of 5
PHYSICAL REVIEW B
89
, 205432 (2014)
Coherent and incoherent thermal transport in nanomeshes
Navaneetha K. Ravichandran and Austin J. Minnich
*
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, California 91125, USA
(Received 12 November 2013; revised manuscript received 17 March 2014; published 27 May 2014)
Coherent thermal transport in nanopatterned structures is a topic of considerable interest, but whether it
occurs in certain structures remains unclear due to a poor understanding of which phonons conduct heat. Here,
we perform fully three-dimensional, frequency-dependent simulations of thermal transport in nanomeshes by
solving the Boltzmann transport equation with an efficient Monte Carlo method. From the spectral information in
our simulations, we show that thermal transport in nanostructures that can be created with available lithographic
techniques is dominated by incoherent boundary scattering at room temperature. Our result provides important
insights into the conditions required for coherent thermal transport to occur in artificial structures.
DOI:
10.1103/PhysRevB.89.205432
PACS number(s): 63
.
22
.
Kn
,
61
.
43
.
Bn
,
68
.
65
.
k
,
66
.
70
.
Df
I. INTRODUCTION
Heat conduction in solids at length scales comparable
to phonon mean free paths (MFPs) and wavelengths is a
topic of considerable interest [
1
3
]. Recently, nanostructured
materials, such as nanowires [
4
,
5
], superlattices [
6
], nanocom-
posites [
7
9
], and all-scale nanostructured materials [
10
],
have demonstrated strongly reduced thermal conductivities
compared to their parent bulk materials. Many of these
nanostructured semiconductor materials show promise for
thermoelectric energy conversion [
4
11
].
Coherent thermal transport, in which the phonon dispersion
is modified by the coherent interference of thermal phonons in
an artificial periodic material, is an active area of research
[
12
17
]. Unlike classical boundary scattering, which only
decreases the relaxation times of phonons, coherent effects
can also alter the group velocity and density of states by
zone folding of phonons, which was originally observed at
specific frequencies using Raman scattering [
18
]. A number
of recent works have studied coherent transport both the-
oretically [
15
,
16
,
19
,
20
] and experimentally. Several recent
experiments have reported that coherent effects can affect
thermal transport in superlattices and nanomeshes [
12
,
14
,
17
].
In particular, exceptionally low thermal conductivities were
reported in silicon nanomeshes (NMs) [
14
,
21
] which consist
of periodic pores in a thin membrane. However, attributing
these low thermal conductivity measurements unambiguously
to coherent effects is difficult because boundary scattering
can also reduce the thermal conductivity of the NMs. There-
fore, despite these experimental observations, the conditions
under which coherent thermal transport can occur remain
unclear.
Computational studies have attempted to provide insight
into coherent transport in artificial periodic nanostructures.
Hao
et al.
[
22
] used two dimensional Monte Carlo simula-
tions of phonon transport to predict a reduction in thermal
conductivity of porous silicon with aligned pores. Jain
et al.
[
23
] used a mean free path sampling algorithm to study
phonon transport in NM-like structures with features larger
than 100 nm, concluding that coherent effects are unlikely to
be the origin of the low thermal conductivity in the structures
*
aminnich@caltech.edu
of Hopkins
et al.
[
21
]. He
et al.
[
24
] simulated thermal
transport in NM-like structures with similar surface area to
volume ratio as in the experiments of Yu
et al.
[
14
]butthe
simulated structures were much smaller than those studied
experimentally. Dechaumphai
et al.
[
25
] used a partially
coherent model, in which phonons with a MFP longer than
the NM neck size were assumed to be coherent, to explain the
observations of Yu
et al.
[
14
] but boundary scattering could
not be rigorously treated in their analysis. Thus, due to several
simplifications and approximations used in these studies,
the questions of which phonons are responsible for heat
conduction in complex structures like NMs and under what
conditions coherent transport can occur in these structures
remain unanswered.
To address these issues, we present fully three-dimensional
simulations of thermal transport in NMs using efficient
numerical solutions of the frequency-dependent Boltzmann
transport equation (BTE). Using the spectral information in
our simulations, we find that coherent thermal transport is
likely to occur at room temperature only in structures with
nanometer critical dimensions and atomic level roughness, and
that boundary scattering dominates transport in structures that
can be created lithographically. Our work provides important
insights into the conditions in which coherent thermal transport
can occur in artificial structures.
II. MODELING
We begin by describing our simulation approach. To gain
spectral insights into the heat conduction in nanostructures we
must solve the frequency-dependent BTE, given by [
26
],
∂e
ω
∂t
+
v
·∇
r
e
ω
=−
e
ω
e
0
ω
τ
ω
(1)
where
e
ω
is the desired distribution function,
ω
is the angular
frequency,
e
0
ω
is the equilibrium distribution function,
v
is
the phonon group velocity,
τ
ω
is the frequency-dependent
relaxation time and
t
is the time variable. The phonon BTE
has been solved for simple nanostructure geometries using
several techniques, such as the discrete ordinate method [
27
],
the Monte Carlo simulation method [
28
], the finite volume
method [
29
], and the coupled ordinates method [
30
]. However,
using these methods to accurately simulate transport in the
large three-dimensional (3D) geometry of the NM is extremely
1098-0121/2014/89(20)/205432(5)
205432-1
©2014 American Physical Society
NAVANEETHA K. RAVICHANDRAN AND AUSTIN J. MINNICH
PHYSICAL REVIEW B
89
, 205432 (2014)
challenging due to computational requirements or due to the
use of simplifying approximations that may not be applicable.
We overcome this challenge by solving the BTE with
an efficient variance-reduced Monte Carlo (MC) algorithm,
achieving orders of magnitude reduction in computational cost
compared to other deterministic or stochastic solvers [
31
,
32
].
Briefly, this technique solves the linearized energy-based BTE
by stochastically simulating the emission, advection, and scat-
tering of phonon bundles, each representing a fixed deviational
energy from an equilibrium Bose-Einstein distribution. The
variance of the simulation is reduced compared to tradi-
tional MC simulations by properly incorporating deterministic
information from the known equilibrium distribution in a
control variates approach. This algorithm enables simulations
of thermal transport directly in the complex 3D geometry of
the NM.
To implement the algorithm, the phonon dispersion is
divided into 1000 frequency bins, and the phonon bundles
are emitted into the simulation domain according to the
appropriate distribution as described by P
́
eraud and Hadjicon-
stantinou [
31
]. Since the scattering operator is linearized in
this approach, the phonon bundles are advected and scattered
sequentially and completely independently of each other.
To compute the thermal conductivity of the NM, we
simulate the thermal transport in a single periodic unit cell of
the NM using periodic heat flux boundary conditions [
22
]as
indicated in Fig.
2
(boundaries 1 and 2). The other two periodic
walls of the NM (boundaries 3 and 4) are modeled as specularly
reflecting boundaries since the unit cell of the NM is symmetric
about its center. The top and bottom boundaries in the out-of-
plane direction and the walls of the NM pores are modeled
as diffusely reflecting mirrors. The thermal conductivity of the
NM is computed by adding up the contribution of the trajectory
of each phonon bundle to the overall heat flux. We terminate the
propagation of phonons after 10 internal scattering events as
the change in thermal conductivity of the NM is less than 0
.
5%
between the tenth and the twentieth internal scattering event.
We use an isotropic Si dispersion along the [100] direction
and phonon relaxation times used by Minnich
et al.
[
33
]. To
validate our simulation, we calculate the thermal conductivity
of an unpatterned silicon thin film doped with boron. We
find that we can explain the reported measurements on
silicon thin films (TFs) in Yu
et al.
’s experiments [
14
]
by assuming that the boundaries scatter phonons diffusely
and using the impurity scattering rate of the form
τ
1
Imp
=
2
×
10
44
ω
4
s
1
, where
ω
is the angular frequency of
phonons. For example, we observe from our simulations
that the reduction in the thermal conductivity of TFs in the
experiments of Yu
et al.
[
14
] at 300 K is primarily due to
boundary scattering, reducing the thermal conductivity from
the bulk value of 148
.
2to23
.
5W
/
m K, with the remainder
of the reduction due to impurity scattering, from 23
.
5to
16
.
8W
/
m K. Recent experiments confirm that diffuse bound-
ary scattering of phonons is a reasonable approximation for
analyzing thermal transport in thin silicon membranes [
34
].
For the NM simulations, we consider both circular and square
pores as the shape of the NM pore in the experiments of Yu
et
al.
[
14
] is somewhere in between. Electron-phonon scattering
is expected to be negligible at the temperatures considered [
35
]
and is not included.
100
150
200
250
300
0
2
4
6
8
10
Temperature (K)
Thermal Conductivity (W/m−K)
Yu et al (2010)
BTE − NM (circle)
BTE − NM (square)
FIG. 1. (Color online) Thermal conductivity of the NM as a
function of temperature. The thermal conductivity of the NM reported
by Yu
et al.
[
14
] (black diamonds) is significantly lower than our
simulation result with square and circular pore geometries.
III. RESULTS
We begin our analysis by computing the thermal con-
ductivity of a NM structure. To facilitate comparisons with
experiment, we simulate the same structure in the experiments
of Yu
et al.
[
14
] with a periodicity
w
=
34 nm, a pore width or
diameter
d
=
11 nm, and an out-of-plane thickness
t
=
22 nm.
Since all the physical walls of the NM are modeled as diffusely
reflecting mirrors, our MC simulations yield the Casimir limit
for the thermal conductivity of the NM, which is the theoretical
lower limit for the thermal conductivity of the NM with
phonons following the unmodified bulk dispersion. It is evident
from our simulation results (Fig.
1
) that the experimentally
measured thermal conductivity of the NM is considerably
lower than the Casimir limit.
We now examine whether coherent transport can explain
this exceptionally low thermal conductivity. According to
Jain
et al.
[
23
], for coherent effects to occur in periodic
nanostructures, long wavelength phonons, which are more
likely to scatter specularly from a rough boundary and retain
their phase, should conduct most of the heat. At present, the
minimum wavelength of phonons that can scatter specularly
from a surface with a given roughness remains unclear,
with estimates for the maximum frequency of these specular
phonons ranging from 0.64 [
36
]to2THz[
12
]. From these
experimental observations, we can infer that coherent effects
could affect phonons below 2 THz, while the remaining part
of the phonon spectrum will still follow the bulk material
dispersion and lifetimes.
Conservatively, let us suppose that phonons with frequency
below 2 THz may be able to follow the new dispersion
corresponding to the phononic crystal. When we assume
that the boundaries of the NM reflect these low frequency
phonons specularly so that their contribution to the relative
fraction of heat transport is maximized, we obtain a thermal
conductivity of 11
.
71 W
/
m K at 300 K and 20
.
8W
/
mKat
90 K as shown in Fig.
2
. Even if we assume that phonons
below 2 THz behave coherently and completely remove their
contribution to heat transport, the thermal conductivity of the
NM reduces to 7
.
5W
/
m K at 300 K and 4
.
12 W
/
mKat
90 K, which is still significantly higher than the measured
205432-2
COHERENT AND INCOHERENT THERMAL TRANSPORT IN . . .
PHYSICAL REVIEW B
89
, 205432 (2014)
FIG. 2. (Color online) Thermal conductivity accumulation (k-
accumulation) versus phonon frequency for a NM that reflects
phonons with frequency less than 2 THz specularly and the rest
diffusely. Even under these conservative assumptions, the reported
measurements cannot be explained even by completely neglecting
the contribution of these low frequency phonons that could undergo
coherent interference.
values of 1
.
95 W
/
m K at 300 K and 1
.
3W
/
m K at 90 K
in the experiments of Yu
et al.
[
14
]. A similar conclusion is
reached if all phonons are scattered diffusely. Therefore, even
under the most conservative assumptions, those modes that
have the possibility to undergo coherent interference do not
carry sufficient heat to explain the measurements.
We now use our simulations to identify the mechanism
responsible for the experimentally observed reduction in
thermal conductivity. Although Yu
et al.
[
14
] assumed that the
NM was completely composed of silicon, in other experiments
[
37
] a thin amorphous oxide layer of about 2 to 3 nm thickness
is clearly visible using transmission electron microscopy, even
though the samples were etched in HF vapor. Other studies
have reported that surface damage can result from the reactive
ion etching (RIE) process [
38
] used to create the pores in
the NM.
The presence of such a disordered layer substantially affects
the phonon transport within the NM. A phonon incident on
the disordered layer from silicon has a probability to be
backscattered at the interface before reaching the solid-air
interface of the NM pores. Even if the phonon penetrates into
the disordered layer, it will get scattered nearly immediately
due to its short MFP in the disordered layer. Therefore, this
disordered layer effectively increases the size of the pore and
reduces the cross-sectional area available for heat conduction.
This increased pore size has an important effect on the
interpretation of experimental measurements. In the experi-
ments of Yu
et al.
[
14
], the thermal conductance of the NM
was measured, and the thermal conductivity was calculated by
assuming that heat effectively flows through channels between
arrays of pores in the NM. If the effective size of the NM pores
is larger than assumed, then the width of the heat transport
channels is reduced, thereby increasing thermal conductivity
for a given thermal conductance of the NM. Therefore, in order
to interpret the experimental measurements of Yu
et al.
[
14
]
and compare with our simulations, the thermal conductivity
of the NM has to be scaled by the ratio of the channel areas
without and with the defective layer.
The large pores also lead to additional phonon boundary
scattering due to increased surface area of the pores. To
account for this effect in our MC simulations, we model
the Si-disordered layer interface as a diffusely reflecting
mirror. This is a reasonable approximation considering that
the microscopic details of phonon scattering at interfaces is
poorly understood [
39
]. The effective pore size is increased by
an amount comparable to the thickness of the oxide layer as
observed in the TEM, which is around 2 to 3 nm [
37
]. We also
include the disordered layer on the top and bottom boundaries
of the NM as they were subjected to many of the same etching
processes as the pores.
We now examine if the increase in the effective pore size
can explain the observed reduction in the thermal conductivity
of the NM. Figure
3(a)
shows that our simulations predict a
considerable reduction in thermal conductivity of the NM for
a disordered layer thickness of just 2 nm, compared to the case
without a disordered layer (Fig.
1
). As shown in Fig.
3(b)
,
FIG. 3. (Color online) Thermal conductivity as a function of temperature for different nanostructures in the experiments of Yu
et al.
[
14
]
and our simulations for (a) a 2-nm disordered layer thickness and (b) a 3
.
5-nm disordered layer thickness. The disordered layer is added to both
the nanowire array (NWA) and the nanomesh (NM) in our simulations. In these two figures, the red dashed line, pink circles, and pink squares
are the MC solutions for the NWA, NM with circular holes, and NM with square holes, respectively. The black diamonds, green triangles,
and blue inverted triangles represent the reported thermal conductivity for the NM, the recalculated thermal conductivity for the NM, and the
recalculated thermal conductivity for the NWA, respectively, from Yu
et al.
’s experiments [
14
].
205432-3
NAVANEETHA K. RAVICHANDRAN AND AUSTIN J. MINNICH
PHYSICAL REVIEW B
89
, 205432 (2014)
0
2
4
6
0.5
0.6
0.7
0.8
Disordered Layer Thickness (nm)
Average Backscattering
NWA
NM (Circular pores)
NM (Square pores)
FIG. 4. (Color online) Fraction of backscattered phonons for the
NWA and the NM with circular and square holes for different disor-
dered layer thicknesses. NMs have a larger fraction of backscattered
phonons than the NWA, explaining their lower thermal conductivity.
we are able to explain the experimental observations with a
3
.
5-nm thick disordered layer. Yu
et al.
[
14
] also reported
the thermal conductivity for another NM with a larger pore
(
d
=
16 nm) at lower temperatures. By following the same
simulation procedure, we are able to explain the measurements
for this NM using a 2-nm thick disordered layer.
Our simulations can also explain the difference in thermal
conductivity between the NM and the nanowire array (NWA)
reported in Yu
et al.
’s experiments [
14
]. In Yu
et al.
’s
experiments [
14
], the reduction in thermal conductivity of the
NM was associated with coherent effects primarily because
of the lower thermal conductivity of the NM compared to the
NWA even though boundary scattering considerations would
predict the opposite trend. However, our simulations predict
that the thermal conductivity of the NM is consistently lower
than that of the NWA without considering any coherent effects.
This difference in thermal conductivity can be explained
by backscattering of phonons at the walls of the NM pores
[
40
]. In the NWA, all of the domain walls are aligned parallel
to the direction of the thermal gradient and 50% of the
incident phonons are backscattered on average. The walls of
the NM pores aligned along the temperature gradient also
backscatter half of the incident phonons. However, the walls
of the NM pores that are not aligned with the temperature
gradient backscatter more than half of the incident phonons.
Since backscattering reduces the contribution of the phonons
to thermal transport, the overall thermal conductivity of the
NM is reduced compared to the NWA.
Figure
4
shows the fraction of backscattered phonons in the
NWA and the NM averaged over all frequencies. We consider
a phonon to be backscattered if it returns to the same wall
from which it was emitted. For Fig.
4
, to isolate the effect
of phonon backscattering from the effects of the difference
in the size of the NWA and the NM, we simulate a NM and
NWA with the same effective transport channel area. For the
NM, we use a periodicity
w
=
34 nm, pore size
d
=
12 nm,
and thickness
t
=
22 nm so that it has an effective transport
channel area of 22
×
22 nm
2
. For the NWA, the cross-sectional
area is 22
×
22 nm
2
. To isolate the effect of the geometry, we
compute the backscattered fraction from those phonons that
do not scatter internally in the domain. As expected for the
NWA, 50% of the phonons are backscattered. For the NM
with circular and square pores, the fraction of backscattered
phonons is 20 to 40% higher than that of the NWA for a range
of disordered layer thickness values used in our simulations.
Therefore, the difference in thermal conductivity between
the NM and NWA can be attributed to the larger fraction
of backscattered phonons in the NM along with the smaller
transport channel area of the actual NM.
We now examine the conditions under which coherent
transport could occur in an artificial structure at room tem-
perature. From the spectral information in our simulations,
we find that most of the heat is carried by phonons with
frequencies around 5 THz at room temperature, corresponding
to a wavelength of about 1 to 2 nm in Si. Therefore, a
secondary periodicity on the order of this value is necessary for
coherent effects to affect thermal transport in the NM. Further,
the surface roughness of an artificial structure must be less than
a few angstroms to preserve the phase of the scattered phonons.
Such fine spatial resolution and atomic scale roughness is diffi-
cult to obtain using lithographic techniques, but could be met in
superlattices with epitaxial interfaces [
17
]. In lithographically
patterned structures, coherent thermal transport is likely to
play a role only at very low temperatures where the dominant
thermal phonon wavelength substantially exceeds the surface
roughness amplitude.
IV. CONCLUSION
In conclusion, we have performed fully three-dimensional
simulations of thermal transport in nanomeshes using efficient
numerical solutions of the frequency dependent BTE. From
the spectral information in our simulations, we find that
incoherent boundary scattering dominates thermal transport
in lithographically patterned structures, and that structures
with nanometer critical dimensions and atomic level roughness
are required for coherent thermal transport to occur at room
temperature. Our results provide important insights into the
conditions under which coherent thermal transport can occur
in artificial structures.
ACKNOWLEDGMENTS
This work is part of the “Light-Material Interactions in
Energy Conversion” Energy Frontier Research Center funded
by the US Department of Energy, Office of Science, Office of
Basic Energy Sciences under Award No. DE-SC0001293.
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