PHYSICAL REVIEW APPLIED
12,
024051 (2019)
Crossover between Electron-Phonon and Boundary-Resistance Limits to Thermal
Relaxation in Copper Films
L. B. Wang
,
1,
*
O.-P. Saira,
2,3
D. S. Golubev,
1
and J. P. Pekola
1
1
QTF Centre of Excellence, Department of Applied Physics, Aalto University, FI-00076 Aalto, Finland
2
Department of Physics and Kavli Nanoscience Institute, California Institute of Technology, Pasadena,
California, USA
3
Computational Science Initiative, Brookhaven National Laboratory, Upton, New York 11973, USA
(Received 26 March 2019; revised manuscript received 29 July 2019; published 23 August 2019)
We observe a crossover from electron-phonon (
e
-ph) coupling limited energy relaxation to that governed
by thermal boundary resistance (phonon-phonon coupling, ph-ph) in copper films at subkelvin tempera-
tures. Our measurement yields a quantitative picture of heat currents, in terms of temperature dependences
and magnitudes, in both
e
-ph and pp limited regimes, respectively. We show that by adding a third layer in
between the copper film and the substrate, the thermal boundary resistance is increased fourfold, consistent
with an assumed series connection of thermal resistances.
DOI:
10.1103/PhysRevApplied.12.024051
I. INTRODUCTION
Investigation of energy relaxation of electrons in normal
metal films is important for understanding the underlying
physics as well as for applications [
1
–
4
]. Especially for
mesoscopic devices at low temperature, where the dom-
inant thermal wavelength
λ
is comparable to the device
dimension, phonons in the films could be two dimen-
sional (2D), and it has been shown both experimentally
and theoretically that the reduced phonon dimension does
affect the energy relaxation of electrons in thin films [
5
–
8
].
Heat transport by phonons, electrons, and photons has
been studied experimentally in mesoscopic devices [
9
–
14
].
From the application point of view, a good understand-
ing of energy relaxation in metal films is important, e.g.,
for calorimetry and bolometry [
15
]. Decreasing the heat
conductance from the metal film absorber to the envi-
ronment enhances the energy resolution, but on the other
hand, it makes the device slower. For a transition-edge
sensor, unaccounted-for thermal boundaries can affect the
noise and energy resolution [
16
]. Finally, for a normal-
metal–insulator–superconductor (
N
-
I
-
S
) junction cooler,
quick thermalization of the secondary electrode is favor-
able in order to increase the cooling efficiency [
17
].
In a heated normal metal film on a dielectric substrate,
electrons within the film relax by electron-electron (
ee
)
interactions, and the energy is dissipated to the environ-
ment mainly by electron-phonon (
e
-ph) coupling to the
film phonons, which is characterized by
e
-ph thermal cou-
pling resistance
R
e
-
ph
. Film phonons are coupled to the
*
libinwang5555@gmail.com
substrate phonons, which are usually considered to consti-
tute the heat bath for the device, by phononic coupling. The
corresponding thermal resistance between phonons in the
film and the substrate is the thermal boundary resistance
R
ph-ph
. If the
ee
interactions are assumed to be much faster
than other processes, the energy relaxation of the elec-
trons in the film is determined by
R
e
-
ph
and
R
ph-ph
, with the
weaker of the two governing the energy relaxation process.
For thin films at low temperatures,
e
-ph coupling strength
is weak and it becomes the bottleneck of the energy relax-
ation. With increasing temperature or film thickness, the
e
-ph coupling gets relatively stronger and the heat trans-
port across the boundary between the film and the substrate
becomes the limitation for the energy relaxation.
Electron-phonon coupling in metal films at low temper-
atures has been actively studied during the last decades. In
particular, the effect of disorder and phonon dimensionality
on the
e
-ph coupling strength have been intensively dis-
cussed [
7
,
18
–
20
]. Thermal boundary resistance between
metals and dielectric substrates has also been well inves-
tigated. Experimental observations can be explained with
either the acoustic mismatch model (AMM) or diffuse mis-
match model (DMM) [
21
]. AMM describes phonon heat
transfer through a flat interface between perfect crystals.
In analogy to Snell’s law for the electromagnetic waves,
only the phonons with the incident angles below the critical
one are transmitted through the interface. The critical angle
is determined by the acoustic properties of the materials
on both sides of the boundary. DMM assumes diffusive
phonon scattering at the interface and, hence, the phonon
transmission probability depends only on the phonon den-
sities of the states and sound velocities on both sides. For
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024051-1
© 2019 American Physical Society
WANG, SAIRA, GOLUBEV, and PEKOLA
PHYS. REV. APPLIED
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024051 (2019)
identical acoustic properties of the two sides, thermal resis-
tance predicted by DMM is about two times larger than
that by AMM, as according to AMM, transmission proba-
bility is unity, and for DMM transmission probability, it is
50%. For a smaller acoustic difference of the two materi-
als, the resistance differs by less than 30% [
21
]. In the case
of solid-solid boundaries, the mismatch in sound veloci-
ties and phonon mode densities is usually small, and the
two models give similar predictions.
Here, we present the experimental results showing the
crossover between
e
-ph and boundary-resistance limited
thermal relaxation in Cu films at subkelvin temperatures.
For Cu film 50 nm in thickness, we find the energy relax-
ation to be limited by
e
-ph coupling in the full temperature
range explored. By increasing the film thickness to 300 nm,
the thermal boundary resistance limits the energy relax-
ation and we are able to quantify the heat transport between
the metal-substrate interface directly from the experiments.
By adding a third thin layer of film between the Cu
film and the substrate, the thermal boundary resistance is
increased fourfold, consistent with the assumption of a
series connection of the thermal boundary resistances.
For a heated metal film on a substrate, the energy flow
is shown in the thermal model in Fig.
1(a)
. Within the
film, the energy flow rate from electrons to phonons is
described by
P
e
-
ph
=
V
(
T
n
e
−
T
n
p
)
.(1)
Here,
T
e
and
T
ph
are the electron and phonon temper-
atures in the film,
V
is the metal volume,
n
=
5for
three-dimensional (3D) clean normal metals, and
is the
material-specific
e
-ph coupling constant [
22
]. The cou-
pling between film phonons and the substrate phonons for
a 3D system is characterized by
P
ph-ph
=
kA
(
T
4
p
−
T
4
s
)
,(2)
where
T
s
is the substrate phonon temperature,
A
is the
contact area, and
k
is the interface-material-dependent
constant, which can be calculated with DMM as
k
=
π
2
120
k
4
B
3
(
1
c
2
1
L
+
2
c
2
1
T
)(
1
c
2
2
L
+
2
c
2
2
T
)
1
c
2
1
L
+
2
c
2
1
T
+
1
c
2
2
L
+
2
c
2
2
T
.(3)
Here,
c
xL
and
c
xT
are the speed of longitudinal and trans-
verse sound on the side
x
of the interface. For small
temperature differences, the
e
-ph thermal coupling resis-
tance is expressed as
R
e
-
ph
=
1
/
5
VT
4
and the thermal
boundary resistance as
R
ph-ph
=
1
/
4
kAT
3
.
T
s
equals the
bath temperature of the refrigerator
T
0
due to the large
substrate-bath contact area.
(a)(b)
(c)
(e)(f)
(d)
FIG. 1.
(a) The thermal model for the energy flow for a heated
metal film on a substrate. (b) False-color SEM image of a sample
together with the measurement setup. (c) Enlargement of the rect-
angular area within the white dashed line in (b) showing the JJ
thermometer connected to the heater and to the large Cu pad (Al,
blue; Cu, brown). (d) IV curve of the JJ thermometer. (e)
I
sw
and
I
r
as a function of the bath temperature without heating applied
to the Cu film. This equilibrium temperature dependence of
I
sw
is used as the calibration for the JJ thermometer. (f)
I
sw
as a func-
tion of
I
H
for bath temperatures from 60 mK (blue) to 340 mK
(red) with 20-mK intervals. The decrease of
I
sw
when passing
I
H
through the Cu film indicates heating of the electrons.
II. JOSEPHSON JUNCTION THERMOMETER
AND MEASUREMENT SETUP
One of the devices used in the experiments is shown in
Figs.
1(b)
and
1(c)
together with the measurement setup.
Cu film (brown) is evaporated on the silicon substrate, with
300 nm silicon oxide on top, by electron beam evapora-
tion. The chamber pressure is kept below 5
×
10
−
7
mbar
during the deposition. Before contacting the Cu film with
superconducting Al (blue), Ar plasma milling is used to
clean the Cu film surface in order to achieve good metal-to-
metal contacts between copper and aluminum. The hybrid
structures with short channel length behave as a prox-
imity Josephson junction (JJ). Switching current
I
sw
is
defined as the bias current when the junction switches
from the superconducting state to the resistive state, shown
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in the IV curve in Fig.
1(d)
. The JJ switches back to
the superconducting state at a biasing current well below
I
sw
, defined as retrapping current
I
r
. The hysteresis of the
IV curve originates from the overheating of the electrons
after switching to the resistive state. The bath tempera-
ture dependence of
I
sw
at zero heating, i.e., in equilibrium,
shown in Fig.
1(e)
, is used as the temperature calibra-
tion for the JJ thermometer [
23
]. The long horizontal Cu
wire between the large Cu pad and JJ thermometer is
used as the heater to elevate electron temperature in the
Cu film. A large pad on top enables us to determine the
size of the Cu film with good relative accuracy. Previ-
ous work has shown that for our devices with dimensions
much smaller than the
e
-ph relaxation length, the whole
film has uniform electron temperature with the power lev-
els applied in our experiments [
24
]. We verify this by
measuring electron temperature with another JJ thermome-
ter located at another end of the Cu pad, shown later in
Fig.
3(b)
. We current-bias the two heater contacts with
opposite polarities. Figure
1(f)
is the measured
I
sw
as a
function of
I
H
for various bath temperatures from 60 to
340 mK in 20-mK steps from top to bottom. Decrease of
I
sw
while increasing
|
I
H
|
indicates heating of the Cu film.
The symmetry of the dependence around zero heating sug-
gests no heating current flows to the thermometer in this
configuration.
III. RESULTS AND DISCUSSION
For electrons in the copper film, superconducting Al
acts as a thermal insulator at sufficiently low temperatures
below its critical temperature
T
c
∼
1 K; the joule power
applied
P
to the film dissipates mainly by
e
-ph coupling.
The ratio of the two series thermal resistances is
γ
=
R
ph-ph
R
e
-
ph
=
5
tT
4
k
.(4)
Here,
t
is the thickness of the Cu film. For a thin film at
sufficiently low temperatures, we expect
R
e
-
ph
to domi-
nate over
R
ph-ph
, so we have the standard situation usually
assumed for thin films, i.e.,
P
=
P
e
-
ph
=
V
(
T
5
e
−
T
5
0
)
.In
Fig.
2
, we plot the experimental results of a sample with
50-nm thin Cu film; a linear dependence versus
T
5
e
−
T
5
0
is clearly seen as expected. From the slope, we obtain
the
e
-ph coupling constant
=
2.1
±
0.1 nWK
−
5
μ
m
−
3
with no temperature dependence within the measurement
interval from 60 to 250 mK, as shown in the inset of
Fig.
2
. The measured value of
is consistent with previ-
ous experiments on Cu films [
25
,
26
]. Thus, the experiment
demonstrates that for the 50-nm Cu film at low tempera-
ture, the energy relaxation of electrons is dominated by the
e
-ph coupling, and the exponent
n
=
5 is consistent with
the theory based on the 3D free electron model [
22
].
Equation
(4)
suggests that if one changes the film thick-
ness or temperature to the point where
R
ph-ph
becomes
FIG. 2. Measured
T
5
e
−
T
5
0
plotted as a function of heating
power
P
for a sample with 50-nm Cu film. The observed linear
dependence is consistent with the prediction of Eq.
(1)
, suggest-
ing that the weak
e
-ph coupling limits the energy relaxation of
the electrons in the Cu film. Inset: The measured
e
-ph coupling
constant
as a function of temperature.
equal to
R
e
-
ph
, a crossover from one energy relaxation
mechanism to another should take place. The crossover
temperature
T
cr
depends on the constants
and
k
as
T
cr
=
4
k
/
5
t
. For perfect contacts between Cu and the sil-
icon substrate, one finds
k
≈
310 WK
−
4
m
−
2
, calculated
with Eq.
(3)
. Hence, the crossover temperature of 0.1 K is
expected in films with the thickness of
t
≈
1.2
μ
m. Recent
experiments have suggested that for films evaporated on a
silicon substrate,
k
is smaller than that predicted for per-
fect contacts [
27
,
28
], which makes it possible to observe
T
cr
≈
0.1 K in somewhat thinner films.
In Fig.
3(a)
, we show the SEM image of a sample with
t
=
300 nm Cu film. Firstly, we deposit 50-nm Cu film
(brown) used in the JJ thermometers, heater, and con-
tact pads. Then, we deposit the 300-nm Cu film (purple).
Before contacting the two copper films, Ar plasma milling
is used to clean the surface of the thin one. The inset
of Fig.
3(b)
shows the thick film covering the thin film.
Electron temperature is measured with two JJ thermome-
ters located at the two ends of the thick Cu film (local,
remote) with a distance of 40
μ
m to check the uniformity
of electron temperature in the thick Cu film while heat-
ing. Figure
3(b)
shows that the two thermometers show
identical temperatures except at the largest applied powers.
The small difference at high
P
originates most likely from
the electron diffusion along with the thick Cu film and is
negligible for the analysis. The data also suggest that the
thermal boundary resistance between the two Cu films is
negligible.
We plot
T
n
e
−
T
n
0
as a function of
P
in Fig.
3(c)
.In
contrast to what was seen in Fig.
2
, a linear dependence
is observed when setting
n
=
4 in the full temperature
range and three different bath temperatures explored. For
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(a)
(b)(c)
FIG. 3. (a) False-color SEM image of a sample with large Cu
film (purple) with dimensions 10
μ
m
×
40
μ
m
×
300 nm. Two
JJ thermometers (local, remote) are located at the two ends of
the Cu film to check the uniformity of the electron temperature
while heating. The inset in (b) is the enlargement of the local
JJ thermometer showing the large Cu film covering the thin Cu
film (brown). (b) Measured electron temperature by the local and
the remote thermometer as a function of
P
. Overlapping of the
two curves suggests that electrons reach thermal equilibrium in
the large Cu film. (c) Measured
T
n
e
−
T
n
0
plotted as a function of
P
with exponent
n
=
4 (triangles,
T
0
=
55, 100, 150 mK, from
dark blue to red) and
n
=
5 (blue dotted,
T
0
=
55 mK). A lin-
ear dependence is observed when plotted with
n
=
4; black lines
are the linear fits. Experimental data show that for the 300-nm
Cu film, thermal boundary resistance limits the energy relaxation
process. The derived interface-material-dependent parameter
k
is
about 60 WK
−
4
m
−
2
, shown in the inset.
comparison, we also show clearly nonlinear dependence
for
n
=
5 and for the bath temperature 55 mK with the blue
dots. From the linear fit of
n
=
4 data, we extract the con-
stant
k
as a function of temperature, which is shown in the
inset of Fig.
3(c)
.Wefind
k
to be about 60 WK
−
4
m
−
2
with a slight increase at high temperatures. The origin
of this increase is unclear. The obtained value of
k
is
consistent with the previous experiments on evaporated
metal films [
27
,
28
], but it is smaller than the predictions
of both AMM and DMM models. This difference may be
explained by imperfect interface quality between the Cu
film and the substrate.
Previous studies showed that for disordered normal
metal films, the exponent
n
deviates from 5 depending
on the type of disorder [
19
,
29
–
32
]. The observed
n
=
5
in 50 nm Cu film indicates the clean limit for the Cu
film. Increasing the film thickness reduces the disorder and
makes it closer to the 3D clean limit. So, the observed
n
=
4 for 300-nm Cu film is not to be ascribed to the
film disorder. Instead, it originates from the fact that
R
ph-ph
dominates over
R
e
-
ph
for thick film and becomes the
bottleneck for heat transport.
As the acoustic mismatch between different materials
reduces the phonon transmission, an enhancement of the
thermal boundary resistance is expected when adding a
third layer of material between the Cu film and the sub-
strate. We fabricate a sample with 3 nm of Ti added
between 50-nm Cu film and the substrate. By simply
considering a series connection of the two interface resis-
tances [
33
,
34
], we find
k
−
1
Cu
−
Ti
−
SiO
2
=
k
−
1
Cu
−
Ti
+
k
−
1
Ti
−
SiO
2
,
where
k
Cu
−
Ti
and
k
Ti
−
SiO
2
are the interface-material-
dependent parameters between Cu/Ti and Ti
/
SiO
2
, respec-
tively. In this way, from the DMM model, we estimate
k
Cu
−
Ti
−
SiO
2
=
0.49
k
Cu
−
SiO
2
, where
k
Cu
−
SiO
2
characterizes
the Cu
/
SiO
2
boundary with its measured value shown in
the inset of Fig.
3(c)
.
In Fig.
4(a)
, we plot measured
T
4
e
−
T
4
0
as a func-
tion of
P
. Linear dependence is observed at temperatures
above 130 mK, suggesting that
R
ph-ph
dominates the energy
relaxation in this temperature range. Deviation from the
linearity when
P
is below around 1 pW, corresponding to
electron temperature rise
T
e
=
T
e
−
T
0
below 1–2 mK,
may be due to the uncertainty of the electron temperature
measurement. Its impact on final conclusion is negligible
because of the small
T
e
and the narrow power range
observed. From the linear fits, we estimate the constant
k
Cu
−
Ti
−
SiO
2
to be about 15 WK
−
4
m
−
2
, which is 25% of
the value measured without the intermediate layer. It is less
(a)(b)
FIG. 4. Measurement results of a sample with a 3-nm Ti layer
added between the 50-nm Cu film and the substrate. (a)
T
n
e
−
T
n
0
as a function of heating power
P
with
n
=
4. The observed linear
dependence above 130 mK indicates that the thermal boundary
resistance limits the energy relaxation. The black line is a guide
to the eye. Inset: Derived
k
as a function of temperature. (b)
T
n
e
−
T
n
0
as a function of heating power
P
with
n
=
5 at 60 mK. Linear
dependence is observed only at temperatures up to about
T
cr
=
120 mK. At higher temperatures, linear dependence is observed
when plotted with
n
=
4, shown in (a) with the black dotted line.
The red line is a guide to the eye.
024051-4
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than 49% expected from the model discussed above. How-
ever, considering imperfect interface quality and crudeness
of the model, which, for example, ignores the fact that
the thermal phonon wavelength is much larger than the
thickness of the Ti film, our result is in a good agreement
with the theory. With the experimentally measured values
of
k
and
, we estimate the crossover temperature to be
T
cr
=
124 mK. At temperatures below
T
cr
,
R
e
-
ph
should
dominate over
R
ph-ph
and the linear dependence of
T
5
e
−
T
5
0
on
P
is expected. In Fig.
4(b)
, we show the measurement
results at the bath temperature of 60 mK. As expected, a
linear dependence is observed at low temperatures, while
for
T
T
cr
, deviations from it become visible. In contrast,
if one plots
T
4
e
−
T
4
0
versus power, the linear dependence
is observed at high temperatures
T
T
cr
, as shown in
Fig.
4(a)
with the black line. Thus, an additional 3-nm thin
Ti layer between the Cu film and the substrate results in a
fourfold increase in the thermal boundary resistance, which
allows us to clearly see the crossover between the two
energy relaxation mechanisms with changing temperature.
One of the open questions is what the influence of
the dimensionality of the film is on the acoustic cou-
pling strength [
1
,
5
,
20
]. It has been shown that even though
the phonons in the film are 2D, the strong coupling of
phonons in the film and the substrate can broaden its
subband structure and make it closer to 3D. For Cu, the
dominant phonon wavelength
λ
is about 200 nm at 0.2 K
when transverse phonons are considered, and it increases
as
∝
T
−
1
when lowering the temperature. Assuming weak
acoustic coupling and phonons in the film to be 2D, a
reduction of the exponent
n
from 5 is expected [
6
,
8
].
The observed
n
=
5 for the 50-nm Cu film suggests
phonons in the film are closer to 3D than 2D, though
λ
is much larger than the film thickness. The experi-
mentally observed weaker acoustic coupling strength than
what the theory predicts is not significant in making the
phonons in the film 2D. Investigations are needed to quan-
tify the effect of the strength of the coupling on the phonon
dimensionality.
IV. CONCLUSION
In conclusion, we experimentally observe the crossover
between the limiting energy relaxation mechanisms in
copper films by changing the film thickness and tem-
perature. We demonstrate that an additional Ti layer
between the Cu film and the substrate enhances the
thermal boundary resistance of the interface fourfold.
This result may be useful for hot-electron calorime-
try and bolometry since it can help in improving the
energy resolution of the detectors [
15
]. Our experimen-
tal results further advance the understanding of energy
relaxation mechanisms in mesoscopic devices and of the
heat transport through the solid-solid interfaces at low
temperatures.
ACKNOWLEDGMENTS
We acknowledge the provision of the fabrication facil-
ities by Otaniemi research infrastructure for micro and
nanotechnologies (OtaNano). This work was performed
as part of the Academy of Finland Centre of Excellence
program (Project No. 312057) and the European Research
Council (ERC) under the European Union’s Horizon 2020
research and innovation program (Grant No. 2742559
SQH).
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