Crossover between electron-phonon and boundary resistance limited 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 Technolog, Pasadena, CA, USA
3
Computational Science Initiative, Brookhaven National Laboratory, Upton, NY 11973
(Dated: March 27, 2019)
We observe a crossover from electron-phonon (ep) coupling limited energy relaxation to that
governed by thermal boundary resistance (pp) in copper films at sub-kelvin temperatures. Our
measurement yields a quantitative picture of heat currents, in terms of temperature dependences
and magnitudes, in both ep 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.
Investigation of energy relaxation of electrons in nor-
mal metal films is important for understanding the un-
derlying physics as well as for applications [1–4]. Espe-
cially for mesoscopic devices at low temperature, where
the dominant thermal wavelength
λ
is comparable to the
device dimension, phonons in the films could be two-
dimensional (2D), and it has been shown both experi-
mentally and theoretically that the reduced phonon di-
mension does affect the energy relaxation of electrons
in thin films [5–8]. Heat transport by phonons, elec-
trons, and photons has been studied experimentally in
mesoscopic devices [9–14]. From the application point
of view, a good understanding 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 environment will en-
hance the energy resolution, but on the other hand,
it makes the device slower. For a transition-edge sen-
sor, unaccounted-for thermal boundaries can affect the
noise and energy resolution [16]. Finally, for a normal-
metal/insulator/superconductor (NIS) 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 (ep) coupling to the film
phonons, which is characterized by ep thermal coupling
resistance
R
ep
. Film phonons are coupled to the sub-
strate 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
pp
. If the ee interactions are assumed to be
much faster than other processes, the energy relaxation
of the electrons in the film is determined by
R
ep
and
R
pp
,
with the weaker of the two governing the energy relax-
ation process. For thin films at low temperatures, ep cou-
pling strength is weak and it becomes the bottleneck of
the energy relaxation. With increasing the temperature
or film thickness the ep coupling gets realtively stronger
and the heat transport across the boundary between the
film and the substrate becomes the limitation for the en-
ergy 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 dimen-
sionality on the ep coupling strength have been inten-
sively discussed [7, 18–20]. Thermal boundary resistance
between metals and dielectric substrates has also been
well investigated. Experimental observations can be ex-
plained with either Acoustic Mismatch Model (AMM) or
Diffuse Mismatch Model (DMM) [21]. AMM describes
phonon heat transfer through a flat interface between
perfect crystals. In analogy to the Snell’s law for the
electromagetic 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 bound-
ary. DMM assumes diffusive phonon scattering at the
interface, and hence the phonon transmission probability
depends only on the phonon densities of the states and
sound velocities on both sides. In the case of solid-solid
boundaries the mismatch in sound velocities and phonon
mode densities is usually small, and the two models give
similar predictions.
Here, we will present the experimental results show-
ing the crossover between ep and boundary resistance
limited thermal relaxation in Cu films at sub-kelvin tem-
peratures. For Cu film with 50 nm in thickness, we found
the energy relaxation to be limited by ep coupling in the
full temperature range explored. By increasing the film
thickness to 300 nm, the thermal boundary resistance
limits the energy relaxation, 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, con-
arXiv:1903.10848v1 [cond-mat.mes-hall] 26 Mar 2019
2
sistent 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
ep
= Σ
V
(
T
n
e
−
T
n
p
)
.
(1)
Here,
T
e
and
T
p
are the electron and phonon tempera-
tures in the film,
V
is the metal volume,
n
= 5 for clean
normal metals and Σ is the material-specific ep coupling
constant [22]. The coupling between film phonons and
the substrate phonons is characterized by
P
pp
=
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
̄
h
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 ep thermal coupling resis-
tance is expressed as
R
ep
= 1
/
5Σ
V T
4
, and the thermal
boundary resistance as
R
pp
= 1
/
4
kAT
3
.
T
s
equals to the
bath temperature of the refrigerator
T
0
due to the large
substrate/bath contact area.
One of the devices used in the experiments is shown in
Fig. 1(b) and (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 evap-
oration. 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 alu-
minium. The hybrid structures with short channel length
behave as a proximity Josephson junction (JJ). Switch-
ing current
I
sw
is defined as the bias current when the
junction switches from the superconducting state to the
resistive state, shown in the IV curve in Fig. 1(d). The
JJ switches back to the superconducting state at a bias-
ing 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 resis-
tive state. Bath temperature dependence of
I
sw
at zero
heating, i.e., in equilibrium, shown in Fig. 1 (e), is used
as the temperature calibration 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. 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
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) Zoom-
in of the rectangular 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 function
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.
temperatures from 60 mK 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 suggests no heating
current flows to the thermometer in this configuration.
For electrons in the copper film, superconducting Al
acts as a thermal insulator at sufficiently low tempera-
tures below its critical temperature
T
c
∼
1 K, the Joule
power applied
P
to the film dissipates mainly by ep cou-
pling. The ratio of the two series thermal resistances is
γ
=
R
pp
R
ep
=
5Σ
tT
4
k
.
(4)
Here,
t
is the thickness of the Cu film. For a thin film
3
at sufficiently low temperatures, we expect
R
ep
to domi-
nate over
R
pp
, so we have the standard situation usually
assumed for thin films, i.e.,
P
=
P
ep
= Σ
V
(
T
5
e
−
T
5
0
). In
Fig. 2, we plot the experimental results of a sample with
50 nm thick Cu film, a linear dependence vs.
T
5
e
−
T
5
0
is
clearly seen as expected. From the slope, we obtain the
ep coupling constant Σ = 2.1
±
0.1 nWK
−
5
μ
m
−
3
with no
temperature dependence within the measurement inter-
val from 60 mK to 250 mK, as shown in the inset of Fig. 2.
The measured value of Σ is consistent with previous ex-
periments on Cu films [24, 25]. Thus, the experiment
demonstrates that for the 50 nm Cu film at low tem-
perature, the energy relaxation of electrons is dominated
by the ep coupling, and the exponent
n
= 5 is consis-
tent with the theory based on three-dimensional (3D)
free electron model [22].
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 consistent with the prediction of Eq. (1),
suggesting that the weak ep coupling limits the energy relax-
ation of the electrons in the Cu film. Inset is the measured
ep coupling constant Σ as a function of temperature.
The Eq. 4 suggests that if one changes the film thick-
ness or temperature to the point where
R
pp
becomes
equal to
R
ep
, a crossover from one energy relaxation
mechanism to another should take place. The crossover
temperature
T
cr
depends on the constants Σ and
k
as
T
cr
= 4
k/
5Σ
t
. For perfect contacts between Cu and
the silicon substrate one finds
k
≈
170 WK
−
4
m
−
2
[21].
Hence the crossover temperature of 0
.
1 K is expected
in films with the thickness of
t
≈
700 nm. Recent ex-
periments suggested that for evaporated films on a sil-
icon substrate
k
is smaller than that predicted for per-
fect contacts [26, 27], 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 as JJ thermometers, heater, and the
contact pads. Then we deposit the 300 nm Cu film (pur-
FIG. 3. (a) False-color SEM image of a sample with large Cu
film (purple) of dimension 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 temper-
ature while heating. Inset in (b) is the zoom-in 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
, over-
lapping of the two curves suggesting electrons reach thermal
equilibrium in the large Cu film. (c) Measured
T
n
e
−
T
n
0
plot-
ted as a function of
P
with exponent
n
= 4 (triangles,
T
0
=
55 mK, 100 mK, 150 mK, from dark blue to red) and
n
= 5
(blue dotted,
T
0
= 55 mK). A linear dependence is observed
when plotted with
n
= 4, black lines are the linear fits. Ex-
perimental 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.
ple). Before contacting the two copper films, Ar plasma
milling is used to clean the surface of the thin one. In-
set of Fig. 3(b) shows the thick film covering the thin
film. Electron temperature is measured with two JJ ther-
mometers located at the two ends of the thick Cu film
(local, remote) with a distance of 40
μ
m to check the uni-
formity of electron temperature in the thick Cu film while
heating. Figure. 3(b) shows that the two thermometers
show identical temperature except at the largest applied
powers. The small difference at high
P
originates most
likely from the electron diffusion along 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 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 comparison, we also show clearly non-linear depen-
4
dence for
n
= 5 and for the bath temperature 55 mK with
the blue dots. From the linear fit of
n
= 4 data we have
extracted the constant
k
as a function of temperature,
which is shown in the inset of Fig. 3 (c). We have found
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 [26, 27], 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.
FIG. 4. Measurement results of a sample with 3 nm Ti layer
added between 50 nm Cu film and the substrate. (a)
T
n
e
−
T
n
0
as 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
temperature up to about
T
cr
= 120 mK. At higher tempera-
tures, linear dependence is observed when plotted with
n
=
4, shown in (a) with the black dotted line. The yellow line is
a guide to the eye.
Previous studies showed that for disordered normal
metal films, the exponent
n
deviates from 5 depending
on the type of disorder [19, 28–30]. The observed
n
=
5 in 50 nm Cu film indicates the clean limit for the Cu
film. Increasing the film thickness will reduce the dis-
order and make 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
pp
dominates over
R
ep
for thick film and becomes
the bottleneck for heat transport.
As the acoustic mismatch between different materials
will reduce 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 substrate. We have fabricated a sample with
3 nm of Ti added between 50 nm Cu film and the sub-
strate. By simply considering a series connection of the
two interface resistances [31, 32], we find
k
−
1
Cu
−
T i
−
SiO
2
=
k
−
1
Cu
−
T i
+
k
−
1
T i
−
SiO
2
, where
k
Cu
−
T i
and
k
T i
−
SiO
2
are the
interface-material-dependent parameters between Cu/Ti
and Ti/SiO
2
, respectively. In this way, from the DMM
model we have estimated
k
Cu
−
T i
−
SiO
2
= 0.38
k
Cu
−
SiO
2
,
where
k
Cu
−
SiO
2
characterises the Cu/SiO
2
boundary.
In Fig. 4 (a), we plot measured
T
4
e
−
T
4
0
as a function of
P
. Linear dependence is observed at temperatures above
130 mK, suggesting that
R
pp
dominates the energy relax-
ation in this temperature range. From the linear fit we
have estimated the constant
k
Cu
−
T i
−
SiO
2
to be about 15
WK
−
4
m
−
2
, which is 25% of the value measured without
the intermediate layer. It is a bit less than 38% expected
from the model discussed above. However, 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 Σ we estimate the crossover temperature to be
T
cr
= 124 mK. At temperatures below
T
cr
,
R
ep
should dom-
inate over
R
pp
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 60 mK. As expected, a linear dependence is
observed at low temperatures, while for
T
>
∼
T
cr
devi-
ations 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 the four-
fold increase in the thermal boundary resistance, which
allows us to clearly see the crossover between the two en-
ergy relaxation mechanisms with changing temperature.
One of the open questions is what the influence of the
dimensionality of the film is on the acoustic coupling
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. As-
suming weak acoustic coupling and phonons in the film
to be 2D, a reduction of the exponent
n
from 5 is ex-
pected [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 large than the film thickness. The ex-
perimentally observed weaker acoustic coupling strength
than what the theory predicts is not significant in mak-
ing the phonons in the film 2D. Investigations are needed
to quantify the effect of the strength of the coupling on
the phonon dimensionality.
5
In conclusion, we have experimentally observed the
crossover between the limiting energy relaxation mech-
anisms in copper films by changing the film thickness
and temperature. We have demonstrated that an addi-
tional Ti layer between the Cu film and the substrate
enhances the thermal boundary resistance of the inter-
face fourfold. This result may be useful for hot-electron
calorimetry and bolometry since it can help in improv-
ing the energy resolution of the detectors [15]. Our ex-
perimental 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.
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 Excel-
lence program (Projects No.312057.) and European Re-
search Council (ERC) under the European Union’s Hori-
zon 2020 research and innovation program.(No. 742559
SQH).
∗
libin.wang@aalto.fi
[1] M. Kanskar, M. N. Wybourne, and K. Johnson, Physical
Review B
47
, 13769 (1993).
[2] D. R. Schmidt, R. J. Schoelkopf, and A. N. Cleland,
Physical Review Letters
93
, 045901 (2004).
[3] F. Giazotto, T. T. Heikkil ̈a, A. Luukanen, A. M. Savin,
and J. P. Pekola, Reviews of Modern Physics
78
, 217
(2006).
[4] E. Pinsolle, A. Rousseau, C. Lupien, and B. Reulet,
Physical Review Letters
116
, 236601 (2016).
[5] J. C. Nabity and M. N. Wybourne, Physical Review B
44
, 8990 (1991).
[6] B. A. Glavin, V. I. Pipa, V. V. Mitin, and M. A. Stroscio,
Physical Review B
65
, 205315 (2002).
[7] S.-X. Qu, A. N. Cleland, and M. R. Geller, Physical
Review B
72
, 224301 (2005).
[8] J. T. Karvonen and I. J. Maasilta, Physical Review Let-
ters
99
, 145503 (2007).
[9] K. Schwab, E. A. Henriksen, J. M. Worlock, and M. L.
Roukes, Nature
404
, 974 (2000).
[10] O. Chiatti, J. T. Nicholls, Y. Y. Proskuryakov, N. Lump-
kin, I. Farrer, and D. A. Ritchie, Physical Review Letters
97
, 056601 (2006).
[11] M. Meschke, W. Guichard, and J. P. Pekola, Nature
444
, 187 (2006).
[12] S. Jezouin, F. D. Parmentier, A. Anthore, U. Gennser,
A. Cavanna, Y. Jin, and F. Pierre, Science (New York,
N.Y.)
342
, 601 (2013).
[13] M. Partanen, K. Y. Tan, J. Govenius, R. E. Lake, M. K.
M ̈akel ̈a, T. Tanttu, and M. M ̈ott ̈onen, Nature Physics
12
, 460 (2016).
[14] A. Tavakoli, K. Lulla, T. Crozes, N. Mingo, E. Collin,
and O. Bourgeois, Nature Communications
9
, 4287
(2018).
[15] J. P. Pekola, P. Solinas, A. Shnirman, and D. V. Averin,
New Journal of Physics
15
, 115006 (2013).
[16] K. M. Kinnunen, M. R. J. Palosaari, and I. J. Maasilta,
Journal of Applied Physics
112
, 034515 (2012).
[17] P. J. Lowell, G. C. O’Neil, J. M. Underwood, and J. N.
Ullom, Applied Physics Letters
102
, 082601 (2013).
[18] J. F. DiTusa, K. Lin, M. Park, M. S. Isaacson, and J. M.
Parpia, Physical Review Letters
68
, 1156 (1992).
[19] A. Sergeev and V. Mitin, Physical Review B
61
, 6041
(2000).
[20] J. M. Underwood, P. J. Lowell, G. C. O’Neil, and J. N.
Ullom, Physical Review Letters
107
, 255504 (2011).
[21] E. T. Swartz and R. O. Pohl, Reviews of Modern Physics
61
, 605 (1989).
[22] F. C. Wellstood, C. Urbina, and J. Clarke, Physical Re-
view B
49
, 5942 (1994).
[23] L. B. Wang, O.-P. Saira, and J. P. Pekola, Applied
Physics Letters
112
, 013105 (2018).
[24] M. L. Roukes, M. R. Freeman, R. S. Germain, R. C.
Richardson, and M. B. Ketchen, Physical Review Letters
55
, 422 (1985).
[25] M. Meschke, J. P. Pekola, F. Gay, R. E. Rapp, and
H. Godfrin, Journal of Low Temperature Physics
134
,
1119 (2004).
[26] S. Rajauria, P. S. Luo, T. Fournier, F. W. J. Hekking,
H. Courtois, and B. Pannetier, Physical Review Letters
99
, 047004 (2007).
[27] L. M. A. Pascal, A. Fay, C. B. Winkelmann,
and
H. Courtois, Physical Review B
88
, 100502 (2013).
[28] S. Y. Hsu, P. J. Sheng, and J. J. Lin, Physical Review
B
60
, 3940 (1999).
[29] M. E. Gershenson, D. Gong, T. Sato, B. S. Karasik, and
A. V. Sergeev, Applied Physics Letters
79
, 2049 (2001).
[30] J. T. Karvonen, L. J. Taskinen, and I. J. Maasilta, Phys-
ical Review B
72
, 012302 (2005).
[31] E. Dechaumphai, D. Lu, J. J. Kan, J. Moon, E. E. Fuller-
ton, Z. Liu, and R. Chen, Nano Letters
14
, 2448 (2014).
[32] Z. Li, S. Tan, E. Bozorg-Grayeli, T. Kodama,
M. Asheghi, G. Delgado, M. Panzer, A. Pokrovsky,
D. Wack, and K. E. Goodson, Nano Letters
12
, 3121
(2012).