of 10
J. Appl. Phys.
127
, 024307 (2020);
https://doi.org/10.1063/1.5132948
127
, 024307
© 2020 Author(s).
Modification of electron-phonon coupling by
micromachining and suspension
Cite as: J. Appl. Phys.
127
, 024307 (2020);
https://doi.org/10.1063/1.5132948
Submitted: 21 October 2019 . Accepted: 18 December 2019 . Published Online: 14 January 2020
Olli-Pentti Saira
, Matthew H. Matheny
, Libin Wang
, Jukka Pekola
, and
Michael Roukes
Modification of electron-phonon coupling by
micromachining and suspension
Cite as: J. Appl. Phys.
127
, 024307 (2020);
doi: 10.1063/1.5132948
View Online
Export Citation
CrossMar
k
Submitted: 21 October 2019 · Accepted: 18 December 2019 ·
Published Online: 14 January 2020
Olli-Pentti Saira,
1,2
,
a)
Matthew H. Matheny,
1
Libin Wang,
3
Jukka Pekola,
3
and Michael Roukes
1
AFFILIATIONS
1
Condensed Matter Physics and Kavli Nanoscience Institute, California Institute of Technology, Pasadena, California 91125, USA
2
Computational Science Initiative, Brookhaven National Laboratory, Upton, New York 11973, USA
3
QTF Centre of Excellence, Department of Applied Physics, Aalto University, FI-00076 Aalto, Finland
a)
Electronic mail:
osaira@bnl.gov
ABSTRACT
Weak electron-phonon interaction in metals at low temperatures forms the basis of operation for cryogenic hot-electron bolometers and cal-
orimeters. Here, we develop a thermometry scheme based on proximity supercurrent to study the thermal response of a thin gold film on a
SiO
2
platform at temperatures below 100 mK. We find that the exponent of the power law describing electron-phonon coupling in the film
drops by approximately 1
=
2 as the platform is micromachined and released from its substrate. This contrasts the conventional theory for
bulklike geometries that predicts integer-valued exponents. We attribute the fractional change to a modified phonon spectrum described by
recent theoretical developments.
Published under license by AIP Publishing.
https://doi.org/10.1063/1.5132948
I. INTRODUCTION
Nanostructures at low temperatures exhibit strong thermal
response. This follows from their minuscule heat capacities and
thermal conductances, both of which diminish in response to the
shrinking of the device dimensio
ns and reduction of the operat-
ing temperature.
1
Consequently, low-temperature nanodevices
excel as thermal detectors, e.g., bolometers and calorimeters, and
various types of heat engines. Quantitative measurements of the
thermal properties of materials and structures constitute a valu-
able tool for scientific and engineering purposes. For detector
applications, thermal characterization is vital for understanding
and optimizing the de
tector performance.
2
A full picture encom-
passing experimentally relevant effects such as reduced dimen-
sionality, material interfaces, and disorder is in its infancy and
has not been tested extensively. In basic research, on the other
hand, thermal transport has been used as a probe of otherwise
elusive states of matter such as strongly interacting quantum Hall
systems
3
and the phonon spectrum in micromachined supports.
4
Several thermal and thermoelectric signatures of Majorana states
have been identified theoretically.
5
7
Despite the above, studies of mesoscopic thermal transport are
far outnumbered by studies of electrical transport. We speculate that
this is because the experimental configurations for quantitative
thermal measurements tend to be complex. The associated techno-
logical challenges can be grouped into two categories. The first chal-
lenge is the measurement of the relevant physical quantities such as
temperature, power, and energy. The measurement should be accu-
rate (i.e., free of systematic biases), precise (i.e., capable of resolving
small differences), and localized (i.e., only probe the targeted thermal
body). A second challenge is ensuring the noninvasiveness of the
measurement scheme. Attaching or operating the thermal probe may
induce additional dissipation, or increase the thermal conductance
or heat capacity of the target body, skewing the results.
In this work, we advance the SNS (superconductor
normal
superconductor) weak link electronic supercurrent thermometry
that has been developed and employed in several earlier studies.
8
11
We show that our implementation of the method meets the criteria
for ideal quantitative thermal characterization. We employ the
method to study the nature of electron-phonon coupling in a
mechanically suspended system that hosts a 3D electron gas and a
quasi-2D phonon system. Our main finding is a modified
electron-phonon heat flow
_
Q
/
T
n
el

T
n
ph
with
n
¼
4
:
54(3), where
T
el
and
T
ph
denote the electron and phonon temperatures, respec-
tively. The approximately half-integer exponent of the power law is
in contrast with the standard theories for bulk systems that predict
integer-valued exponents.
12
Very recent theoretical analysis
13
of the
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127,
024307-1
Published under license by AIP Publishing.
metal-dielectric bilayer system has predicted a
plateau region,
where 4
:
5
,
n
,
5 over a wide range of temperatures where the
phonon system undergoes a 2D to 3D transition. Our work
appears to confirm this prediction. An exponent
n

4
:
5 has also
been seen in an earlier experimental study
14
,
15
that employed dif-
ferent materials, sample geometry, and thermometry method from
ours. We also analyze the electron-phonon heat flow in an unsus-
pended sample, and the phononic heat flow along the wide but
thin support legs of the membrane. The obtained exponents in
these two cases,
n
¼
5
:
07(3) and 3
:
01(4), respectively, are close to
the expected theoretical values of 5 and 3.
II. METHODS
A. Physics of supercurrent thermometry
Any physical phenomenon with a temperature dependence
can be used as a thermometer after calibration against a known
thermometer. This is called secondary thermometry. Here, we
employ proximity superconductivity induced in a metallic SNS
weak link.
16
,
17
The switching current offers a virtually ideal elec-
trical characteristic for steady-state thermometry using a simple
readout circuit. The temperature range of the sensor is limited,
but different temperatures can be targeted by choosing an appro-
priate length for the weak link.
The electrical behavior of the SNS weak link can be analyzed
in terms of the RCSJ (
RC
-shunted junction) model.
18
,
19
When
probed with a low-frequency current waveform, the junction stays
essentially in a zero-voltage state until the probe current exceeds
the temperature-dependent critical current of the junction.
Crucially, the dissipation in the zero-voltage state is vanishingly
small. In contrast, after the junction enters a finite-voltage state,
positive electro-thermal feedback brought upon by the current bias
rapidly heats up the junction to temperatures where the critical
current vanishes. Hence, the observed voltage is
V
¼
R
N
I
, where
R
N
is the normal-state resistance.
A basic characterization of an underdamped weak link can be
performed by determining its switching current histogram. We fix
the shape and duration of a current waveform, and determine the
probability of observing a voltage pulse as a function of the ampli-
tude. In
Fig. 1(a)
, we have used a single cycle of a 100 ms period sin-
ewave to probe an SNS weak link with an approximate separation of
1.5
μ
m between the superconducting electrodes at temperatures
ranging from 8.8 mK to 120 mK. The sample geometry is described
in more detail in Sec.
III A
. To quantify the width of the histograms
[
Fig. 1(b)
], we evaluate the standard deviation of the switching
current, which is found to be less than 16 nA at all temperatures.
To characterize the effect of temperature on the position of the
switching histogram, we evaluate the mean switching current at each
temperature point [
Fig. 1(c)
, markers]. The temperature dependence
obtained in this manner can be reproduced by a low-temperature
expansion of the supercurrent of a diffusive SNS weak link,
8
,
20
eR
eff
I
c
=
E
Th
¼
a
1

b
exp

aE
Th
3
:
2
k
B
T


,
(1)
where
I
c
is the critical current at temperature
T
,
R
eff
is the effec-
tive normal-state resistance,
E
Th
denotes the Thouless energy
(defined below),
a
¼
10
:
82,
b
¼
1
:
3 are additional numerical con-
stants from the theory, and
e
is the elementary charge. A fit yields
R
eff
¼
2
:
75(3)
Ω
and

h
=
E
Th
¼
0
:
204(2) ns. The fitted values can be
compared with theoretical predictions based on the weak link geom-
etry and materials. First, the Thouless energy is given by
E
Th
¼

hD
=
L
2
, where
D
is the diffusion constant and
L
is the length
of the weak link. From the measured low-temperature resistivity of
the Au film
ρ
¼
1
:
28

10

8
Ω
m, we evaluate the mean free path
l
e
¼
50
:
8 nm and the diffusion constant
D
¼
v
F
l
e
=
3
¼
237 cm
2
s

1
,
where
v
F
¼
1
:
40

10
6
m
=
s is the Fermi velocity in Au. Solving for
the length yields
L
¼
2
:
2
μ
m. The discrepancy between the effective
L
and the physical electrode separation (1
:
5
μ
m) can be partially
attributed to the complex geometry of the Au part [see
Fig. 3(a)
],
which stems from the requirements of the electron-phonon coupling
experiment. Second, a direct
I
-
V
measurement yields
R
N
¼
0
:
27
Ω
.
In a rectangular wire geometry, the ratio
R
N
=
R
eff

0
:
1 could be
interpreted as a measure of the interface transparency.
9
,
21
However,
in the present geometry, there is additional normal metal that shunts
the supercurrent link, thus lowering
R
N
but not contributing to
I
c
.
The single-shot temperature resolution of the detector can be
calculated by dividing the width of the histogram by the local tem-
perature responsivity
dI
mean
=
dT
jj
, which peaks at 70 nA
=
mK at
T
¼
70 mK. The detector is capable of resolving temperature differ-
ences of the order of 0.2 mK in its most sensitive range of tempera-
tures with a single readout pulse.
FIG. 1.
Basic characterization of diffusive SNS links. (a) Cumulative
switching probability histograms for an
L
¼
1
:
5
μ
m weak link at different
temperatures from 8.8 mK to 120 mK in approximately 10 mK steps. (b) A
high-resolution scan of histogram at the base temperature for both positive
(solid) and negative (dashed) polarities. (c) The mean switching current
I
50
extracted from the histograms in panel (a) as a function of temperature
(markers), and the theoretical critical current using fitted parameter values
(line). (d) Measured base-temperature
I
c
R
N
product for three different wire
lengths and an exponential fit.
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Another characteristic prediction of the diffusive theory is the
strong length dependence of the induced superconductivity. We
investigate this by determining the base temperature
I
c
R
N
product
for SNS wires of 500 nm width and of varying length [
Fig. 1(d)
].
The data are consistent with an exponential decay with a character-
istic length of 440 nm.
The position and width of the histogram can be weakly affected
by factors such as the probing waveform, current and voltage noise,
and the decision threshold for the detection of voltage pulses.
However, as long as the same excitation and detection methods are
used throughout the experiment, secondary thermometry incurs no
systematic bias from these nonidealities. Two-level composite pulses
have been used in previous works to greatly improve the temporal
accuracy of the supercurrent probing.
22
This work deals exclusively
with steady-state quasiequilibrium thermometry, thereby obviating
the need for more complex pulse sequences.
Since a hysteretic Josephson junction acts as a wide-bandwidth
threshold detector for current noise, we estimate the magnitude of
temperature-dependent current noise that reaches that junction.
The twisted-pair measurement lines had two RC filter banks with
R
¼
600
Ω
and
RC
ðÞ

1
¼
2
π

100 kHz each at the 4 K and base
temperature stage of the fridge. Hence, Johnson-Nyquist noise
from outside of the mixing chamber stage should be negligible.
Between the final RC filter and the sample, there is a powder filter
segment consisting of 20 cm of resistive constantan twisted pair
wire (
R

14
Ω
per wire) embedded in a lossy Stycast/Cu powder
dielectric. Based on room-temperature characterization of a similar
filter box, we estimate the powder filter segment to strongly attenu-
ate signals above
f
c
¼
100 MHz. We then estimate the worst-case
rms current amplitude of noise from the sample box wiring reach-
ing the Josephson junction to be
I
rms
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4
k
B
Tf
c
=
(2
R
)
p
¼
5 nA,
where we substituted
T
¼
0
:
12 K corresponding to the maximum
temperature encountered in this work.
B. Heat conductance measurements
The subject of study in steady-state thermal measurements is
the heat flow between two thermal bodies in quasiequilibrium, i.e.,
assuming they both have well-defined internal temperatures. We
denote the bodies by
a
,
b
and their temperatures by
T
a
and
T
b
from
here on. For elementary, continuous systems of interest to meso-
scopic physics, the heat flow can be generally written in the form
_
Q
a
!
b
¼
h
(
T
a
)

h
(
T
b
),
where
h
is a monotonically increasing function. In a limited temper-
ature range, one often finds a
power law
_
Q
a
!
b
¼
A
(
T
n
a

T
n
b
),
(2)
FIG. 2.
Electrical schematic for measurements reported in this letter. The
output voltage of the HEAT(

) source is always the negative of the output
voltage of the HEAT(
þ
) source. The amplifiers are high-input impedance
(
R
in
.
1000 M
Ω
) differential voltage preamplifiers.
FIG. 3.
Verification experiments.
Electron-phonon coupling on the bulk sub-
strate:
(a) SEM image of a device nominally identical to the measured one. (b)
Electron temperature as a function of heating power and a power law fit.
Phononic thermal conductance of a micromachined structure:
(c) SEM image of
the measured device. (d) Three thermal conductance datasets. Data from the
small membrane [device in panel (c), magenta markers], and the large mem-
brane [device in
Fig. 4(b)
, red and blue markers corresponding to two indepen-
dent measurements]. Thermal conductance obtained as the local numerical
derivative of the measured heating characteristic. Lines are power-law fits to the
data with a background heating offset.
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024307-3
Published under license by AIP Publishing.
where the exponent
n
often conveys information about the dimen-
sionality of the microscopic physics of energy transport. Equivalently,
one can study the thermal conductance
G
(
T
)
¼
@
_
Q
a
!
b
@
T
a




T
a
;
T
b
¼
T
¼
h
0
(
T
)
¼
nAT
n

1
,
(3)
where the last equality holds for any power law. Throughout this
paper, we will use exponent
n
to refer to the heat flow power law
[Eq.
(2)
]. Consequently, the exponent for the thermal conduc-
tance will be
n

1.
Accurate experimental investigations of the above relations
require independent determination of the three quantities
_
Q
a
!
b
,
T
a
,
and
T
b
. In some cases, one can substitute in place of
T
b
the reading
of another thermometer
T
c
. This is possible if the thermal link
between
b
and
c
is strong enough to disallow significant temperature
differences
T
b

T
c
jj
, or if a nonzero direct thermal link between
b
and
c
exists, and
_
Q
b
!
c
¼
0 is known to sufficient accuracy.
We will discuss how supercurrent thermometry allows one to
approach the ideal thermal measurement in practice with a relatively
simple measurement setup. The electrical connections for a single-
body electron thermometry experiment are illustrated in
Fig. 2
.The
metallization pattern on the chip, corresponding to the
Sample
sub-
circuit in the diagram, can be seen in
Fig. 3(a)
. Our metallic samples
containing no tunnel junctions present low electrical impedances to
the measurement circuitry. Referring to the labels of
Fig. 2
,thelargest
resistance on the chip is the heater wire,
R
heat

10
Ω
. For direct elec-
tron heating and thermometry, a galvanic connection between the
heater and thermometer is required. The electrical resistance
R
i
of
this connection is less than 1
Ω
.Inthefollowing,weshowthatthe
amount of current flowing through this connection is negligible.
Due to the low impedance of the sample, it is straightforward
to realize an accurate low-frequency current bias using external
voltage sources and large bias resistors. Operating the voltage sources
in such a manner that
V
heat
(
+
)
¼
+
V
heat
ensures that the heating
current
I
heat
¼
V
heat
=
R
bias
,
heat
flows through the resistor
R
heat
.Tothe
precision within which equal and opposite current biasing is realized,
no
current flows through
R
i
or the Josephson junction (JJ) modeling
the SNS weak link. Similarly, the probe source biases the Josephson
junction with the current
I
probe
¼
V
probe
=
R
bias
,
probe
,andonlyasmall
fraction of it flows through
R
i
or
R
heat
. More quantitatively, the rela-
tive leakage through
R
i
is 2
R
cold
=
R
bias
,
heat

2

10

3
, and the
leakage through
R
heat
is half of that.
We can verify the accuracy of the current-cancellation
scheme
in situ
:Ifafraction
α
of
I
heat
were to flow through
R
i
and
the JJ element, one would observe a switching current
I
c
+
α
I
heat
that depends on the polarity of the heating current. We find
j
α
j
,
1%. Consecutively, the maximum leakage current encoun-
tered in the experiment is less than 2 nA, contributing to a
maximum of 4

10

18
W of unaccounted heating. To further
null the effect of the heating current leakage, and any other unac-
counted current offsets, we always test all four combinations of
polarities for the bias and the heating currents.
The voltage drop
V
heat
over the heating wire can be accurately
measured with a high-input impedance voltage preamplifier. The
heating power in the target electron gas can be then evaluated as
P
¼
I
heat
V
heat
. Since the heating wire is a resistive element with a
linear, temperature-independent
I
-
V
characteristic, we forego the
voltage measurement after the resistance
R
heat
has been determined
and use
P
¼
R
heat
I
2
heat
instead. Finally, we note that our use of leads
made of only superconducting materials ensures that all on-chip
dissipation takes place in the normal metal reservoirs, and that
electronic heat conductance along the leads is suppressed to a neg-
ligible level. Since the JJ used for temperature measurement stays in
zero-voltage state until an output voltage is generated, there is no
spurious heating from the operation of the thermometer that
would need to be included in the model.
C. Sample fabrication
The devices for this study were fabricated on a lightly p-doped
(resistivity 10
100
Ω
cm) silicon substrate with 300 nm of thermally
grown SiO
2
on its surface. The normal-state (3 nm Ti, 50 nm Au)
and superconducting (5 nm Ti, 65 nm Al) films were deposited
separately using e-beam lithography, e-beam evaporation, and
lift-off processes. In the normal-state film, the superconducting
character of the Ti adhesion layer can be expected to be completely
suppressed by the inverse proximity effect. The surface of the Au
film was cleaned with
in situ
Ar ion milling before the supercon-
ducting contacts were formed. To create the mechanically sus-
pended structures, the SiO
2
film was first patterned using e-beam
lithography and Ar ion milling. Finally, the SiO
2
platform and the
metallic structures were released by removing the underlying Si
with an isotropic XeF
2
etch.
III. THERMAL MEASUREMENTS
A. Electron-phonon coupling in bulk
Early work on the measurement of current-induced disequili-
brium between electrons and phonons was carried out using local
noise thermometry.
23
We realize a similar nonsuspended device
geometry with heater and thermometer connections [
Fig. 3(a)
,
only one half of the device was tested]. The power law fit yields
n
¼
5
:
07(3),
A
¼
2
:
23(1)

10

8
WK

n
[
Fig. 3(b)
]. The fitted
exponent is close to the value
n
¼
5 corresponding to a clean-limit
3D metal.
24
We note that the calculation presented in Ref.
25
predicts a slightly higher temperature-dependent exponent for a
thin metallic film on a bulk dielectric. The nominal volume of
the gold island is
V
¼
57
μ
m
2

50 nm
¼
2
:
9
μ
m
3
.Thefitted
A
yields an electron-phonon coupling constant
Σ
¼
A
=
V
¼
7
:
7

10
9
WK

n
m

3
. We have neglected the effect of the Ti adhe-
sion layer, which makes up 4% of the thickness of the normal-
state film. The only previously reported measurement of
Σ
for Au that we are aware of is 2
:
2
...
3
:
3

10
9
WK

5
m

3
from
Ref.
26
, which employed a much thinner and more disordered
film (
d
¼
11
:
2 nm, mean free path 2
:
5 nm) compared to ours.
B. 2D phononic thermal conductance
We have studied two different membrane designs. The
smaller membrane [
Fig. 3(c)
] has four support legs with a nominal
width of
w
¼
2
μ
m and length
l
¼
7
:
5
μ
m. The geometry factor is
4
w
=
l
¼
1
:
07. We find
n
¼
2
:
98
+
0
:
03,
A
¼
2
:
46(5)

10

10
WK

n
[magenta data in
Fig. 3(d)
].
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127,
024307-4
Published under license by AIP Publishing.
The larger membrane [
Fig. 4(b)
] has four support legs with a
nominal width of 4
:
5
μ
m and length 10
:
25
μ
m. We find
n
¼
3
:
05(2),
A
¼
2
:
62(2)

10

10
WK

n
using the Right heater/
Left thermometer combination [red data in
Fig. 3(d)
], and
n
¼
3
:
01(2),
A
¼
2
:
37(2)

10

10
WK

n
using the Left heater/
Right thermometer combination [blue data in
Fig. 3(d)
].
The fitted exponents are very close to the integer value
n
¼
3
(corresponding to
κ
/
T
2
). This would indicate that the phonons
are quantized in the thickness direction but not in the lateral direc-
tion. The prefactor
A
appears to be independent of the geometry
factor, indicating that phonon scattering in the support legs is
weak. Phononic thermal conductance in suspended and patterned
structures with a similar geometry has been analyzed theoretically
in detail in Refs.
27
and
28
.
C. Electron-phonon coupling in a thin suspended
system
To study the electron-phonon coupling on a suspended mem-
brane, we have fabricated a device that contains two symmetric heater-
thermometer structures.
Figure 4(a)
shows a schematic of the device
with labeled components;
Fig. 4(b)
is a scanning electron micrograph
of the completed device. The upward buckling of the platform is
caused by stresses in the SiO
2
layerfromthefilmgrowthprocess(see
Appendix A
). The thermal block diagram that forms the basis of the
quantitative analysis is shown in
Fig. 4(c)
. Base temperature
I
-
V
characteristics of the four electrical components of the device are
shown in
Figs. 4(d)
4(g)
. The temperature calibration curves of
the two supercurrent thermometers are shown in
Fig. 4(h)
.
The critical currents after suspending the membrane are a
factor 3
4 smaller than in the sample with bulk dielectrics. We
suspect that this is caused by the degradation of the SNS weak link
due to the additional processing steps and aging of the sample.
While the principle of supercurrent thermometry is still applicable,
the temperature range where digital switching dynamics are
observed is restricted to below 80 mK.
We wish to study the electron-phonon heat flow that we pos-
tulate to follow a power law
_
Q
¼
A
(
T
n
el

T
n
ph
)

P
0
,
(4)
where the excess heating term
P
0
accounts for the background
heat loads that are beyond experimental control and are responsi-
ble for the saturation of the thermometer signal at low bath tem-
peratures and low heating powers. Details of the thermometer
calibration and data analysis are presented in
Appendix B
.The
dual-heater, dual-thermometer configuration allows us to inde-
pendently measure the quantities
_
Q
,
T
el
,and
T
ph
appearing in
the above equation. By swapping the roles of the left and right
halves of the nominally symmetric design, we obtain two inde-
pendent measurements of the power law. We find
n
¼
4
:
55(1),
A
¼
3
:
78(1)

10

9
WK

n
using Right heater data [red markers
in
Figs. 5(a)
and
5(b)
] and
n
¼
4
:
55(1),
A
¼
3
:
76(1)

10

9
WK

n
using Left heater data [blue markers in
Figs. 5(a)
and
5(b)
]. The
two independent measurements of the exponent and the magnitude
of the electron-phonon heat flow are remarkably close to each
other. The exponent is in agreement with earlier studies by
Karvonen and Maasilta,
14
,
15
where they obtained
n

4
:
5 at temper-
atures up to 0
:
4 K employing quasiparticle thermometry on a SiN
membrane. The fitted value for the excess heating term is
0
:
9

10

16
Wand

2
:
1

10

16
W for the Right and Left heater
data, respectively. A negative
P
0
indicates that the background
heating was higher in the phonon temperature measurement config-
uration compared to the electron temperature measurement.
The theoretical work that most closely models our experimen-
tal system is that by Anghel and co-workers,
13
,
29
,
30
wherein they
consider a sheet consisting of finite-thickness metal and dielectric
layers. In Ref.
13
, in particular, they extend their analysis to cover
the range of temperatures where the phonon dimensionality
changes from two to three. A key parameter of their model is the
crossover temperature, for which they give the homogeneous media
approximation
T
C


hc
t
=
(2
k
B
d
), where
c
t
is the transversal (shear
wave) speed of sound and
d
is the total thickness of the membrane.
We estimate
T
C
for our device as
T
C


h
2
k
B
d
SiO2
c
t
,SiO2
þ
d
Ti
c
t
,Ti
þ
d
Au
c
t
,Au


1
¼
30 mK
:
FIG. 4.
Experiment for studing electron-phonon coupling on a suspended mem-
brane. (a) Annotated 3D rendering of a sample. (b) SEM image of the measured
device. (c) Thermal block diagram of the experiment. (d)
(f) Base temperature
I-V characteristics of the thermometers (top panels) and heating elements
(bottom panels) on the platform. (h) Observed temperature dependence of the
two critical current thermometers: left (blue markers), right (red). The thickness
of the line indicates the standard deviation (2
σ
) evaluated from 30 measure-
ments per temperature point.
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Published under license by AIP Publishing.
Hence, our experiment covers the range
T
=
T
C
¼
0
:
7
...
2
:
7. In the
numerical calculation of Ref.
13
, the local power-law exponent
@
ln
_
Q
@
ln
T
el
is found to lie between 4.5 and 4.7 for
T
=
T
C
¼
2,
...
, 10,
constituting the plateau region.
The quality of our experimental data does not permit a reli-
able extraction of local exponents. However, the global exponent
given by an unweighted fit is dominated by the data at higher tem-
peratures, and, therefore, measures primarily the plateau exponent.
To verify this, we have performed a separate power law fit using
only the data where
T
el

2
T
C
¼
59 mK, which yields
n
¼
4
:
56(2)
using Right heater data, and
n
¼
4
:
51(2) using Left heater data.
IV. CONCLUSIONS
We have used an SNS critical current thermometer to study
electron-phonon coupling in a micromechanical platform.
Comparing the measurements before and after micromachining
and suspension of the platform, we find the electron-phonon
power law to be affected by the change in the local phonon spec-
trum. In particular, we observe an exponent 4
:
48
,
n
,
4
:
59 (2
σ
coverage) in the suspended membrane in the plateau temperature
region corresponding to the phononic 2D-to-3D crossover, in
agreement with a recent theoretical prediction.
In a larger context, our results exemplify how the notion of
electron-phonon coupling as a bulk property of materials breaks
down at low temperatures in microfabricated structures. For
example, an explicit accounting of the geometry-dependent
phonon mode spectrum might be needed for accurate modeling
of suspended transition-edge
sensors. We have also demon-
strated the applicability of supercurrent thermometry for accu-
rate quantitative thermal characterization of nanostructures. We
envision similar techniques to be immediately useful in investi-
gations of, e.g., novel van der Waals materials that hold great
potential for future detector applications.
31
ACKNOWLEDGMENTS
We thank D.-V. Anghel for important discussions during the
preparation of the manuscript. This material is based upon work
supported by, or in part by, the U.S. Army Research Laboratory
and the U.S. Army Research Office under Contract Nos.
W911NF-13-1-0390 and W911NF-18-1-0028, and the Academy of
Finland under Contract No. 312057.
APPENDIX A: SIMULATION OF PLATFORM
DEFORMATION
We estimate the residual stress in the films, as it has the
potential to affect the electronic and phononic structure, and hence
the thermal properties. A literature value for thermally grown SiO
2
is 2

10
10
dyn cm

2
compressive,
32
and for evaporated metals,
one finds 2
...
9

10
9
dyn cm

2
tensile.
33
The stress forces
become evident in the deformation of the structure after the under-
lying Si substrate is removed. Ignoring the effect of the metal films
and using
E
¼
70 GPa as Young
s modulus of SiO
2
, we predict an
engineering strain
ε
¼
0
:
029. The observed distortion is well
reproduced by a finite-element model where the nonsuspended sur-
roundings of the membrane are rigidly compressed by that
amount. The result of the finite-element analysis is shown in
Fig. 6
.
At the cryogenic temperatures of the experiment, the stress
and strain will be different due to thermal contraction. We did not
possess the capability to directly measure the deformation of the
membrane in the cryostat. Instead, we note the literature values of
the total linear thermal contraction (
l
0K

l
300 K
)
=
l
300 K
of the con-
stituent materials:
34

22

10

5
for Si, 8

10

5
for SiO
2
,

324

10

5
for Au, and

415

10

5
for Al. All of these are at
least an order of magnitude smaller than the strain induced by the
FIG. 5.
Electron-phonon power law. (a) The observed heating characteristic at
the base temperature for all four heater and thermometer combinations. (b)
Comparison of the applied heating power and the calculated electron-power
heat flow using the power law in Eq.
(4)
with best-fit parameters.
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intrinsic stress in the SiO
2
film and, therefore, will not significantly
alter the deformation observed in the SEM micrograph.
APPENDIX B: SECONDARY THERMOMETRY
In this section, we consider possible systematic errors that
arise in the measurements of thermal properties of nanostruc-
tures performed with secondary t
hermometers. By definition, a
primary thermometer is one that does not need calibration
against other thermometers. Primary thermometers are based on
a physical measurands with an
apriori
known temperature
dependence. A secondary thermometer can be constructed from
any temperature-dependent measurand by calibrating it against a
known thermometer. In this work, we use extensively secondary,
local electron thermometry that is based on the critical current
I
c
of an SNS weak link.
In principle, it is possible to calculate the
I
c
(
T
) dependence
from physical properties of the weak link. In practice, however, one
cannot independently and accurately determine all factors affecting
the measured critical current. Instead, one calibrates the empirical
I
c
against the cryostat thermometer, which is assumed to be trace-
able to primary temperature standards. Many practical thermome-
ters are nonlinear and suffer from loss of sensitivity at either end of
their usable temperature range. Both of these apply to the critical
current thermometer. For the following analysis, we only assume
the existence of a measurand
M
(
T
) that is a monotonic function of
the temperature
T
of the mesoscopic system of interest.
Virtually all low-temperature mesoscopic systems are suscepti-
ble to background heating. We use the term background heating to
refer to an inflow or energy to the system under study that is not
under the control of the experimentalist. For bolometerlike elec-
tronic systems, such as ours, a common source of background
heating is electronic noise that originates from hotter temperature
stages of the refrigerator, or from room temperature electronics.
Although filtering and shielding of the sample and experimental
wiring helps, the heating typically cannot be completely eliminated.
The question we wish to address here is whether systematic
errors can be avoided in a measurement that employs a nonlinear sec-
ondary thermometer and is subject to nonzero background heating.
1. Single thermal body
We denote the temperature of the mesoscopic electron system
by
T
e
. We assume that the experimentalist has control over the
cryostat temperature
T
b
and can apply an additional heating
P
ext
to
the electron system. We will assume that the background heating is
independent of both
T
e
and
T
b
, which is reasonable for external
noise sources and has magnitude
P
0
. We model the experiment
under steady state heating as
P
ext
þ
P
0
¼
h
(
T
e
)

h
(
T
b
),
(B1)
M
¼
M
(
T
e
),
(B2)
where the unknown function
h
describes the heat flow between
the electron system and the surrounding thermal bath. For a 3D
electron system cooled only by electron-phonon coupling, one
would have
h
(
T
)
¼
Σ
VT
5
. The thermometer measurand
M
is a
function of
T
e
only.
In the first step of the experiment, one performs a calibration
of
M
(
T
e
) by setting
P
ext
¼
0 and sweeping
T
b
. In all physically rele-
vant scenarios,
h
(
T
) grows at least quadratically (superlinear
growth is sufficient for the argument) and hence
T
e
tends to
T
b
at
high temperatures even in the presence of finite
P
0
. We establish a
temperature calibration
by associating the value
M
(
T
e
) of the meas-
urand with the temperature
T
b
.
Continuing the experiment, one obtains readings of the sec-
ondary electron thermometer by measuring
M
and inverting the
temperature calibration relation. It is evident that
T
meas
obtained
in this manner differs from the true
T
e
. Within our model, they
are related by
h
(
T
meas
)
¼
h
(
T
e
)

P
0
:
(B3)
To finalize our analysis, we rewrite the heat balance law Eq.
(B1)
in terms of
T
meas
instead of
T
e
.Wefind
P
ext
¼
h
(
T
meas
)

h
(
T
b
)
:
(B4)
In what appears to be a fortuitous coincidence, the background
heating term has been eliminated.
2. Three thermal bodies
The thermal model for the full experiment presented in this
work includes the two electron reservoirs and the phonon system
as independent thermal bodies. We write
P
ext
þ
P
0,
e
1
¼
h
e
1
(
T
e
1
)

h
e
1
(
T
p
),
(B5)
P
0,
e
2
¼
h
e
2
(
T
e
2
)

h
e
2
(
T
p
),
(B6)
FIG. 6.
Simulated deformation of the approximate large membrane geometry
under isotropic strain
ε
¼
0
:
029 applied to the support frame. The lateral size
of the mesh is approximately 2
μ
m. To be compared with
Fig. 4(b)
.
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024307-7
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P
0,
p
¼
h
e
1
(
T
p
)

h
e
1
(
T
e
1
)

þ
h
e
2
(
T
p
)

h
e
2
(
T
e
2
)

þ
h
p
(
T
p
)

h
p
(
T
b
),
(B7)
M
1
¼
M
1
(
T
e
1
),
(B8)
M
2
¼
M
2
(
T
e
2
)
:
(B9)
Above, subscripts
e
1 and
e
2 refer to the two electron reservoirs and
p
to the phonon system. We allow each of the three heat links to be
described by a different power law
h
i
(
T
)
¼
A
i
T
n
i
. The experiment
does not allow direct phonon heating or phonon thermometry. The
two mesoscopic thermometers
M
1
and
M
2
sense the local elec-
tronic temperatures
T
e
1
and
T
e
2
. To determine the electron-phonon
coupling in reservoir
e
1, it is sufficient to apply electron heating is
only to
e
1, as we will demonstrate shortly. Repeating the previous
analysis, we model the full experimental protocol in two steps.
First, to model temperature calibration, we zero
P
ext
and find
the self-consistent solution for (
T
e
1
,
T
e
2
,
T
p
) as a function of
T
b
.
These triplets should be interpreted as follows: After the calibration
for the two independent electron thermometers has been carried
out, the electron thermometer for reservoir
e
1(
e
2) indicates tem-
perature
T
b
when the true physical temperature is
T
e
1
(
T
e
2
). In the
following, we will denote this temperature reading by
T
meas
,
e
1(
e
2)
.
Note that the apparent temperatures are always lower than the
physical temperature if any of the
P
0
terms is finite.
Second, to model the phonon heating step, we sweep
P
ext
at a
fixed
T
b
, and solve for self-consistent (
T
e
1
,
T
e
2
,
T
p
). Utilizing the
temperature calibration model derived above, the simulated heater
sweep experiment consists of the triplets (
P
ext
,
T
meas
,
e
1
,
T
meas
,
e
2
).
The preceding single-body analysis is directly applicable to
the extraction of the phononic power law
h
p
(
T
). To make the
analogy exact, we note that Eqs.
(B5)
(B7)
yield for the phononic
thermal balance
P
ext
þ
P
0,
tot
¼
h
p
(
T
p
)

h
p
(
T
b
),
(B10)
where we have introduced
P
0,
tot
¼
P
0,
e
1
þ
P
0,
e
2
þ
P
0,
P
.Importantly,
Eq.
(B10)
is of the same form as Eq.
(B1)
.Notingthat
M
2
is
determined uniquely by
T
p
(since
P
0,
e
2
is fixed), it fulfills the con-
dition for a nonlinear secondary thermometer for
T
p
.Thiscom-
pletes the analogy, implying that a power law
h
p
(
T
)canbe
determined exactly by fitting a model
P
ext
¼
A
p
T
meas
,
e
2
ðÞ
n
p
þ
B
(B11)
to the data.
For the analysis of the electron-phonon power law, Eqs.
(B5)
(B7)
can be algebraically manipulated to yield the exact relation
P
ext
¼
A
e
1
T
meas
,
e
1
ðÞ
n
p
þ
C
½
n
e
=
n
p

T
meas
,
e
2
ðÞ
n
p
þ
C
½
n
e
=
n
p
no
, (B12)
where the free parameter
C
stands for
P
0,
tot
=
A
p
. The phonon heat
law exponent
n
p
can be determined independently in the manner
described above and is, therefore, considered a known value.
3. Experimental data analysis
Experimental data with a single thermal body is analyzed by
fitting it with a model
P
ext
¼
AT
meas
ðÞ
n
þ
B
:
(B13)
In light of the preceding analysis, assuming that the true
h
(
T
)isa
power law, it is accurately reproduced by the first term of the fitted
model, and the constant term
B
equals

AT
n
b
.
For the three-body case, we find that the theoretically exact
algebraic expression
(B12)
leads to numerical instabilities when
used to fit experimental data. Instead, we use a simpler phenome-
nological model
P
ext
¼
A
[
T
meas
,1
ðÞ
n

T
meas
,2
ðÞ
n
]
þ
B
(B14)
and verify the validity of the fit by observing small residuals.
In conclusion, our analytical studies show that thermal power
laws can be determined from mesoscopic experiments using sec-
ondary thermometers even in the presence of unknown back-
ground heating offsets. We have verified the above analytical
conclusions with numerical simulations. However, experimental
statistical noise can introduce systematic biases when parameters
of nonlinear models are fitted.
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