pyrolytic graphite is only ~6 Wm-1
K-1
at room
temperature [82] (Fig. 2.3a). For graphene, it is often attached to a substrate or embedded in a
FIG. 2.6: Experimental thermal interface conductance G vs. temperature for SLG/SiO2 by Chen
et al. [48] (open purple diamond), FLG/SiO2 by Mak et al. [44] (open purple square), CNT/SiO2
by Pop et al. [102] (solid purple right-triangle), Au/SLG by Cai et al. [36] (solid gold diamond),
Au/Ti/SLG/SiO2 (solid blue circle) and Au/Ti/graphite (solid orange circle) by Koh et al. [45],
interfaces of graphite with Au (solid magenta square), Al (solid gray up-triangle), Ti (solid green
asterisk) by Schmidt et al. [103], interfaces of Al/SLG/SiO2 without treatment (open black up-
triangle), with oxygen treatment (Al/O-SLG/SiO2, open green up-triangle), and with hydrogen
treatment (Al/H-SLG/SiO2, open red up-triangle) by Hopkins et al. [46].
100 100010
100
50
20
40
G (
MW
m-2
K -
1)
T (K)
300
Al/O-SLG/SiO2
Au/SLG
Au/Graphite
Au/Ti/SLG/SiO2
Al/SLG/SiO2
Al/H-SLG/SiO2
SLG/SiO2
CNT/SiO2
Ti/Graphite
FLG/SiO2
Au/Ti/Graphite
21
medium for potential applications. Heat conduction along the cross-plane direction is character-
ized by the thermal interface/boundary conductance between graphene and adjacent materials,
which could become a limiting dissipation bottleneck in highly scaled graphene devices and
interconnects [23, 104-107].
The thermal interface conductance across graphene/graphite and other materials has been
measured by using 3ω method [48], time-domain thermoreflectance (TDTR) technique [44-47,
103, 108], and Raman-based method [36, 43]. Most experimental data available to date are
shown in Fig. 2.6, and they are consistent with each other in general, given the variations of
sample qualities and measurement techniques. Chen et al. [48] and Mak et al. [44] showed the
thermal interface conductance per unit area of graphene/SiO2 is G ~ 50-100 MWm-2
K-1
at room
temperature, with no strong dependence on the FLG thickness. Their values are close to that of
CNT/SiO2 [102], reflecting the similarity between graphene and CNT. Schmidt et al. [103]
measured G of the graphite/metal interfaces, including Au, Cr, Al, and Ti. Among them, the
graphite/Ti has the highest G ~120 MWm-2
K-1
, and the graphite/Au interface has the lowest G
~30 MWm-2
K-1
near room temperature. Their G of graphite/Au is consistent with the value by
Norris et al. [108] and values of SLG/Au by Cai et al. [36] and FLG/Au by Ermakov et al. [43].
Koh et al. [45] later measured heat flow across the Au/Ti/N-LG/SiO2 interfaces with the layer
number N=1−10. Their observed room-temperature G is ~25 MWm-2
K-1
, which shows a very
weak dependence on the layer number N and is equivalent to the total thermal conductance of
Au/Ti/graphite and graphene/SiO2 interfaces acting in series. This indicates that the thermal
resistance of two interfaces between graphene and its environment dominates over that between
graphene layers. Interestingly, Hopkins et al. [46] showed the thermal conduction across the
Al/SLG/SiO2 interface could be manipulated by introducing chemical adsorbates between the Al
22
and SLG. As shown in Fig. 2.6, their measured G of untreated Al/SLG/SiO2 is ~30 MWm-2
K-1
at room temperature, in agreement with Zhang et al. [47]. The G increases to ~42 MWm-2
K-1
for oxygen-functionalized graphene (O-SLG), while decreases to ~23 MWm-2
K-1
for hydrogen-
functionalized graphene (H-SLG). These effects were attributed to changes in chemical bonding
between the metal and graphene, and are consistent with the observed enhancement in G from
the Al/diamond [109] to Al/O-diamond interfaces [110].
2.4 Thermoelectric Properties of Graphene
Thermoelectric materials can convert waste heat into electricity by the Seebeck effect and
use electricity to drive electronic cooling or heating by the Peltier effect. Thermoelectric devices
are all-solid-state devices with no moving parts, thus are silent, reliable and scalable. However,
they only find limited applications due to their low efficiency. The efficiency of a thermoelectric
material is determined by the thermoelectric figure of merit (ZT), which typically is defined as
2S
ZT Tk
(2.1)
where σ is the electrical conductivity, S is the Seebeck coefficient [also called thermoelectric
power (TEP) or thermopower], T is the absolute temperature, and the thermal conductivity k =
ke+kl have contributions from electrons (ke) and lattice vibrations (kl). ke is usually extracted
based on the Wiedemann-Franz Law ke/σ = L0T, where the Lorenz number L0 is equal to
2.44×10-8
WΩK-2
for free electrons. This law does not always hold. For example, ke becomes
zero for a delta-shaped transport distribution [111]. However, in graphene ke is negligible with
respect to kl [112-116], similar to CNTs [71, 117]. Currently, the state-of-art commercial
thermoelectric materials, like Bi2Ti3, have room-temperature ZT around 1 [3].
23
Thermoelectric transport in graphene has been experimentally investigated in the past five
years [118-129]. The Seebeck coefficient S and electrical conductance Ge of graphene can be
measured against the gate voltage Vg (thus, carrier density nc) simultaneously via a widely-used
microfabricated structure (inset of Fig. 2.7a), which was developed by Small et al. [130] to
measure thermoelectric transport in CNTs. Figure 2.7 shows typical results of measured Ge and S
as a function of Vg in graphene at different temperatures [118]. The Seebeck coefficient S shows
two peaks near the Dirac point (charge neutrality point) and changes its sign across the Dirac
point as the majority carrier switches from electron to hole. The room-temperature peak values
of S for SLG and BLG are observed to be ~50-100 µVK-1
in different experiments [118-122].
For high carrier density nc (i.e., high |Vg|), the measured Seebeck coefficient scales as S~|nc|-1/2
for SLG due to its linear dispersion [119], while S~1/|nc| for BLG due to its hyperbolic disper-
FIG. 2.7: (a) Electrical Conductance Ge and (b) thermopower TEP of a graphene sample as a
function of back gate voltage Vg for T = 300 K (square), 150 K (circle), 80 K (up triangle), 40 K
(down triangle), and 10 K (diamond). Upper inset: SEM image of a typical device for thermoe-
lectric measurements, scale bar is 2 µm. Lower inset: TEP values taken at Vg = −30 V (square)
and −5 V (circle). Dashed lines are linear fits to the data. Adapted from Ref. [118].
Conduct
ance
a
b
24
sion [121], consistent with theories. Importantly, the simultaneous measurements of Ge and S
enable testing the validation of the semiclassical Mott relation [131]:
2 2 1
3 | | F
geBE E
e g
dVdGk TS
e G dV dE
(2.2)
where kB is the Boltzmann constant, e is the electron charge, and EF is the Fermi energy. For
SLG the measured S shows a linear T dependence (inset of Fig. 2.7b) and matches calculated S
from measured Ge by Eq. 2.2 [118-120], indicating an agreement with the Mott relation. For
BLG, however, the agreement only holds for high carrier density; for low carrier density there is
an obvious difference between measured and calculated S as well as a deviation from the linear T
dependence at high temperature [121, 122]. This failure of the Mott relation was attributed to the
low Fermi temperature in BLG.
The thermoelectric properties of materials can be also probed by using a conducting tip to
measure the thermoelectric voltage between the sample and tip, induced by a given temperature
difference between them. By employing atomic force microscopy (AFM) and scanning tunneling
microscopy (STM) techniques, Cho et al. [132] and Park et al. [133] measured the thermopower
of epitaxial graphene on SiC, respectively. The advantage of this method is the simultaneous
imaging of the sample structure and thermoelectric signals with a spatial resolution of atomic-
scale. Since the Seebeck coefficient relies on the sample local density of states (LDOS) near the
Fermi energy, and LDOS can be quite different in the presence of boundaries and disorders [134-
138], thermoelectric imaging allows us to probe grain boundaries, wrinkles, defects, and impuri-
ties in graphene, which may not be reflected in topography images [132, 133].
For practical applications, the Seebeck coefficient and power factor σS 2
of graphene should
be improved. Some experimental efforts have been made in this direction. Wang et al. [121]
observed enhanced S below room temperature in a dual-gated BLG device, resulting from the
25
opening of a band gap by applying a perpendicular electric field on BLG. Additionally, the
Seebeck coefficient and power factor of FLG could be enhanced at high temperature (>500 K)
by molecular attachments [139] and oxygen plasma treatment [140], attributed to the band gap
opening. By constructing the c-axis preferentially oriented nanoscale Sb2Te3 film on monolayer
graphene, both S and σ were increased, benefiting from a highway for carriers provided by
graphene [141]. From a practical point of view, Hewitt et al. [142] focused on maximizing the
power output of FLG/polyvinylidene fluoride composite thin films by considering the absolute
temperature, temperature gradient, load resistance, and physical dimensions of films.
26
CHAPTER 3
BALLISTIC TO DIFFUSIVE CROSSOVER OF HEAT FLOW IN GRA-
PHENE RIBBONS*
3.1 Introduction
The thermal properties of graphene are derived from those of graphite, and are similarly ani-
sotropic. The in-plane thermal conductivity of isolated graphene is high, ~2000 Wm-1
K-1
at room
temperature, due to the strong sp2 bonding and relatively small mass of carbon atoms [21, 23, 24,
58]. Heat flow in the cross-plane direction is nearly a thousand times weaker, limited by van der
Waals interactions with the environment (for graphene) [45] or between graphene sheets (for
graphite) [21, 23]. Recent studies have suggested that the thermal conductivity of graphene is
altered when in contact with a substrate through the interaction between vibrational modes
(phonons) of graphene and those of the substrate [49, 54, 97, 101]. However an understanding of
heat flow properties in nanometer scale samples of graphene [or any other two-dimensional (2D)
materials] is currently lacking.
By comparison, most graphene studies have focused on its electrical properties when con-
fined to scales on the order of the carrier mean free path (MFP) [105, 143-149]. For example,
these have found that “short” devices exhibit near-ballistic behavior [145], Fabry-Perot wave
interference [146], and “narrow” nanoribbons display a steep reduction of charge carrier mobility
[144, 147]. Previous studies do exist for heat flow in three-dimensional (3D) structures like
nanowires and nanoscale films. For instance, ballistic heat flow was observed in suspended
GaAs bridges [150] and silicon nitride membranes [151] at low temperatures, of the order 1 K.
* This chapter is reprinted from M.-H Bae
†/Z. Li
†, Z. Aksamija, P. N. Martin, F. Xiong, Z.-Y. One, I.
Knezevic, and E. Pop, “Ballistic to Diffusive Crossover of Heat Flow in Graphene Ribbons,” Nature Communica-
tion, vol. 4, p. 1734 (2013). Copyright 2014, Macmillan Publishers Limited. †Denotes equal contribution.
27
Conversely, suppression of thermal conductivity due to strong edge scattering effects was noted
in narrow and rough silicon nanowires [80, 81], up to room temperature. Yet such effects have
not been studied in 2D materials like graphene, and ballistic heat conduction has not been
previously observed near room temperature in any material.
In this work we find that the thermal properties of graphene can be tuned in nanoscale de-
vices comparable in size to the intrinsic phonon MFP. (By “intrinsic” thermal conductivity or
phonon MFP we refer to that in large samples without edge effects, typically limited by phonon-
phonon scattering in suspended graphene, and by substrate scattering in supported graphene, here
λ ≈ 100 nm at room temperature). The thermal conductance of “short” quarter-micron graphene
reaches up to 35% of theoretical ballistic upper limits [96]. However, the thermal conductivity of
“narrow” graphene nanoribbons (GNRs) is greatly reduced compared to that of “large” graphene
samples. Importantly, we uncover that nanoengineering the GNR dimensions and edges is
responsible for altering the effective phonon MFP, shifting heat flow from quasi-ballistic to
diffusive regimes. These findings are highly relevant for all nanoscale graphene devices and
interconnects, also suggesting new avenues to manipulate thermal transport in 2D and quasi-one-
dimensional (1D) systems.
3.2 Experimental Method
3.2.1 Fabrication and Characterization of Graphene Nanoribbons
Graphene monolayers were deposited on SiO2/Si (~290 nm/0.5 mm) substrates by mechani-
cal exfoliation from natural graphite. Graphene thickness and GNR edge disorder were evaluated
with Raman spectroscopy [45, 152, 153]. Samples were annealed in Ar/H2 at 400 °C for 40
minutes. We used two approaches to define and fabricate graphene nanoribbon (GNR) arrays
with pitch ~150 nm and varying widths: one with a PMMA mask (Fig. 3.1a), the other with an
28
Al mask (Fig. 3.1b) [154]. Double poly(methyl methacrylate) (PMMA) layers (PMMA 495K
A2/PMMA 950K A4) were coated on the Si/SiO2 substrate. For the electron (e)-beam lithogra-
phy, we used 30 keV e-beam accelerating voltage. After opening 40 nm wide PMMA windows,
we etched the graphene exposed through the windows with an oxygen plasma, creating GNRs of
width W (Fig. 3.1a). This PMMA mask method was used for the W ≈ 130 nm, ~85 nm, and ~65
nm wide GNRs. For the narrower ~45 nm GNRs we used Al masks (Fig. 3.1b). In this case, after
opening the PMMA windows, instead of plasma etching, we deposited 30 nm thick Al and
obtained ~45 nm wide Al strips on graphene. After plasma etching of exposed graphene and Al
etching (type A, Transene Company) we obtained ~45 nm wide GNRs. Figure 3.2 shows the
atomic force microscopy (AFM) images of fabricated GNR arrays with W ≈ 130 nm, 85 nm, 65
nm, and 45 nm, respectively. The bottom and top regions of Figs. 3.2a and 3.2c correspond to the
un-etched pristine graphene.
FIG. 3.1: Process to define the GNR widths. (a) PMMA mask method. (b) Al mask method.
Plasma etching
W
Al deposition and lift-off Plasma etching
Al etching W W
a
b
PMMA
grapheneSiO2
Al
29
To characterize the prepared GNRs, we performed Raman spectroscopy with a 633 nm
wavelength laser (~1 µm spot size) as shown in Fig. 3.3. Even before patterning into GNRs, we
selected only monolayer graphene flakes, identifiable through their 2D (G’) to G Raman peak
ratio, and through a single fitted Lorentzian to their 2D (G’) peak. The unpatterned graphene
samples had no identifiable D peak, indicative of little or no disorder [155]. On the other hand,
the GNR arrays showed a pronounced D band consistent with the presence of edge disorder
[152]. The peak intensity of the D band with respect to that of the G band increases with narrow-
er GNR width. Because the edges of graphene serve as defects by breaking the translational
symmetry of the lattice, the larger fraction of the edge in narrow GNRs will enhance the D peak
[156]. The inset of Fig. 3.3a quantitatively shows the behavior by calculating the ratio of inte-
grated D band (ID) to G band (IG), ID/IG, as a function of GNR width (symbols). The width
FIG. 3.2: Atomic force microscopy (AFM) images of GNR arrays. (a) W ~ 130 nm GNRs. (b) W
~ 85 nm GNRs. (c) W ~ 65 nm GNRs. Inset: AFM image near metal electrodes. (d) W ~ 45 nm
GNRs. The axis units are given in microns on each panel.
graphene
a
d
0.0 0.2 0.4 0.6 0.8 1.0 1.2
c
0.0 0.2 0.4 0.6 0.8 1.0 1.2
[µm]b
30
dependence of the peak ratio follows a relation of ID/IG = cW-1
with c = 210 nm (dashed line),
which is consistent with previous reports of GNR characterization [95, 152]. Figure 3.3b shows
the D, G, and D’ peaks in detail of fabricated GNR arrays and un-patterned graphene with 633
nm wavelength laser. Figures 3.3c-g show the Raman 2D band spectra (scattered points) for the
samples. All 2D bands are fit by a single Lorentzian peak (solid red curves) with ~2650 cm-1
peak position, which is consistent with previous reports of monolayer graphene and GNRs [152].
FIG. 3.3: Raman spectra of GNR arrays and un-patterned graphene. (a) Raman signal for W ~
130, 85, 65, 45 nm GNRs and un-patterned graphene. Each spectrum is vertically offset for
clarity. Inset is the ID/IG ratio as a function of GNR width, consistent with the enhanced role of
edge disorder in narrower GNRs [105, 152, 153]. (b) Zoomed-in D, G, and D’ bands of all
samples. (c-g) 2D bands with a single Lorentzian fit for all samples, consistent with the existence
of monolayer graphene.
a
2400 2600 2800
Inte
nsity (
a.
u.)
Raman shift (cm-1)
2400 2600 2800
Raman shift (cm-1)
2400 2600 2800
Raman shift (cm-1)
2400 2600 2800
Raman shift (cm-1)
2400 2600 2800
Raman shift (cm-1)
W=45 nm GNRW=65 nm GNRW=85 nm GNRW=130 nm GNR
wide graphene
1300 1400 1500 1600 1700
Inte
nsity (
a.u
.)
Raman shift (cm-1)
45 nm GNR
65 nm GNR
85 nm GNR
130 nm GNR
Graphene
D
G
D’
b c
d e f g
0 100 2000
2
4
6
8
W (nm)
I D/
I G
DG
2D (G’)W=45 nm
W=65 nm
W=85 nm
W=130 nm
graphene
1500 2000 2500
Raman Shift (cm-1)
Inte
nsity (a
.u.)
~ 1/W
31
3.2.2 Thermometry Platform and Measurements
To fabricate thermometry platform for thermal measurements, electron (e)-beam lithography
was used to pattern the heater and sensor thermometers [54, 157] as long, parallel, ~200-nm-
wide electrodes with current and voltage probes, with a separation of L ≈ 260 nm. Electrodes
were deposited by successive evaporation of SiO2 (20 nm) for electrical insulation and Ti/Au
(30/20 nm) for temperature sensing. Figure 3.4 illustrates scanning electron microscopy (SEM)
images of our several experimental test structures, showing graphene and GNR arrays supported
FIG. 3.4: Measurement of heat flow in graphene ribbons. (a) Scanning electron microscopy
(SEM) image of parallel heater and sensor metal lines with ~260 nm separation, on top of
graphene sample (colorized for emphasis). A thin SiO2 layer under the metal lines provides
electrical insulation and thermal contact with the graphene beneath. (b) Similar sample after
graphene etch, serving as control measurement for heat flow through contacts and SiO2/Si
underlayers. (c) Heater and sensor lines across array of graphene nanoribbons (GNRs). (d)
Magnified portion of array with GNR widths ~65 nm; inset shows atomic force microscopy
(AFM) image of GNRs. Scale bars of (a-d) are 2 μm, 1 μm, 2 μm and 1 μm, respectively. (e)
Three-dimensional (3D) simulation of experimental structure, showing temperature distribution
with current applied through heater line.
a
b
∆T (K)
c d
e
1010.1
290 nm SiO2 Silicon
graphene
SiO2
32
on a SiO2/Si substrate, as well as bare SiO2 with graphene etched off by an oxygen plasma
(however, graphene still exists under the metal electrodes, consistent with the other samples).
Figure 3.5 shows the measurement set-up for the SiO2 sample (Fig. 3.4b) as an example. To
block environmental noise including electrostatic discharge, π-filters with a cut-off frequency of
2 MHz were inserted across all measurement lines. To control the temperature, Physical Property
Measurement System (PPMS, Quantum Design) was used with a temperature range of 10 K –
363 K. Inside the PPMS, the vacuum environment is always a few ~10-3
Torr, rendering convec-
tive heat losses negligible.
The measurement proceeds as follows. In the heater, we apply a sinusoidal voltage with fre-
quency lower than 2 mHz through a standard resistor of 1 kΩ to flow current with a range of
±1.5 mA, generating a temperature gradient across the sample. To obtain the response of the
FIG. 3.5: Scanning electron micrograph (SEM) of thermometry platform and measurement
configuration. Scale bar is 4 µm. Image taken of sample after graphene was etched off, and after
all electrical and thermal measurements were completed. Dark region around “part 1” is substrate
charging due to previous SEM imaging performed to obtain Fig. 3.4b in the main text.
VH
Lock-in amplifier
2.147 kHz
1 k
Ω
< 2
mH
z
π-filter
sensor
heater
1 MΩ
33
sensor (Fig. 3.6a), we measured its resistance change by a standard lock-in method with excita-
tion frequency 2.147 kHz and root-mean-square (rms) current 1 μA (carefully chosen to avoid
additional heating). All electrical measurements were performed in a four-probe configuration.
Both electrode resistances are calibrated over the full temperature range for each sample (Fig.
3.6b), allowing us to convert measured changes of resistance into changes of sensor temperature
ΔTS as a function of heater power PH (Fig. 3.6c). We sometimes found that the electrical re-
sistance of the sensor slowly drifted (increased) with time at room temperature. However, this
effect was stabilized after annealing the sample at 363 K for 5 min, eliminating resistance drift at
room temperature. Therefore, this behavior could be related to the absorbed water on the metal
electrodes.
3.2.3 Experimental Data and Error
Figure 3.6a shows the measured sensor resistance change, ∆RS, as a function of the power
applied to the heater, PH, at T = 100 K for the SiO2 sample (Fig. 3.4b). The black (for negative
heater current, IH) and red (for positive IH) lines overlap with each other, indicating the meas-
urement is symmetric and reliable. The calibration for sensor and heater resistance vs. tempera-
ture is shown in Fig. 3.6b; thus, sensor heating due to heater power ∆RS can be converted to a
temperature rise, ∆TS, as shown in Fig. 3.6c by using the resistance-temperature calibration
curve. The fitted slope of the ∆TS vs. PH curve in Fig. 3.6c is 0.01797 K/µW, which is then used
for the extraction of thermal properties through simulations (see Section 3.3). Figure 3.6d shows
the measured ratio of PH to ∆TS for all representative samples as a function of ambient tempera-
ture from 20 K to 300 K. The uncertainty of the electrical thermometry measurement is ~2%
(Tables 3.1 and 3.2), which is comparable to the symbol size on this plot. Thus, although data are
available down to 20 K, the values are distinguishable without ambiguity only when T ≥ ~70 K,
34
which is the main temperature range shown in Section 3.5.
We note that PH/ΔTS shown in Fig. 3.6d is not the thermal conductance through graphene,
because ΔTS is the temperature rise in the sensor, not the temperature drop from the heater to
sensor, and PH is the heat generated in the heater, not the one flowing in graphene. The thermal
FIG. 3.6: Measurement process. (a) Sensor resistance change, ∆RS as a function of heater power,
PH at T =100 K for the SiO2 sample (Fig. 3.5). Red and black lines are taken with current flow in
opposing direction. (b) Calibration of sensor and heater resistances as a function of temperature.
The inset shows the R-T curve and slope of the sensor near T = 100 K. (c) Converted sensor
temperature rise, ∆TS as a function of heater power, PH at T = 100 K from (a) and (b). The slope
of the fitted red line is ∆TS/PH = 0.01797 K/µW, which is later used to extract the thermal
properties of the SiO2 layer (see Fig. 3.9). (d) Measured ratio of heater power to sensor tempera-
ture rise for all representative samples. The uncertainty of these data is ~2% (Tables 3.1 and 3.2),
comparable to the symbol size. Although this plot shows all raw data taken, the values can be
distinguished without ambiguity only at T ≥ 70 K, which is the temperature range displayed in
the Figs. 3.12 and 3.16.
0 50 100 150 200 250 300 350
45
50
55
60
65
70
R (
Oh
m)
T (K)
Sensor
Heater
0 20 40 60 80 100 120 1400
50
100
150
200
R
S (
m
)
PH (W)
0 20 40 60 80 100 120 1400.0
0.5
1.0
1.5
2.0
2.5
T
S (
K)
PH (W)
90 95 100 105 11051.5
52.0
52.5
53.0
53.50.0866 Ω/K
ΔTS/PH = 0.01797 K/µW
T=100 K
T=100 K
0 50 100 150 200 250 3000
10
20
30
40
50
60
70
80
90
100
SiO2 (part 2)
GNR, W = 45 nm
GNR, W = 65 nm
GNR, W = 85 nm
GNR, W = 130 nm
Graphene (L=0.26 um)
PH/
TS (W
/K)
T (K)
a b
c ddistinguishable
35
conductance of the graphene cannot be immediately extracted from our raw data, due to heat
leakage into the substrate (a drawback of the substrate-supported thermometry method). Instead,
we employ 3D simulations to carefully account for all heat flow paths and, by comparison with
the experiments, to obtain the thermal conductance of the graphene samples (see Section 3.4).
Figure 3.7a shows the sensor resistance as a function of count number (time) without apply-
ing current to the heater at T = 102 K. The standard deviation of the scattered data points is δR =
3.1 mΩ, which corresponds to δT ~ 36 mK by using calibration coefficient 0.0866 Ω/K obtained
in Fig. 3.6b. Thus, the error of the temperature reading is ±36 mK, primarily due to slight ambi-
ent temperature fluctuations in the PPMS (consistent with a fluctuation of ±30 mK of the dis-
played temperature on the PPMS monitor). Zooming into the circled region of Fig. 3.7a, we note
a resistance fluctuation δR = 0.17 mΩ, corresponding to a temperature uncertainty ±2 mK due to
electrical measurement instruments. Therefore, during the time scales of most of our measure-
ments our temperature accuracy is limited by the ambient temperature control of the PPMS
rather than by the electrical measurements themselves.
FIG. 3.7: Measurement error and thermal steady-state. (a) Sensor resistance as a function of
count number (time) at background T = 102 K. (b) Heater power, PH and corresponding re-
sistance change in sensor, ΔRS as a function of time.
0 500 1000 1500 2000 2500
53.385
53.390
53.395
53.400
RS (
)
count number
0 2 4 6 8 10 12 14 160
20
40
60
80
100
120
Time (min.)
PH (W
)
0
50
100
150
200
R
S (
m
)
a b
36
The sweep speed of the heater power is chosen to be sufficiently slow to reach thermal
steady-state between the heater and sensor. Figure 3.7b shows the heater power (PH) sweep with
time and corresponding resistance change in the sensor, ∆RS. Data shown here correspond to the
linear ramp in Fig. 3.6a. After ~15 minutes, the heater power reaches its maximum, and the
change of sensor resistance follows the same trend without delay, indicating that the thermal
steady-state between the heater and the sensor is established during the entire sweep process. If
the sweep speed of the heater power is too fast to reach the steady-state, the data point at PH ~
110 µW in Fig. 3.6a will deviate from the linear trend. We also verified this by a comparison
between the corresponding constant DC power and the above methods.
3.3 Model to Analyze Experimental Data
3.3.1 3D Simulation to Extract Thermal Properties
To extract the thermal properties of graphene, GNRs, or the SiO2 substrate from the meas-
ured ∆TS vs. PH, we use a commercial software package (COMSOL) to set up a three-
dimensional (3D) finite element method (FEM) model of the entire structure. A typical setup is
shown in Fig. 3.8, where only a half of the sample is included due to the symmetry plane which
bisects the region of interest. The size of the Si substrate is 100×50×50 μm3, covered by a 290
nm thick SiO2 layer. While the simulated Si substrate is slightly smaller than the actual Si chip
employed in practice (to manage computational complexity and meshing), the size of the simu-
lated structure has been carefully chosen and verified to reproduce all heat flow through the
substrate itself. Figure 3.8b shows the zoomed-in structure containing the core area of the
thermometry, with GNRs, heater, and sensor highlighted by different colors. A more zoomed-in
structure is shown in Fig. 3.8c, where from top to bottom different layers are 40 nm metal, 25 nm
top oxide insulator, GNRs, 290 nm bottom oxide, and silicon, respectively.
37
To perform the simulation, the bottom surface and three side surfaces (except symmetry
plane) of the Si substrate are held at the ambient temperature, i.e. isothermal boundary condition.
Other outer surfaces of the whole structure are treated as insulated, i.e. adiabatic boundary
condition. The Joule heating in the heater is simulated by applying a power density within the
heater metal, and the stationary calculation is performed to obtain the temperature distribution in
the steady state, as shown in Figs. 3.4e and 3.8d typically. After calculating the average tempera-
ture rise in the measured segment of the sensor, we obtain the simulated value of (∆TS/PH). Thus,
the simulation effectively fits the thermal conductance (G) of the test sample between heater and
thermometer. The thermal conductivity (k = GL/A) of the test sample is thus an effective fitting
parameter within the FEM simulator, ultimately adjusted to yield the best agreement between the
FIG. 3.8: 3D Finite element method (FEM) model. (a) Whole structure of 3D FEM model. (b)
Zoomed-in structure to show the core area of the thermometry. (c) More zoomed-in structure to
show different layers. (d) Typical distribution of temperature rise due to heating in simulations
which matches with measurements.
Si
50
μm
GNRs
a b
c∆T (K)
101
100
10-1
10-2
Silicon290 nm SiO2
GNR
SensorHeaterd
38
simulated and measured ∆TS vs. PH. This fitting process is implemented by using MATLAB to
interface directly with the COMSOL software, taking ~0.5 hour on a single desktop computer to
converge to a best-fit value at a single temperature point for a typical calculation.
Before performing a substantial amount of calculations, a series of optimizations were car-
ried out. First, the mesh was optimized. Due to the extreme ratio of the graphene/GNRs thickness
(~0.34 nm) to their typical in-plane dimensions (~10 μm), this subdomain was optimized using a
swept mesh strategy rather than the typical free mesh. Other subdomains were optimized careful-
ly using the free mesh strategy, and in the bottom oxide and Si substrate the mesh size grows
gradually from the heating region to the boundaries. Second, the real substrate size is about
8×8×0.5 mm3, which can be regarded as a semi-infinite substrate relative to the small heating
region (~10 μm). In FEM modeling, however, we have to select a finite size for the substrate due
to the computational limitation. By choosing the distance from the center of the heater to the side
and bottom surfaces of the substrate as a testing variable, we found 50 μm is large enough to
model this 3D heat spreading, consistent with the recent work by Jang et al. [54]. Third, the
length of the six probe arms attached to the heater and sensor (see Fig. 3.8b) was chosen as 2 μm
(shorter than their real counterparts), which was found to be sufficiently long to mimic any
peripheral heat loss. Fourth, it was confirmed that the simulated ∆TS/PH is independent of the
power PH applied in the heater, which means the final results do not rely on the choice of the
power PH.
3.3.2 Uncertainty Analysis
We can estimate the uncertainty of our analysis with the classical partial derivative method:
2
ixki
i i
uus
k x
(3.1)
39
where uk is the total uncertainty in the extracted thermal conductivity k, uxi is the uncertainty in
the i-th input parameter xi, and the dimensionless sensitivities si are defined by
(ln )
(ln )
ii
i i
x k ks
k x x
. (3.2)
The partial derivatives are evaluated numerically by giving small perturbation of each parameter
around its typical value and redoing the extraction simulation to obtain the change of k. To
highlight the relative importance of each input parameter, we define their absolute contributions
Table 3.1: Uncertainty analysis for heat flow in the unpatterned “short” graphene. Example of
calculated sensitivities and uncertainty analysis for the thermal conductivity of the graphene
sample GS1 at 300 K. The extracted k is 319 Wm-1
K-1
and its overall uncertainty is 19%.
Units Values x i
Uncertainty
u xi
u xi /x i
Sensitivity
s i
Contribution
c i =|s i |×u xi /x i
c i2/Σc i
2
Expt. Sensor response ΔT S/P H K/μW 0.01635 0.0002 1.2% 4.49 5.5% 8.6%
k ox 1.267 0.04 3.2% 4.25 13.4% 51.6%
k Si 115 10 8.7% 0.55 4.8% 6.6%
k met 55 4 7.3% 0.34 2.5% 1.8%
R gox 1.15E-08 2.0E-09 17.4% -0.12 2.1% 1.3%
R oxs 9.92E-09 3.0E-09 30.2% -0.25 7.6% 16.5%
R mox 1.02E-08 3.0E-09 29.5% 0.05 1.5% 0.7%
t box 288 1 0.3% -5.65 2.0% 1.1%
t tox 20 1 5.0% -0.01 0.1% 0.0%
t met 50 2 4.0% 0.35 1.4% 0.5%
D met 494 5 1.0% 5.38 5.4% 8.5%
D tox 486 5 1.0% 2.39 2.5% 1.7%
Wmet 186 4 2.2% -0.16 0.3% 0.0%
W tox 224 4 1.8% -0.17 0.3% 0.0%
Half length of H/S
electrodesL HS/2 5.86 0.03 0.5% 0.04 0.0% 0.0%
Distance of 2
Voltage probesD pVV 4.23 0.02 0.5% 3.75 1.8% 0.9%
Distance of
Current and
Voltage probes
D pIV 1.04 0.02 1.9% 0.003 0.0% 0.0%
Geo
metr
ica
l
Thickness of
bottom and top
SiO2, and metal
nmDistance of H/S
metal lines and
H/S SiO2 lines
Width of metal and
SiO2 lines
μm
Input parameters (T = 300 K)
Th
erm
al
Thermal
conductivity of
SiO2, Si, metal
W/m/K
TBR of
graphene/SiO2,
SiO2/Si, metal/SiO2
interfaces
m2K/W
40
as ci = |si|×(uxi/xi), and relative contributions as ci2/Σci
2. The uncertainty analysis for extracting
kox and Roxs from the SiO2 control experiment is performed in the same way.
Table 3.1 summarizes the sensitivities and uncertainty analysis for the unpatterned graphene
(GS1) at 300 K. All input parameters can be separated into 3 classes: experimental data, thermal
parameters, and geometric parameters. Their uncertainties are our estimates considering both
random and systematic errors, and those of experimental and thermal parameters are updated
appropriately as the temperature changes. The calculated sensitivities show that the graphene
thermal conductivity is the most sensitive (|si| > 2) to the measured sensor response (ΔTS/PH),
thermal conductivity (kox) and thickness (tbox) of bottom SiO2, center-to-center distances between
metal lines of heater and sensor (Dmet), between top SiO2 lines of heater and sensor (Dtox), and
between two voltage probes (DpVV). These findings are consistent with previous work by W.
Jang et al. [54] using similar substrate-supported thermometry structures. The input parameters
with the greatest relative uncertainty (uxi/xi) are all three thermal boundary resistances (TBRs),
thicknesses of top SiO2 (ttox) and metal (tmet), and the thermal conductivity of the Si substrate
(kSi) and metal (kmet). The combined effects of both sensitivities and relative uncertainties show
that five largest contributions (ci > ≈ 5%) to the overall uncertainty of our thermal measurement
are from ΔTS/PH, kox, kSi, Roxs, and Dmet. In slight contrast to Jang et al. [54], we find that uncer-
tainties introduced by Roxs, Rmox and tmet are non-negligible for our structure and should be
considered. On the other hand, geometric parameters related to the shape and size of the gra-
phene sheet have very small sensitivities (|si| < 0.001), so their contributions are negligible in
uncertainty analysis and not listed in Table 3.1.
41
For the extraction of GNR thermal conductivity, an example of calculated sensitivities and
uncertainty analysis at 300 K is summarized in Table 3.2 (here for the sample with W ~ 65 nm).
Compared with Table 3.1, we have two more parameters: the thermal conductivity of outer
graphene connected to GNRs (kg) and the width of GNRs (WGNR). The parameters with the
largest sensitivities (|si| > 5) are the same as those in Table 3.1, but their values increase because
Table 3.2: Uncertainty analysis for GNR thermal conductivity. Example of calculated sensitivi-
ties and uncertainty analysis for the thermal conductivity of GNRs with W = 65 nm at 300 K.
The extracted k is 101 Wm-1
K-1
and its overall uncertainty is 60%.
Units Values x i
Uncertainty
u xi
u xi /x i
Sensitivity
s i
Contribution
c i =|s i |×u xi /x i
c i2/Σc i
2
Expt. Sensor response ΔT S/P H K/μW 0.0124 0.0002 1.6% 13.22 21.3% 12.9%
k ox 1.267 0.04 3.2% 10.38 32.8% 30.5%
k Si 115 10 8.7% 2.61 22.7% 14.6%
k met 22 2 9.1% 0.68 6.2% 1.1%
k g 320 60 18.8% 0.001 0.0% 0.0%
R gox 1.15E-08 2.0E-09 17.4% -0.04 0.6% 0.0%
R oxs 9.92E-09 3.0E-09 30.2% -1.00 30.3% 26.1%
R mox 1.02E-08 3.0E-09 29.5% 0.11 3.4% 0.3%
t box 291 1 0.3% -22.51 7.7% 1.7%
t tox 25 1 4.0% -0.17 0.7% 0.0%
t met 40 2 5.0% 0.70 3.5% 0.3%
D met 517 5 1.0% 14.73 14.2% 5.8%
D tox 509 5 1.0% 14.25 14.0% 5.6%
Wmet 218 4 1.8% -0.64 1.2% 0.0%
W tox 266 4 1.5% -2.27 3.4% 0.3%
WGNR 65 3 4.6% -0.80 3.7% 0.4%
Half length of H/S
electrodesL HS/2 5.77 0.03 0.5% 0.20 0.1% 0.0%
Distance of 2
Voltage probesD pVV 4.17 0.02 0.5% 7.80 3.7% 0.4%
Distance of
Current and
Voltage probes
D pIV 1.04 0.02 1.9% -0.286 0.5% 0.0%
Geo
metr
ica
l
Thickness of
bottom and top
SiO2, and metal
Distance of H/S
metal lines and
H/S SiO2 lines
μm
Width of metal
lines, top SiO2
lines, and GNRs
nm
Input parameters (T = 300 K)
Th
erm
al
TBR of
graphene/SiO2,
SiO2/Si, metal/SiO2
interfaces
m2K/W
Thermal
conductivity of
SiO2, Si, metal, and
outter graphene
W/m/K
42
the total width of the GNR array is smaller than that of the unpatterned graphene. Due to the
significant increase of sensitivities, the total uncertainty increases from 19% for unpatterned
graphene to 60% for GNRs, while the input parameters with the greatest contributions (ci > 10%)
to the total uncertainty are ΔTS/PH, kox, kSi, Roxs, Dmet, and Dtox, the same as those for graphene
along with Dtox. From the 60% total uncertainty, approximately 21% is due to measurement
uncertainty and the remainder from geometric and temperature-dependent variables as listed in
Table 3.2. [we note that the geometric parameters related to the shape and size of GNRs and
graphene outside the heater and sensor region have very small sensitivities (|si| < 0.01) and are
not listed in Table 3.2, which is also consistent with the GNR results being insensitive to kg.]
For other GNR samples with different widths, the total uncertainties gradually increase from
22% to 83% at 300 K as W decreases from ~130 to ~45 nm, as less heat flows in the GNR array
rather than the substrate. As the temperature decreases, the uncertainties of all graphene and
GNR thermal conductivities also increase due to either increased sensitivities or increased
relative uncertainties of input parameters.
As mentioned earlier and shown in Tables 3.1 and 3.2, all input parameters are classified in-
to three groups, not all of which need to be included when comparing relative GNR thermal
conductivities (e.g. with width or temperature). For instance, Fig. 3.12b compares the thermal
conductivities of the samples at different temperatures considering the contributions of experi-
mental and thermal parameters to the error bars. Similarly, Fig. 3.12d compares thermal conduc-
tivities of different samples at the same temperature, considering the contributions of experi-
mental and geometric parameters to the error bars. Different samples share the same thermal
parameters and these uncertainties would only shift all k values in the same direction without
affecting their relative values. In the end, the differences are relatively subtle, and within the
43
fitting capabilities of our models, all based on the initial data above 70 K which are clearly
distinguishable from one another as seen from the raw thermometry in Fig. 3.6d.
3.4 Results
Based on experimental method described above, we performed heat flow measurements
from 20 to 300 K on unpatterned graphene (Fig. 3.4a), control samples with the graphene etched
off (Fig. 3.4b), and arrays of GNR widths W ≈ 130, 85, 65 and 45 nm (Figs. 3.2 and 3.4c-d). To
obtain the thermal properties of our samples we used the above mentioned 3D simulations of the
structures with dimensions obtained from measurements by SEM and AFM, as shown in Figs.
3.4e and 3.8. The results for the SiO2 control sample, unpatterned graphene, and GNR arrays are
presented in the following.
3.4.1 Thermal Properties of the SiO2 Underlayer
To validate our thermometry approach, we first carefully focused on a control experiment to
measure the thermal properties of the SiO2 underlayer supporting our samples. The sample was
first prepared as described before, including the graphene under the heater and thermometer
electrodes, to reproduce the thermal contacts encountered in all samples. However, the exposed
graphene was then etched by oxygen plasma, leaving the bare SiO2 as shown in Figs. 3.4b and
3.5. This allows us to obtain the thermal properties of the parallel heat flow path through the
contacts, supporting SiO2 and substrate. Measurements were performed on part 1 and part 2 of
the metal electrodes (see Fig. 3.5), and the analyzed sensor and heater temperature rises normal-
ized by heater power as a function of the ambient temperature are shown in Fig. 3.9a.
44
To compare our 3D simulations to this experimental data set, we needed to fit the thermal
properties of the SiO2 layer (which dominate), and to a lesser degree those of the SiO2-Si TBR at
the bottom of the SiO2 layer (which also plays a role). While the thermal conductivity of SiO2 is
well-known and easy to calibrate against [30, 158], to the best of our knowledge no consistent
data for the TBR of the SiO2-Si interface (Roxs) exist as a function of temperature. Two recent
studies [159, 160] suggested Roxs ~ 5−7×10-8
m2KW
-1 at room temperature, but some earlier
efforts [161-165] found the total TBR of metal-SiO2-Si interfaces as low as ~1−3×10-8
m2KW
-1,
putting an upper bound on Roxs without being able to separate it from the total TBR. Due to this
contradiction, we set out to obtain the temperature-dependent Roxs, treating it as another fitting
parameter of our simulations in addition to the thermal conductivity of SiO2 (kox). Other thermal
parameters well characterized in the literature are the thermal conductivity of highly doped Si
[166-168], thermal boundary resistances of the graphene-SiO2 interface [44, 48] and the Au-Ti-
SiO2 interfaces [45]. In addition, the effective thermal conductivity of the metal electrodes
(Au/Ti) was calculated from the measured electrical resistance according to the Wiedemann-
FIG. 3.9: Control experiment to extract SiO2 thermal properties. (a) Sensor and heater tempera-
ture rise normalized by heater power from measurements taken at part 1 and part 2 of the SiO2
sample (see Fig. 3.5). (b) Extracted thermal conductivity of SiO2 from two measurements
compared with well-known data from Cahill [158]. The green solid line is the polynomial fit up
to the 7th order to our data. (c) Extracted thermal boundary resistance of the SiO2-Si interface,
Roxs. The green solid line is the fit to our data by Eq. 3.3.
0 50 100 150 200 250 300 3500.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4 Cahill (1990)
Polynomial Fit
Part 1
Part 2
ko
x (
Wm
-1K
-1)
T (K)10 100
10-8
10-7
Part 1
Part 2
Fit
Roxs (
m2K
/W)
T (K)
a b c
0 50 100 150 200 250 300
0.01
0.1
1
Part 1
Part 2
T
/PH (
K/
W)
T (K)
Sensor
Heater
45
Franz Law, where an average Lorentz number Lz = 2.7×10-8
WΩK-2
is used for Au/Ti electrodes
[169]. (all parameters were allowed to vary within known experimental bounds, leading to the
uncertainty analysis in Section 3.3.2.)
Our extracted kox and Roxs of two data sets (part 1 and 2) are shown in Figs. 3.9b and 3.9c,
respectively. Our kox data are in a good agreement with well-established values reported by
Cahill [158], and the typical uncertainty is ~5% at most temperatures. By fitting our kox data with
a polynomial up to the 7th
-order, we obtained a smooth dependence of kox on T (green solid line),
and this was used to extract the thermal properties of our graphene and GNRs. Our extractions
suggest Roxs ~ 10-8
m2KW
-1 at room temperature, in agreement with the reported upper bound in
Refs. [161-165]. For the subsequent thermal analysis, our Roxs data are best fit by a simple
expression,
4
9
2.25
1.046 10( ) 9.67 10
( 13.4)oxsR T
T
m2K/W (3.3)
as shown by the green solid line in Fig. 3.9c. Thus, this control experiment demonstrates the
feasibility and reliability of our thermometry platform, also giving the first report of the tempera-
ture-dependent TBR of SiO2-Si interfaces.
3.4.2 Quasi-Ballistic Thermal Transport in Short Graphene
After demonstrating the feasibility of our thermometry method by the control sample, we
next examined three “short” graphene samples (GS1, 2, 3) which were not patterned into GNRs;
all had length L ~ 260 nm and width W ~ 12 μm between the heater and sensor thermometers
(corresponding to schematic in Fig. 3.10a). Besides GS1 which is shown in Fig. 3.4a of the main
paper, the SEM image of another sample (GS2) is shown in Fig. 3.11a. The third one (GS3)
broke after measurements at two temperature points, which are nevertheless listed among the
46
data in Fig. 3.11. Figure 3.11b displays raw data taken as ratio of heater power to sensor temper-
ature (PH/ΔTS) for all three samples; the corresponding data of the SiO2-only sample (part 2 of
Fig. 3.9a) is also plotted here for comparison. The presence of graphene notably “heats” the
sensor (higher ΔTS) and is distinguishable from the SiO2-only sample all the way down to ~20 K
(although the GNRs become harder to distinguish below ~70 K as mentioned earlier). The
extracted graphene thermal conductivities are shown in Fig. 3.11c, and the three samples show
very similar values. They all decrease from ~300 Wm-1
K-1
at 300 K to ~10 Wm-1
K-1
at 30 K and
show similar temperature dependence. The data of the “long” graphene with L ~10 μm from Seol
et al. [49] are also plotted in Fig. 3.11c (black dots).
FIG. 3.10: Schematic of size effects and different heat flow regimes. (a) Diffusive heat transport
in “large” samples with dimensions much greater than the intrinsic phonon MFP (L, W ≫ λ).
This regime corresponds to the samples measured in both substrate-supported [49] and suspend-
ed graphene [58] studies to date. (b) Quasi-ballistic heat flow in “short but wide” samples (L ~ λ
and W ≫ λ). These correspond to our geometry shown in Fig. 3.4a, with L ≈ 260 nm and W ≈ 12
μm. (c) Return to a diffusive heat transport regime as the sample width is narrowed down, and
phonon scattering with edge roughness (of rms Δ) begins to dominate. These correspond to our
arrays of GNRs from Figs. 3.4c-e (L ≈ 260 nm and W varying from 45 nm to 130 nm). A fourth
regime (long L, narrow W) is not shown here, but it can be easily understood from the above.
‘Large’ L, W » λ
(diffusive)
‘Short’ L ~ λ, W » λ
(quasi-ballistic)
‘Short and narrow’
L, W ~ λ
(diffusive)
λ W
Δ
λ
a b
c
λL
HOT
COLD
47
Next we compare the sample thermal conductance to the theoretical ballistic limits in Fig.
3.11d, recalling the relationship between conductance and conductivity, G = kA/L, where A is the
cross-sectional area of heat flow, A = WH, with W and H = 0.335 nm as the width and thickness
of the graphene samples, respectively. The theoretical ballistic conductance per unit cross-
sectional area Gball/A of graphene (solid line in Fig. 3.11d) is calculated by using the full phonon
dispersion of graphene and integrating over the entire 2D Brillouin zone (see Sections 5.4.1 and
5.3.2). For convenience, we note that the Gball/A of graphene can be approximated analytically as
FIG. 3.11: Data of three unpatterned graphene samples (all L ~ 260 nm). (a) SEM image of the
unpatterned graphene sample GS2 (colorized for emphasis). (b) Measured ratio of heater power
to sensor temperature rise of all three graphene samples compared with that of SiO2 sample from
the control experiment (Fig. 3.9). (c) Extracted graphene thermal conductivity of all our “short”
samples (L ~ 260 nm) compared with the “long” graphene (L ~ 10 μm) reported by Seol et al.
[49]. (d) Thermal conductance per unit area compared to theoretical ballistic limit of graphene.
10 1001
10
100
1000 Graphene (L~10 m)
Graphene (GS1)
Graphene (GS2)
Graphene (GS3)
k (
Wm
-1K
-1)
T (K)
~T2
~T1.5
~T
Seol (2010)
10 100
10
100 SiO
2 (part 2)
Graphene (GS1)
Graphene (GS2)
Graphene (GS3)
PH/
TS (W
/K)
T (K)
a b
c d
10 100
107
108
109
1010
Graphene (L~10 m)
Graphene (GS1)
Graphene (GS2)
Graphene (GS3)
G/A
(W
m-2K
-1)
T (K)
GS2
Seol (2010)
48
Gball/A ≈ [1/(4.4 × 105 T
1.68) + 1/(1.2 × 10
10)]
-1 WK
-1m
-2 over the temperature range 1-1000 K, as
a fit to full numerical calculations. As shown in Fig. 3.11d, our three “short” graphene samples
all display on average ~35% of the ballistic conductance limit, indicating they reach a quasi-
ballistic conduction regime (schematic in Fig. 3.10b). The data for the 10-μm “long” sample
from Seol et al. [49] show <2% of ballistic limit on average, indicating a diffusive transport
regime (schematic in Fig. 3.10a) as would be expected for a sample much longer than the phonon
MFP (W, L ≫ λ). Both percentages are consistent with a simple estimation of transmission
probability ~λbs/(λbs + L) using their own lengths and back scattering mean free path λbs = (π/2)λ ~
160 nm (see Section 3.4.3) where the intrinsic phonon MFP λ ~ 100 nm for most temperatures.
3.4.3 Length dependence of thermal conductivity
To understand different transport behaviors in short and long graphene, we recall that in the
ballistic limit (L ≪ λ) the conductance rather than the conductivity approaches a constant at a
given temperature [29, 112, 170], Gball(T). Nevertheless, the thermal conductivity is the parame-
ter typically used for calculating heat transport in practice, and for comparing different materials
and systems. Thus, the well-known relationship k = (G/A)L imposes the conductivity k to be-
come a function of length in the ballistic regime and to decrease as L is reduced. This situation
becomes evident when we plot the thermal conductivity in Figs. 3.11c and 3.12b, finding k ≈ 320
Wm-1
K-1
for our “short” and wide samples at room temperature (schematic Fig. 3.10b), almost a
factor of two lower than the large graphene [49] (schematic Fig. 3.10a). We note that both
unpatterned samples here and in Ref. [49] were supported by SiO2, showed no discernible
defects in the Raman spectra, and the measurements were repeated over three samples (Fig. 3.11)
with similar results obtained each time.
49
The transition of thermal conductivity from diffusive to ballistic can be captured through
FIG. 3.12: Thermal conduction scaling in GNRs. (a) Thermal conductance per cross-sectional
area (G/A) vs. temperature for our GNRs (L ≈ 260 nm, W as listed, see Fig. 3.10c), a “short”
unpatterned sample (L ≈ 260 nm, W ≈ 12 μm, see Fig. 3.10b), and a “large” sample from Seol et
al. [49] (L ≈ 10 μm, W ≈ 2.4 μm, see Fig. 3.10a). The short but wide graphene sample attains up
to ~35% of the theoretical ballistic heat flow limit [29, 112, 170]. (b) Thermal conductivity for
the same samples as in (a) (also see Fig. 3.14a). (c) Thermal conductivity reduction with length
for “wide” samples (W ≫ λ), compared to the ballistic limit (kball = GballL/A) at several tempera-
tures. Symbols are data for our “short” unpatterned graphene samples (Figs. 3.4a and 3.10b), and
“large” samples from Seol et al. [49] (Fig. 3.10a). Solid lines are model from Eq. 3.4. (d) Ther-
mal conductivity reduction with width for GNRs, all with L ≈ 260 nm (Figs. 3.4c-d and 3.10c).
Solid symbols are experimental data from (b), open symbols are interpolations for the listed
temperature; lines are fitted model from Eq. 3.5. The thermal conductivity of plasma-etched
GNRs in this work appears lower than that estimated for GNRs from unzipped nanotubes [144]
at a given width, consistent with a stronger effect of edge disorder [153]. Also see Fig. 3.13a.
100 200 300 40010
7
108
109
G/A
(W
m-2K
-1)
T (K)50
10 100 1000 100000
50
100
150
200
250
300
350
k (
Wm
-1K
-1)
W (nm)10 100 1000 10000
0
100
200
300
400
500
600
700
k (
Wm
-1K
-1)
L (nm)
ballistic
limit kballT = 300 K
T = 190 K
T = 150 K
T = 70 K
L ~ 260 nm
100 K
c d
large graphene
(L ~ 10 μm)
Seol (2010)
W~130 nm
W~85 nm
W~65 nm
W~45 nm
~T1.5
b
100 200 300 40010
100
1000
k (
Wm
-1K
-1)
T (K)50
~T1.5
~T
W~130 nm
W~85 nm
W~65 nm
W~45 nm
large graphene (L ~ 10 μm)
Seol (2010)
short graphene
(L ~ 260 nm)
a
Seol (2010)
50
simple models [171], similar to the apparent mobility reduction during quasi-ballistic charge
transport observed in short-channel transistors [172, 173]:
1 1
ball
ball diff
1 1 1.
( / 2)b bb
GAk L
LG k A L
(3.4)
The first equality is a “three-color” model with b the phonon branch (longitudinal acoustic,
LA; transverse, TA; flexural, ZA), ball
bG calculated using the appropriate dispersion [29], and ∑
diff
bk = kdiff the diffusive thermal conductivity. At T = 300 K, kdiff ≈ 560 ± 50 Wm-1
K-1
, and diff
bk =
148, 214, 198 Wm-1
K-1
for b = ZA, TA, LA modes, respectively [49]. A simpler “gray” approx-
imation can also be obtained by dropping the b index, k(L) ≈ [A/(LGball) + 1/kdiff]-1
, where Gball/A
≈ 4.2 × 109 WK
-1m
-2 at room temperature [96] (see Section 3.4.2). The second expression in Eq.
3.4 is a Landauer-like model [171, 174], with π/2 accounting for angle averaging [175] in 2D to
obtain the phonon backscattering MFP.
We compare the simple models in Eq. 3.4 with the experiments in Fig. 3.12c and find good
agreement over a wide temperature range. The Eq. 3.4 also yields our first estimate of the
intrinsic phonon MFP in SiO2-supported graphene, λ ≈ (2/π)kdiff/(Gball/A) ≈ 90 nm at 300 K and
115 nm at 150 K. (The same argument estimates an intrinsic phonon MFP λ ≈ 300-600 nm in
freely suspended graphene at 300 K, if a thermal conductivity 2000-4000 Wm-1
K-1
is used [21,
23, 24, 58].) This phonon MFP is the key length scale which determines when the thermal
conductivity of a sample becomes a function of its dimensions, in other words when L and W are
comparable to λ. Based on Fig. 3.12c, we note that ballistic heat flow effects should become non-
negligible in all SiO2-supported graphene devices shorter than approximately 1 μm.
3.4.4 Width dependence of thermal conductivity
Figure 3.12a also displays in-plane thermal conductance per area (G/A) for our measured
51
GNRs (L ≈ 260 nm, W ≈ 130, 85, 65 and 45 nm). As the width decreases, thermal conduction
returns from quasi-ballistic to diffusive regimes gradually (corresponding to schematic Fig.
3.10c). In parallel, Fig. 3.10 summarizes schematics of the size effects and all three transport
regimes discussed, corresponding to the samples measured in Fig. 3.12.
We now turn to the width-dependence of heat flow in narrow GNRs. Our experimental data
in Figs. 3.12b and 3.12d show a clear decrease of thermal conductivity as the width W is reduced
to a regime comparable to the intrinsic phonon MFP. For instance, at room temperature k ≈ 230,
170, 100, and 80 Wm-1
K-1
for GNRs of W ≈ 130, 85, 65 and 45 nm, respectively, and same L ≈
260 nm. To understand this trend, we consider k limited by phonon scattering with edge disorder
[176, 177] through a simple empirical model with a functional form suggested by previous work
on rough nanowires [178, 179] and GNR mobility [180]:
1
eff
1 1,
( )
n
k W Lc W k L
. (3.5)
Here Δ is the rms edge roughness and k(L) is given by Eq. 3.4 [k(L) = 346, 222, 158, 37 Wm-1
K-1
for T = 300, 190, 150, 70 K, respectively]. The solid lines in Fig. 3.12d show good agreement
with our GNR data (L ≈ 260 nm) using Δ = 0.6 nm and a best-fit exponent n = 1.8 ± 0.3, as well
as c fitted to be 0.04019, 0.02263, 0.01689, and 0.00947 Wm-1
K-1
for T = 300, 190, 150, and 70
K, respectively. Note that we cannot assign overly great physical meaning to the parameter c
because the empirical model can only fit Δn/c, not Δ or c independently. The simple model
appears to be a good approximation in a regime with Δ ≪ W, where the data presented here were
fitted. However it is likely that this simple functional dependence would change in a situation
with extreme edge roughness [81], where the roughness correlation length (which cannot be
directly quantified here) could also play an important role.
52
Nevertheless, the nearly W-squared dependence of thermal conductivity in narrow GNRs
with edge roughness is consistent with previous findings for rough nanowires [178, 179], and
also similar to that suggested by theoretical studies of GNR electron mobility [180]. The precise
scaling with Δ is ostensibly more complex [176, 177] than can be captured in a simple model, as
it depends on details of the phonon dispersion, the phonon wave vector, and indirectly on tem-
perature. However the Δ estimated from the simple model presented above is similar to that from
extensive numerical simulations below, and to that measured by transmission electron microsco-
py (TEM) on GNRs prepared under similar conditions [153]. Thus, the simple expressions given
above can be taken as a practical model for heat flow in substrate-supported GNRs with edge
roughness (Δ ≪ W) over a wide range of dimensions, corresponding to all size regimes in Fig.
3.10.
It is instructive to examine some similarities and differences between our findings here vs.
previous results regarding size effects on charge carrier mobility in GNRs with dimensions
comparable to the phonon or electron MFP. Figure 3.13 displays the width W dependence of our
GNR thermal conductivity side-by-side with the electrical mobility measured by Yang and
Murali [147] on similar samples (Fig. 3.13a is re-plot of Fig. 3.12d, here using log axis). Both
the GNR thermal conductivity and electrical mobility show a similar trend with W, starting to
decrease significantly when scattering becomes edge limited. However it is apparent that their
fall-off occurs at different critical widths: W ~ 200 nm for thermal conductivity and W ~ 40 nm
for electrical mobility. Above these widths, the thermal conductivity is limited by phonon-
substrate scattering, while the electrical mobility is limited by electron scattering with substrate
impurities. The difference between their critical W is consistent with the intrinsic phonon MFP
λph ~ 100 nm being approximately five times larger than the intrinsic electron MFP [104] λel ~ 20
53
nm in SiO2-supported graphene. (We note phonons are entirely responsible for the thermal
conductivity of these GNRs, with a negligible electronic contribution [144]). Thus, the fall-off of
thermal conductivity and electrical mobility corresponds approximately to GNR widths approx-
imately twice the phonon and electron MFPs. Interestingly, this also suggests a GNR width
regime (~40 < W < ~200 nm) where the thermal conductivity is reduced from intrinsic values but
the electrical mobility is not yet affected by edge disorder. This suggests the possibility of
manipulating heat and charge flow independently in such narrow edge-limited structures. Further
control of such behavior could also be achieved if substrates with different roughness, impurity
density, and vibrational (phonon) properties are used.
3.5 Discussion and BTE Calculations
3.5.1 Low-T Scaling and Comparison with CNT
FIG. 3.13: Width dependence of GNR thermal conductivity and electrical mobility. (a) Thermal
conductivity k vs. W at several temperatures from this study (L ~ 260 nm); same plot as Fig.
3.12d but on a log-log scale. (b) Electrical mobility μ vs. W at room temperature for several layer
(L) GNRs, adapted from Yang et al. [147]. Solid line is a fit to data points with μ = (1/0.0163W3
+ 1/3320)-1
. Both data sets show a decreasing trend as W is narrowed, but with different fall-offs.
10 100 1000 1000010
100
1000
k (
Wm
-1K
-1)
W (nm)
T = 300 K
T = 190 K
T = 150 K
T = 70 K
a
10 100 100010
2
103
104
ML
BL
3-5L
6-8L
mob
ility
(cm
2V
-1s
-1)
W (nm)
bedge limited
impurity limited
edge limited
substrate limited
54
Figure 3.14a shows the extracted thermal conductivity of our “short” graphene (GS1) and
GNRs for the full temperature range measured, down to ~20 K (however, we recall that GNR
measurements are challenging to distinguish below ~70 K, as previously mentioned). We can fit
the thermal conductivity as k = αTβ below ~200 K, and the obtained power β is shown as an inset
of Fig. 3.14a. We find that β decreases from ~1.6 for the unpatterned graphene to ~1 for narrow
GNRs. However, we note that this does not necessarily mark a transition to one-dimensional (1-
D) phonon flow, as the GNRs here are much wider than the phonon wavelengths (few nm).
Thus, the simple model is given as a convenient analytic estimate, and the exponent β represents
the complex physical behavior of GNR heat flow due to the increasing heat capacity (which
scales as ~T1.5
) and the slightly decreasing phonon MFP in this T range.
We also compare our extracted graphene and GNR thermal conductance with carbon nano-
tubes (CNTs) in Fig. 3.14b. We perform this comparison with the calculated ballistic upper limit,
with a single-wall carbon nanotube (SWCNT) by M. Pettes et al. [75], and with multi-wall
FIG. 3.14: Complete data sets including low-T range and comparison with CNTs. (a) Extracted
thermal conductivity of graphene (GS1) and GNRs, same as Fig. 3.12b but with low-T (20~60
K) data included for completeness. Long graphene data are from Seol et al. [49]. Inset: power
exponent β vs. GNR width fit from k = αTβ (see text). (b) Thermal conductance (G/A) of our data
compared with those of long graphene [49], ballistic limit, SWCNT of M. Pettes et al. [75], and
MWCNTs from studies of P. Kim et al. [71], M. Pettes et al. [75], and M. Fujii et al. [181].
10 1001
10
100
1000 Long Graphene (L~10 m)
Short Graphene (L~0.26 m)
GNR, W = 130 nm
GNR, W = 85 nm
GNR, W = 65 nm
GNR, W = 45 nm
k (
Wm
-1K
-1)
T (K)10 100
106
107
108
109
1010
MWCNT
MWCNT
SWCNT
MWCNT
G/A
(W
m-2K
-1)
Long Graphene (L~10 m)
Short Graphene (L~0.26 m)
GNR, W = 130 nm
GNR, W = 85 nm
GNR, W = 65 nm
GNR, W = 45 nm
T (K)
a b
~T2
~T1.5
~T
, Seol (2010)
Kim (2001)
Fujii (2005)
Pettes (2009)10 100 1000 10000
1.0
1.2
1.4
1.6
W (nm)
, Seol (2010)
55
carbon nanotubes (MWCNTs) by P. Kim et al. [71], M. Pettes et al. [75], and M. Fujii et al.
[181]. We find that our short graphene data (red filled dots) and P. Kim’s MWCNT data [71]
(open black squares) have the highest values, both reaching up to ~35% of the ballistic limit of
graphene.
3.5.2 Effects of Contacts
We note that the 3D simulations automatically include all known contact resistance effects,
including those of the graphene-SiO2 and SiO2-metal interfaces, matched against data from the
literature and our control experiments (see Section 3.4.1). To provide some simple estimates, the
contact thermal resistance (per electrode width) is RC ≈ 0.7 m∙KW-1
, the “wide” unpatterned
graphene thermal resistance is RG ≈ 2.5 m∙KW-1
, and that of the GNR arrays is in the range RGNR
≈ 4 to 32 m∙KW-1
(from widest to narrowest). The graphene is not patterned under the electrodes,
thus the contact resistance remains the same for all samples. The 3D simulations also account for
heat spreading through the underlying SiO2, and our error bars include various uncertainties in
all parameters (see Section 3.3.2).
We also consider the effects of measurement contacts and how they relate to the interpreta-
tion of sample length in the quasi-ballistic heat flow regime. As in studies of quasi-ballistic
electrical transport [172, 173], we defined the “channel length” L as the inside edge-to-edge
distance between the heater and thermometer electrodes (Fig. 3.10c). Simple ballistic theory
assumes contacts with an infinite number of modes, and instant thermalization of phonons at the
edges of the contacts. The former is well approximated here by electrodes two hundred times
thicker than the graphene sheet, however phonons may travel some distance below the contacts
before equilibrating. The classical, continuum analog of this aspect is represented by the thermal
transfer length (LT) of heat flow from the graphene into the contacts [129], which is automatical-
56
ly accounted in our 3D simulations (Fig. 3.4e). However, a sub-continuum perspective [182]
reveals that phonons only thermally equilibrate after traveling one MFP into the graphene under
the contacts. Previous measurements of oxide-encased graphene [54] had estimated a thermal
conductivity kenc = 50-100 Wm-1
K-1
, which suggests a phonon MFP λenc = (2kenc/π)/(Gball/A) ≈ 8-
15 nm under the contacts. This adds at most 12% to our assumption of edge-to-edge sample
length (here L ≈ 260 nm), a small uncertainty which is comparable to the sample-to-sample
variation from fabrication, and to the size of the symbols in Fig. 3.12c. (The relatively low
thermal conductivity of encased monolayer graphene [54] is due to scattering with the SiO2
sandwich, although some graphene damage from the SiO2 evaporation [183] on top is also
possible.)
3.5.3 BTE Calculations
To gain deeper insight into our experimental results, we employ a numerical solution of the
Boltzmann transport equation (BTE) with a complete phonon dispersion [177, 184]. Our ap-
proach is similar to previous work [28, 49, 184], but accounting for quasi-ballistic phonon
propagation and edge disorder scattering in short and narrow GNRs, respectively. We obtain the
thermal conductivity by solving the Boltzmann transport equation in the relaxation time approx-
imation including scattering at the rough GNR edges [177]. The simulation uses the phonon
dispersion of an isolated graphene sheet, which is a good approximation for SiO2-supported
graphene within the phonon frequencies that contribute most to transport [101], and at typical
graphene-SiO2 interaction strengths [97]. (However, we note that artificially increasing the
graphene-SiO2 coupling, e.g. by applying pressure [185], could lead to modifications of the
phonon dispersion and hybridized graphene-SiO2 modes [97].) We assume a graphene monolay-
er thickness H = 0.335 nm, and a concentration of 1% 13
C isotope point defects [21, 49]. The
57
FIG. 3.15: Computed phonon scattering rates at T = 300 K for a GNR with W = 65 nm, L = 260
nm, and Δ = 0.65 nm. Substrate contact patch radius is a = 8.75 nm. Rates are plotted as a
function of energy for each scattering mechanism and their total; however, each dot represents
one phonon mode. Note that angle-dependent mechanisms like contact and edge roughness
scattering have additional dependence on the angle of the phonon velocity vector, which can lead
to different rates for the same value of phonon energy.
0 0.05 0.1 0.15 0.210
8
109
1010
1011
1012
Energy (eV)
Scattering r
ate
(s-1
)
0 0.1 0.20.05 0.1510
8
109
1010
1011
1012
Energy (eV)
Scattering r
ate
(s-1
)
0 0.05 0.1 0.15 0.210
5
107
109
1011
1013
1015
Energy (eV)
Scattering r
ate
(s-1
)
0 0.1 0.20.05 0.1510
8
109
1010
1011
1012
Energy (eV)
Scattering r
ate
(s-1
)
0 0.05 0.1 0.15 0.210
10
1011
1012
1013
1014
Energy (eV)
Scattering r
ate
(s-1
)
0 0.1 0.20.05 0.1510
10
1011
1012
1013
1014
Energy (eV)
Scattering r
ate
(s-1
)
Edge Roughness
Substrate Isotopes
Total
LA
a b
c d
e f
Umklapp
Contacts
ZO LO TOZA
TA
58
interaction with the substrate is modeled through perturbations to the scattering Hamiltonian [49]
at small patches where the graphene is in contact with the SiO2, with nominal patch radius a =
8.75 nm. Anharmonic three-phonon interactions of both normal and umklapp type are included
in the relaxation time. An equivalent 2D ballistic scattering rate [171, 175] ~2vx/L is used in the
numerical solution (x is direction along graphene) to account for transport in short GNRs.
Figure 3.16a finds good agreement of thermal conductivity between our measurements and
the BTE model across all samples and temperatures. We obtained the best fit for GNRs of width
130 and 85 nm with rms edge roughness Δ = 0.25 and 0.3 nm, where the gray bands in Fig. 3.16a
correspond to ±5% variation around these values. For GNRs of widths 65 and 45 nm the gray
bands correspond to edge roughness ranges Δ = 0.35-0.5 and 0.5-1 nm, respectively. We note
that, unlike the empirical model of Eq. 3.5, the best-fit BTE simulations do not use a unique
value of edge roughness Δ. This could indicate some natural sample-to-sample variation in edge
roughness from the fabrication conditions, but it could also be due to certain edge scattering
physics (such as edge roughness correlation [81] and phonon localization [186]) which are not
yet captured by the BTE model.
Figure 3.16b examines the scaling of MFPs by phonon mode, finding they are strongly re-
duced as the GNR width decreases below ~200 nm, similar to the thermal conductivity in Fig.
3.12d. The frequency-dependent phonon MFP is directly obtained from the calculated total
scattering rate (e.g. Fig. 3.15f). The MFP for each phonon mode is calculated as an average over
the entire frequency spectrum, weighted by the frequency-dependent heat capacity cb(ω) and
group velocity vb(ω),
( ) ( ) ( )
( ) ( )
b b b
b
b b
c v d
c v d
. (3.6)
59
We note that LA and TA modes, which have larger intrinsic MFPs, are more strongly affected by
GNR edge disorder. On the other hand ZA modes are predominantly limited by substrate scatter-
ing and consequently suffer less from edge disorder, consistently with recent findings from
FIG. 3.16: Insights from numerical simulations. (a) Comparison of Boltzmann transport model
(lines) with experimental data (symbols, same as Fig. 3.12b, but here on linear scale). (b) Com-
puted scaling of phonon mean free path (MFP) vs. width, for two sample lengths as listed. LA
and TA phonons with long intrinsic MFP are subject to stronger size effects from edge scattering
than ZA modes, which are primarily limited by substrate scattering. (c) Estimated contribution of
modes to thermal conductivity vs. edge roughness Δ, for a wide sample and a narrow GNR. Like
W, changes in Δ also more strongly affect LA and TA modes, until substrate scattering begins to
dominate. (d) Phonon MFP vs. energy (frequency) for a large sample and a narrow GNR. Low-
frequency modes are strongly affected by substrate scattering, such that effects of edge rough-
ness are most evident for ℏω > 15 meV. Panels (b-d) are all at room temperature. Also see Figs.
3.15 and 3.17.
0 50 100 1501
10
100
1000
Energy (meV)
mfp
(nm
)
101
102
103
104
0
50
100
150
200
W (nm)
mfp
(nm
)
0 100 200 3000
100
200
300
400
500
600
700
T (K)
k (
Wm
-1K
-1)
130 nm
85 nm
65 nm
large graphene (L ~ 10 μm)
Seol (2010)
short graphene
(L ~ 260 nm)
GNRs
GNR W ~
45 nm
a b
TA long
LA long
ZA long
ZA short
long: L=10 μm
short: L=260 nm (Δ~0.65 nm)
0.1 10.510
1
102
103
(nm)
k (
Wm
-1K
-1)
GNR
W ~ 65 nm
TA wide
LA wide
LA GNR
TA GNR
ZA wide
ZA GNR
total wide
total GNR
Seol (2010)
wide: W=2 μm, L=10 μm
GNR: W=65 nm, L=260 nm
c d
TA large
LA large
LA GNR
TA GNR
ZA large
Energy ħω (meV)
ZA GNR
large: W=2 μm, L=10 μm
GNR: W=65 nm, L=260 nm (Δ=0.65 nm)
60
molecular dynamics (MD) simulations [97, 101].
Increasing edge disorder reduces phonon MFPs (Fig. 3.17d), and the thermal conductivity is
expected to scale as shown in Fig. 3.16c. In the BTE model the edge roughness scattering is
captured using a momentum-dependent specularity parameter,
2 2 2( ) exp 4 sin Ep q q , (3.7)
where q is the magnitude of the phonon momentum and θE is the angle between the direction of
phonon momentum and the edge. This specular parameter indicates that small wavelength (large
q) phonons are more strongly affected by line edge roughness. However, as Δ increases the
specularity parameter saturates, marking a transition to fully diffuse edge scattering, and also to a
regime where substrate scattering begins to dominate long wavelength phonons in substrate-
supported samples. This transition cannot be captured by the simplified Δn dependence in the
empirical model of Eq. 3.5.
To further illustrate such distinctions, Fig. 3.16d displays the energy (frequency ω) depend-
ence of phonon MFPs for a “small” GNR and a “large” SiO2-supported graphene sample (corre-
sponding to Figs. 3.10c and 3.10a, respectively). Low-frequency substrate scattering (propor-
tional to ~1/ω2) dominates the large sample [49, 97], while scattering with edge disorder affects
phonons with wavelengths comparable to, or smaller than, the roughness Δ. Therefore, larger Δ
can affect more long-wavelength (low energy) phonons, but only up to Δ ~ 1 nm, where the
effect of the substrate begins to dominate in the long wavelength region. (Also seen in Fig.
3.16c.) Such a separation of frequency ranges affected by substrate and edge scattering could
provide an interesting opportunity to tune both the total value and the spectral components of
thermal transport in GNRs, by controlling the substrate and edge roughness independently.
61
We also show that the lattice thermal conductivity (Fig. 3.17a) and the phonon MFP (Fig.
3.17b) both scale with the length of the GNR in “short” ribbons due to the MFP being limited to
about half the length (τball = ½L/|vb(q)|). A difference exists between wide (W = 2 μm) and narrow
FIG. 3.17: Dimension scaling behaviors. (a) Variation of lattice thermal conductivity and (b)
mean free path with the length of graphene samples for a wide (W=2 μm) and narrow (W=65 nm)
ribbon, showing that the length effect is more pronounced in wide ribbons. Phonons in narrow
ribbons suffer more scattering at the rough edges; hence, the effect of length is weaker in narrow
GNRs. (c) Dependence of thermal conductivity on ribbon width shows good agreement with
experimental data (symbols). Our model shows that thermal conductivity in short (L=260 nm)
ribbons is independent of width when ribbons are wide (W>L), but strongly dependent on width
when they are narrow (W<L), consistent with strong diffuse scattering at the rough edges. (d)
Effect of rms edge roughness Δ is confirmed in the dependence of the MFP, where we can see
that the phonon MFP in wide ribbons is largely independent of Δ, while the MFP in narrow
ribbons decreases with increasing Δ, indicating the strong role of edge roughness in thermal
transport in narrow GNRs.
10 100 1000 10,0000
100
200
300
400
500
600
700
W (nm)
k (
Wm
-1K
-1)
TA long
LA long
LA short
TA short
ZA longZA short
Total long
Total short
0.1 10.510
1
102
LER (nm)
mfp
(nm
)
TA wide
LA wide
LA narrow
TA narrow
ZA wide
ZA narrow
Total wide
Total narrow
101
102
103
104
101
102
103
L (nm)
k (
Wm
-1K
-1)
TA wide
LA wide
LA narrowTA narrow
ZA wide
ZA narrow
Total wide
Total narrow
GNR W=65 nmSeol (2010)
graphene
101
102
103
104
101
102
L (nm)
mfp
(nm
)
TA wide
LA wide
LA narrow
TA narrow
ZA wide
ZA narrow
Total wide
Total narrow
a b
c d
Δ
Seol (2010)
62
(W = 65 nm) ribbons due to the presence of edge roughness scattering in the narrow ribbons. In
the wide ribbons, the most significant scattering mechanism is substrate scattering, which limits
the phonon MFP to around 100 nm (the value to which the total MFP converges for “long”
ribbons in Fig. 3.17b). We note that this value is very nicely consistent with the average sub-
strate-limited (or intrinsic) MFP deduced through a simple comparison with the ballistic con-
ductance limits in the main text (Fig. 3.12 and surrounding discussion).
For a sample length L ~ 200 nm, or approximately twice the intrinsic MFP, the length-
dependent ballistic scattering rate is comparable to the substrate scattering rate and the effective
thermal conductivity and phonon MFP become one-half of the substrate-limited values (Fig.
3.17a-b for “wide” sample). However, in narrow GNRs, the diffuse scattering at the edges limits
the phonon MFP to approximately one-half of the width W (the edge-roughness-limited MFP
also depends on the rms edge roughness, as shown in Fig. 3.17d). Consequently, thermal
transport in narrow GNRs is mostly diffusive until the length is reduced to values comparable to
the width, and the transport again becomes quasi-ballistic (shown by solid lines in Fig. 3.17a-b).
3.6 Summary
In conclusion, we investigated heat flow in SiO2-supported graphene samples of dimensions
comparable to the phonon MFP. Short devices (L ~ λ, corresponding to Fig. 3.10b schematic)
have thermal conductance much higher than previously found in micron-sized samples, reaching
35% of the ballistic limit at 200 K and 30% (~1.2 GWK-1
m-2
) at room temperature. However,
narrow ribbons (W ~ λ, corresponding to Fig. 3.10c schematic) show decreased thermal
conductivity due to phonon scattering with edge disorder. Thus, the usual meaning of thermal
conductivity must be carefully interpreted when it becomes a function of sample dimensions.
The results also suggest powerful means to tune heat flow in 2D nanostructures through the
63
effects of sample width, length, substrate interaction and edge disorder.
64
CHAPTER 4
SUBSTRATE-SUPPORTED THERMOMETRY PLATFORM FOR NANO-
MATERIALS LIKE GRAPHENE, NANOTUBES, AND NANOWIRES*
4.1 Introduction
To probe nanoscale thermal conduction, different measurement techniques have been devel-
oped, such as the 3ω method, scanning thermal microscopy, time-domain thermoreflectance, and
various bridge platforms [30]. Among them, using a microfabricated suspended bridge can
directly measure in-plane heat flow through nanostructures by electrical resistance thermometry
[69, 187-189]. This suspended thermometry has good measurement accuracy and has been
widely used for nanofilms [189-191], one-dimensional (1D) materials like nanowires (NWs)
[187, 188] and nanotubes [73, 76], as well as two-dimensional (2D) materials like hexagonal
boron nitride [192] and graphene [51, 52]. A major drawback of this platform is that the fabrica-
tion is complicated and the test sample has to be either fully suspended [51, 69, 73, 76, 187-192]
or supported by a suspended dielectric (e.g. SiNx) membrane [52, 78]. This limits the diversity of
measureable materials and makes the suspended platform fragile.
To overcome these limitations, substrate-supported platforms could be preferred and have
been recently employed in thermal studies of Al nanowires [157], encased graphene [54] and
graphene nanoribbons in our previous work [55]. Nanomaterials are almost always substrate-
supported in nanoscale electronics [193], thus a substrate-supported platform has the advantage
of testing devices including extrinsic substrate effects [1, 106], which could be different from
their intrinsic thermal properties (probed by suspended platforms) [21, 23, 24]. Substrate-
* This Chapter is reprinted from Z. Li, M.-H. Bae, and E. Pop, “Substrate-Supported Thermometry Platform for
Nanomaterials Like Graphene, Nanotubes, and Nanowires,” Applied Physics Letters, vol. 105, p. 023107 (2014).
Copyright 2014, American Institute of Physics.
65
supported thermometry platforms could also readily be incorporated in industrial mask designs
and fabrication processes as thermal test structures in addition to existing electrical test struc-
tures. Thus, measuring the thermal conduction of nanomaterials on a substrate is crucial from a
practical viewpoint.
In this work, we critically examine the applicability and limitations of nanoscale thermal
measurements based on a substrate-supported platform utilizing electrical resistance thermome-
try. As a prototype, the thermal conductivity of graphene on a SiO2/Si substrate was experimen-
tally tested. The fabrication of this supported platform is much easier than that of suspended
platforms, but as a trade-off the thermal conductivity extraction is slightly more challenging and
must employ a three-dimensional (3D) heat conduction simulation of the test structure. Through
careful uncertainty analysis, we find that the supported platform can be optimized to improve the
measurement accuracy. The smallest thermal sheet conductance that can be measured by this
method within a 50% error is ~25 nWK-1
at room temperature, which means the supported
platform can be applied to nanomaterials like carbon nanotube (CNT) networks, nanotube or
nanowire arrays, and even a single Si nanowire. Additionally, it is suitable for materials which
cannot be easily suspended, like many polymers, and the substrate is not limited to SiO2/Si but
can be extended to other substrates such as flexible plastics.
4.2 Thermometry Platform Demonstration
4.2.1 Platform Structure and Measurement
Figure 4.1a shows a scanning electron microscopy (SEM) image of a typical supported
thermometry platform, here applied to a monolayer graphene sample. (In general, the sample to
be measured is prepared on a SiO2/Si substrate, though this is not always necessary, as we will
show below.) Then, two parallel, long metal lines with at least four probe arms are patterned by
66
electron-beam (e-beam) lithography as heater and sensor thermometers. If the sample is conduc-
tive (here, graphene), then the heater and sensor must be electrically insulated by a thin SiO2
layer, as seen in the Fig. 4.1b cross-section. To perform measurements, a DC current is passed
through one metal line (heater) to set up a temperature gradient across the sample, and the
electrical resistance changes of both metal lines (heater and sensor) due to the heating are
monitored. After temperature calibration of both metal line resistances the measured changes in
resistance (ΔR) can be converted into changes in temperature of the heater and sensor, ΔTH and
ΔTS, as a function of heater power PH.
FIG. 4.1: (a) Scanning electron microscopy image of supported thermometry platform designed
to measure thermal conductivity of a graphene sample (purple) on a SiO2/Si substrate. (b, c) 2D
and 3D finite element models used to simulate heat conduction in the supported thermometry
platform, respectively. In the 2D model, only the cross-section is included and the zoom-in
shows the typical temperature distribution with heating current applied through the heater. (d)
Zoomed-in temperature distribution around heater and sensor obtained from 3D simulation,
which matches with measured temperature. The white dashed lines indicate the outline of the
sample, which is highlighted by the red line and rectangle in (b) and (c), respectively. The
detailed shape and size of the sample will not affect the simulation.
a
c
b
d
sample
2 μm
2D Model
2LS = 6 μm
LS
= 3
μm
tbox
Si
sensorheater
sample
SiO2LHS
SiO2
Si
∆T (K)10-2 10-1 100 101
3D Model
LS
= 2
0 μ
m
3D Model
thin SiO2
zx
y
LDmet
sample
67
As the sample is not suspended, a control experiment should be performed after removing
the exposed parts of the sample and repeating the above measurements to independently find the
thermal properties of the heat flow path through the contacts and substrate. Etching away the
sample is important, rather than simply performing the measurement without the original sample,
because it preserves the sample portion beneath the heater and sensor electrodes, i.e. the same
contact resistance in both configurations. Using this approach, in previous work [55] we obtained
the thermal conductivity of the underlying SiO2 within 5% error of widely known values, which
also helps support the validity of this approach.
4.2.2 Model to Extract Thermal Conductivity
We note that the fabrication of the supported platform can be performed with greater yield
than that of suspended platforms, but the thermal conductance between the heater and sensor
cannot be obtained analytically as in the suspended case, due to non-negligible heat leakage into
the substrate. Therefore, numerical modeling of such heat conduction must be employed to
extract the thermal conductivity of the sample. For comparison, we considered both 2D and 3D
finite element models of the sample, which are implemented by a commercial software package
(COMSOL). In the 2D model, only the cross-section of the platform is simulated, and the Si
substrate size is chosen as 2LS×LS (Fig. 4.1b). In the 3D model, half of the platform needs to be
simulated due to the symmetry plane which bisects the region of interest, and the Si substrate
size is chosen as 2LS×LS×LS (Fig. 4.1c). To perform the simulation, in both 2D and 3D models
the bottom and side boundaries (except symmetry plane in 3D) of the Si substrate are held at the
ambient temperature, i.e. isothermal boundary condition. Other outer boundaries of the whole
structure are treated as insulated, i.e. adiabatic boundary condition. Joule heating is simulated by
applying a power density within the heater metal, and the calculation is performed to obtain the
68
temperature distribution in steady state, as shown in Figs. 4.1b and 4.1d for the 2D and 3D
models, respectively. After calculating the average temperature rises in the measured segments
of the heater and sensor, we obtain the simulated values of ΔTH and ΔTS vs. PH. Then, we can
match these with the measured values by fitting the thermal conductance G of the test sample
between the heater and sensor. The thermal conductivity of the sample, k = GL/A, can then be
extracted. Here L is the sample length, i.e. the heater-sensor separation and A is the cross-
sectional area, i.e. the sample width W times thickness h (see Figs. 4.1b and 4.1d).
To correctly obtain the sample thermal conductivity, the simulated size LS of the Si substrate
must be carefully chosen because the real Si chip is ~0.5 mm thick and several mm wide. The
chip dimensions are semi-infinite compared to the small heating region (~10 μm) and cannot be
fully included due to computational grid limits. Thus, the simulated LS should be large enough to
model the heat spreading and give a converged value of the extracted k. For the 2D and 3D
models, the extracted thermal conductivities as a function of simulated LS from our graphene
measurement [55] at 270 K are shown in Fig. 4.2a. Here, for the 3D model, we further consid-
ered two cases: heater and sensor with probe arms (as shown in Fig. 4.1d), and without probe
FIG. 4.2: (a) Extracted graphene thermal conductivity as a function of the Si substrate size LS for
different models. (b) Simulated temperature profiles along the sensor from 3D models with and
without probes. Two solid lines are obtained by using the same graphene k. Changing from solid
to dash red lines corresponds to the red arrow in (a). (c) Extracted graphene k converges as the
simulated probe length Lprobe increases [corresponding to the red arrow in (a)].
0 1 2 3 4 5 60.0
0.2
0.4
0.6
0.8
1.0
T
s (
K)
y (m)
a cb
0 20 40 60 80 10050
100
150
200
250
300
350
Simulated LS (m)
Extr
acte
d k
(W
m-1K
-1)
3D with probe
3D without probe
2D
3D with probe
3D without probe
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6150
200
250
300
Extr
acte
d k
(W
m-1K
-1)
Simulated Lprobe (m)
T=270 K LS=50 μmSimulated Lprobe
W/2
69
arms (as shown in Fig. 4.3a), because the latter has a more direct correspondence to the 2D
model (reflected by the similar extracted k for small LS). Comparing these two cases also allows
us to test how many details of the electrode geometry should be included.
It is clear that the extracted k by the 2D model continues decreasing as the simulated LS in-
creases, whereas the two 3D models give converged k when the simulated LS is sufficiently large
(≥ 50 μm). The 2D model is insufficient because it neglects the heat spreading along the y-
direction perpendicular to the 2D plane. Although the heater and sensor length (~10 μm or
similar to the sample width) are long compared to their separation LHS (~0.5 μm), we find that
3D heat spreading about 10 μm away from the heating center cannot be neglected (see Fig. 4.3b).
As the simulated LS increases, the effect of neglecting the heat spreading along the y-direction
becomes stronger in the 2D model. Thus, we find that the 3D simulation is preferable in order to
fully capture all heat spreading effects due to the finite size of the sample.
FIG. 4.3: (a) Structure and temperature distribution for the 3D model without probe arms, which
can be regarded as the one extruded from the 2D model, and hence has a better correspondence
to the 2D model than the 3D with probe model. (b) Evolvement of temperature isosurface shape
in 3D simulation. When heat spreads out from the heater, the isosurface is close to a cylinder at
the beginning (inset), behaving as 2D heat conduction, while it changes to a sphere after ~10 μm
from the center, indicating heat spreading along the third direction cannot be ignored, which is
the reason why the 2D model fails.
b
LS
= 2
0 μ
m
∆T (K)
101
100
SiO2Silicon
∆T (K)10-2 10-1 100 101
a
70
Figure 4.2a also shows that simulating the effect of heat loss through the voltage probe arms
is necessary. As shown in Fig. 4.2b, if the same k of the sample is used, the simulated tempera-
ture rise along the sensor for the “3D with probe” case (solid red line) is lower than that for the
“3D without probe” case (solid black line), and in the former case there are temperature dips at
the points where the probe arms are connected due to heat leakage through them. Figure 4.2c
shows that the extracted k increases and saturates gradually as the simulated length of the probe
arms Lprobe increases, indicating Lprobe ≥ 1.5 μm is sufficiently long to catch its effect and this
converged value finally provides a correct k of the sample.
4.3 Uncertainty Analysis and Optimization
4.3.1 Uncertainty Analysis of Measurements
Next, we turn to the estimation of uncertainty in the extracted thermal conductivity k, which
can be accomplished by the classical partial derivative method: uk/k = [Σi (si×uxi/xi)2]1/2
, where uk
is the total uncertainty of extracted thermal conductivity k, uxi is the estimated uncertainty for
each input parameter xi of the simulation, and the sensitivity si is defined by si = (xi/k)∂k/∂xi =
∂(lnk)/∂(lnxi). The sensitivity is evaluated numerically by giving a small perturbation for each
input parameter around its typical value and redoing the extraction simulation to obtain the
change of k [54]. To highlight the relative importance of each input parameter, we define its
absolute contribution as ci = |si|×(uxi/xi), and a relative contribution as ci2/Σci
2. As an example, the
calculated sensitivities and uncertainty analysis for the extracted k in Fig. 4.2c are shown in
Table 4.1. The total uncertainty in this case is ~20.8%, and it mainly arises from the contribu-
tions (ci > 5%) of the thermal conductivity of bottom SiO2 (kox), thermal boundary resistance
(TBR) of the SiO2/Si interface (Roxs), measured sensor response (ΔTS/PH), thermal conductivity
of Si substrate (kSi), and heater-sensor midpoint separation (LHS). The values and their uncertain-
71
ties of kox and Roxs are from our previous measurements [55]. The thermal conductivity of metal
lines (kmet) is calculated from the measured electrical resistance according to the Wiedemann-
Franz Law. kSi is well-known data from Ref. [168]. The TBR values and their uncertainties for
the graphene/SiO2 interface (Rgox) and metal/SiO2 interface (Rmox) are based on measurements in
Refs. [48] and [45], respectively. These two interface resistances have small contributions to the
total uncertainty (ci ≤ 2%) due to their small sensitivities (si). Even when their TBR values
Table 4.1: Calculated sensitivities and uncertainty analysis for the extracted graphene k from our
measurement at 270 K [correspond to the converged value in Fig. 4.2c with LS = 50 µm and
Lprobe = 1.6 µm]. The total uncertainty is ~20.8%.
Units Values x i
Uncertainty
u xi
u xi /x i
Sensitivity
s i
Contribution
c i =|s i |×u xi /x i
c i2/Σc i
2
Sensor response ΔT S/P H 0.01623 0.0003 1.8% 4.41 8.2% 15.4%
Heater response ΔT H/P H 0.14080 0.001 0.7% 2.42 1.7% 0.7%
k ox 1.213 0.04 3.3% 4.32 14.2% 46.9%
k Si 127 15 11.8% 0.46 5.5% 6.9%
k met 49 4 8.2% 0.40 3.3% 2.5%
R gox 1.15E-08 2.0E-09 17.4% -0.11 1.8% 0.8%
R oxs 9.99E-09 3.5E-09 35.0% -0.25 8.8% 17.7%
R mox 1.02E-08 3.5E-09 34.3% 0.06 2.0% 0.9%
t box 294 1 0.3% -5.90 2.0% 0.9%
t tox 25 1 4.0% -0.02 0.1% 0.0%
t met 50 2 4.0% 0.40 1.6% 0.6%
Distance of H/S L HS 505 5 1.0% 5.09 5.0% 5.9%
D met 199 4 2.0% -0.15 0.3% 0.0%
D tox 243 4 1.6% -0.31 0.5% 0.1%
Half length of H/S
metal linesL met/2 5.9 0.04 0.7% 0.19 0.1% 0.0%
Distance of 2
Voltage probesD pV 4.21 0.02 0.5% 3.58 1.7% 0.7%
Distance of
Current and
Voltage probes
D pIV 1.04 0.02 1.9% -0.12 0.2% 0.0%
Geo
metr
ica
l
Thickness of
bottom and top
SiO2, and metal
nm
Width of metal and
top SiO2 lines
μm
Input parameters (T = 270 K)
Th
erm
al
Thermal
conductivity of
SiO2, Si, metal
W/m/K
TBR of
graphene/SiO2,
SiO2/Si, metal/SiO2
interfaces
m2K/W
Expt. K/μW
72
change by a factor of 2 (i.e., uxi =100%), their contributions become ci = 10% and 6%, which
only changes the total uncertainty from ~20.8% to ~23% and ~21.5%, respectively. Thus, their
contributions are not shown in the following results.
4.3.2 Optimization of the Platform
The accuracy of our supported thermometry platform can be optimized through two im-
portant geometric parameters: i) the center-to-center distance between the heater and sensor
(LHS) and ii) the bottom insulator (here oxide) thickness (tbox) (see Fig. 4.1b). If LHS is too large
then too much of the heater power is dissipated into the substrate; if it is too short, then the
temperature drop between heater and sensor is not large compared with the temperature variation
under the heater/sensor. If tbox is too thin, significant heat leakage will occur into the substrate; if
it is too thick, then its lateral thermal conductance will dominate the heat flow between heater
and sensor, overwhelming that of the supported sample.
The optimized values of LHS and tbox can be found by monitoring the uncertainty change
FIG. 4.4: (a, b) Estimated uncertainty of extracted sample thermal conductivity as a function of
the heater-sensor midpoint separation LHS and the bottom oxide thickness tbox, respectively. This
gives the optimized design with LHS = 600 nm and tbox = 300 nm for the partly-etched sample
case (inset of b, red part indicates the sample). (c) Estimated uncertainty of extracted sample k
increases as its thermal sheet conductance G□ decreases for LHS = 600 nm and tbox = 300 nm,
showing the measureable G□ (blue region) by this SiO2/Si supported thermometry platform. Inset
is a schematic for CNT/NW array measurements, where green lines are CNTs/NWs, and red and
blue lines are heater and sensor, respectively.
200 400 600 800 10000
5
10
15
20
25
30
35
Un
ce
rta
inty
(%
)
LHS
(nm)0 200 400 600
0
10
20
30
40
50
Un
ce
rta
inty
(%
)
tbox (nm)10 100
0
20
40
60
80
100
Uncert
ain
ty (
%)
G (nWK
-1)
total
ΔTS
Roxs
kSi
kox
LHS
DpV
tbox
a b c
tbox=300 nm
LHS=600 nm
73
(due to sensitivity change) of extracted sample k, and the results are shown in Figs. 4.4a and
4.4b. Here we consider T = 300 K and thin film sample of thermal sheet conductance G□ = kh =
100 nWK-1
, and assume the sample is partly etched off so that only the part between the heater
and sensor is preserved (see the inset of Fig. 4.4b). The optimization is calculated at heater and
sensor linewidth Dmet = 200 nm, and only the input parameters whose contributions ci are larger
than 2% are included. From the estimated total uncertainty, we find that the optimized values of
LHS and tbox are ~ 600 nm and ~300 nm, respectively, leading to the minimized uncertainty
~18%. The non-etched case has the similar results but with a slightly higher uncertainty ~25%
(see Figs. 4.5a and 4.5b). By looking at the uncertainty contributed by each input parameter, it is
clear that the optimization is achieved mainly due to the competition between the measured
sensor response (ΔTS) and the role of the bottom oxide (kox), as we explained above. Additional
calculations (not shown) indicate that narrower heater and sensor linewidths (~100 nm) give
almost the same optimized values of LHS and tbox, but slightly lower total uncertainty.
FIG. 4.5: Results for the sample non-etched case. (a, b) Optimize the heater-sensor distance LHS
and the bottom oxide thickness tbox, respectively. The optimized LHS and tbox are almost the same
as those for the sample non-etched case (Fig. 4.4), but with a slightly higher uncertainty ~25%.
(c) Estimated uncertainty of extracted sample k increases as its thermal sheet conductance G□
decreases for LHS = 600 nm and tbox = 300 nm, showing the measureable G□ within a 50% error
(blue region) is G□ ≥ 32 nWK-1
. Inset is a schematic for CNT/NW array measurements, where
green lines are CNTs/NWs, and red and blue lines are heater and senor, respectively.
200 400 600 800 10000
10
20
30
40
50
60
70
Uncert
ain
ty (
%)
LHS
(nm)
0 100 200 300 400 500 6000
10
20
30
40
50
60
Un
ce
rta
inty
(%
)
tbox
(nm)10 100
0
20
40
60
80
100
Un
ce
rta
inty
(%
)
G (nWK
-1)
a b ctotal
ΔTS
Roxs
kSi
kox
LHS
tbox
DpV
tbox=300 nm LHS=600 nm
74
4.4 Applicability of the Platform
4.4.1 Measureable Thermal Sheet Conductance
After optimizing our supported thermometry platform, the next question concerns the small-
est in-plane thermal conductance that can be sensed by this method. To address this, we calculate
the uncertainty change of extracted thermal conductivity as a function of the sample thermal
sheet conductance G□ = kh. When the sample is partly etched to match the dimensions of the
heater-sensor width and spacing (inset of Fig. 4.4b) the results at T = 300 K are shown in Fig.
4.4c. As expected, the estimated measurement uncertainty increases as G□ decreases. If we set
the maximum uncertainty to ~50%, the sensible range of G□ enabled by this platform is G□ > 25
nWK-1
(the blue region in Fig. 4.4c). For samples that are not etched to conform to the heater-
sensor width and separation (e.g., Fig. 4.1a), the uncertainty is slightly higher, and the smallest
sensible G□ within a 50% error is ~32 nWK-1
(see Fig. 4.5c). This requirement could be satisfied
in most thin film materials, such as polymer films (k > 0.2 Wm-1
K-1
and h > 200 nm) [194] and
CNT networks (k > 20 Wm-1
K-1
and h > 2 nm) [195, 196], etc.
4.4.2 Nanotube or Nanowire Array
We emphasize that this supported thermometry platform can be applied not only to thin
films but also to arrays of quasi-one-dimensional materials (inset of Fig. 4.4c). In our previous
work [55], we had shown its application to graphene nanoribbon arrays. Here we give estima-
tions of minimum array density required to apply the platform to carbon nanotube and Si nan-
owire arrays. For single-wall CNT arrays, assuming array density p (the number of CNTs per
unit width), the equivalent thermal sheet conductance is G□ = k(πdδ)p, where k, d, and δ are the
thermal conductivity, diameter, and wall thickness of single-wall CNTs, respectively. Then, the
array density is given by p = G□/(kπdδ). Considering single-wall CNTs with k = 1000 Wm-1
K-1
, d
75
= 2 nm, and δ = 0.34 nm, as well as G□ = 25 nWK-1
(the best case), we obtain the density re-
quired for CNT array measurements is p ≥ 12 μm-1
. This CNT array density is achievable exper-
imentally today, as some studies [197, 198] have demonstrated CNT densities up to ~50 μm-1
.
For NW arrays and for thicker multi-wall CNTs the array density is given by p =
G□/(kπd2/4), where k and d are the thermal conductivity and diameter of the NWs, respectively.
For 20 nm diameter Si NWs [77] with k ≈ 7 Wm-1
K-1
, the required array density is p ≥ 11 μm-1
;
for 50 nm diameter Si NWs, smooth and rough edges lead to k ≈ 25 and 2 Wm-1
K-1
, respectively
[77, 79], and the required array density is p ≥ 0.5 μm-1
and 6 μm-1
. Nanowire arrays can be
fabricated much denser than these required densities [78, 199]. The above estimations indicate
that the supported thermometry platform can be easily applied to CNT arrays and Si NW arrays
with both smooth and rough edges. In addition, such arrays do not require uniform spacing.
4.4.3 Single Si Nanowire
We note that the minimum array density for 50 nm diameter smooth Si NWs is very low
(~0.5 μm-1
), which implies that it is possible to measure a single Si NW by using this platform.
To confirm this idea, we performed the simulation with just one NW between the heater and
sensor (inset of Fig. 4.6a). To achieve the best measurement accuracy, we first optimize the
dimensions of the heater and sensor, that is, the midpoint distance between them (LHS) and the
distance between two voltage probe arms (DpV) (inset of Fig. 4.6a). The calculated uncertainty
contributed from the measured sensor temperature rise (ΔTS) as a function of LHS and DpV is
shown in Fig. 4.6a. In the calculation, the bottom oxide thickness (tbox) and electrode linewidth
(Dmet, see Fig. 4.1b) are chosen as 300 nm and 200 nm, respectively, and a Si NW with d = 50
nm and k = 25 Wm-1
K-1
is used. The minimum of the uncertainty indicates the optimized struc-
ture is LHS = 600 nm and DpV = 1000 nm. By using these values, the total uncertainty as a
76
function of kA (A is the NW cross-sectional area) is calculated and shown in Fig. 4.6b. For highly
conductive NWs (kA > 5×10-14
WmK-1
) the measurement uncertainty is around 60%, indicating
that obtaining an estimate of the thermal properties of a single NW is possible. However, this
also indicates that it is not possible to measure an individual single-wall CNT with the supported
platform because its kA is low (< 2×10-14
WmK-1
), although it may be possible to measure one
multi-wall CNT as long as the kA condition above is satisfied.
4.4.4 Plastic Substrate
Before concluding, we note that the supported thermometry platform is not limited to
SiO2/Si substrates, but could be extended to thermally insulating plastic substrates like Kapton,
polyethylene terephthalate (PET) and so on. Here, we consider measurements with a 25 μm thick
Kapton substrate on a heat sink (inset of Fig. 4.7a). We note that results for other Kapton thick-
ness or PET are similar. In simulations, the ambient (isothermal) boundary condition should be
applied to at least one surface of the substrate, but due to its low thermal conductivity the outer
FIG. 4.6: (a) Optimizing the design for measuring a single nanowire (see inset) by estimating the
uncertainty of extracted nanowire thermal conductivity as a function of LHS and DpV. Here only
the uncertainty contributed from the measured sensor temperature rise (ΔTS) is calculated. The
optimized design is LHS = 600 nm and DpV = 1000 nm. (b) Estimated uncertainty of extracted
nanowire k increases as its cross-sectional thermal conductance kA decreases.
400 600 800 1000 0
2000
4000
30
40
50
60
70U
ncert
ain
ty (
%)
10-14
10-13
0
20
40
60
80
100
120
140 total
TS
Wmet
kSi
kox
tbox
Dmet
kmet
DpVV
dwire
Un
ce
rta
inty
(%
)
kA (WmK-1)
a b
DpV
NW
LHS DpV=1000 nm
LHS=600 nm
total
ΔTS
Dmet
kSi
kox
tbox
LHS
kmet
DpV
dwire
77
surfaces of the plastic substrate cannot reach the ambient temperature. Thus, a heat sink with
high thermal conductivity is added for this purpose. On the other hand, from a practical
viewpoint, measurements are generally performed in vacuum and on a chuck for ambient
temperature control, that is, on a heat sink. Since the plastic substrate is generally tens of microns
thick, its background thermal conductance will be typically larger than that of the sample; thus
the sample should be trimmed (etched) leaving just the portion between the heater and sensor
(inset of Fig. 4.7b), otherwise it will be difficult to sense the difference between the sample
measurement and the control experiment without the sample, resulting in a large uncertainty. By
using kps = 0.37 Wm-1
K-1
for Kapton (DuPontTM
Kapton® MT) and G□ = 100 nWK-1
for the
sample, we calculate the measurement uncertainty as a function of the heater-sensor distance
LHS, as shown in Fig. 4.7a. The minimized uncertainty is ~ 38% at LHS = 1.2 μm. With this
optimized structure, we further calculate the uncertainty change as a function of the sample
thermal sheet conductance (Fig. 4.7b), and find the smallest sensible G□ within a 50% error for a
FIG. 4.7: (a) Optimizing the heater-sensor midpoint separation LHS with the platform supported
by a plastic substrate on a heat sink (see inset). The estimated uncertainty of extracted sample k
is minimized at LHS = 1.2 µm for a 25 μm thick Kapton substrate. (b) Estimated uncertainty of
extracted sample k increases as its thermal sheet conductance G□ decreases, giving the sensible
range (blue) of G□ by this platform. Inset shows the optimized structure of the platform applied
to the plastic substrate, with LHS = 1.2 µm and the sample (red) only between heater and sensor.
10 1000
20
40
60
80
100
Uncert
ain
ty (
%)
G (nWK
-1)
0 1 2 30
10
20
30
40
50
60
70
80
Un
ce
rta
inty
(%
)
LHS (m)
a btotal
ΔTS
kps
kmet
tmet
LHS
DpV
plastic sub
heat sink
78
Kapton substrate is ~60 nWK-1
, which is ~2.5 times higher than for the optimized SiO2/Si
substrate. Correspondingly, the required density for CNT and NW arrays will be also 2.5 times
higher, which remains achievable in experiments.
4.5 Summary
In conclusion, we demonstrated that a relatively simple, substrate-supported platform can be
used to measure heat flow in nanoscale samples like graphene, CNT or NW arrays. This platform
requires fewer fabrication efforts and is useful for materials which are difficult to suspend, but
the sample thermal conductivity must be extracted by 3D finite element analysis. Based on
careful uncertainty analysis we find the platform design can be optimized and the smallest
thermal sheet conductance measurable by this method within 50% error is estimated to be ~25
nWK-1
at room temperature. This thermometry platform can also be applied to individual nan-
owires, and can be implemented both on SiO2/Si (or similar) and flexible plastic substrates.
79
CHAPTER 5
BALLISTIC THERMAL CONDUCTANCE AND LENGTH-DEPENDENT SIZE
EFFECTS IN LAYERED TWO-DIMENSIONAL MATERIALS
5.1 Introduction
Stimulated by extensive studies of graphene, enormous interest is being generated in the
properties of other two-dimensional (2D) layered materials, such as hexagonal boron nitride (h-
BN) and transition-metal dichalcogenides (TMDs, e.g., MoS2, and WS2). They exhibit remarka-
ble physical and chemical properties, and are promising candidates for applications in electron-
ics, optoelectronics, and photonics [9, 18, 19, 143]. Understanding thermal properties of materi-
als is important for improving thermally limited performance in devices and efficiency of energy
conversion [1, 5, 106]. The thermal properties of these layered materials are unique and highly
anisotropic, including high in-plane but very low cross-plane thermal conductivities, due to
strong in-plane chemical bonding and weak interlayer van der Waals interactions [23, 24]. Like
other materials, as sample dimensions are comparable to the average phonon mean free path
(MFP) λ, size effects of thermal transport become important [5]. However, compared to bulk
properties our understanding in this scale is relatively lacking in general (not limited to layered
materials).
First, if the sample length L is shorter than average phonon MFP λ, heat conduction will be
ballistic (i.e., no scattering) and governed by an upper limit, ballistic thermal conductance (Gball),
but the corresponding values for these layered materials (even for most materials) are still
unknown except for graphene [29, 55]. Second, in the range of L > λ, how thermal conductivity k
evolves from the ballistic to diffusive regime as L increases is not well established, and when it
will eventually enter the diffusive regime is not clear. Particularly, some theoretical models
80
predicted that in low-dimensional momentum-conserving systems k will diverge with L, i.e., ~Lα
for one-dimension (1D) [200, 201] and ~logL for 2D [201-203]. Whereas, in what length scale
the divergence might occur and should be experimentally tested has not been carefully discussed.
Third, as an important characteristic length, MFP itself has not been estimated consistently. For
example, the “textbook” phonon MFP value for Si is ~40 nm at room temperature [204], ob-
tained from the kinetic theory by using the sound velocity as phonon group velocity [205].
However, by analyzing accumulative thermal conductivity as a function of MFP, phonons with
MFP longer than 1 µm contribute ~40% to the bulk k in Si [206-209], so a “median” MFP of 0.5-
1 µm is suggested [204]. Thus, it is crucial to systematically study these L-dependent size effects
as well as the proper estimation of phonon MFP.
In this work, we study ballistic-diffusive thermal conduction in monolayer graphene, h-BN,
MoS2, and WS2, as well as their three-dimensional (3D) bulk counterparts. Based on full phonon
dispersions obtained from ab initio simulations, we calculate the in-plane (for monolayer and
bulk) and cross-plane (for bulk) ballistic thermal conductance Gball of these materials for the first
time. Due to their stronger chemical bonding, graphene/graphite and h-BN show higher Gball than
MoS2 and WS2 above ~100 K in general. From our calculated Gball, we can obtain L-dependent
thermal conductivity by using a phenomenological ballistic-diffusive model, which shows good
agreement with very recent measurements [57] of k vs. L up to 9 µm in suspended graphene. We
also show that a proper estimation of the overall MFP λ should be a rigorous average of all
phonon modes, which can be condensed to an expression just including Gball and diffusive
thermal conductivity kdiff. We point out that the kinetic theory can reach the same result as long
as the correctly averaged phonon group velocity is used. Importantly, for anisotropic layered
materials, 2D and 1D forms of the kinetic theory should be used for in-plane and cross-plane,
81
respectively. Based on the ballistic-diffusive model, k eventually converges to its diffusive value,
kdiff, around L ~ 100λ, much longer than the commonly assumed length. Thus, whether k is
divergent in low-dimensional materials should be examined beyond ~100λ to distinguish from
the intrinsic increase of k due to the ballistic-to-diffusive transition. All these results broaden our
understanding of thermal transport in low-dimensions and short length scales, and will help
guide the use of these layered materials for tailored applications.
5.2 Phonon Dispersions by First-Principles Calculations
The equilibrium atomic structures are calculated within density functional theory (DFT) us-
ing projector-augmented wave (PAW) potentials [210], as implemented in the VASP code [211].
The local density approximation (LDA) is used for the exchange-correlation functional [212].
The plane-wave basis set with kinetic energy cutoff of 400 eV is used except for h-BN (450 eV
Fig. 5.1: Atomic structures for graphite with AB stacking (a), h-BN with AA' stacking (b), and
TMD with 2H-phase (c), where three atomic layers are plotted for each of them. Gray lines
indicate primitive cells for their bulk, and lattice constants a and c are labeled. Unrepeatable
atoms within the primitive cells are highlighted by bigger and darker colored balls: 4 atoms for
graphite and h-BN, 6 atoms for TMDs. (d) Schematic of their Brillouin zone with high-symmetry
lines highlighted. For their 2D monolayers, the material thickness is generally defined as c/2, and
the 2D BZ is simply a regular hexagon (see inset of Fig. 5.2a).
A
MK
LHΓ
ca
d
bgraphite h-BN TMD
A
B
A
A
A’
A
N
B
C
S, Se
Mo, W
A
A’
A
c
c
a
a a
c
82
is used). The convergence for energy is chosen as 10-8
eV between two consecutive steps, and
Table 5.1: Optimized lattice constants from our DFT calculations compared to experimental
values from literature [26, 213-220].
Experiments DFT simulations
a (Å) c (Å) a (Å) c (Å)
graphite [26, 213-216] 2.46 6.71 2.446 6.414
h-BN [217, 218] 2.50 6.66 2.490 6.488
MoS2 [219] 3.15 12.29 3.121 12.087
WS2 [220] 3.153 12.323 3.123 12.115
Fig. 5.2: Calculated phonon dispersions along high-symmetry lines for monolayer graphene (a),
h-BN (b), MoS2 (c), and WS2 (d). For graphene, red circles [25] and blue triangles [26] are
experimental data plotted for comparison. Other experimental data are also shown for h-BN
[221] and MoS2 [219]. The 2D Brillouin zone for monolayers is shown as the inset in (a).
0
50
100
150
200
Phonon E
(m
eV
)
0
10
20
30
40
50
60
Phonon E
(m
eV
)
0
10
20
30
40
50
60 Phonon E
Phonon E
(m
eV
)
0
50
100
150
200
Phonon E
(m
eV
)
LO
LA
TA
TO
ZA
ZO
a b
c d
Γ M
K
graphene h-BN
MoS2 WS2
83
full geometry relaxation is carried out until remaining forces on atoms smaller than 0.01 eV/Å.
For monolayers, a vacuum spacing of 20 Å is used to prevent interlayer interactions.
After obtaining relaxed lattice constants and atomic structures, the interatomic force con-
stants are calculated within a supercell using the frozen-phonon approach [223]. For gra-
phene/graphite and h-BN, the plane-wave kinetic energy cutoff is increased to 1000 eV, and the
supercell size is 5×5×1 and 5×5×2 for their monolayers and bulk, respectively. For MoS2 and
WS2, the energy cutoff is still 400 eV, and the supercell size is 7×7×1 and 4×4×3 for their
monolayers and bulk, respectively. Last, phonon dispersions are calculated by the PHONOPY
Fig. 5.3: Calculated phonon dispersions along high-symmetry lines for bulk graphite (a), h-BN
(b), MoS2 (c), and WS2 (d). Insets in (a) and (b) are zoom-in dispersions along the cross-plane
direction. For graphite, green diamonds [222], red circles [25], and blue triangles [26] are
experimental data plotted for comparison. Other experimental data are also shown for h-BN
[221] and MoS2 [219].
0
50
100
150
200
Ph
on
on
E (
me
V)
0
10
20
30
40
50
60
Phonon E
(m
eV
)
0
10
20
30
40
50
60
Phonon E
(m
eV
)
0
50
100
150
200
Phonon E
(m
eV
)
M KA
graphite h-BN
MoS2 WS2
a b
c d
0
10
20
A 0
5
10
15
A
LO1,LO2
TO1,TO2
ZO1,ZO2
84
package [224] interfaced with VASP.
The atomic lattices of bulk graphite, h-BN, and TMDs are shown in Figs. 5.1a-c. Graphite is
well known to crystalize in the AB stacking, while h-BN favors the AA' stacking [217, 225]. For
TMDs, the most common 2H phase [18, 19] (similar to the AA' stacking in h-BN) is considered
Fig. 5.4: (a) Brillouin zone of calculated bulk layered materials. High-symmetry lines and points
are highlighted. M1 and L1 are mid-points of lines Γ–M and A–L, respectively. (b-d) Zoomed-in
cross-plane phonon dispersions along Γ–A for graphite, h-BN, and MoS2. (e-h) Calculated in-
plane phonon dispersions on the ALH plane and cross-plane dispersions along Γ–A and M1–L1
directions for graphite, h-BN, MoS2, and WS2. For h-BN cross-plane (M1–L1), optical branches
with high phonon group velocity are highlighted in red and labeled. In (b-h) symbols are experi-
mental data for graphite (blue triangles [26] and green diamonds [222]), h-BN [221] and MoS2
[219].
0
10
20
30
40
50
60
L
Phonon E
(m
eV
)
H M1
L1
0
10
20
30
40
50
60
H
Phonon E
(m
eV
)
M1
L1
L
0
50
100
150
200
M1
Phonon E
(m
eV
)
L1
L H
0
50
100
150
200
M1
L1
Phonon E
(m
eV
) L H
a
0
5
10
15
20
Ph
ono
n E
(m
eV
)
A
graphite h-BN
MoS2 WS2
e f
g h
LO2
A
M1K
L1H
Γ
M
L
LO1
ZO2
ZO1
b d
0
5
10
15
20
A
cgraphite h-BN
0
2
4
6
8
A
MoS2
LA
TA(2)
TO’(2)
LO’
LA
TA(2)
TO’(2)
LO’
LO1,LO2
TO1,TO2
ZO1,ZO2
85
here. The Brillouin zone (BZ) of their bulk is shown in Fig. 5.1d. Table 5.1 summarizes our
relaxed lattice constants, and experimental results [26, 213-220] are listed as well for compari-
son. Our calculations show that using LDA gives underestimated lattice constants than experi-
mental values, but it gives better agreement with experimental phonon dispersions than using
generalized gradient approximation (GGA).
Figure 5.2 shows the calculated phonon dispersions along high-symmetry lines in BZ (see
inset) for monolayer graphene, h-BN, MoS2 and WS2. The phonon dispersions of their bulk are
shown in Fig. 5.3. Our calculations are in excellent agreement with the experimental data [25,
26, 219, 221, 222] plotted as symbols in Figs. 5.2 and 5.3. For monolayer graphene and h-BN,
there are six phonon branches due to two atoms in their primitive cells (Figs. 5.1a-b); whereas,
there are nine phonon branches for monolayer MoS2 and WS2 because of three atoms in their
primitive cells (Fig. 5.1c). For the bulk counterparts, each branch in their monolayer phonon
dispersions splits into two branches because two layers form a primitive cell in bulk (Fig. 5.1).
The two branches are distinguishable on the ΓMK plane, but they become degenerate on the
ALH plane in BZ (see Figs. 5.4e-h). Moreover, the dispersion along the cross-plane direction
appears in bulk materials (see Fig. 5.4). For cross-plane dispersions along the Γ–A direction,
transverse acoustic (TA) and transverse optical (TO’) branches are double degenerate, as shown
in Figs. 5.4b-d. Interestingly, some high-frequency optical branches (ZO1, ZO2, LO1, LO2) of h-
BN have significantly nonzero cross-plane group velocity, different from other layered materials,
which can be noticed along the M1–L1 direction (Fig. 5.4a) for example (see Figs. 5.4e-h). Thus,
when these modes in h-BN are excited as temperature T increases, they will provide extra
contributions to ballistic thermal conductance Gball, resulting in a “bump” in Gball versus T of h-
BN (see Fig. 5.6b).
86
5.3 Formalism of Thermal Physical Quantities
5.3.1 Heat Capacity
After obtaining full phonon dispersion in the Brillouin zone (BZ), heat capacity can be cal-
culated by
V ,
1( )b
b q
fC q
V T
, (5.1)
where the material volume V=WLH, a product of width W, length L, and hight H; ħ is the re-
duced Planck constant; q and ( )b q are phonon wavevector and frequency (b denotes different
branches), respectively; f=1/[exp(ħωb/kBT)-1] is the Bose-Einstein distribution; T is temperature
and kB is the Boltzmann constant. The sum Σ is over different branches (b) and all wave vectors (
q ) in the entire Brillouin zone. In practice, the sum over q is converted to an integral, and the
general expressions for one-dimension (1D), two-dimension (2D), and three-dimension (3D) are
given by
V
BZ
1
2
b
b
fC dq
A T
, (1D) (5.2a)
2
V 2 BZ
1
(2 )
b
b
fC dq
H T
, (2D) (5.2b)
3
V 3 BZ
1
(2 )
b
b
fC dq
T
, (3D) (5.2c)
where the integral is calculated in the entire BZ. For 2D and 3D, considering isotropic and
anisotropic cases, the expressions change to
V
1
2
b
b
fC qdq
H T
, (isotropic 2D) (5.3a)
2
V 2
1
2
b
b
fC q dq
T
, (isotropic 3D) (5.3b)
87
V 2
1
4
b
zb
fC dq qdq
T
, (anisotropic layered 3D) (5.3c)
These equations will be used in the determination of pre-factors for averaged phonon group
velocity v in Section 5.3.5. Figure 5.5 shows our calculated heat capacity from Eq. 5.2b (mono-
layer) and Eq. 5.2c (bulk) for graphite, h-BN, MoS2, and WS2.
5.3.2 Ballistic Thermal Conductance
Ballistic thermal conductance Gball can be calculated from full phonon dispersion without
any approximation by
ball ,
1( ) | ( ) |
2
b b
nb q
fG q v q
L T
, (5.4)
Fig. 5.5: Calculated heat capacity as a function of temperature for graphite (a), h-BN (b), MoS2
(c), and WS2 (d) from Eq. 5.2b (for monolayer) and Eq. 5.2c (for bulk).
1 10 100 100010
-3
10-2
10-1
100
101
102
103
Cv (
JK
-1kg
-1)
T (K)
101
102
103
104
105
106
Cv (
JK
-1m
-3)
1 10 100 100010
-3
10-2
10-1
100
101
102
103
Cv (
JK
-1kg
-1)
T (K)
101
102
103
104
105
106
107
Cv (
JK
-1m
-3)
1 10 100 100010
-3
10-2
10-1
100
101
102
103
Cv (
JK
-1kg
-1)
T (K)
101
102
103
104
105
106
107
Cv (
JK
-1m
-3)
1 10 100 100010
-3
10-2
10-1
100
101
102
103
Cv (
JK
-1kg
-1)
T (K)
101
102
103
104
105
106
Cv (
JK
-1m
-3)
~T3
~T2
~T1
~T3
~T2
~T1
h-BN
MoS2 WS2
graphene/graphitea b
c d
~T3
~T2
~T1
~T3
~T2
~T1
88
where ( )b
nv q = ˆ ( )b
qn q is phonon group velocity along the direction n (a normalized vector).
After converting the sum to an integral and dividing it by the cross-sectional area A=WH to get a
value independent of material sizes, the general expressions of Gball/A for 1D-3D are given by
ball
BZ
1( ) | ( ) |
4
b b
b
G fdq q v q
A A T
, (1D) (5.5a)
2ball
2 BZ
1( ) | ( ) |
8
b b
nb
G fdq q v q
A H T
, (2D) (5.5b)
3ball
3 BZ
1( ) | ( ) |
16
b b
nb
G fdq q v q
A T
, (3D) (5.5c)
In our calculations, ( )b
zv q is taken for cross-plane (3D); ( )b
xv q and ( )b
yv q are taken for in-plane
(2D and 3D) and they show almost the same results (within 2%), demonstrating the isotropy of
in-plane thermal conduction. The average of two results using ( )b
xv q and ( )b
yv q is adopted for in-
plane Gball/A.
For isotropic 2D and 3D, the velocity along a certain direction, ( )b
nv q can be expressed in
terms of the radial velocity scalar vb(q) as ( )b
nv q = vb(q)cosφ and ( )b
nv q = vb(q)sinθcosφ, respec-
tively, i.e., ( )b
nv q is the projector of vb(q) along the direction of temperature gradient. For aniso-
tropic 3D layered materials, we have 2( )b
xv q +2( )b
yv q =2
|| ( , )b
zv q q with || ( , )b
zv q q as the in-plane
velocity scalar, and | ( )b
zv q |= ( , )b
zv q q with ( , )b
zv q q as the cross-plane velocity scalar. Thus, Eq.
5.5b-c in different cases change to
2ball
2 0
2
1| ( )cos |
8
1 ( )
2
b b
b
b b
b
G fqdq d v q
A H T
fqdq v q
H T
, (isotropic 2D) (5.6a)
89
22ball
3 0 0
2
2
1sin | ( )sin cos |
16
1 ( )
8
b b
b
b b
b
G fq dq d d v q
A T
fq dq v q
T
, (isotropic 3D) (5.6b)
2ball, ||
||3 0
||3
1| ( , ) cos |
16
1 ( , )
4
b b
z zb
b b
z zb
G fdq qdq d v q q
A T
fdq qdq v q q
T
, (layered 3D in-plane) (5.6c)
2ball,
3 0
2
1| ( , ) |
16
1 ( , )
8
b b
z z zb
b b
z zb
G fdq qdq d v q q
A T
fdq qdq v q q
T
, (layered 3D cross-plane) (5.6d)
These equations will be used in the determination of pre-factors for averaged phonon mean free
path (MFP) λ and group velocity v in Sections 5.3.4 and 5.3.5.
5.3.3 Diffusive Thermal Conductivity
In the Boltzmann transport equation (BTE) approach with the relaxation time approxima-
tion, the diffusive thermal conductivity along a transport direction n is generally given by
2
diff tot,
1( )[ ( )] ( )b b b
nb q
fk q v q q
V T
, (5.7)
where ( )b
tot q is phonon relaxation time. After converting the sum of q to an integral, the general
expressions of kdiff for 1D-3D are then given by
2
diff tot
1( )[ ( )] ( )
2
b b b
b BZ
fk dq q v q q
A T
, (1D) (5.8a)
2 2
diff tot2
1( )[ ( )] ( )
(2 )
b b b
nb BZ
fk dq q v q q
H T
, (2D) (5.8b)
3 2
diff tot3
1( )[ ( )] ( )
(2 )
b b b
nb BZ
fk dq q v q q
T
. (3D) (5.8c)
For isotropic 1D-3D, phonon MFP is λb(q)=v
b(q)
b
tot . For anisotropic 3D layered materials, in-
90
plane and cross-plane phonon MFPs are || ||( , ) ( , )b b b
z z totq q v q q and ( , ) ( , )b b b
z z totq q v q q ,
respectively. Similar to the treatment of ( )b
nv q in Gball/A, kdiff in different cases can be further
expressed as
22
diff tot2 0
1[ ( )cos ]
4
1 ( ) ( )
4
b b b
b
b b b
b
fk qdq d v q
H T
fqdq v q q
H T
, (isotropic 2D) (5.9a)
22 2
diff tot3 0 0
2
2
1sin [ ( )sin cos ]
(2 )
1 ( ) ( )
6
b b b
b
b b b
b
fk q dq d d v q
T
fq dq v q q
T
, (isotropic 3D) (5.9b)
22
diff, || || tot3 0
|| ||2
1[ ( , )cos ]
(2 )
1 ( , ) ( , )
8
b b b
z zb
b b b
z z zb
fk dq qdq d v q q
T
fdq qdq v q q q q
T
, (layered 3D in-plane) (5.9c)
22
diff, tot3 0
2
1[ ( , )]
(2 )
1 ( , ) ( , )
4
b b b
z z zb
b b b
z z zb
fk dq qdq d v q q
T
fdq qdq v q q q q
T
, (layered 3D cross-plane) (5.9c)
These equations will be used in the determination of pre-factors for averaged phonon MFP λ in
Section 5.3.4.
5.3.4 Averaged Phonon Mean Free Path
If we look at the ratio of kdiff to Gball/A, based on Eqs. 5.5a, 5.6, 5.8a, and 5.9, we can find
that besides a pre-factor the ratio is just the defined average of phonon MFPs with all modes
weighted properly, which is shown below for all cases:
diff
ball
| ( ) |
2 2 ( ) 2/
| ( ) |
( )b b
bb
b b
b
bs
b
fdq v q
k T qfG A
dq v qT
q
, (1D) (5.10a)
91
diff
ball
( )( )
( )/ 2 2 2
( )
b b
bb
bs
b
b b
b
fqdq v q
k T qfG A
qdq v qT
q
, (isotropic 2D) (5.10b)
2
diff
2ball
( ()4 4 4
( )/ 3 3 3
)
( )
b b
bb
b
b
b
bsb
qf
q dq v qk T q
fG Aq dq v q
T
, (isotropic 3D) (5.10c)
||diff, ||
|| ||
ball, ||||
|| ( , )( , )
( , )/ 2 2 2
( , )
b b
z zbb
z bsb b
z
z
b
b
z
fdq qdq v q qk T q q
fG Adq qdq
q q
v q qT
,
(layered 3D in-plane) (5.10d)
diff,
ball,
( , )
2 2 ( , ) 2/
(
, )
)
(
,
b b
z zbb
z bsb b
b
z
z
zb
fdq qdq v q qk T q q
fG Adq qdq v q q
T
q q
,
(layered 3D cross-plane) (5.10e)
Thus, in general we have
diffbs
ball /
k
G A , (5.11)
which is the rigorous and correct average of phonon MFPs of all modes. The pre-factor βλ is 1/2,
2/π, and 3/4 for isotropic 1D, 2D, and 3D materials, consistent with Ref. [175]. Interestingly, for
layered 3D materials the in-plane pre-factor is 2/π, same as isotropic 2D, and the cross-plane pre-
factor is 1/2, same as 1D.
5.3.5 Averaged Phonon Group Velocity
When we look at the ratio of Gball/A to Cv, based on Eqs. 5.2a, 5.3, 5.5a, and 5.6, we can find
that besides a pre-factor the ratio is just the defined average of phonon group velocity with all
modes weighted properly, which is shown below for all cases:
92
ball
V
/ 1 1 1( )
2
( )
2 2
b
bb
b
b
b fdq
G A T v q vf
v
Cd
q
qT
, (1D) (5.12a)
ball
V
/ 1 1 1(
( )
)
b
bb
b
b
b fqdq
G A T v q vfC
q
v q
dqT
, (isotropic 2D) (5.12b)
2
ball
2V
/ 1 1 1( )
4 4 4
( )b
bb
b
b
b fq dq
G A T v q vfC
dq
v
qT
q
, (isotropic 3D) (5.12c)
ball, ||
|| ||
V
||/ 1 1)
1( ,
( , )
bb
zbb
zb
z
z
b
fdq qdqG A T v q q
v q
vfC
dq q
q
dqT
,
(layered 3D in-plane) (5.12d)
ball,
V
/ 1 1 1( , )
2 2 2
( , )b
zbb
zb
zb
b
z
fdq qdqG A T v q q v
f
v q q
Cdq qdq
T
,
(layered 3D cross-plane) (5.12e)
Thus, in general we have
ball
V
/v
G Av
C , (5.13)
which is the rigorous and correct average of phonon group velocity of all modes. The pre-factor
βv is 2, π, and 4 for isotropic 1D, 2D, and 3D materials. Particularly, for layered 3D materials the
in-plane and cross-plane pre-factors are π and 2, respectively, similar to the choice of βλ. We note
that to use Eq. 5.13 the units of Cv should be JK-1
m-3
.
After having averaged phonon MFP λ and group velocity v, we can examine if they can lead
93
to a consistent result with the kinetic theory. From Eqs. 5.11 and 5.13, we have
ball diffdiff V V diff
V ball
/1 1
/
vv
G A kk C v C k
d d C G A d
, (5.14)
where d=1−3 is the material dimension. Based on our derived pre-factors βv and βλ, we find that
βvβλ /d=1 holds for isotropic 1D, 2D, and 3D, meaning the reproduced kinetic theory. Important-
ly, for anisotropic layered 3D materials d has to be 2 and 1 for in-plane and cross-plane, respec-
tively, to hold the kinetic theory. This is a very important conclusion which has not been pointed
out before.
5.4 Results and Discussion
5.4.1 Ballistic Thermal Conductance
Based on the obtained phonon dispersions, the monolayer and bulk Gball/A of layered mate-
rials are calculated by Eqs. 5.5b and 5.5c, respectively. The results of Gball/A [including mono-
layer, bulk in-plane (||) and cross-plane ()] as a function of T for graphene/graphite, h-BN,
MoS2, and WS2 are shown in Fig. 5.6. For each material, its monolayer Gball/A is higher than that
of bulk (||) at low T, but they will become almost overlapped at high T, because the in-plane
phonon dispersions of monolayer and bulk are nearly degenerate except at low frequencies. For
bulk (), its Gball/A is similar to that of bulk (||) at low T, but the former is about ten times lower
than the latter at high T, reflecting the significant anisotropy in heat conduction. For better
comparison among four kinds of materials, Gball/A of the most widely-used semiconductor
material, silicon [174], is also plotted (dashed line) in each panel of Fig. 5.6 as a reference.
Above ~100 K, graphene, graphite (||), h-BN monolayer and bulk (||) show much larger Gball/A
than Si (Figs. 5.6a-b), due to their very strong in-plane covalent bonds and more dispersive
phonon dispersions, whereas, the monolayer and bulk (||) of MoS2 and WS2 show smaller Gball/A
94
than Si at T > 100 K (Figs. 5.6c-d) due to relatively weak metal-sulfur bonds. All cross-plane
Gball/A of four materials are lower than that of Si because of their weak van der Waals interac-
tions between layers. The room-temperature Gball/A values and Debye temperature ΘD [219, 226,
227] of these materials are listed in Table 5.2.
In the high T limit, all Gball/A vs. T curves flatten out (saturate) when all phonon modes are
fully excited. The temperatures at which Gball/A reaches 90% of its saturated (maximum) value
are nearly the same for the monolayer and bulk (||) of each material. They are ~1150 K, ~1060 K,
~320 K and ~310 K for graphite, h-BN, MoS2 and WS2, respectively, which are roughly 60% of
Fig. 5.6: Calculated ballistic thermal conductance per unit cross-sectional area as a function of
temperature for graphene/graphite (a), h-BN (b), MoS2 (c), and WS2 (d). Each panel includes
Gball/A of their monolayer, bulk (||), and bulk (), as well as that of silicon [174] for comparison.
1 10 100 100010
3
104
105
106
107
108
109
1010
Gball/
A (
WK
-1m
-2)
T (K)
1 10 100 100010
3
104
105
106
107
108
109
Gball/
A (
WK
-1m
-2)
T (K)1 10 100 1000
103
104
105
106
107
108
109
Gball/
A (
WK
-1m
-2)
T (K)
1 10 100 100010
3
104
105
106
107
108
109
1010
Gball/
A (
WK
-1m
-2)
T (K)
graphene/graphite h-BN
MoS2 WS2
~T 3
~T 2.5
~T 2
~T 1.5
graphite (||)graphite ()
bulk (||)bulk ()
bulk ()bulk (||)
bulk ()
Si Si
Si Si
a b
c d
bulk (||)
~T 3
~T 2.5
~T 2
~T 1.5
~T 3
~T 2.5
~T 2
~T 1.5
~T 3
~T 2.5
~T 2
~T 1.5
95
their Debye temperature ΘD (Table 5.2). The corresponding temperatures for bulk () are ~780
K, ~1270 K, ~345 K and ~360 K for graphite, h-BN, MoS2 and WS2, respectively. Interestingly,
there is a “bump” in Gball/A of h-BN bulk () before it flattens out, different from other bulk ()
curves. The reason is that the high frequency optical branches (ZO1, ZO2, LO1, LO2) of bulk h-
BN have noticeable non-flatten dispersions along the cross-plane direction (see Fig. 5.4f),
leading to non-zero phonon velocity vz, unlike the nearly zero vz in other bulk materials. Thus,
when temperature increases and these phonon modes start to contribute to cross-plane conduc-
tion, a significant increase (bump) appears in the Gball/A vs. T curve.
In the low T limit, all Gball/A vs. T curves show power law scaling, ~T n
(Fig. 5.6). For the
monolayers, the power exponent is n ≈ 1.59−1.68, which is a combined effect of n = 1.5 from the
quadratic ZA branch and n = 2 from the linear TA and LA branches [29, 55]. The power law
only applies to monolayers below ~90 K for graphene and h-BN, and below ~40 K for MoS2 and
WS2. For bulk (||), the power exponent is n ≈ 2.91−2.96, which is a combined effect of n = 2.5
Table 5.2: Our calculated room-temperature Gball/A of interested layered materials, as well as
their Debye temperature ΘD from literature [219, 226, 227]. Since Debye temperature ΘD is
actually a function of temperature T [219, 226], here we mainly list their values of high T limit
for our discussion, but values of low T limit for MoS2 and WS2 (in parentheses) are listed due to
lack of high T limit value for WS2.
Debye
temperature
ΘD (K)
Gball/A (GWK-1
m-2
)
monolayer bulk (||) bulk ()
graphite [226] 1930 4.37 4.34 0.294
h-BN [226] 1740 3.85 3.65 0.291
MoS2 [219, 227] 570 (253) 0.88 0.83 0.135
WS2 [227] (210) 0.72 0.67 0.114
96
from the ZA branch (if purely quadratic) [29] and n = 3 from linear TA and LA branches (see
Supporting Information Section 7). The reason that linear TA and LA branches lead to n = 3 for
bulk layered materials can be understood by rewriting Eq. 5.5c as
ball
3
2 2
3 2
1( ) | ( ) |
16
( ) | ( ) |16 ( 1)
b b b
nb
xb bB
n xb
G fd D v
A T
k T x edxD v
e
, (5.15)
where x=ħωb/kBT is a dimensionless number. For TA and LA branches at low-frequency, ( )b
nv
is constant, and Db(ω) ω
2 [228], which will give additional ~T
2 when the integral is converted
to a dimensionless one. Thus, TA and LA branches will have a ~ T 3
scaling in Gball/A at low T.
The reason that the overall n is closer to 3 than 2.5 is that the real ZA branch for bulk has a
linear component (not purely quadratic) at very low frequency [171], resulting in n larger than
2.5. The power law for bulk (||) is valid only below ~35 K for graphite and h-BN, and below ~20
K for MoS2 and WS2. For bulk (), the power exponent is n ≈ 2.73−2.89 with the same physical
reason as bulk (||), and it applies only below ~10 K for graphite and h-BN, and below ~7 K for
MoS2 and WS2.
Calculated ballistic thermal conductance is a useful quantity in the investigation of heat con-
duction in materials. It reflects the intrinsic anisotropic conduction and gives the upper limit of
heat flow at a certain temperature. Any measured and calculated thermal conductivity k can be
formally converted to G/A=k/L, and compared to Gball/A, from which we can know if obtained
thermal conductivity satisfies the law of quantum mechanics (i.e., below the ballistic limit) [22]
and what percentage of ballistic conduction it reaches [55, 57]. More importantly, it can be used
to predict thermal conductivity as a function of transport length L and also to estimate the
average phonon MFP λ.
97
5.4.2 Length-Dependent Thermal Conductivity
We first discuss the L-dependent thermal conductivity. In the ballistic regime (L ≪ λ), since
the conductance rather than the conductivity approaches a constant, the ballistic thermal conduc-
tivity kball = (Gball/A)L becomes linearly dependent on length L. In the diffusive regime (L ≫ λ),
the conductivity is generally independent of length L (the case of k divergent with L will be
discussed later), becoming a constant kdiff (diffusive thermal conductivity). Therefore, in the
intermediate ballistic-diffusive regime (i.e., quasi-ballistic regime), the thermal conductivity
should increase with L and gradually converge to kdiff, similar to the mobility change during
quasi-ballistic charge transport observed in short-channel transistors [172, 173]. This transition
can be captured through a phenomenological model, called ballistic-diffusive (BD) or quasi-
ballistic (QB) model [55, 171]:
1 1
ball diff ball diff
1 1 1 1
( / ) ( / )b bb
k LG A L k G A L k
, (5.16)
where the first expression is a “1-color” model, and the second one is a branch-resolved “multi-
color” model taking into account different phonon branch contributions separately. Apparently,
such L-dependent k expressions can correctly reproduce both ballistic and diffusive regimes
stated above. As we will show later, this model is essentially a Landauer-like model and it can
yield an expression of diffusive thermal conductivity consistent with the kinetic theory.
We first apply this model to the thermal conductivity of graphene supported on SiO2. Figure
5.7a shows its k (T=300 K) as a function of L from both experimental measurements [49, 55,
229] (solid symbols) and Boltzmann transport equation (BTE) calculations (open symbols,
calculations follow the method described in Ref. [49], including the use of an empirical potential
to describe the atomic interactions). By using our calculated Gball/A = Σb ball
bG /A = 4.37 GWK-1
m-2
98
for graphene as well as kdiff = Σb diff
bk = 578 Wm-1
K-1
from BTE simulations, the 1-color model
(solid blue line) and 6-color model (solid orange line) as well as its components of each branch
(dash-dot lines) are shown in Fig. 5.7a. They are in good agreement with experimental data and
BTE simulations, indicating that the BD model can provide reasonable L-dependent k in the
intermediate regime. We find that the simple 1-color model is sufficiently good to give almost
the same result as the 6-color model, although the latter provides more information about the
contribution of each branch. For supported graphene, besides three acoustic branches, the
flexural optical (ZO) branch also has a notable contribution, but other optical (TO and LO)
branches have negligible contributions (<1%). This is because the ZO branch has relatively low
phonon frequency, and hence larger thermal population.
We have demonstrated the BD model can successfully describe the k change with L in sup-
ported graphene, next we focus on the L-dependent k in suspended graphene. Since graphene is a
2D material, and some theoretical studies predicted in isolated low-dimensional (1D and 2D)
momentum-conserving systems k will diverge with L, i.e., ~Lα for 1D [200, 201] and ~logL for
2D [201-203]. However, other works [85, 230, 231] argued that disorder and higher-order three-
phonon scattering may eliminate the divergence, and no experiments have confidently observed
divergent k yet. For supported graphene discussed above, the ~logL divergence does not appear
due to phonon scattering with substrate vibrational modes [97, 98], but for suspended graphene it
is still an open question.
Very recently, Xu et al. [57] systematically measured k of suspended graphene as a function
of L, and their data at T=300 K (green squares) are shown in Fig. 5.7b. We find that our 1-color
BD model (Eq. 5.16) can fit their data well with the fitting parameter kdiff = 1790 Wm-1
K-1
(see
solid line in Fig. 5.7b). If assuming the divergence of k ~ logL for large L, the BD model can be
99
changed by adding a logarithmic term to capture this effect:
1
ball diff 0 0
1 1
( / ) ln( / 1)k L
G A L k k L L
, (5.17)
Fig. 5.7: Thermal conductivity k as a function of length L at room temperature. (a) L-dependent k
for graphene supported on SiO2. The 1-color (solid blue line) and 6-color (solid orange line) BD
models from Eq. 5.16 show good agreement with BTE calculations (open circles) and experi-
mental data of Bae et al. [55] (solid triangle) and Seol et al. [49] (solid diamonds). The dash-dot
lines are components in the 6-color model. (b) L-dependent k for suspended graphene. The 1-
color BD model (blue solid line) from Eq. 5.16, log model A (grey dash-dot line) from Eq. 5.17,
and log model B (brown dot line) are used to fit experimental data of Xu et al. [57]. (c) L-
dependent k for SWCNTs. Experimental data of Yu et al. [72], Pop et al. [83], Wang et al. [232],
and Li et al. [233] are plotted. Dash lines represent the 1-color BD model fitted to individual data
points, and the yellow band shows the overall range covering plotted data. In (a-c), the short-
dash line represents the ballistic upper limit kball = (Gball/A)L. (d) L-dependent k for different
materials obtained from the 1-color BD model using calculated Gball/A and experimental kdiff in
literature (see text).
101
102
103
104
0
500
1000
1500
2000
2500
k (
Wm
-1K
-1)
L (nm)
10-1
100
101
102
103
104
10-2
10-1
100
101
102
103
104
k (
Wm
-1K
-1)
L (nm)
101
102
103
104
0
100
200
300
400
500
600
700
k (
Wm
-1K
-1)
L (nm)
Yu (2005)
Li (2009)
Pop (2006)
SWCNT
a b
c d
graphite (||)
graphene (suspended)
MoS2 (1L)
h-BN (||)
graphite ()
h-BN ()
Si
Wang (2007)
kball
graphene(on SiO2)
Seol (2010)
Bae (2013)
BTE
101
102
103
104
105
102
103
104
k (
Wm
-1K
-1)
L (nm)
TALAZA
ZO
TO+LO
Xu (2014)
kball
graphene(suspended)
100
where k0 and L0 are parameters with units of thermal conductivity and length, respectively. We
call this model as the “log model A” and it can reproduce the ballistic limit correctly as well, i.e.,
k → (Gball/A)L when L→0. By choosing kdiff = 1590 Wm-1
K-1
, k0 = 130 Wm-1
K-1
, and L0 = 1.1
µm, the log model A can also fit the data of Xu et al. [57] well (see dash-dot line in Fig. 5.7b).
Here we also propose another log model B:
ball ln 1B
B
G Lk L L
A L
, (5.18)
where LB is a length parameter. Similar to the log model A, the log model B can also reproduce
the correct ballistic behavior at short L, that is, k → (Gball/A)L when L→0. The best fit of this
model to experimental data of Xu et al. [57] by using LB = 100 nm is shown as the dot line in
Fig. 5.7b. The fit of the log model B is not as good as that of the log model A, but both are worth
to be employed and tested. It is clear that the convergent BD model and divergent log model A
are distinguishable only at L > 10 µm, and below 10 µm the increase of k with L mainly results
from the ballistic-to-diffusive transition (or quasi-ballistic effect). However, currently available
data of k are only up to 9 µm, so further measurements beyond 10 µm are required to eventually
show whether thermal conductivity is divergent in suspended graphene. For 1D it is predicted
that k diverges with L as ~Lα, so similar to Eq. 5.17 we propose divergent k(L) with a power law
term for 1D:
1
ball diff
1 1
( / )k L
G A L k L
, (5.19)
where kdiff, γ, and α can be fitting parameters to fit experimental data or theoretical calculations.
In Fig. 5.7c we also plot experimental thermal conductivity data [72, 83, 232, 233] for sus-
pended single-wall carbon nanotubes (SWCNTs) with similar diameters as a function of length
at room temperature. There is no systematic measurement of k versus L for SWCNTs, and the
101
few available data do not fall in a single trend, so here we only use the 1-color BD model (Eq.
5.16) to fit to individual data points (dash-dot lines) and give a “band” (yellow area) to show the
k range as L increases (Fig. 5.7c). As shown by Mingo and Broido [29], SWCNTs have the same
Gball/A as graphene above ~200 K, so the used room temperature Gball/A for SWCNTs is 4.37
GWK-1
m-2
(same as graphene) in the BD model. The range of obtained kdiff is ~2500−8500 Wm-
1K
-1, and thermal conductivity keeps increasing with length up to tens of microns. A more
general expression of k including both length and temperature dependence for SWCNTs is
provided in our previous study of short channel (L < 100 nm) CNT devices [193], where heat
conduction is nearly ballistic.
For graphite, h-BN, MoS2, and Si, their L-dependent k at 300 K obtained from the 1-color
BD model are summarized in Fig. 5.7d, including monolayer (1L), bulk in-plane (||) and bulk
cross-plane () k of each material where the corresponding measured kdiff are known. For sus-
pended graphene, most Raman measurements report k in the range of ~2000−4000 Wm-1
K-1
for
L ~ 1−10 µm [23, 24], so here we use kdiff = 4000 Wm-1
K-1
in the BD model to plot its L-
dependent k if assuming convergent k(L). In Fig. 5.7d, other kdiff used in the BD model are all
from experimental measurements in the literature: kdiff = 2000, 6, 400, 2, 35, 2.5, and 150 Wm-
1K
-1 for graphite (||) [82], graphite () [82], h-BN (||) [234], h-BN () [235], MoS2 (1L) [236],
MoS2 () [237], and Si [168], respectively. These layered materials show a strong anisotropy in
thermal conductivity, and their values span a wide range, more than three orders of magnitude.
The estimated k-L dependence given here will be helpful for understanding heat conduction at
scales where size effects take place.
5.4.3 Estimation of Average Phonon Mean Free Path
Next we turn to the estimation of phonon MFP λ in terms of the calculated Gball/A. We note
102
that the underlying physics of the BD model is Landauer transport theory, in which the conduct-
ance (or conductivity) is given by the ballistic conductance (or conductivity) multiplied by a
transmission coefficient [24, 174, 238]:
1
ball bs
bs ball diff
1 1
( / )
Gk L L
A L G A L k
, (5.20)
where the transmission coefficient is given in terms of the transport length L and averaged
phonon back-scattering MFP λbs, that is, λbs/(L+λbs). A simple rearrangement of the first expres-
sion will reproduce the BD model, meanwhile yielding a relation λbs = kdiff/(Gball/A). This relation
can be directly derived from the definition of averaged phonon MFP with all modes weighted
properly (see Section 5.3.4). The common phonon MFP λ is shorter than the back-scattering
MFP λbs, and they are related with a factor βλ, as shown by Eq. 5.11.
As long as we know reliable kdiff from calculations or experimental measurements, the over-
all phonon MFP can be estimated based on Eq. 5.11, because Gball/A can be calculated without
approximations. Take graphite as an example, using its widely-accepted values of measured kdiff
[23, 82] and our calculated Gball/A, the obtained phonon in-plane and cross-plane MFPs (λ|| and
λ) as a function of temperature are shown as solid lines in Fig. 5.8a. They decrease rapidly as T
increases and differ by two and one orders of magnitude at low and high T, respectively. At room
temperature, λ|| = 290 nm and λ = 10 nm. The latter corresponds to ~30-layers thickness in
graphite, which means the cross-plane heat conduction in multi-layer graphene is already in the
ballistic regime. Thus, the advantage of the single average MFP (Eq. 5.11) is giving information
in a concise way, compared to mode-dependent MFPs, and it helps us understand when to
consider size effects and make corrections to thermal conductivity in a simple way, which is
quite useful especially from a device point of view.
103
We point out that the traditional way to estimate average phonon MFP is through the kinetic
theory, kdiff = (1/d)Cvvλ., where d=1−3 is the material dimension, Cv is the heat capacity, and v is
the average phonon group velocity. Different from Eq. 5.11, using this method requires not only
kdiff and Cv but also careful calculations for averaged values of phonon group velocity v. Howev-
er, most studies simply used the sound velocity vs (low-frequency group velocity) averaged
among acoustic branches as v. This will underestimate phonon MFP for two reasons. First, the
sound velocity is only valid for low-frequency phonons and hence for low T. For high T, even
room temperature, the effective group velocity is much smaller. Second, for the flexural ZA
mode in layered materials, its low-frequency group velocity approaches zero, and cannot be
simply included in the usual way [239]. Thus, only including TA and LA modes will overesti-
mate v.
Take graphite as an example, through the sound velocity average [239], we have v|| = [2/(
TA
s,||v -2+
LA
s,||v -2)]
1/2 = 15.8 km/s and v = [3/(2
TA
s,v
-2+
LA
s,v
-2)]
1/2 = 2.0 km/s, where
TA
s,||v , LA
s,||v , TA
s,v ,
Fig. 5.8: (a) Phonon MFP λ as a function of temperature for graphite (||) and (). The kinetic
theory with sound velocity gives underestimated λ (dash lines) compared to the correct results
(solid lines) from Eq. 5.11. (b) Room temperature phonon MFP λ for different materials obtained
from Eq. 5.11 using calculated Gball/A and experimental kdiff in literature (see text).
10 100 100010
0
101
102
103
104
Ph
on
on
MF
P
(n
m)
T (K)
graphite ()
graphite (||)
100
101
102
103
104
Ph
on
on
MF
P
(n
m)
gra
ph
ite
(||)
gra
ph
en
e
(su
sp
en
de
d)
Mo
S2
(1L
)
h-B
N (
||)
gra
ph
ite
(
)
h-B
N (
)
Si
Mo
S2
()
gra
ph
en
e
(on S
iO2)
T = 300 Ka b
104
and LA
s,v are from our calculated phonon dispersion. Using these values with Cv from our calcula-
tions (see Section 5.3.1) and kdiff given previously, the obtained λ|| and λ as a function of tem-
perature from the kinetic theory are shown as dash lines in Fig. 5.8a. We note that since graphite
is anisotropic, the equations for in-plane and cross-plane should be taken as 2D (d=2) and 1D
(d=1) forms of the kinetic theory, respectively, similar to the choice of βλ (shown in Section
5.3.5). As expected, the obtained λ|| (=145 nm at 300 K) from the kinetic theory is lower than that
from our method (Eq. 5.11) over the whole T range, mainly due to neglecting the small group
velocity of ZA mode in the average. The λ (=1.7 nm at 300 K) from the kinetic theory is also
underestimated, except for low T (<30 K), the range in which the sound velocity is valid. The
consistence between two methods for λ at low T also indicates that the 1D form of the kinetic
theory for cross-plane is correct.
To obtain correct average phonon MFP λ by the kinetic theory, the group velocity v should
be an average weighted by the heat capacity, as shown by Eq. 5.13 in Section 5.3.5. Substituting
Eq. 5.13 into the kinetic theory, the phonon MFP is then given by
diff diff
V ball /v
k d kd
C v G A
. (5.21)
We can find d/βv ≡ βλ for all cases, so when the correct v is used, the kinetic theory can give the
same estimation of MFP as our model (Eq. 5.11). Figure 5.9a shows the calculated v in terms of
Eq. 5.13 as a function of T for graphene, graphite (||), and graphite (). In the whole temperature
range, the maximum of v for graphite (||) and graphene is ~8 km/s, about two times smaller than
the averaged sound velocity shown above (15.8 km/s). For graphite (), the averaged v at 300 K
is only ~345 m/s, almost six times smaller than 2 km/s from the sound velocity average, and it
reaches 2 km/s only below ~30 K, i.e., the T range where the sound velocity is valid. The calcu-
105
lated v [including monolayer, bulk (||), and bulk ()] based on Eq. 5.13 for h-BN, MoS2, and
WS2 are also shown in Fig. 5.9.
After demonstrating the BD model and kinetic theory are consistent in the MFP estimation,
we apply Eq. 5.11 to other interesting materials whose measured kdiff are available. As shown in
Fig. 5.8b, the estimated λ at 300 K are ~3, 9, 10, 25, 70, 90, 100, 290, 580 nm for h-BN (),
MoS2 (), graphite (), MoS2 (1L), h-BN (||), graphene (on SiO2), Si, graphite (||), and graphene
(suspended), respectively. The used kdiff are listed previously. We note that Eq. 5.11 gives the
“averaged” MFP including contributions from all phonon modes, but some modes (e.g., optical)
have very small contributions to thermal conductivity (heat conduction) and very short MFPs
Fig. 5.9: Calculated phonon group velocity as a function of temperature for graphite (a), h-BN
(b), MoS2 (c), and WS2 (d) from Eq. 5.13. Each material includes its monolayer, bulk (||), and
bulk ().
1 10 100 1000100
1000
10000
v (
m/s
)
T (K)1 10 100 1000
100
1000
10000
v (
m/s
)
T (K)
graphene/graphite h-BN
MoS2 WS2
graphite (||)
graphite ()
bulk (||)
bulk (||)monolayer
bulk ()
bulk (||)monolayer
a b
c d
bulk ()
bulk ()
1 10 100 1000100
1000
10000
v (
m/s
)
T (K)1 10 100 1000
100
1000
10000
v (
m/s
)
T (K)
106
(<10 nm for in-plane and <1 nm for cross-plane). This means that those modes contributing to
heat conduction significantly should have MFPs longer than the value given by Eq. 5.11. Indeed,
by analyzing accumulative thermal conductivity as a function of MFP, it is found that phonons
with MFP longer than 1 µm and 100 nm contribute ~40% to the total thermal conductivity for Si
[206-209] and cross-plane graphite [240] at 300 K, respectively. Thus, a “median” MFP of 0.5-1
µm and ~100 nm (much larger than our “averaged” numbers, 100 nm and 10 nm) is suggested to
explain the strong size effects of thermal transport in Si membrane [204] and graphite (cross-
plane) [240], respectively. However, we emphasize that the value given by Eq. 5.11 is a rigorous
average with all phonon modes weighted properly (see Section 5.3.4) to represent the overall
MFP, which should be used in the kinetic theory and to understand quasi-ballistic thermal
transport. By comparing our estimated MFPs in Fig. 5.8b and k(L) in Fig. 5.7d, we can find that
when L < λ, heat conduction enters the ballistic regime (i.e., the linear region in Fig. 5.7d) as
expected. Whereas, when L is shorter than the suggested “median” MFP (~10λ), k just starts to
decrease rapidly, but is not in the ballistic regime, meaning the suggested “median” MFP is more
like a characteristic length below which the size effects (not ballistic transport) take place.
Interestingly, beyond 10λ thermal conductivity still increases slowly and will eventually reach
kdiff (i.e., become fully diffusive) at L ~ 100λ (Fig. 5.7d), which is much longer than the com-
monly assumed length. The whole ballistic-to-diffusive transition takes more than two orders of
magnitude in length (λ−100λ) to complete. This also indicates that any obtained k increase in this
range might arise from the intrinsic ballistic-to-diffusive transition, and whether k is divergent in
low-dimensional materials should be examined beyond 100λ to draw a realistic conclusion.
5.5 Summary
In conclusion, we calculated the ballistic thermal conductances Gball of graphene/graphite, h-
107
BN, MoS2, and WS2 based on full phonon dispersions obtained from ab initio simulations. From
the calculated Gball, we obtained L-dependent k by using a phenomenological ballistic-diffusive
model. In particular, for suspended graphene recent measurements [57] of k vs. L can be fitted by
both the convergent BD model and divergent log model, and the two models only differ beyond
L~10 µm. We also showed that the rigorously averaged phonon MFP λ is simply determined by
Gball and kdiff, and that the kinetic theory can reach the same result as long as a proper phonon
group velocity is used. We emphasize that for anisotropic layered materials, 2D and 1D forms of
the kinetic theory should be used for in-plane and cross-plane, respectively. Based on the calcu-
lated λ and the BD model, we find that k will not fully converge to kdiff until after L~100λ, much
longer than the commonly assumed length. Thus, to verify predictions of divergent k in low-
dimensional systems, simulations and measurements should be performed beyond L~100λ.
108
CHAPTER 6
CONCLUSIONS AND OUTLOOK
6.1 Conclusions
In this dissertation, we investigated thermal transport in graphene and several other 2D lay-
ered materials, with a focus on nanoscale ballistic conduction and size effects. We combined
experimental measurements, finite element simulations, and theoretical calculations to give a
comprehensive study.
First, we measured heat conduction in nanoscale graphene by a substrate-supported ther-
mometry platform. Short, quarter-micron graphene samples reach ~35% of the ballistic thermal
conductance limit up to room temperature. In contrast, patterning similar samples into nanorib-
bons (GNRs) leads to a diffusive heat flow regime that is controlled by ribbon width and edge
disorder. In the edge-controlled regime, the GNR thermal conductivity scales with width approx-
imately as ~W1.8±0.3
, being about 100 Wm-1
K-1
in 65-nm-wide GNRs, at room temperature. Thus,
the usual meaning of thermal conductivity must be carefully interpreted when it becomes a
function of sample dimensions. These findings are highly relevant for all nanoscale graphene
devices and interconnects, also suggesting new avenues to manipulate thermal transport in 2D
and quasi-one-dimensional systems.
We also examined the possibility of using this supported platform to measure other materials
through finite element simulations. The platform geometry is optimized based on uncertainty
analysis. The smallest thermal sheet conductance that can be sensed by this method within a 50%
error is ~25 nWK-1
at room temperature, indicating this platform can be applied to most thin
films like polymer and nanotube networks, as well as nanomaterials like nanotube/nanowire
arrays, even a single Si nanowire. Moreover, the platform can also be extended to plastic sub-
109
strates (not limited to the SiO2/Si substrate), and could find wide applicability in circumstances
where fabrication challenges and low yield associated with suspended platforms must be avoid-
ed.
Last, we calculated in-plane (for monolayer and bulk) and cross-plane (for bulk) ballistic
thermal conductances Gball of graphene/graphite, h-BN, MoS2, and WS2, based on full phonon
dispersions from first-principles approach. We find that the anisotropy of thermal transport in
these materials comes from both anisotropic Gball and phonon MFPs, not just the effect of one of
them. A proper estimation of the overall MFP λ should be a rigorous average of all phonon
modes, which can be condensed to an expression just including Gball and diffusive thermal
conductivity kdiff. We point out that the kinetic theory can reach the same result as long as the
correctly averaged phonon group velocity is used. We emphasize that for anisotropic layered
materials, 2D and 1D forms of the kinetic theory should be used for in-plane and cross-plane,
respectively. Based on the calculated λ and the BD model, we find that k will not fully converge
to kdiff until after L~100λ, much longer than the commonly assumed length. Thus, to verify
predictions of divergent k in low-dimensional systems, simulations and measurements should be
performed beyond L~100λ to distinguish from the intrinsic increase of k due to the ballistic-to-
diffusive transition.
These results represented a comprehensive study of thermal conduction in 2D layered mate-
rials in micro and nanoscale where ballistic conduction and size effects take place. We extended
experimental methods to measure nanoscale thermal conduction and showed that thermal con-
ductivity should be redefined in nanoscale and could be tuned effectively. This dissertation
broadens our understanding of thermal conduction and how to manipulate it to reach the re-
quirements for potential applications like thermal management and thermoelectric conversion.
110
6.2 Future Work
There has been an increasing interest of thermoelectric properties in low-dimensional
nanostructures. Thermoelectric transport of graphene has been experimentally studied, showing a
maximum Seebeck coefficient of S ~ 100 μV/K at room temperature (Section 2.4). However, the
thermoelectric properties of GNRs are still unknown in experiments. As pointed out in Section
3.4.4, in the intermediate width range of GNRs (40 nm < W < 200 nm), compared with graphene,
GNR thermal conductivity k is suppressed significantly while electrical conductivity σ is not
affected. Thus, GNRs may have higher ZT than graphene. The Seebeck coefficient of GNRs is
suggested to be measured to investigate the effects of carrier density and edge scattering on their
thermoelectric properties, which will be useful for potential applications in thermoelectric
devices and energy conversion.
Besides graphene, other 2D materials like h-BN and TMDs also show outstanding electrical
properties and promising applications in electronics and optoelectronics. They attracted a lot of
studies about its electronic transport properties, but few focused on their thermal properties.
Although our current study has extended to the thermal transport of these 2D materials in a
theoretical frame (Chapter 5), there are very few experimental works available in the community.
Therefore, it is crucial to measure thermal conduction of these materials in the future, which will
help to better understand the performance of devices made of them. In addition, TMDs are
predicted to have higher thermopower than graphene due to the presence of band gaps, so their
thermoelectric properties are worth to investigate experimentally.
As mentioned in Chapter 5, whether thermal conductivity diverges with transport length in
low-dimensional materials is still an open question. To clarify this issue, experimental measure-
ments should be carried out on 1D and 2D materials with length beyond ~100 times of their
111
phonon MFP. Furthermore, studying the thermal rectification effect of nanostructures composed
of 2D materials is important and helpful to develop thermal diodes and phononics in the future.
112
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