Thermal boundary conductance across rough interfaces probed by molecular dynamics

Abstract

We report on the influence of the interfacial roughness on the thermal boundary conductance between two solids, using molecular dynamics. We show evidence of a transition between two regimes, depending on the interfacial roughness: When the roughness is small, the boundary conductance is constant, taking values close to the conductance of the corresponding planar interface. When the roughness is larger, the conductance becomes larger than the planar interface conductance and the relative increase is found to be close to the increase of the interfacial area. The cross-plane conductivity of a superlattice with rough interfaces is found to increase in a comparable amount, suggesting that heat transport in superlattices is mainly controlled by the boundary conductance. These observations are interpreted using the wave characteristics of the energy carriers. We characterize also the effect of the angle of the asperities and find that the boundary conductance displayed by interfaces having steep slopes may become important if the lateral period characterizing the interfacial profile is large enough. As a result, triangular-shaped interfaces may be used to enhance the conductance of planar interfaces by a factor greater than three. Finally, we consider the effect of the shape of the interfaces and show that the sinusoidal interface displays the highest conductance because of its large true interfacial area. All of these considerations are relevant to the optimization of nanoscale interfacial energy transport.

Full text

PHYSICAL REVIEW B 89, 054309 (2014)
Thermal boundary conductance across rough interfaces probed by molecular dynamics
Samy Merabia*
Institut Lumi`
ere Mati`
ere, UMR5306 Universit´
e Lyon 1-CNRS, Universit´
e de Lyon, F-69622 Villeurbanne cedex, France
Konstantinos Termentzidis
LEMTA CNRS-UMR7563, Universit´
e de Lorraine, F-54506 Vandoeuvre les Nancy, France
(Received 14 March 2013; revised manuscript received 29 January 2014; published 25 February 2014)
Wereport on the influence of the interfacial roughness on the thermal boundary conductance between two solids,
using molecular dynamics. We show evidence of a transition between two regimes, depending on the interfacial
roughness: When the roughness is small, the boundary conductance is constant, taking values close to the
conductance of the corresponding planar interface. When the roughness is larger, the conductance becomes larger
than the planar interface conductance and the relative increase is found to be close to the increase of the interfacial
area. The cross-plane conductivity of a superlattice with rough interfaces is found to increase in a comparable
amount, suggesting that heat transport in superlattices is mainly controlled by the boundary conductance. These
observations are interpreted using the wave characteristics of the energy carriers. We characterize also the effect
of the angle of the asperities and find that the boundary conductance displayed by interfaces having steep slopes
may become important if the lateral period characterizing the interfacial profile is large enough. As a result,
triangular-shaped interfaces may be used to enhance the conductance of planar interfaces by a factor greater than
three. Finally, we consider the effect of the shape of the interfaces and show that the sinusoidal interface displays
the highest conductance because of its large true interfacial area. All of these considerations are relevant to the
optimization of nanoscale interfacial energy transport.
DOI: 10.1103/PhysRevB.89.054309 PACS number(s): 68.35.Ja,07.05.Tp,44.10.+i
I. INTRODUCTION
The existence of a finite thermal boundary resistance
between two solids has important practical consequences, es-
pecially in the transport properties of nanostructured materials.
When the distance between interfaces becomes submicronic,
heat transfer is mainly controlled by the interfacial phonon
transmission, which in turn governs the thermal boundary
resistance. In certain applications, such as electro-optical
modulators [1], optical switching devices [2], and pressure
sensors [3], a low resistance is desired to enhance energy flow.
In thermoelectric devices, on the contrary, a large resistance
is preferable so as to generate large barriers for a wide class
of phonons [4–6]. Two strategies may be followed in order
to tune the value of the boundary resistance between two
solids. Either the solid/solid interaction is changed through the
coupling with a third body, which is usually a self-assembled
monolayer [7,8], or the other possibility is to modulate
the interfacial roughness [9]. This latter direction has been
illustrated experimentally through chemical etching [10–12].
However, a theoretical model describing the effect of the
interfacial roughness on the thermal boundary conductance
at room temperature is still lacking [13,14]. Note that the role
of the interfacial roughness on the Kapitza conductance was
identified a long time ago, in the context of liquid helium/metal
interfaces at very low cryogenic temperatures [15,16]. At
these temperatures, the phonon coherence length may be
comparable with the typical heights of the interface, leading
to strong phonon scattering which is put forward to explain
the high values of the conductance experimentally reported, as
compared with the classical acoustic mismatch theory which
*samy.merabia@univ-lyon1.fr
assumes planar interfaces [17]. Such considerations have
received less attention for room-temperature solids, probably
because in this case the phonon coherence length is very small.
Understanding the role of the interfacial roughness also
has important consequences in the transport properties of
superlattices, which are good candidates for thermoelectric
conversion materials, thanks to their low thermal conductivity
[18]. Designing superlattices with rough interfaces has been
recently achieved, opening an avenue for reducing the thermal
conductivity in the direction perpendicular to the interfaces
[19]. Again, the physical mechanisms at play in the heat-
transport properties of rough superlattices have not been
elucidated so far. Molecular dynamics offers a privileged route
to understand the interaction between the energy carriers in a
solid and the asperities of the interface [20–22]. In this paper,
we use molecular dynamics to probe interfacial heat transfer
across a model of rough interfaces. Because of the difficulty to
determine the temperature jump across a nonplanar interface,
we have used thermal-relaxation simulations, which enable
one to compute the thermal boundary resistance characterizing
rough interfaces. In Sec. II, we describe the structures used to
probe the conductance of rough interfaces. In Sec. III,weex-
plain the methodology retained to extract the thermal boundary
conductance from molecular dynamics. The simulation results
are presented and discussed in Sec. IV. We first concentrate on
model interfaces made of isosceles triangles. For these model
interfaces, we present the results for the thermal conductance
as a function of the interfacial roughness and interpret the
results using a simple acoustic model in Sec. IV B.Inthe
following section, we characterize the effect of the angles of
the asperities. Finally, in Sec. IV D, we have appraised the
effect of the interfacial shape. We discuss the consequences of
this work in Sec. V.
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SAMY MERABIA AND KONSTANTINOS TERMENTZIDIS PHYSICAL REVIEW B 89, 054309 (2014)
II. STRUCTURES AND SAMPLE PREPARATION
We will consider a model of rough interfaces, constructed
from two perfect fcc Lennard-Jones solids whose interface
is orientated along the crystallographic [100] direction, as
represented in Fig. 1. We introduce some two-dimensional
(2D) roughness in the xz plane, where xand zdenote,
respectively, the [100] and [001] directions. As we use
periodic boundary conditions in all spatial directions, the
system studied is similar to a superlattice. The dimension
in the ydirection has been fixed to 10 a0, where a0is the
lattice parameter, while the dimension in the zdirection—the
superlattice period—has been varied between 5 and 40 a0.All
of the atoms of the system interact through a Lennard-Jones
(LJ) potential, VLJ(r)=4[(σ/r)12 −(σ/r)6], truncated at
a distance 2.5σ. A single set of energy and diameter
σcharacterizes the interatomic interaction potential. As a
result, the two solids have the same lattice constant a0.To
introduce an acoustic mismatch between the two solids, we
have considered a difference between the masses of the atoms
of the two solids, characterized by the mass ratio mr=m2/m1.
In all of the following, we will use mr=2, which has been
shown to give an impedance ratio typical of the interface
between Si and Ge [22]. From now on, we will use real
FIG. 1. (Color online) (a) Diagram showing the different pa-
rameters used as nomenclature for the triangular-shaped interfaces.
(b)–(d) Schematic representation of the different parameters that have
been tested: (b) interfacial height at a constant value of the angle
α=45 deg, (c) angle αat a constant value of the interface height
h, and (d) angle αand interfacial height hat a constant value of the
interfacial area A.
units where =1.67 ×10−21 J; σ=3.4×10−10 m, and
m1=6.63 ×10−26 kg, where these different values have been
chosen to represent solid argon. With this choice of units,
the unit of time is τ=mσ 2/ =2.14 ps and the unit of
interfacial conductance is G=kB/(τσ2)56 MW/K/m2.
The different interfaces have been prepared as follows: First
the structures have been generated by mapping the space
with fcc structures using the lattice parameter of the fcc
LJ solid at zero temperature [23]: a0(T=0K)=1.5496σ.
The structures have been equilibrated at the two final finite
temperatures T=40 and T=18 K using a combination of
a Berendsen, a Nos´
e Hoover thermostat, and a barostat at 0
atm [24]. The total equilibration time lasts one million time
steps which corresponds to a total time of 4,28 ns with a time
step dt =4.28 fs. The equilibrium lattice parameters have
been found to be a=1.579σat T=40 K and a=1.5563σ
at T=18 K. In this paper, we consider different types of
rough interfaces, as represented in Fig. 1. The first one
consists of triangular-shaped interfaces having a constant
angle α=45 deg with respect to the xy plane and a variable
height h[Fig. 1(b)]. In the second type of interface analyzed,
we keep constant the interfacial height hand we vary the
angle α[Fig. 1(c)]. In the third case, both the angle and
the height are varied, keeping constant the interfacial area
A. Finally, the effect of the shape of the interfaces has also
been appraised considering totally rough interfaces, small
triangles juxtaposed on triangular interfaces, and square- and
wavy-shaped interfaces. This analysis covers all of the possible
parameters that might be involved in the geometry of the
interfaces with the aim to quantify their influence on phonon
interfacial transport.
III. THERMAL BOUNDARY CONDUCTANCE FROM
THERMAL-RELAXATION SIMULATIONS
In this section, we briefly review the methodology adopted
to probe the interfacial conductance between two solids, using
relaxation simulations. Generally speaking, there are different
methods to extract the boundary conductance from molecular
simulations: one method consists of applying a net heat flux
qacross the system through the coupling of two energy
reservoirs and measuring the finite-temperature jump T
across the interface [25,26]. This allows one to compute the
interfacial conductance GK=q/T. For the rough interfaces
that we will consider in the following, it may be difficult to
clearly identify a temperature jump, especially if the roughness
is large. On the other hand, relaxation simulations do not
require one to spatially resolve the temperature field in the
vicinity of the interface, and for this reason they are well
adapted to the determination of the conductance of imperfect
interfaces. The principle is akin to the thermoreflectance
technique and consists of instantaneously heating one of
the two solids and recording the temporal evolution of the
temperature of the hot solid [27–29]. The conductance GK
is then obtained from the time τcharacterizing the thermal
relaxation of the hot solid,
GK=3N1kB
4A0τ,(1)
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THERMAL BOUNDARY CONDUCTANCE ACROSS ROUGH . . . PHYSICAL REVIEW B 89, 054309 (2014)
0 20 40 60 80 100 120 140 160
Time
(
ps
)
0.0001
0.001
E (eV)
FIG. 2. (Color online) Energy decay of the heated solid obtained
using thermal-relaxation simulations. Dashed lines show the expo-
nential fit. The parameters are total length =40 a0, temperature
T=40 K, and mass ratio mr=2.
where N1is the number of atoms of solid 1, kBis the
Boltzmann’s constant, A0is the interfacial area projected
along the mean normal vector to the interface (corresponding
to direction zon Fig. 1), and the factor 4 accounts for
the presence of two interfaces. Because the temperature of
the heated solid may display some oscillations which may
make the determination of the time constant τdifficult, we
have instead used the decay of the energy E1to extract τ,
Em
1=
j∈1
1
2mv2
j+
j,k∈1
V(rj−rk),(2)
where the first term is the kinetic energy and the second
term corresponds to the interatomic potential, which here is
supposed to be pairwise. An example of the time decay of the
energy during thermal relaxation of the hot solid is displayed
in Fig. 2, showing that the exponential decay hypothesis is
reasonable.
In practice, after equilibration of the system, we have
heated one of the two solids and followed the thermal
relaxation of the system at constant energy and volume.
The heating is performed by instantaneously rescaling the
velocities of the hot solid by the same factor. In this paper,
we have used a constant value of the temperature increment
of T =18 K, and checked also that using a smaller value
(T =8 K) does not significantly change the measured
conductances. To remove any contribution stemming from
internal phonon scattering in the heated solid, we have run
in parallel simulations across the interface between identical
solids and calculated the corresponding internal resistance.
The Kapitza conductance calculated in this paper has been
obtained after having subtracted this internal resistance,
1/GK=1/G12 −1/G11,(3)
where G12 is the conductance measured for the interface
between solid 1 and solid 2, and G11 is the conductance
measured between identical solids using Eq. (1). This lat-
ter conductance is typically twice as large as G12:for
a flat interface equilibrated at T=18 K, one measures
G12 =143 MW/m2/K and G11 =336 MW/m2/K resulting
in a conductance GK248 MW/m2/K. Note also that Eq. (3)
assumes that internal phonon-phonon scattering in the hot solid
is the same in the interfacial and in the no-interface system,
an assumption which should be verified in the limit of thick
solid media. The procedure described above has been followed
for all of the systems studied in this paper. Finally, for the
simulations discussed in this paper, we have used between five
and ten independent configurations, depending on the system
size, to determine the value of the Kapitza conductance and
the error bar has been found to be typically 15%.
IV. RESULTS
In this section, we present the simulation results obtained
using the relaxation simulations, as detailed above. We will
successively study the effect of the interfacial roughness, the
angle of the asperities, and the shape of the interface. A
summary of the different parameters that will be varied is
depicted in Fig. 1.
A. Effect of the superlattice period and number of periods
In this section, we first quantify finite-size effects in the
determination of the conductance of rough interfaces, as
measured by Eq. (1). It is important to note that the system
simulated is not a single isolated interface, but rather a
superlattice because of the periodic boundary conditions. It
is thus relevant to assess the influence of the superlattice
period on the thermal conductance as measured by Eq. (1).
To this end, we will consider a model of rough interfaces,
made of isosceles triangles, as depicted in Fig. 1.InFig.3,
we report the conductance of triangular-shaped interfaces
having a fixed roughness height of 32 monolayers (MLs) and
a varying period. For the two temperatures considered, the
thermal conductance is found to decrease with the system size
for short periods, and then saturates for periods larger than
30 nm. The increase of the conductance for thin layers may
FIG. 3. (Color online) Thermal boundary conductance of
isosceles-triangular-shaped interfaces having a roughness height
h=32 MLs, as a function of the period pdefined in Fig. 1. Top:
T=40 K; bottom: T=18 K. The mass ratio is mr=2.
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SAMY MERABIA AND KONSTANTINOS TERMENTZIDIS PHYSICAL REVIEW B 89, 054309 (2014)
01234
No. of periods
0
100
200
300
400
500
600
700
800
900
Kapitza Conductance (MW/m2 K)
FIG. 4. (Color online) Thermal boundary conductance of
isosceles-triangular-shaped interfaces having a roughness height
h=32 MLs and a period 40 a0, as a function of the No. periods of
the superlattice. The temperature is T=40 K and the mass ratio is
mr=2.
be explained because long wavelength phonons will not see
two independent interfaces, but rather a single one. A similar
trend has also been reported in lattice dynamics simulations
[30] and Green’s-function calculations [31]. For thick layers,
the conductance measured is constant and has converged to
the value characterizing an infinitely thick film. We remark
also that the conductance is higher at high temperatures.
Generally speaking, the thermal boundary conductance is
found to increase with temperature, a trend often attributed
to the existence of inelastic phonon scattering at the interface
[25,32]. This behavior is consistent with our simulation data.
In the following, we will fix the period to 20 a030 nm
as it leads to moderate finite-size effects as already found
for superlattices with planar interfaces [33]. Finally, since
the system we simulate is akin to a superlattice because of
the periodic boundary conditions, it is important to probe the
effect of the number of periods on the measured conductance.
Figure 4quantifies the effect of the number of interfaces on
the conductance measured in thermal-relaxation simulations.
From this figure, we can conclude that within error bars, the
number of interfaces has a mild effect on the conductance that
we calculate. This is the behavior expected, as we probe a
quantity characterizing the interface solely, independently of
the number of interfaces.
B. Effect of the interfacial roughness
We will now concentrate on the influence of the interfacial
roughness on the thermal boundary conductance. We will
consider rough interfaces having an angle αfixed at α=
45 deg, while the height hof the interface is increased so
as to change the interfacial roughness, as seen in Fig. 1(b).
For the following, it is important to keep in mind that when
varying the height of the interface hat a constant value of the
angle, the total interfacial area remains constant and larger by
a factor 1/cos α=√2 than its corresponding projection on
the horizontal xy plane.
0 1020304050 60
interfacial height (ML)
100
200
300
400
500
Kapitza Conductance (MW/m2 K)
T=18 K
0 1020304050 60
interfacial hei
g
ht
(
ML
)
100
200
300
400
500
600
Kapitza Conductance (MW/m2 K)
T=40 K
FIG. 5. (Color online) Thermal boundary conductance of
isosceles-triangular-shaped interfaces as a function of the interfacial
roughness here measured in monolayers (1ML =a0/20.75 nm).
Top: T=18 K; bottom: T=40 K. We have also indicated the
conductance of the corresponding planar interfaces (ML =0) and
very rough interface (ML =60). The horizontal dashed lines show
the conductance obtained after rescaling the conductance of the
planar interface by the true interfacial area. The solid lines denote the
theoretical model (5) with the parameter ξ=0.2. The parameters
are total length =40 a0and mass ratio mr=2.
Figure 5displays the evolution of the measured Kapitza
conductance as a function of the interfacial roughness for two
temperatures. The conductance of a planar interface, which
corresponds to the value h=0, has been also indicated for
the sake of comparison. Two regimes are to be distinguished,
depending on the roughness of the interface h. When the
height is smaller than typically 20 monolayers (MLs), the
conductance seems to be constant or slightly decreases with
the roughness, taking values close to the planar interface
conductance. When the interfacial height becomes larger, the
conductance suddenly increases and tends to saturate for very
rough surfaces.
Interestingly, the increase of the conductance between
planar and very rough surfaces is found to be close to the
increase of the interfacial area. This is materialized in Fig. 5,
where we have shown with dashed lines the value of the
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THERMAL BOUNDARY CONDUCTANCE ACROSS ROUGH . . . PHYSICAL REVIEW B 89, 054309 (2014)
h
λ
h
λ
FIG. 6. (Color online) Schematic representation of the
roughness-induced phonon scattering. Top: case of a small
roughness. The huge majority of incoming phonons see the interface
as a plane, and the transmitted heat flux is proportional to the
projected interface area. Bottom: case of a large roughness. Most of
the phonons have a wavelength larger than the interfacial roughness,
and the transmitted heat flux is proportional to the true surface
area. For the sake of the representation, we have not drawn the
reflected waves. Note also that the phonon wavelength is generally
not conserved at the passage of the interface.
conductance obtained by multiplying the conductance of a
planar interface by a factor 1/cos α. On the other hand, we
have reported in a previous study [20] that the cross-plane
thermal conductivity of superlattices with rough interfaces
is greater than the conductivity of perfect superlattices by
a factor between 1.3 and 1.5, which encompasses the value
√21.41. This reinforces the message according to which
the thermal conductivity of a superlattice is mainly controlled
by the Kapitza resistance exhibited by the interfaces, which in
turn seems to be primarily governed by the interfacial area.
We now give some qualitative elements to understand the
previous simulation results regarding the influence of the
interfacial roughness on the Kapitza conductance. At this
point, it is important to have in mind that in the situations that
we have modeled, the energy carriers are phonons which are
classically populated. A given phonon mode is characterized
among others by its wavelength λ, which may take practically
any value between an atomic distance 2a0and the simulation
box length L[34]. First, let us concentrate on the case of a
small roughness h, as shown in Fig. 6. In this case, the majority
of phonon modes have a wavelength larger than hand they see
the interface as a planar one: the transmitted heat flux is then
controlled by the projected area. On the other hand, when the
interface is very rough, most of the phonons have a wavelength
smaller than the height h, obviously the phonons no longer feel
the interface as planar, phonon scattering becomes completely
incoherent, and the transmitted heat flux is controlled by the
true surface area.
To put these arguments on quantitative grounds, we will
consider the following expression of the thermal conductance,
inspired by the classical acoustic mismatch model (AMM)
model [35]. We introduce a mode-dependent fraction ψ(λ),
which depends on the considered wavelength and which is
equal to 1 when the wavelength is supposed to be small
compared with the interfacial roughness h, and equal to 0
in the opposite case. We define a dimensionless parameter ξ,
such that
ψ(λ)=1ifλ<ξh,
ψ(λ)=0 otherwise.(4)
The parameter ξwill be the adjustable fitting parameter of
the model. The interfacial conductance is then supposed to be
given by
GK=3
2ζρkBc1ωDmin
0
g(ω)ψ(λ)dωI12
+A
A0ωDmin
0
g(ω)[1 −ψ(λ)]dωI12,(5)
where ρis the crystal number density, c1is the average sound
velocity in medium 1, and ωDmin is the Debye frequency of
the softer solid. The parameter ζis a scaling factor which
accounts for the tendency of the AMM model to overpredict the
measured Kapitza conductance [33]. The integral I12 involves
the angular-dependent transmission coefficient,
I12 =1
0
4Z1μ1Z2μ2
(Z1μ1+Z2μ2)2μ1dμ1,(6)
where Zi=ρm
iciare the acoustic impedances of the two
solids, ρm
iis the mass density, and μ1=cos θ1is a shorthand
notation to denote the cosine of the phonon incident angle
[33]. Finally, the quantity A/A0is the ratio of the true
interfacial area over the projected one. The physical motivation
of Eq. (5) is simple: phonons having a wavelength λlarger
than ξh contribute to a transmitted heat flux proportional
to the projected area A0, while phonon modes having a
wavelength smaller than ξh contribute to the transmitted heat
flux proportionally to the true surface area. We have compared
the prediction of Eq. (5) to the simulation results discussed
before. To this end, we have assumed Debye solids, with
a vibrational density of states g(ω)=ω2/(2π2c3
1) and, for
the sake of consistency, the mode-dependent wavelength λ
has been taken to be simply related to the frequency ω:
λ=2πc1/ω. Figure 5compares the predictions of Eq. (5)
to the simulations results. The values of the planar interface
conductance have been rescaled by a factor ζ=3 and 4 at
the temperatures T=18 and T=40 K, respectively. These
correction factors account for the fact that the simple AMM
model relies on several assumptions, i.e., Debye solids and
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SAMY MERABIA AND KONSTANTINOS TERMENTZIDIS PHYSICAL REVIEW B 89, 054309 (2014)
elastic scattering, which may lead to a discrepancy with the
molecular dynamics (MD) value. We have chosen the two
factors because they yield good agreement with the MD value
for smooth interfaces. Apart from this rescaling, the parameter
ξhas been treated as the only fitting parameter. Figure 5
shows that a good agreement is found using the value ξ=0.2
for the two temperatures considered. The small value of the
fitting coefficient may be understood in the following way:
Consider a given phonon mode. If its wavelength is larger
than the roughness h, the effective scattering area would be
the projected one. On increasing the roughness, hwill become
comparable with λand the interface will strongly scatter the
considered phonon in all directions. This will contribute a to
slight decrease of the conductance, as compared with the planar
case, in agreement with the simulation data points. It is only
when the roughness becomes very large as compared with the
wavelength λhthat interfacial scattering becomes again
negligible and the transmitted energy is proportional to the
true area. The fitting procedure concludes that this regime is
reached when the wavelength becomes smaller than typically
one-fifth of the interfacial roughness.
C. Effect of the angle of the asperities
So far, we have considered the case where the angle αwas
constant. We now discuss the effect of varying the slope of
the model interfaces on the interfacial energy transfer. First,
we will change the angle at a fixed value of the interfacial
height h, as represented in Fig. 1(c). Figure 7shows the
evolution of the Kapitza conductance as a function of the
angle, at the two considered temperatures. The constant height
hused here corresponds to the regime of large roughnesses in
terms of Fig. 5discussed before. We have also indicated the
conductance of a planar interface, for the sake of comparison.
The evolution of the conductance with the asperities angle
0 20406080
angle (deg)
100
200
300
400
500
Kapitza Conductance (MW/m2 K)
T=40 K
T=18 K
FIG. 7. (Color online) Thermal boundary conductance as a func-
tion of the angle of the asperities α. The height of the asperities
is fixed here and equal to 24 ML. The points corresponding to
α=0 denote the conductance of a planar interface. The solid lines
show the interfacial conductance rescaled by the true surface area:
GK=GK(α=0)/cos α. The other parameters are total length =40
a0and mass ratio mr=2.
seems to be nonmonotonous: first, it increases for low angles,
reaches a maximum for an asperities angle between 30 and 45
degrees, and then decreases when the angle becomes large. In
particular, the conductance for an angle greater than 60 deg
becomes smaller than the planar interface conductance. This
is all the more remarkable as in this latter case, where the true
surface area may increase by a factor of four as compared
to the planar interface. This discrepancy is best shown after
comparing the simulation results with the rescaled conduc-
tance GK(α=0)/cos α, which accounts for the increased
surface area induced by the asperities. It is immediately clear
that for the lowest values of the asperities angles, α=30
and 45 deg, the rescaled conductance seems to reasonably
describe interfacial energy transfer. On the other hand, at large
values of α, the theoretical expression greatly overestimates
interfacial transport. Two phenomena may explain the poor
conductance reported: first, on increasing the angle, phonon
multiple scattering and backscattering may contribute to
diminish interfacial transmission. This has been evidenced by
Rajabpour et al. using Monte Carlo ray tracing calculations
[21]. Second, for the steep slope interfaces considered here,
the effective surface area seems to be the projected one, not the
true area, even if the height of the asperities is large. This may
be understood qualitatively because for steep interfaces, even
if the height is large, the lateral correlation length l=h/ tan α
may become comparable with the phonon wavelengths, and the
effective interfacial area becomes the projected one. For these
steep interfaces, the regime where the transmitted heat flux is
controlled by the true surface area should occur at a very large
value of the interfacial height h. To verify this assessment, we
have run simulations where the true surface area has been
kept constant [cf. Fig. 1(d)]. The results are displayed in
Fig. 8, which concludes a different scenario as compared to
the evolution shown previously in Fig. 7. The evolution of
the conductance with the angle is no longer nonmonotonous
as previously observed, but it rather increases monotonously
with α. For the relatively small values of the angles α,the
0 20406080100
an
g
le
(
de
g)
200
400
600
800
1000
1200
Kapitza Conductance (MW/m2 K)
T=40 K
T=18 K
FIG. 8. (Color online) Same as Fig. 7, but for a constant value
of the true surface area. The interfacial heights are, respectively,
h=21,34,43, and 45 monolayers for the asperities angles α=
25.8,45,64, and 71.7 deg. The solid lines show the interfacial con-
ductance rescaled by the true surface area: GK=GK(α=0)/cos α.
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conductance measured may even exceed the rescaled one. We
have no interpretation for these large values reported here.
Further increasing the angle α, the simulation data take values
close to the scaled conductances GK(α=0)/cos α. Note, in
particular, that the increase of the conductance is pretty large,
overpassing the conductance of a planar interface by a factor
larger than three. In this regime, and for these steep interfaces,
it is highly probable that the regime of rough interfaces,
in terms of the previous discussion, has been reached: heat
transmission becomes controlled by the true surface area.
These large enhancements of the Kapitza conductance open
the way to design interfaces with tailored interfacial energy
transport properties.
D. Effect of the shape of the interfaces
We end the presentation of the results with a discussion
of the effect of the shape of the interface on the boundary
conductance. All of the previous discussion concentrated on a
model of triangular interfaces, and it is worth asking how gen-
eral are the conclusions drawn from the study of the particular
type of surfaces. To appraise this question, we have considered
different shapes of the interfaces, as depicted in Fig. 9.The
common characteristic of these surfaces is the mean interfacial
height, here fixed at a value h=12 MLs. Different morpholo-
gies have been designed, ranging from the random interface to
the case of the squarelike surface and wavy interface obtained
by a sinusoidal modulation of the interfacial height.
Figure 10 compares the interfacial conductance for the
different shapes shown before, at two different temperatures.
The relatively large values reported at the highest temperature
may be explained by inelastic phonon scattering taking place
between two interfaces. The shape of the interface seems
to considerably affect interfacial transport: random interfaces
display a conductance practically equal to the planar interface.
Rough interfaces may transfer energy slightly better than
planar interfaces depending on the temperature. On the
other hand, wavy and squarelike interfaces tend to favor
energy transmission, with the wavy pattern displaying the
highest conductance among the different shapes analyzed.
These results may be interpreted qualitatively: random- and
rough-shaped interfaces display a distribution of length scales,
FIG. 9. (Color online) Illustration of the different interfacial
shapes simulated, namely, random, rough, square, and wavelike
interfaces.
200
300
400
500
Kapitza Conductance (MW/m2 K)
T=40 K
T=18 K
smooth
random
rough
squares
wavy
FIG. 10. (Color online) Thermal boundary conductance for the
different interfacial shapes represented in Fig. 9. The height of the
different interfaces is fixed here and equal to 12 ML. The other
parameters are total length =40 a0and mass ratio mr=2.
which tend to promote phonon scattering. Even if the global
height his large, in terms of a triangular-shaped interface, the
effective area for the phonons is not the true surface area, but
rather the projected area, because his not the only relevant
roughness parameter and the interaction between incident
phonons and small length-scale asperities tend to diminish the
effective transmission area. On the other hand, regular-shaped
patterns do not display such a distribution of length scales,
and interfacial heat transport becomes controlled by the true
surface area: as soon as the majority of phonon modes has a
wavelength greater than the single length hcharacterizing the
interfacial morphology, one enters into a “large roughness”
regime, where the energy transport becomes governed by the
true surface area and the conductance is increased as compared
with the planar case. The interfacial conductance is found to
be the highest for the wavy interface because it has the greatest
surface area.
V. CONCLUSION
In summary, we have concentrated on the role of the
interfacial roughness on energy transmission between solid
dielectrics. Thanks to the versatility of the molecular dynamics
simulations technique, we have conceived a model of rough
surfaces and probe their ability to conduct heat. The scenario
emerging from the simulations is the following: when the
roughness introduced is small, most of the phonons see
the interface as a planar one and the effective surface area
contributing to the transmitted heat flux is the projected area,
not the true one. In this regime, one does not expect a Kapitza
conductance much different from the planar interface. On
the other hand, when the roughness becomes large enough,
typically 20 monolayers in our case, most of the phonons
propagating towards the interface are incoherently scattered
and the effective surface area becomes the true surface area.
This latter case may differ significantly from the projected
one, and this is the reason why the boundary conductance of
rough interfaces may be greatly enhanced, as compared to
planar interfaces. This has been demonstrated in this work
with the example of triangular-shaped interfaces displaying
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steep slopes: provided the lateral dimensions characterizing
the interfacial roughness are large enough, the increase of the
conductance may be threefold. On the other hand, we have
probed the conductance of randomly rough interfaces and
shown that they display, in general, conductances comparable
or smaller than atomic planar interfaces. This difference of
behavior is explained by the distribution of length scales
displayed by the randomly rough surfaces, in comparison with
our model patterned surfaces.
The roughness analyzed in this paper was large compared
to the lattice constants. The case of atomic roughness has
been more widely addressed in the literature, and wave-packet
simulations [36] give a clear picture of the effect of small
atomic roughness on phonon transmission: long wavelength
phonons see the interface as ideal and do not contribute to
the change of the thermal boundary conductance. On the
other hand, short wavelength phonons strongly interact with
the small-scale roughness, and the corresponding change in
phonon transmission is found to depend on the structure of the
interface: for regular-shaped patterned interfaces, constructive
wave interference leads to enhanced transmission, thereby
increasing the boundary conductance [37]. Random atomic
roughness promotes incoherent phonon scattering, reducing
the thermal conductance. These observations are consistent
with MD results concerning the cross-plane conductivity of
superlattices with rough interfaces: for regularly patterned
interfaces, the cross-plane conductivity is slightly greater
than ideal superlattices [20] and the boundary conductance is
enhanced [38]. When the roughness is random, the cross-plane
conductivity shows a small decrease as compared with planar
interfaces [39,40]. The small amplitude of the reduction is
related to the small proportion of energy carriers affected by the
atomic roughness. Small reductions have been also reported
for Si/Ge superlattices with one layer of interfacial mixing
in the incoherent regime of transport [41]. If it is reasonable
to rationalize such variations in terms of an atomic interfacial
roughness, it is less clear for superlattices having a thicker
mixed layer. Large enhancements have been observed in this
latter case using MD [25]. Further work is clearly needed to
understand if part of these enhancements is explained by the
large-scale interfacial roughness [37].
Most of the results reported here concern regular-shaped
patterned interfaces. MD results seem to conclude that these
patterned interfaces are good candidates to enhance the
intrinsic boundary conductance between two semiconductors.
An enhancement by a factor of three has been reported for
triangular-shaped interfaces; see Fig. 8. On the other hand,
randomly rough surfaces should be considered if one prefers
to reduce the Kapitza conductance between two solids [10].
In particular, in the context of superlattices, randomly rough
interfaces should be designed if one aims to tailor materials
with the lowest cross-plane thermal conductivity.
We have also introduced a simple model to rationalize the
variations of the thermal boundary conductance as a function of
the interfacial height of our model of rough interfaces. Further
analytical work is clearly desired to understand the interplay
between the interface morphology and energy interfacial
transport. This will enable the definition of new directions
for the design of interfaces with optimized energy transport
properties and with a relative low cost.
ACKNOWLEDGMENTS
Simulations have been run at the “Pole Scientifique de
Mod´
elisation Num´
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