PHYSICAL REVIEW APPLIED
10,
054068 (2018)
Quasiballistic Thermal Transport from Nanoscale Heaters and the Role of the
Spatial Frequency
Xiangwen Chen,
1,
†
Chengyun Hua,
2,
†
Hang Zhang,
3
Navaneetha K. Ravichandran,
4
and
Austin J. Minnich
1,
*
1
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, California 91125,
USA
2
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
3
Institute of Engineering Thermophysics, Chinese Academy of Sciences, Beijing 100190, China
4
Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA
(Received 24 August 2017; revised manuscript received 24 May 2018; published 29 November 2018)
Quasiballistic heat conduction from nanoscale heat sources of size comparable to phonon mean free
paths has recently become of intense interest both scientifically and for its applications. Prior work has
established that, in the quasiballistic regime, the apparent thermal properties of materials depend both on
intrinsic mechanisms and the characteristics of the applied thermal gradient. However, many aspects of
this regime remain poorly understood. Here, we experimentally study the thermal response of crystals to
large thermal gradients generated by optical heating of nanoline arrays. Our experiments reveal the key
role of the spatial frequencies and Fourier series amplitudes of the heating profile for thermal transport in
the quasiballistic regime, in contrast to the conventional picture that focuses on the geometric dimensions
of the individual heaters. Our work provides the insight needed to rationally mitigate local hot spots in
modern applications by manipulating the spatial frequencies of the heater patterns.
DOI:
10.1103/PhysRevApplied.10.054068
I. INTRODUCTION
Heat conduction by phonons is of both fundamental and
practical importance, playing a key role in modern appli-
cations ranging from thermoelectrics [
1
–
5
] to electronic
devices [
6
–
10
]. The limiting regimes of heat conduction
are well understood. If thermal gradients occur over length
scales much longer than mean free paths (MFPs), heat
conduction occurs by diffusion as described by Fourier’s
law. In the opposite limit of an extreme thermal gradient
over which no scattering occurs, heat conduction occurs by
phonon radiation in an exact analogy to blackbody radia-
tion [
11
]. The intermediate quasiballistic regime, in which
some phonons are ballistic but some undergo scattering
events, has recently become of intense interest [
12
–
23
].
This regime occurs in practice much more frequently than
the completely ballistic regime because phonons possess a
very broad MFP spectrum [
24
] and often not all phonons
are ballistic for experimentally achievable heating length
scales.
Koh and Cahill first reported variations of thermal con-
ductivity with modulation frequency in time-domain ther-
moreflectance (TDTR) experiments that they attributed to
*
aminnich@caltech.edu
†
These authors contributed equally to this work.
quasiballistic transport [
13
]. Subsequently, a number of
works reported observations of quasiballistic effects in sys-
tems including lithographically patterned nanoline arrays
on sapphire [
14
], silicon at cryogenic temperatures [
15
]as
well as at room temperature [
18
,
19
], and thin silicon mem-
branes at room temperature [
25
]. A recent work reported
the existence of a collective-diffusive regime [
20
], again
involving nanolines with variable periods. These effects
have been used to map the spectral thermal conductiv-
ity of crystals [
16
,
17
]. Meanwhile, numerous models have
been proposed to explain the various observations, includ-
ing two-channel models with ballistic and diffusive modes
[
26
,
27
], approximate solutions of the Boltzmann trans-
port equation (BTE) [
23
,
28
], a superdiffusive formalism
[
21
,
22
], and numerical methods [
29
].
Despite these prior studies, many aspects of the qua-
siballistic regime remain poorly understood, with prior
works drawing contradictory conclusions. For example,
Hu
et al
.[
16
] and Zeng
et al
.[
17
] report large and mono-
tonic increases in thermal resistance as the dimensions
of individual heaters in patterned arrays become smaller
than MFPs. However, other works suggest that size effects
should not play a role in closely spaced patterned heaters
due to the lack of in-plane thermal gradient [
30
]. Overall,
a comprehensive understanding of the thermal response of
solids to large temperature gradients is lacking, impacting
2331-7019/18/10(5)/054068(11)
054068-1
© 2018 American Physical Society
CHEN
et al.
PHYS. REV. APPLIED
10,
054068 (2018)
efforts to enhance heat dissipation in modern devices that
possess nanoscale heat sources.
Here, we experimentally study quasiballistic transport in
crystals generated by optically heating metallic nanoline
arrays. Our experiments reveal the key role of the spatial
frequencies and Fourier series amplitudes of the heating
profile for thermal transport in the quasiballistic regime, in
contrast to the conventional picture in which the character-
istic dimensions of the individual heaters play the central
role. In addition, our work provides the insight needed
to rationally mitigate local hot spots in modern applica-
tions by manipulating the spatial frequencies of the heater
patterns.
II. METHODS
A. TDTR measurements and fitting model
We use two-tint TDTR [
31
] to study heat conduction
in
c
-sapphire with patterned aluminum line arrays as trans-
ducer. Briefly, the pump pulse train, at a 76-MHz repetition
rate and wavelength of 785 nm, is amplitude-modulated at
η
0
and directed to the sample to provide a heat impulse.
The change in reflectance of the aluminum lines due to the
temperature change is detected by a reflected probe beam
with wavelength near 785 nm but spectrally distinct from
the pump using sharp-edged optical filters. Sapphire is cho-
sen as the substrate as it is transparent to the 785-nm laser
used in the experiments so that only the aluminum lines
absorb the incident pump light, as shown in Fig.
1(a)
.The
pump and probe beams are fixed with 1
/
e
2
diameters of
30 and 10
μ
m, respectively. Both beam sizes are measured
using a home-built two-axis knife-edge beam profiler. We
use a mechanical delay line with up to 14 ns of total delay
time. All the measurements are performed in an optical
cryostat (JANIS ST-500) under high vacuum of 10
−
6
Torr.
The experimental data consist of in-phase and out-of-phase
signal versus delay time as measured by a rf lock-in ampli-
fier (Zurich Instruments HF2LI). The signal is converted
into amplitude and phase for fitting and presentation, as
shown in Figs.
1(e)
and
1(f)
.
To fit the nanoline data, we assume that both the pump
and probe spot sizes are much larger than the linewidth and
period, allowing us to consider the nanoline heating profile
as spatially periodic along the
x
axis, as shown in Figs.
1(a)
and
1(b)
.The
y
-axis direction is considered as infinite. As a
result, the heat transport can be described using a 2D model
in Cartesian coordinates. The square wave profile of the
lines is described mathematically by imposing a square-
wave heating at the Al transducer layer surface and setting
the in-plane thermal conductivity of the transducer layer
κ
r
=
0. The solution then follows exactly the same deriva-
tion as given in Refs. [
32
]and[
33
]. Further details of the
fitting model are available in Ref. [
17
]. The only differ-
ence between the present model and this prior model is
that we allow some of the frequency components of the
TDTR signal to have different thermal properties rather
than forcing all frequency components to have the same
value of the fitting parameter as in the traditional model.
B. Sample fabrication and characterization
The sample consists of aluminum nanoline arrays with
an area of 60
×
60
μ
m
2
fabricated on
c
-sapphire substrates
as shown schematically in Fig.
1(a)
, using a standard
electron-beam lithography and lift-off process. The width
of the lines
w
varies between 50 nm and 1.5
μ
m, while the
period of line array
L
ranges from around 1.5 to 4.5 times
the corresponding linewidth; the duty cycle is defined
as
w
/
L
. Owing to the spatial periodicity of the heating
pattern, it can be represented as a Fourier series with
discrete spatial-frequency components of a square wave
as shown in Fig.
1(b)
. Single-side polished
c
-sapphire
substrates from University Wafers are first cleaned in
Nanostrip, followed by sonication in acetone and isopropyl
alcohol (IPA), and then rinsed with IPA and dried with
dry N
2
. Approximately 100-nm poly(methyl methacrylate)
(PMMA) is spun on the substrate, followed by baking
at 180
◦
C for 5 min. After cooling, a conductive layer of
AquaSave (Mitsubishi Rayon Co. Ltd.) is spun on the resist
as an anticharging layer for e-beam lithography. The resist
is exposed to the electron beam in a Leica/Vistec EBPG
5000+ electron beam writer operating at 100-kV acceler-
ating voltage and a 5-nm spot size. After patterning, the
AquaSave is removed with a rinse of DI water and then the
sample is dried with N
2
. Development is performed at 4
◦
C
in a 1:3 mixture of methyl isobutyl ketone (MIBK) and IPA
for 65 s and terminated in IPA for 1 min. The patterns are
subjected to a brief dose of oxygen plasma to clean the
PMMA residues after development. Al film 42 nm thick
is deposited with a Lesker LAB Line
E
-beam evapora-
tor at a pressure of 7
×
10
−
7
Torr. Lift-off is performed
in dichloromethane at room temperature for 15 min.
The dimensions of aluminum line arrays, including the
linewidth, period, and height, are characterized by atomic
force microscopy (AFM) (Bruker Dimension ICON) and
scanning electron microscopy (SEM). For smaller lines,
the cross section is also characterized by transmission elec-
tron microscopy (TEM) (FEI Tecnai TF-20). The TEM
samples are prepared with a standard focused ion beam
(FIB) lift-out technique and FIB milling for thinning to
<
100 nm after TDTR measurements are performed. AFM
and cross-section TEM images of the fabricated lines are
presented in Figs.
1(c)
and
1(d)
. The linewidth is computed
using AFM measurements corrected for the tip radius and
these measurements are confirmed with the cross-section
TEM measurements for select lines. Accurate determina-
tion of linewidths is essential as the TDTR measurements
are exceedingly sensitive to this parameter; see Sec. 4 and
Fig. S8 in the Supplemental Material [
34
]
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QUASIBALLISTIC THERMAL TRANSPORT...
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(a)(b)
(c)(d)
(e)(f)
FIG. 1. Schematic of experimental geometry and example data from nanoline arrays. (a) Schematic of sample geometry. (b) Spatial
heating profile on the substrate in real space (top) and versus spatial frequency (bottom). In the experiment, the probe beam that
measures the thermal response has an identical geometry. (c) Representative AFM topography of 627-nm line arrays with a period of
1.2
μ
m. The dashed green line indicates the cross section used for the height profile shown in the inset. (d) TEM cross-section profile
of a 62-nm-width line. (e),(f) Representative TDTR amplitude and phase experimental data and best fit using a standard heat diffusion
model [
16
] on line arrays at (e) 5.3-MHz modulation frequency,
w
=
1
μ
m, and
L
=
2
μ
m at 294 K and (f) 3.1 MHz,
w
=
117 nm, and
L
=
200 nm at 150 K. In (e), the best-fit curve matches the data and yields a fitted thermal conductivity and interface conductance of
38 W m
−
1
K
−
1
and 196 MW m
−
2
K
−
1
, respectively. In (f), the phase fitting is poor. In this situation, fitting to amplitude [
16
] or phase
[
17
] will give different results.
C. Spectral BTE calculation
Thermal transport in the nanoline array system is
described by the two-dimensional spectral BTE under the
relaxation time approximation (RTA),
∂
g
ω
∂
t
+
v
ω
,
x
∂
g
ω
∂
x
+
v
ω
,
z
∂
g
ω
∂
z
=−
g
ω
+
f
0
(
T
0
)
−
f
0
(
T
)
τ
ω
+
Q
ω
(
x
,
z
,
t
)
4
π
,(1)
f
0
(
T
)
=
1
4
π
ω
D
(ω)
f
BE
(
T
)
≈
f
0
(
T
0
)
+
1
4
π
C
ω
T
,(2)
where
g
ω
=
ω
D
(ω)
[
f
ω
(
x
,
t
,
μ)
−
f
0
(
T
0
)
] is the devia-
tional distribution function;
f
0
=
f
0
(
x
,
t
)
is the equilibrium
distribution function;
μ
=
cos
(θ )
is the directional cosine;
v
ω
,
x
and
v
ω
,
z
are the phonon group velocities in the
x
(in-
plane) and
z
(cross-plane) directions; and
τ
ω
is the phonon
relaxation time. For a line array heating pattern,
Q
ω
(
x
,
z
,
t
)
,
054068-3
CHEN
et al.
PHYS. REV. APPLIED
10,
054068 (2018)
the spectral volumetric heat generation, is formulated as
follows
Q
ω
(
x
,
z
,
t
)
=
Q
ω
(
z
,
t
)
rect
(
x
)
,(3)
where rect
(
x
)
represents a square wave.
Assuming a small temperature rise,
T
=
T
−
T
0
, rela-
tive to a reference temperature,
T
0
, the equilibrium distri-
bution is proportional to
T
, as shown in Eq.
(2)
. Here,
is the reduced Planck constant,
ω
is the phonon fre-
quency,
D
(ω)
is the phonon density of states,
f
BE
is the
Bose-Einstein distribution, and
C
ω
=
ω
D
(ω)(∂
f
BE
/∂
T
)
is the mode specific heat. The volumetric heat capacity
is then given by
C
=
∫
ω
m
0
C
ω
d
ω
and the Fourier thermal
conductivity
κ
=
∫
ω
m
0
κ
ω
d
ω
, where
κ
ω
=
1
3
C
ω
v
ω
ω
and
ω
=
τ
ω
v
ω
is the phonon MFP. To close the problem,
energy conservation is used to relate
g
ω
to
T
, given by
∫∫
ω
m
0
[
g
ω
(
x
,
t
)
τ
ω
−
1
4
π
C
ω
τ
ω
T
(
x
,
t
)
]
d
ω
d
=
0,
(4)
where
is the solid angle and
ω
m
is the cutoff fre-
quency. Note that summation over phonon branches is
implied without an explicit summation sign whenever an
integration over phonon frequency or MFP is performed.
Since a typical laser spot size is much larger (at least
10–100 times larger) than the linewidth/period of the line
array, the in-plane direction parallel to the lines (the
y
-axis
direction) is assumed to be infinite. As a result, heat trans-
port in the line array is reduced to a two-dimensional heat
transfer problem as shown in Fig.
1(a)
. The in-plane direc-
tion perpendicular to the lines is also assumed to be infinite
and we neglect any effect of the finite pump size. There-
fore, a Fourier transform can be applied to the in-plane
direction perpendicular to the lines and Eq.
(1)
becomes
∂
̃
g
ω
∂
t
+
i
ξ
x
̃
g
ω
v
ω
,
x
+
v
ω
,
z
∂
̃
g
ω
∂
z
=−
̃
g
ω
τ
ω
+
1
4
π
C
ω
̃
T
(ξ
x
,
z
,
t
)
+
Q
ω
(
z
,
t
)
4
π
+∞
∑
n
=−∞
2sin
(π
nw
/
L
)
n
,(5)
where
∑
+∞
n
=−∞
[2 sin
(π
nw
/
L
)/
n
] is the Fourier transform
of a square wave and
ξ
x
is the Fourier variable in the
x
direction.
ξ
x
is discrete and takes the value of 2
π
n
/
L
,
where
L
is the period length and
n
is an integer.
The calculation can be divided into three parts: the
transducer layer, substrate, and interface. In the transducer
layer, the transport is considered only in the cross-plane
direction. Therefore,
ξ
x
=
0. The BTE in the transducer
layer can be reformulated as a Fredholm integral equation
of the second kind and solved using the method of Ref.
[
35
]. The solution in the substrate can be obtained using the
multidimensional Green’s function to the BTE [
36
]. The
solutions in the two layers depend on each other through
the interface conditions that enforce conservation of heat
flux. The detailed discussion and derivation exactly follow
those given in Ref. [
37
] excepting the use of the multidi-
mensional Green’s function in this work. The dispersion
and relaxation times for sapphire are approximated from
first-principles calculations by Lucas Lindsay for Si [
38
];
the relaxation times are divided by 4 to more closely match
the thermal conductivity of sapphire.
III. RESULTS AND DISCUSSION
A. Observing quasiballistic heat conduction
Experimental amplitude and phase data from the lock-in
amplifier for 1-
μ
m lines with 50% duty cycle at room tem-
perature are given in Fig.
1(e)
. We fit these data using a
traditional multilayer heat diffusion model [
16
,
32
,
33
] with
the substrate thermal conductivity and interface conduc-
tance between the lines and substrate as fitting parameters.
We see in Fig.
1(e)
that the fitting quality is excellent and
yields a thermal conductivity of 38 W m
−
1
K
−
1
, which is in
good agreement with the literature value for sapphire [
39
,
40
]. In contrast, Fig.
1(f)
presents the amplitude and phase
data for 117-nm lines at 150 K. Although the fitted ampli-
tude reasonably matches the experimental data, the phase
fit is quite poor. This discrepancy indicates that heat diffu-
sion theory is failing to describe key aspects of the trans-
port dynamics, complicating the interpretation of the data.
B. Interpreting the TDTR measurement
The Boltzmann transport equation is the most rigorous
formalism to analyze the measured data, but a simplified
model can provide insight into the experimental data with-
out requiring extensive microscopic input. To identify this
model, we examine the TDTR signal more closely. The
transfer function
Z
(
t
)
that relates the measured amplitude
and phase of the surface temperature to a spatially periodic
input surface heat flux is given by
Z
(
t
)
=
−∞
∑
m
=−∞
−∞
∑
n
=−∞
H
(η
0
+
m
η
s
,
ξ
x
,
n
)
e
im
η
s
t
Q
(ξ
x
,
n
)
(6)
where
ξ
x
,
n
=
2
π
n
/
L
are the in-plane spatial frequencies of
the periodic heating pattern with period
L
,
η
0
is the mod-
ulation frequency,
η
s
is the laser pulse repetition rate, and
t
is the time delay of the probe relative to the pump. Here,
n
and
m
are integers. Equation
(6)
represents the time-
domain TDTR signal as a double Fourier series, allowing
us to observe that the transfer function is composed of a
frequency response function,
H
(η
,
ξ)
, evaluated at discrete
temporal and spatial frequencies weighted by the Fourier
series amplitudes of the heat source,
Q
(ξ
x
,
n
)
. The weights
Q
(ξ
x
,
n
)
are given by the Fourier series components of the
square wave, as shown in Fig.
1(b)
.
The inadequacy of the traditional model to explain the
experiments implies that the surface-temperature response
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of the sample versus spatial and temporal frequency differs
from that predicted by Fourier’s law. To gain insight
into this discrepancy, we calculate the surface-temperature
response
H
(η
,
ξ)
as a function of temporal and in-plane
spatial frequency of the applied surface heat flux for a
thin film on substrate geometry with multidimensional
heat conduction using the spectral BTE (see Sec.
II C
).
The amplitude and phase of the surface thermal response
versus temporal frequency at different spatial frequencies
computed from Fourier’s law and the BTE are shown
in Figs.
2(a)
and
2(b)
. The calculations demonstrate that
the BTE response agrees with the Fourier law calcula-
tion at smaller spatial frequencies but does not agree for
larger spatial frequencies. As the poor fitting of Fig.
1(f)
shows, the two fitting parameters of substrate thermal
conductivity and interface conductance in the traditional
model cannot explain these various differences at all of
the temporal and spatial frequencies present in the TDTR
signal. Therefore, a new model is needed to interpret the
experiments.
We introduce such a model by making several
observations. In the weak quasiballistic regime where
relevant timescales are far longer than phonon relaxation
times, we have previously shown that apparent thermal
properties can be rigorously defined [
41
], but these prop-
erties may not be constant with variations in spatial fre-
quencies of the heater pattern. For instance, our analytical
solution of the BTE for a semi-infinite domain shows
that the thermal conductivity depends primarily on the
magnitude of the spatial frequencies
ξ
of the heating pat-
tern rather than the temporal frequency [
36
], where
ξ
=
√
ξ
2
x
+
ξ
2
y
+
ξ
2
z
. In the line array pattern Fig.
1(a)
,
ξ
x
cor-
responds to the in-plane direction. For the square-wave
heater,
ξ
x
=
2
π
n
/
L
, where
L
is the period and
n
is an inte-
ger, as shown in Fig.
1(b)
.
ξ
y
=
0 due to the uniformity
along the
y
direction. The cross-plane spatial frequency
ξ
z
is determined by the temporal heating frequencies in
the modulated heating beam. For a given modulation
frequency, the
ξ
z
are fixed and, hence, our experiment
(a)(b)
(c)(d)
Four
FIG. 2.
Model to interpret TDTR data. (a) Amplitude and (b) phase of the thermal response of a thin film on a substrate versus tem-
poral frequency for three spatial frequencies calculated using Fourier’s law (lines) and BTE (symbols). (c) Refit of experimental data
for the case in Fig.
1(f)
with the four-parameter fitting model. The best fit gives an excellent fit to the data with
κ
0
=
142 W m
−
1
K
−
1
,
κ
1
=
4Wm
−
1
K
−
1
,
G
0
=
137 MW m
−
2
K
−
1
,
G
1
=
155 MW m
−
2
K
−
1
, respectively. (d) Centerline temperature profile versus depth
into the sample for a line array at a single temporal frequency of 10 MHz calculated using the four-parameter model (dashed lines)
and the exact solution from the BTE (solid lines). The four-parameter model reasonably explains the spatial temperature profile into
the sample.
054068-5
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primarily probes the effect of the in-plane spatial frequency
ξ
x
on thermal conductivity set by the square-wave period.
Additionally, we expect that the interface conductance will
primarily depend on the cross-plane spatial frequencies
ξ
z
that are, in turn, set by the temporal heating frequency (see
Sec. 2 in the Supplemental Material [
34
]).
Given these observations, we construct a four-parameter
model. Two parameters are thermal conductivities, one for
zero spatial frequency
κ
0
=
κ(ξ
x
,0
)
, corresponding to spa-
tially uniform heating, and one for all higher-order spatial
frequencies
κ
1
=
κ(ξ
x
,
|
n
|
>
0
)
, where
ξ
x
,
|
n
|
>
0
are integer mul-
tiples of the fundamental spatial frequency
ξ
x
,
F
=
2
π/
L
.
The other two are interface conductances, one at the mod-
ulation frequency
G
0
=
G
(η
0
)
and one for higher-order
temporal frequencies
G
1
=
G
(η
0
+
m
η
s
)
|
m
|
>
0
. If the trans-
port occurs by diffusion, thermal properties are constant
for all spatial frequencies and
κ
0
=
κ
1
and
G
0
=
G
1
.The
model is not a two-channel model as has been reported
in the past [
26
], but rather allows for the possibility that
the spatial frequency components of the TDTR thermal
response may be described by different thermal properties
due to quasiballistic effects.
As in the traditional procedure, these parameters are
obtained by fitting to the measured surface-temperature
response of the sample. Applying the four-parameter
model to the experimental data of Fig.
1(f)
yields an
excellent fit, as in Fig.
2(c)
. Of course, adding additional
parameters to a model could lead to satisfactory fitting
even if the underlying model is not physical. We provide
support that our four-parameter model accurately describes
the thermal transport in the system by computing the exact
cross-plane spatial temperature profile in the sample using
the BTE and comparing it to that obtained from four-
parameter model with the best-fit parameters. The result
is shown in Fig.
2(d)
and demonstrates that the four-
parameter model agrees nicely with the exact BTE result
despite its simplicity, supporting the physical validity of
our model. In the following discussion, we use these BTE
simulations to compute TDTR data sets as a comparison
with the experimental data; we term these simulations as
synthetic data and the same four-parameter model is used
to fit the synthetic data to obtain thermal conductivities and
interface conductances exactly as in experiment.
C. Results using four-parameter model
We now apply this model to our measurements on nano-
lines. As the apparent thermal conductivity of the substrate
is our primary interest, we defer discussion of interface
conduction to Sec. 2 of the Supplemental Material [
34
].
When we measure the thermal conductivities (
κ
0
and
κ
1
)
versus modulation frequency from 0.5 to 15 MHz, little
dependence on modulation frequency is observed (see Sec.
3 and Fig. S3 in the Supplemental Material [
34
]), confirm-
ing our assumption that the measured thermal conductivity
primarily depends on in-plane spatial frequency. For the
measurements in all the figures below, the error bar indi-
cates the standard deviation from measurements taken at
multiple modulation frequencies between 1 and 10 MHz
and the uncertainty from fitting (see Secs. 4 and 5 in the
Supplemental Material [
34
]), unless otherwise stated.
We first examine the thermal conductivities versus
period with the duty cycle fixed at 50% at room temper-
ature and 150 K in Fig.
3(a)
. The thermal conductivity at
zero spatial frequency,
κ
0
, is independent of period and
agrees with the bulk value of the thermal conductivity of
sapphire at the relevant temperatures [
39
]. The thermal
conductivity at higher spatial frequencies,
κ
1
, decreases
with decreasing period for periods smaller than 2
μ
mat
room temperature. This observation indicates that phonon
MFPs are on the scale of hundreds of nanometers and that
the thermal resistance to heat flow at these spatial frequen-
cies is larger than predicted by Fourier’s law, in line with
prior work [
16
]. At 150 K,
κ
1
is less than
κ
0
as well as the
bulk thermal conductivity even at period 3
μ
m, the max-
imum period used in the experiments, indicating that the
mean free paths are longer than 3
μ
m at 150 K.
We present
κ
1
versus period normalized to
κ
0
at the
specified temperatures in Fig.
3(b)
. The trend of increasing
thermal conductivity with increasing period is reproduced
by the synthetic TDTR data. Both the experimental and
synthetic data show the physically intuitive result that the
departure from the bulk thermal conductivity is larger at
lower temperatures where phonon MFPs are longer than at
room temperature.
While Figs.
3(a)
and
3(b)
show the relationship between
κ
1
and period
L
at a fixed duty cycle of 50%, Fig.
3(c)
shows how the
κ
1
varies with fixed linewidth
w
and chang-
ing period
L
. The figure shows that
κ
1
increases as the
period increases for a fixed linewidth of approximately 115
nm at different temperatures. Again, the synthetic TDTR
data show a similar trend. In Fig.
3(d)
, at 150 K, the
κ
1
for different line array periods with a duty cycle of
67% are compared to those with a 50% duty cycle. For
the same period, the patterns with a smaller duty cycle
(smaller linewidth) have higher
κ
1
. Generally, these two
figures indicate that
κ
1
decreases as the lines become closer
together.
D. Role of the spatial frequency
The observed trends of thermal conductivity
κ
1
with
geometrical properties of the heater pattern are difficult to
interpret using the conventional notion that characteristic
dimensions of the individual heaters are the key parameters
that govern thermal transport. Under this assumption, the
separation of the heater lines of a given width should have
no impact on the thermal properties, yet a clear dependence
is observed in Figs.
3(c)
and
3(d)
.
054068-6
QUASIBALLISTIC THERMAL TRANSPORT...
PHYS. REV. APPLIED
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054068 (2018)
(a)(b)
(c)(d)
150 K
FIG. 3. Thermal conductivities of nanoline arrays. (a) Thermal conductivity
κ
0
(open symbols) and
κ
1
(filled symbols) versus period
at 294 K (blue circles) and 150 K (red squares) for nanoline arrays with a 50% duty cycle. (b)
κ
1
normalized to
κ
0
versus period at
different temperatures from measurements (symbols) and synthetic TDTR data (dotted lines). (c)
κ
1
normalized to
κ
0
from measure-
ments (symbols) and synthetic TDTR data (dotted lines) versus period for nanoline arrays with
w
≈
115 nm at various temperatures.
(d)
κ
1
normalized to
κ
0
versus period for duty cycles of 50% and 67%, respectively. The temperature is 150 K. In (c), the linewidth is
fixed and the period changes, while in (d), the linewidth and period both change so as to keep the duty cycle fixed. In both (c) and (d),
the synthetic TDTR data show similar trends. The synthetic data are the TDTR data sets computed from the BTE and then fit with the
four-parameter model to obtain thermal conductivities and interface conductances.
These apparent inconsistencies can be eliminated by
instead considering the spatial frequencies of the heating
pattern as the key parameters. More precisely, we identify
the key dimensionless parameter as
ξ
x
,
F
, where
is the
phonon MFP and
ξ
x
,
F
is the fundamental spatial frequency
[
36
]. This parameter is very analogous to the familiar
Knudsen number if the fundamental spatial frequency is
written as 2
π/
L
.
Using this dimensionless parameter, the observed trends
can be explained simply by considering the spatial fre-
quencies of the heater pattern. For instance, the trend of
increasing
κ
1
with increasing period, with duty cycle held
constant at 50% as in Fig.
3(b)
, can be explained by recog-
nizing that an increasing period
L
implies a decreasing
ξ
x
,
F
and thermal conductivity increases with decreasing spatial
frequency [
36
,
42
]. More subtle trends such as the increase
of
κ
1
with period for a fixed linewidth in Fig.
3(c)
can also
be rationalized by the same explanation. The dependence
of
κ
1
on duty cycle in Fig.
3(d)
reflects the dependence of
the overall thermal response on the relative weights of the
nonzero in-plane spatial frequencies in the heating pattern.
In this case, a duty cycle of 67% has larger weights on
higher spatial frequencies than the 50% duty cycle, result-
ing in a slightly lower thermal conductivity. However, our
measurements show that the primary parameter that gov-
erns
κ
1
is the fundamental spatial frequency set by the
period of the heating pattern.
To further demonstrate the importance of the fundamen-
tal spatial frequency, in Fig.
4
, we plot all of our data on
nanoline arrays versus the fundamental spatial frequency
of the heating pattern. At each temperature, the data nicely
collapse onto a single curve that decreases monotonically
as the fundamental spatial frequency increases, exactly
as predicted by theory [
36
]. The trend is reproduced by
the synthetic TDTR data for line arrays with a 50% duty
cycle.
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et al.
PHYS. REV. APPLIED
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IV. HEAT DISSIPATION FOR DIFFERENT
HEATER GEOMETRIES
The identification of the critical role of the spatial fre-
quency allows us to examine several subtle points regard-
ing quasiballistic thermal transport from nanoscale heat
sources. First, the conventional view holds that as an indi-
vidual heat source, either isolated or in a pattern, becomes
far smaller than mean free paths, a local hot spot will
form at the heater due to the ballistic thermal resistance.
However, as qualitatively suggested previously [
30
], this
argument fails to account for the lack of in-plane gradient
if heat sources in a pattern are sufficiently close together.
In this situation, the thermal transport that occurs will be
identical to that from a continuous thin film heater.
We experimentally examine this prediction by consid-
ering a qualitative measure of the thermal resistance as
an apparent substrate thermal conductivity obtained by fit-
ting the amplitude component of the TDTR signal using
the model of Ref. [
16
]. We perform this fitting for nano-
line arrays of variable period in Fig.
5(a)
. We find that
the apparent thermal conductivity with the nanoline array
obtained experimentally has only weak dependence on
the period, indicating that the thermal resistance of the
substrate has little dependence on the in-plane thermal gra-
dient. This weak period dependence is also observed in the
synthetic TDTR data. Hence, the heater pattern conducts
heat nearly identically as would a continuous thin film.
Analysis of the spatial frequencies of the nanoline heater
pattern allows us to explain this observation. As shown
in Fig.
1(b)
, the nanoline heater pattern can be expressed
as a Fourier series of discrete spatial frequencies and
corresponding weights, and the overall thermal response
depends on these weights and the relative magnitudes of
the thermal responses at the discrete spatial frequencies.
Importantly, the thermal response at zero spatial frequency
makes the largest contribution to the overall thermal
response and, by definition, it is unaffected by the linewidth
of the heating pattern, no matter how small. At higher spa-
tial frequencies, the thermal response amplitude decreases
dramatically with increasing spatial frequency. Addition-
ally, the weights only depend on the duty cycle of the
pattern and also decrease with increasing spatial frequency.
The weak dependence of the nanoline apparent thermal
conductivity on period can thus be understood from the rel-
atively small contribution of thermal responses at higher
spatial frequencies compared to that at zero spatial fre-
quency (dc). Therefore, considering the superposition of
thermal responses shows that the overall thermal response
of the nanoline pattern will be only weakly affected by its
geometry, exactly as observed in experiment. This depen-
dence will weaken as the linewidth decreases and hence
the fundamental spatial frequency increases.
On the other hand, consider a Gaussian spot heater cre-
ated by a focused laser beam. As shown in Fig.
5(a)
, in this
FIG. 4.
Normalized thermal conductivity versus spatial fre-
quency.
κ
1
normalized to
κ
0
at each temperature versus spatial
frequency at different temperatures from measurements (open
symbols) and synthetic TDTR data (dotted lines) for line arrays
with a 50% duty cycle. Measurements for line arrays with other
duty cycles are also shown with filled symbols. For all the mea-
surements, the thermal conductivity decreases monotonically as
the spatial frequency increases. At a given temperature, the duty
cycle has only a little effect on the thermal conductivity for a
given spatial frequency, indicating that spatial frequency is the
key parameter. The synthetic TDTR data show similar trends as
those observed experimentally.
situation, we observe an obvious decrease in apparent ther-
mal conductivity at characteristic length scales far larger
than those achieved in the nanoline arrays. Again, this
trend is reproduced by synthetic TDTR data. To understand
this observation, we must examine the overall thermal
response to the Gaussian heater, given as
Z
(
t
)
=
∫
∞
0
∞
∑
m
=−∞
H
(η
+
m
η
s
,
ξ
r
)
e
im
η
s
t
Q
(ξ
r
)ξ
r
d
ξ
r
,(7)
where
ξ
2
r
=
ξ
2
x
+
ξ
2
y
is the in-plane spatial Fourier variable
in polar coordinates and
Q
(ξ
r
)
=
exp [
−
(
r
2
0
+
r
2
1
)ξ
2
r
/
8],
where
r
0
and
r
1
are the 1
/
e
2
radii of the pump and probe
beams, respectively. At a given
ξ
r
, the overall thermal
response, as shown in Fig.
5(b)
, is weighted by
Q
(ξ
r
)ξ
r
.
Because of the factor of
ξ
r
in the weights, the dc compo-
nent (
ξ
r
=
0) contributes nothing to the overall response
and the response peaks at a nonzero
ξ
r
, at which the
deviation from the Fourier response may be prominent.
Therefore, for an isolated heater localized in more than
one spatial dimension, heat dissipation can be substan-
tially impeded compared to the predictions of Fourier’s
law, in line with prior work [
15
,
19
,
30
,
43
]. This ballistic
resistance can be mitigated by placing heaters sufficiently
close such that the dc component once again contributes to
the thermal response.
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