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
(Dated: April 1, 2014)
Abstract
Coherent thermal transport in nanopatterned structures is a topic of considerable interest, but
whether it occurs in certain structures remains unclear due to poor understanding of which phonons
conduct heat. Here, we perform the first fully three-dimensional, frequency-dependent simulations
of thermal transport in nanomeshes by solving the Boltzmann transport equation with a novel,
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.
1
arXiv:1403.7647v1 [cond-mat.mtrl-sci] 29 Mar 2014
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], nanocomposites [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 coher-
ent 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 theoret-
ically [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 arti-
ficial periodic nanostructures.
Hao et al
[22] used two dimensional Monte Carlo simulations
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 of
Ref. [21].
He et al
[24] investigated NM-like structures with similar surface area to volume
ratio as in Ref. [14] but the simulated structures were much smaller than those studied
experimentally.
Dechaumphai et al
[25] used a partially coherent model, in which phonons
with MFP longer than the NM neck size were assumed to be coherent, to explain the obser-
vations of Ref. [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
2
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 this issue, we present the first 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.
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 and
τ
ω
is the frequency-dependent
relaxation time.
The phonon BTE has been solved for simple nanostructure geometries using several tech-
niques such as the discrete ordinate method [27], 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 3D geometry of the NM is extremely
challenging due to computational requirements or due to the use of simplifying approxima-
tions that may not be applicable.
We overcome this challenge by solving the BTE with an efficient variance-reduced Monte
Carlo 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 scattering of
phonon bundles, each representing a fixed deviational energy from an equilibrium Bose-
Einstein distribution. The variance of the simulation is reduced compared to traditional MC
by properly incorporating deterministic information from the known equilibrium distribution
in a control variates approach. This algorithm enables the first simulations of thermal
transport directly in the complex 3D geometry of the NM.
3
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 appro-
priate distribution as described in Ref. [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 the inset of 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
from Ref. [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 assuming 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 the NM simulations, we consider both circular and square pores as the shape
of the NM pore is somewhere in between. Electron-phonon scattering is expected to be
negligible at the temperatures considered [34] and is not included.
We begin our analysis by computing the thermal conductivity of a NM structure. To
facilitate comparisons with experiment, we simulate the same structure in Ref. [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.
4
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 nanos-
tructures, 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 mini-
mum phonon wavelength that can scatter specularly from a surface with a given roughness
remains unclear, with estimates ranging from 0.64 THz [35] to 2 THz [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.
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: Thermal conductivity of the NM
as a function of temperature. The
thermal conductivity of the NM
reported in Ref. [14] (black diamonds) is
significantly lower than our simulation
result with square and circular pore
geometries.
FIG. 2: Thermal conductivity
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.
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
5
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 300K and 20
.
8 W/m-K at 90K 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
.
5 W/m-K
at 300K and 4
.
12 W/m-K at 90K, which is still significantly higher the measured values of
1
.
95 W/m-K at 300K and 1
.
3 W/m-K at 90K in
Yu et al
’s experiments. 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 [36] a thin amorphous oxide layer
of about 2
−
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 [37] 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 experiments 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. There-
fore, in order to interpret the experimental measurements in Ref. [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.
6
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 approxi-
mation considering that the microscopic details of phonon scattering at interfaces is poorly
understood [38]. The effective pore size is increased by an amount comparable to the thick-
ness of the oxide layer as observed in TEM, which is around 2
−
3 nm [36]. 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.
FIG. 3: Thermal conductivity as a function of temperature for different nanostructures
from experiments in Ref. [14] and our simulations for (a) 2 nm disordered layer thickness
and (b) 3
.
5 nm disordered layer thickness. The disordered layer is added to both the NWA
and the 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 Ref. [14].
We now examine if the increase in the effective pore size can explain the observed reduc-
tion 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),
7
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 NWA. 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 [39]. 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 phonon to thermal transport, the overall
thermal conductivity of the NM is reduced compared to the NWA. Figure 4 shows the
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: Fraction of backscattered phonons for the NWA and the NM with circular and
square holes for different disordered layer thicknesses.
8
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
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
−
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 arti-
ficial structure at room temperature. 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
−
2 nm in Si. Therefore, a secondary
periodicity on the order of this value is necessary for coherent effects to affect thermal trans-
port in the NM. Further, the surface roughness of an artificial structure must be less than
a few
̊
A
to preserve the phase of the scattered phonons. Such fine spatial resolution and
atomic scale roughness is difficult to obtain using lithographic techniques, but could be met
in superlattices with epitaxial interfaces [17]. In lithographically patterned structures, co-
herent thermal transport is likely to play a role only at very low temperatures where the
dominant thermal wavelength substantially exceeds the surface roughness amplitude.
In conclusion, we have performed the first fully three-dimensional simulations of ther-
mal 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
9
structures.
This work is part of the ’Light-Material Interactions in Energy Conversion’ Energy Fron-
tier Research Center funded by the U.S. Department of Energy, Office of Science, Office of
Basic Energy Sciences under Award Number de-sc0001293.
∗
aminnich@caltech.edu
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