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Fresnel transmission coefficients for thermal phonons at solid
interfaces
Chengyun Hua
a
,
1
Xiangwen Chen
a
,
1
Navaneetha
K. Ravichandran,
1
and Austin J. Minnich
1,
1
Division of Engineering and Applied Science
California Institute of Technology, Pasadena, California 91125,USA
(Dated: September 28, 2015)
Abstract
Interfaces play an essential role in phonon-mediated heat conduction in solids, impacting appli-
cations ranging from thermoelectric waste heat recovery to heat dissipation in electronics. From
a microscopic perspective, interfacial phonon transport is described by transmission and reflection
coefficients, analogous to the well-known Fresnel coefficients for light. However, these coefficients
have never been directly measured, and thermal transport processes at interfaces remain poorly
understood despite considerable effort. Here, we report the first measurements of the Fresnel trans-
mission coefficients for thermal phonons at a metal-semiconductor interface using ab-initio phonon
transport modeling and a thermal characterization technique, time-domain thermoreflectance. Our
measurements show that interfaces act as thermal phonon filters that transmit primarily low fre-
quency phonons, leading to these phonons being the dominant energy carriers across the interface
despite the larger density of states of high frequency phonons. Our work realizes the long-standing
goal of directly measuring thermal phonon transmission coefficients and demonstrates a general
route to study microscopic processes governing interfacial heat conduction.
a
These authors equally contributed to this work.
1
arXiv:1509.07806v1 [cond-mat.mes-hall] 25 Sep 2015
Transport across interfaces in heterogeneous media is a fundamental physical process
that forms the basis for numerous widely used technologies. For example, the reflection
and transmission of light at interfaces, as described by the Fresnel equations, enables wave-
guiding with fiber-optics and anti-reflection coatings, among many other functionalities.
Interfaces also play an essential role in phonon-mediated heat conduction in solids.
1
Material
discontinuities lead to thermal phonon reflections that are manifested on a macroscopic scale
as a thermal boundary resistance (TBR), also called Kapitza resistance,
R
k
, that relates the
temperature drop at the interface to the heat flux flowing across it. TBR exists at the
interface between any dissimilar materials due to differences in phonon states on each side
of the interface.
2
Typical interfaces often possess defects or roughness which can lead to
additional phonon reflections and hence higher TBR.
TBR plays an increasingly important role in devices, particularly as device sizes decrease
below the intrinsic mean free paths (MFPs) of thermal phonons.
1
At sufficiently small length
scales, TBR can dominate the total thermal resistance. For instance, the effective thermal
conductivity of a superlattice can be orders of magnitude smaller than that of the constituent
materials due to high TBR.
3–6
This physical effect has been used to realize thermoelectrics
with high efficiency
7,8
and dense solids with exceptionally low thermal conductivity.
9
On the
other hand, TBR can lead to significant thermal management problems
10–12
in applications
such as LEDs
13,14
and high power electronics.
12,15
Thus both scientifically and for applications, a fundamental understanding of thermal
transport across solid interfaces is essential. In principle, Fresnel transmission coefficients
can also be used to provide a microscopic description of thermal phonon transport at in-
terfaces owing to the similarities between photons and phonons. However, despite decades
of work, the microscopic perspective of heat transport across interfaces remains poorly de-
veloped compared to that available for photons. Today, interfaces are most often studied
using macroscopic measurements of TBR or thermal conductivity. For example, numerous
works have studied interfacial thermal transport by observing the temperature dependence
of the thermal conductivity
16
or interface conductance,
G
= 1
/R
k
17–21
or by correlating
changes in bonding strength and interface conductance.
22,23
However, these experimental
approaches provide limited information about the transmission coefficients because the ob-
servable quantities are averaged over all phonons and thus obscure the microscopic processes
of interest.
2
As a result, considerable uncertainty exists as to the values of the phonon transmission
coefficients at solid interfaces. Commonly used analytical models for the transmission co-
efficients include the gray model, in which the transmission coefficients are a constant for
all phonon modes,
24–26
the acoustic mismatch model
27,28
and the diffuse mismatch model
(DMM),
29–31
yielding incompatible predictions for the transmission coefficients. Further,
their predictions are in poor agreement with experiment, and none are able to account
variations in the atomic structure of actual interfaces. Atomistic methods such as molec-
ular dynamics
32–40
and atomistic Green’s functions
41–45
have been extensively applied to
obtain the transmission coefficients, predicting the phonon transmission coefficients to de-
crease with increasing phonon frequency. However, no direct experimental verifications of
these predictions have been reported. Prior experimental observations have reached con-
flicting conclusions, with one reporting that the transmission coefficients follow the trend of
atomistic calculations
16
while another reaching the opposite conclusion that high frequency
phonons are the dominant heat carriers across the interface.
46
Therefore, despite consid-
erable experimental and theoretical study, the phonon transmission coefficients at actual
interfaces remain unclear.
Here, we report the first measurements of the Fresnel transmission coefficients for thermal
phonons at a solid interface. Our approach is based on applying our recent advances in ab-
initio phonon transport modeling based on the phonon Boltzmann transport equation (BTE)
to interpret measurements from the time-domain thermoreflectance (TDTR), a widely-used
optical experiment in the thermal sciences. Our measurements reveal that low frequency
phonons are nearly completely transmitted across solid boundaries while high frequency
phonons are nearly completely reflected, leading to an interfacial heat flux distribution
that is dominated by low to mid frequency phonons. Further, our approach demonstrates a
general route to directly experimentally study the microscopic transport processes governing
interfacial heat conduction.
TDTR WITH AB-INITIO PHONON TRANSPORT MODELING
Our measurement is based on the TDTR experiment, an optical pump-probe technique
that is used to characterize thermal properties on micron length scales. In this experiment,
a femtosecond pulsed laser beam is split into a pump and a probe beam. The pump pulse
3
train is modulated at a frequency from 1 to 15 MHz to enable lock-in detection, and is then
used to impulsively heat a metal film coated on the sample. The transient temperature
decay at the surface is detected as a change in optical reflectance by the probe beam.
47
Typically, this temperature decay curve is fitted to a standard multilayer heat diffusion
model based on Fourier’s law with the substrate thermal conductivity and metal-substrate
interface conductance as fitting parameters.
48,49
This method is now a standard thermal
metrology technique.
Recently, considerable interest has focused on quasiballistic heat conduction in TDTR
in which a lack of phonon scattering in the substrate leads to the breakdown of Fourier’s
law. Many experimental reports have demonstrated clear evidence of quasiballistic heat
transport in different material systems in the form of thermal properties that appear to
deviate from their bulk values.
50–55
In this work, we interpret these effects as fundamentally
originating from the non-equilibrium phonon distribution emerging from the interface. As
illustrated in Fig. 1(a), when MFPs are much shorter than the characteristic length scale
of the thermal gradient, information about the phonon distribution at the interface is lost
due to scattering. On the other hand, if some phonons have sufficiently long MFPs, the
non-equilibrium distribution penetrates into the substrate and affects the resulting heat
conduction, thereby providing direct information about the spectral phonon distribution at
the interface.
While the required data is straightforward to obtain, the key to extracting the trans-
mission coefficients is to rigorously interpret the data with the ab-inito phonon transport
modeling based on the BTE. It is this step that has long impeded efforts to study interfaces
due to the extreme cost of solving the BTE for the TDTR experiment. A number of sim-
plified models
55–61
have been proposed to explain these observations. However, all of these
models make various approximations that limit the information that can be extracted from
the measurements.
In this work, we overcome this challenge using two recent advances we reported for
rigorously solving the spectral BTE under the relaxation time approximation (RTA), with
no additional simplifications, that yield a factor of 10
4
speedup compared to existing methods
and allows the first ab-initio phonon transport modeling of TDTR free of artificial parameters
or simplifications of the phonon dispersion. First, we have obtained an analytical solution of
the spectral BTE in a semi-infinite substrate subject to an arbitrary heating profile.
62
Second,
4
we have employed a series expansion method to efficiently solve the spectral, one-dimensional
(1D) BTE in a finite layer, suitable for the transducer film.
63
In this work, these two solutions
are combined using a spectral interface condition
57
that expresses the conservation of heat
flux at each phonon frequency. Following the conclusions of prior computational works
17,64
that the energy transmission at the interfaces considered here is elastic, we enforce that
phonons maintain their frequencies as they transmit through the interface; direct electron-
phonon coupling and inelastic scattering is neglected (see Supplementary Information).
The details of the calculation are in the Supplementary Information. The only inputs to
our calculation are the phonon dispersions and lifetimes, calculated using density functional
theory (DFT) with no adjustable parameters by Jes ́us Carrete and Natalio Mingo, and the
only unknown parameter is the spectral transmission coefficients across the interface. We
adjust the spectral transmission coefficients to obtain the best fit of the simulated surface
temperature decay to the experimental measurement. The reflection coefficients are specified
by energy conservation once the transmission coefficients are known.
57
MEASUREMENTS OF PHONON TRANSMISSION COEFFICIENTS
We demonstrate our transmission coefficient measurements by performing a TDTR mea-
surement of an Al film on Si substrate with the native oxide removed by Hydrofluoric acid
prior to Al deposition, yielding a clean interface. The TEM image in Fig. 1(b) shows the
interface thickness is less than 0.5 nm. The amplitude and phase of a typical signal from the
lock-in amplifier are given in Figs. 1(c) & (d). To begin, we follow the typical procedure of
fitting data with the standard heat diffusion model to extract thermal conductivity
k
and
interface conductance
G
.
65,66
We obtain
G
280 MW/m
2
-K and
k
140 W/m-K, which
is in good agreement with prior works and literature values for the thermal conductivity of
Si.
46,52
Due to the good agreement, this measurement is typically taken as evidence that phonon
MFPs in silicon are small compared to the thermal length
α
f
0
1
μ
m, where
α
is the
thermal diffusivity of silicon and
f
0
is the modulation frequency. However, a number of
ab-initio calculations, by different groups and with different computational packages, clearly
show that a substantial amount of heat is carried by phonons with MFPs exceeding 1
μ
m.
67,68
This prediction has recently been experimentally confirmed by Cuffe
et al
using
5
FIG. 1.
Measurements and simulations of TDTR experiments on Al/Si with a clean
interface.
(a) Schematic of the principle underlying the measurement of transmission coefficients.
If the characteristic length scale of the thermal transport is much longer than the phonon MFPs,
information about the interfacial distribution is lost due to strong scattering. If some MFPs are
comparable to the thermal length scale, the non-equilibrium distribution at the interface propa-
gates into the substrate where it can be detected. (b) TEM image showing the clean interface
of a Al/Si sample with the native oxide removed. The interface thickness is less than 0
.
5 nm.
Experimental TDTR data (symbols) on this sample at
T
= 300 K for a modulation frequency
f
= 10
.
2 MHz along with the (c) amplitude and (d) phase fit to the data from the BTE simula-
tions (dashed lines), demonstrating excellent agreement between simulation and experiment. (e)
Apparent thermal conductivity and (f) apparent interface conductance of the experiments (sym-
bols) and BTE simulations (lines) versus modulation frequency. These quantities are extracted by
fitting the data and simulations to a thermal diffusion model. The pairs of solid lines denoted BTE
bounds correspond to the uncertainty in the measured transmission coefficients plotted in Fig. 2.
6
thermal measurements on variable thickness silicon membranes.
69
This fact implies that
quasiballistic transport should be readily observable in a typical TDTR experiment on Si,
despite the seemingly correct thermal properties measured.
This apparent contradiction can be resolved by considering the spectral profile of the
transmission coefficients. Our BTE calculations reveal that the measurements from TDTR
on this sample strongly depend on the spectral profile of the transmitted phonon spectrum,
a dependence that does not occur in the heat diffusion regime. This dependence allows the
transmission coefficients to be directly obtained from the TDTR data by adjusting them to
fit the surface temperature data from the lock-in amplifier in a procedure exactly analogous
to that used to measure bulk thermal properties.
We performed the fitting of transmission coefficients by adjusting them until the simulated
surface temperature curve and experimental data from the lock-in amplifier matched each
other. This comparison for all samples can be found in Supplementary Information. To
more compactly report the data, we further process the BTE results by fitting the simulated
surface temperature decay curves to the same heat diffusion model used in the experiments to
extract the apparent thermal conductivity and interface conductance at different modulation
frequencies. If these two parameters at each modulation frequency match, then the fitting
curves will also match, enabling a compact comparison of the simulation and experimental
data sets. However, we emphasize that the use of the heat diffusion model is for ease of
comparison only and was not used in the transmission coefficient measurement.
An example of the measurement process for the data shown in Figs. 1(e) & (f) is given
in Fig. 2. Prior works
57,70
used a constant transmission coefficient profile that explained the
apparent interface conductance. However, we find that this profile predicts a modulation-
frequency dependent apparent thermal conductivity that becomes as low as 100 W/m-K,
in strong disagreement with experiment. Similarly, other commonly used models such as
the DMM predict the wrong trend of thermal conductivity and interface conductance as a
function of modulation frequency. (See Supplementary Information)
The only transmission coefficient profile that is able to simultaneously match the appar-
ent thermal conductivity and interface conductance in Figs. 1(d) & (e) from the experiment
is shown in Fig. 2(a). The figure shows that the transmission coefficient from Si to Al for
longitudinal phonons, T
Si
Al
(
ω
), starts at unity, its maximum possible value and decreases
steadily to near zero for high phonon frequencies. The transmission coefficient profiles for
7
Phonon frequency (THz)
2
4
6
8
10
Transmission coefficient
0.0
0.2
0.4
0.6
0.8
1.0
Constant T
Si->Al
Measured T
Si->Al
Transmission coefficient
Constant T
Si->Al
Measured T
Si->Al
a)
Phonon wavelength (nm)
0.5
1.0
1.5
2.0 2.5 3.0
0.0
0.2
0.4
0.6
0.8
1.0
b)
Phonon frequency (THz)
2
4
6
8
10
Interface heat flux (MW/m
2
)
0.0
0.1
0.2
0.3
0.4
Measured T
Si->Al
Constant T
Si->Al
0.5
1.0
1.5
2.0 2.5 3.0
Accumulation (normalized)
0.0
0.2
0.4
0.6
0.8
1.0
Constant T
Si->Al
Measured T
Si->Al
Al/Si with clean interface at 300 K
c)
d)
FIG. 2.
Transmission coefficients and spectral heat flux at the interface.
Transmission
coefficients of longitudinal phonons T
Si
Al
(
ω
) (blue shaded region) versus (a) phonon frequency
and (b) phonon wavelength, along with the constant transmission coefficient profile (green dashed
line) that gives the same interface conductance as the measured T
Si
Al
(
ω
). The boundaries of the
shaded region correspond to the BTE bounds in Figs. 1(e) and (f). Our measurement shows that
low frequency phonons have a much higher transmission probability across the interface than high
frequency phonons. (c) Spectral heat flux with the measured (blue shaded region) and constant
(green dashed line) transmission coefficient profiles across the interface versus phonon frequency.
(d) Normalized accumulative interface conductance with the measured (blue shaded region) and
constant (green dashed line) transmission coefficient profile versus phonon wavelength. Contrary
to the prediction of the constant transmission coefficient profile, low-frequency/long-wavelength
phonons carry a significant amount of heat across the interface.
the other polarizations have similar shapes, and so throughout the paper we plot only the
longitudinal transmission coefficients for simplicity. The transmission coefficients from Al to
8
Si, T
Al
Si
(
ω
) are calculated by satisfying the principle of detailed balance; the relationship
between T
Si
Al
(
ω
) and T
Al
Si
(
ω
) reflect the differences in density of states and group ve-
locity between the two materials. The transmission coefficients for each side of the interface
and for the other polarizations are given in the Supplementary Information.
Our measured transmission coefficient profile thus indicates that low frequency, long
wavelength phonons are transmitted to the maximum extent allowed by the principle of
detailed balance, while high frequency, short wavelength phonons are nearly completely re-
flected at the interface. This result is consistent with many prior works in the literature.
First, our result qualitatively reproduces the physically expected behavior that transmission
coefficient increases with decreasing phonon frequency because phonons with sufficiently
long wavelength do not perceive the atomistic disorder at the interface. The absence of
acoustic reflections in our TDTR signal confirm the good acoustic match between the two
materials and support our finding that long-wavelength phonons are mostly transmitted at
the interface. Second, the measured transmission coefficient profile agrees with the experi-
mental studies of polycrystalline silicon by Wang
et al
,
16
which suggested that transmission
coefficient should decrease with increasing frequency. Finally, the measurement also agrees
with numerous molecular dynamics and atomic Green’s function calculations, essentially all
of which predict a decreasing transmission coefficient with increasing phonon frequency.
42–45
Our work is thus able to provide direct experimental confirmation of these predictions for the
first time while eliminating other possibilities for the transmission coefficients that appear
in the literature.
Using this transmission coefficient profile, we plot the spectral interfacial heat flux versus
phonon frequency and accumulative heat flux versus phonon wavelength in Figs. 2(c) & (d),
respectively. In contrast to the prediction by the constant transmission coefficient profile,
our results show that most of interfacial heat flux is carried by low to mid frequency phonons,
with the contribution from high frequencies strongly reduced due to their small transmission
coefficients. This observation highlights the importance of our ab-initio phonon transport
modeling approach, as prior works neglected the contribution of these low-frequency phonons
to interfacial heat flux.
58
In fact, we are unable to explain the magnitude of the observed
interface conductance without the contribution of phonons of frequency less than 3 THz.
Similarly, we find that we can only explain the measurements using the exact phonon dis-
persion for Al computed from DFT; simple dispersion relations such as Debye model cannot
9
Modulation freq. (MHz)
0
2
4
6
8
10
Thermal conductivity (W/m-K)
80
100
120
140
160
BTE bounds
300 K
BTE bounds
350 K
BTE bounds
400 K
Modulation freq. (MHz)
0
2
4
6
8
10
Interface conductance (MW/m
2
-K)
240
260
280
300
320
300 K
350 K
400 K
BTE bounds
Al/Si with clean interface
a)
b)
Al/SiGe with clean interface at 300 K
Ref. 50
FIG. 3.
TDTR measurements on Al/Si at different temperatures and Al/SiGe.
(a) Ap-
parent thermal conductivity and (b) apparent interface conductance versus modulation frequency
from experiments (symbols) and simulations (shaded regions) for Al on Si with a clean interface
at 300 K, 350 K and 400 K. (c) Apparent thermal conductivity and (d) apparent interface con-
ductance versus modulation frequency from experiments (circles: this work; triangle: Ref. 50) and
simulations (shaded regions) for Al on SiGe with a clean interface. The magnitude and trend of
the experimental data are reproduced using the same transmission coefficient profile as in Fig. 2.
explain the data because they underestimate the contribution of low frequency phonons to
thermal transport.
Since the energy transmission at the interfaces is considered elastic, the transmission
coefficient profile in theory should be independent of temperature. To confirm the robustness
10
of the measured transmission coefficients, we conducted experiments at 350 K and 400 K
and compared the experimental results with the calculations using the same transmission
coefficient profile measured at 300 K. As shown in Figs. 3 (a) & (b), the transmission
coefficient profile measured at 300 K yields excellent agreement between simulations and
experiments at higher temperatures.
To further support our measurements, we additionally measure the transmission coeffi-
cients for Al on SiGe. While this material has an additional point defect scattering mecha-
nism compared to pure Si, we expect the transmission coefficients to be nearly the same given
that the host lattice is unchanged. Figs. 3(c) & (d) plot the apparent thermal conductivity
and interface conductance, demonstrating that our BTE modeling with the exact transmis-
sion coefficient profile shown in Fig. 2(a) agrees well with the measured apparent thermal
conductivity and interface conductance for Al/SiGe. This result confirms that the measured
transmission coefficients for Si and SiGe substrates are indeed the same. Importantly, the
apparent thermal conductivity of SiGe differs from its actual thermal conductivity, mea-
sured by Thermtest as 50
.
7
±
0
.
5 W/m-K using the transient plane source method on a
bulk sample. This discrepancy highlights the strong influence of the transmission coefficient
profile on the measurements from TDTR.
So far, the measurements indicate that low frequency phonons are nearly completely
transmitted through a clean interface. We next seek to determine what types of interfacial
disorder can reflect these modes. We conducted additional measurements for Al on Si with
a native oxide layer (thickness
1 nm as shown in a TEM image in Fig. 4(a)) and Si with
thermally grown oxide layer (thickness
3.5 nm as shown in a TEM image in Fig. 4(b)).
Since the oxide layers are sufficiently thin to neglect their thermal capacitance, we can treat
them as part of the interface that modifies the transmission coefficient profile in our current
BTE model.
By again fitting the BTE results to the measurement as in Figs. 4(c) & (d), we are
able to find the transmission coefficient profiles for these two cases as shown in Figs. 4(e)
& (f). Compared to a clean interface, the transmission coefficients for Al on Si with a
native oxide are reduced for all phonon modes. However, low-frequency phonons experience
a larger reduction in transmission coefficient than do high-frequency phonons, which have
transmission coefficients close to zero even at a clean interface. When the roughness of the
interface increases with a thicker oxide layer, the transmission coefficient keeps decreasing
11