Modification of electron-phonon coupling by micromachining and suspension
Olli-Pentti Saira,
1, 2,
a)
Matthew H. Matheny,
1
Libin Wang,
3
Jukka Pekola,
3
and Michael Roukes
1
1)
Condensed Matter Physics and Kavli Nanoscience Institute, California Institute of Technology, Pasadena,
CA 91125
2)
Computational Science Initiative, Brookhaven National Laboratory, Upton, NY 11973
3)
QTF Centre of Excellence, Department of Applied Physics, Aalto University, FI-00076 Aalto,
Finland
(Dated: 23 October 2019)
Weak electron-phonon interaction in metals at low temperatures forms the basis of operation for cryogenic
hot-electron bolometers and calorimeters. Standard power laws, describing the heat flow in the majority of
experiments, have been identified and derived theoretically. However, a full picture encompassing experimen-
tally relevant effects such as reduced dimensionality, material interfaces, and disorder is in its infancy, and
has not been tested extensively. Here, we study the electron-phonon heat flow in a thin gold film on a SiO
2
platform below 100 mK using supercurrent thermometry. We find the power law exponent to be modified
from 5
.
1 to 4
.
6 as the platform is micromachined and released from its substrate. We attribute this change
to a modified phonon spectrum. The findings are compared to past experiments and theoretical models.
I.
INTRODUCTION
Nanostructures at low temperatures exhibit strong
thermal response. This follows from their minuscule heat
capacities and thermal conductances, both of which di-
minish in response to the shrinking of the device dimen-
sions and reduction of the operating temperature
1
. Con-
sequently, 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 valuable tool for scientific and engineering purposes.
For detector applications, thermal characterization is vi-
tal for understanding and optimizing the detector perfor-
mance
2
. Recently, in basic research, thermal transport
has been used as a probe of otherwise elusive states of
matter such as strongly interacting quantum Hall sys-
tems
3
, and the phonon spectrum in micromachined sup-
ports
4
. Several thermal and thermoelectric signatures of
Majorana states have been identified theoretically
5–7
.
Despite the above, studies of mesoscopic thermal trans-
port are far outnumbered by studies of electrical trans-
port. We speculate this is because the experimental con-
figurations for quantitative thermal measurements tend
to be complex. The associated technological challenges
can be grouped into two categories. The first challenge is
the measurement of the relevant physical quantities tem-
perature, power, and energy. The measurement should
be accurate (
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 chal-
lenge is ensuring the non-invasiveness of the measure-
ment scheme. Attaching or operating the thermal probe
may induce additional dissipation, or increase the ther-
mal conductance or heat capacity of the target body,
skewing the results.
a)
Electronic mail: osaira@bnl.gov
In this work, we advance the SNS 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 elec-
tron gas and a quasi-2D phonon system. Our main find-
ing is a modified electron-phonon heat flow
̇
Q
∝
T
n
el
−
T
n
ph
with
n
≈
4
.
6, where
T
el
and
T
ph
denote the electron and
phonon temperatures, respectively. Very recent theoret-
ical analysis
12
of the 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 that employed
different materials, sample geometry, and thermometry
method from ours
13,14
.
II.
METHODS
A.
Physics of supercurrent thermometry
Any physical phenomenon with a temperature depen-
dence can be used as a thermometer after calibration
against a known thermometer. This is called secondary
thermometry.
Here, we employ proximity supercon-
ductivity induced in a metallic SNS (superconductor–
normal–superconductor) weak link
15,16
. The switching
current offers a virtually ideal electrical characteristic for
steady-state thermometry using a simple readout circuit.
The temperature range of the sensor is limited, but dif-
ferent temperatures can be targeted by choosing an ap-
propriate length for the weak link.
The electrical behavior of the SNS weak link can be
analyzed in terms of the RCSJ model
17,18
. When probed
with a low-frequency current waveform, the junction
stays essentially in a zero-voltage state until the probe
arXiv:1910.10126v1 [cond-mat.mes-hall] 22 Oct 2019
2
1
1.5
2
Length (
m)
1
10
I
c
R
n
(
V)
Fit:
l = 0.44
m
A = 63
V
-0.02
0
0.02
I - I
50
(
A)
0
0.5
1
Switching probability
5
10
15
Current (
A)
0
0.5
1
Switching probability
(a)
(c)
(d)
0
40
80
120
6
8
10
12
T (mK)
I
50
(μA)
(b)
FIG. 1. Basic characterization of diffusive SNS links. (a)
Switching probability histograms at different temperatures
for an
l
= 1
.
5
μ
m weak link. (b) A high-resolution scan of
histogram at 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 cur-
rent using fitted parameter values (line). (d) Measured base-
temperature
I
c
R
N
product for three different wire lengths,
and an exponential fit.
current exceeds the temperature-dependent critical cur-
rent of the junction. Crucially, the dissipation in the
zero-voltage state is vanishingly small. After the junction
enters a finite-voltage state, positive electro-thermal feed-
back brought upon by the current bias rapidly heats up
the junction to a temperature where
I
c
= 0 and
V
=
R
n
I
.
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 amplitude. In Fig. 1(a),
we have used a single cycle of a 100 ms period sinewave
to probe an SNS weak link with an approximate separa-
tion of 1.5
μ
m between the superconducting electrodes at
temperature ranging from 10 mK to 120 mK. The sam-
ple 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 quantify 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
8,19
E
Th
=
~
D/L
2
eR
eff
I
c
/E
Th
=
a
(
1
−
be
−
aE
Th
/
(3
.
2
k
B
T
)
)
(1)
where
E
Th
denotes the Thouless energy,
D
is the diffusion
constant,
L
is the physical length of the weak link,
R
eff
is the normal-state resistiance,
I
c
is the critical current,
T
is the temperature, and
a
= 10
.
82,
b
= 1
.
3 are nu-
merical constants. A separate four-wire characterization
yields
D
= 240 cm
2
s
−
1
for the Au film, and
R
N
= 0
.
27 Ω
for this particular weak link. The other parameters ap-
pearing in the theory can be fitted to be
L
= 2
.
2
μ
m and
R
eff
= 2
.
7 Ω [Fig. 1(c), line]. In a rectangular wire ge-
ometry, the ratio
R
N
/R
eff
≈
0
.
1 could be interpreted as
a measure of the interface transparency
9,20
. However, in
the more complex geometry employed here, there is ad-
ditional 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 temperature responsivity
|
dI
mean
/dT
|
, which
peaks at 70 nA
/
mK at
T
= 70 mK. The detector is ca-
pable of resolving temperature differences of the order of
0.2 mK in its most sensitive range of temperatures with
a single readout pulse.
Another characteristic prediction of the diffusive the-
ory is the strong length dependence of the induced su-
perconductivity. We investigate this by determining the
base temperature
I
c
R
N
product for SNS wires of 500 nm
width and varying length [Fig. 1(d)]. The data is con-
sistent with an exponential decay with a characteristic
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 de-
tection of voltage pulses. However, as long as the same
excitation and detection methods are used throughout
the experiment, secondary thermometry incurs no sys-
tematic bias. Two-level composite pulses have been used
in previous works to greatly improve the temporal accu-
racy of the supercurrent probing
21
. This work deals ex-
clusively with steady-state quasi-equilibrium thermome-
try, 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 es-
timate the magnitude of temperature-dependent current
noise that reaches that junction. The twisted-pair mea-
surement lines had two RC-filter banks with
R
= 600 Ω
and (
RC
)
−
1
= 2
π
×
100 kHz each at the 4K 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 re-
sistive 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
3
HEAT(+)
Cryostat
V
heat
R
bias
R
cold
V
SNS
Sample
1.2 kΩ
10 MΩ
10 MΩ
100 kΩ
1.2 kΩ
1.2 kΩ
1.2 kΩ
1.2 kΩ
1.2 kΩ
1.2 kΩ
1.2 kΩ
HEAT(-)
Probe
FIG. 2. Electrical schematic for measurements reported in
this letter. The output voltage of the HEAT(
−
) source is
always the negative of output voltage of the HEAT(+) source.
The amplifiers are high input-impedance (
R
in
>
1000 MΩ)
differential voltage preamplifiers.
box, we estimate the powder filter segment to strongly
attenuate signals above
f
c
= 100 MHz. We then estimate
the worst-case rms current amplitude of noise from the
sample box wiring reaching the Josephson junction to be
I
rms
=
√
4
k
B
Tf
c
/
(2
R
) = 5 nA, where we substituted
T
= 0
.
12 K corresponding to the maximum temperature
encountered in this work.
B.
Heat conductance measurements
The subject of study in steady-state thermal measure-
ments is the heat flow between two thermal bodies in
quasi-equilibrium,
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
=
h
(
T
a
)
−
h
(
T
b
)
,
where
h
is a monotonically increasing function. In a lim-
ited temperature range, one often finds a
power law
̇
Q
a
→
b
=
A
(
T
n
a
−
T
n
b
)
,
(2)
where the exponent
n
often conveys information about
the dimensionality of the microscopic physics of energy
transport. Equivalently, one can study the thermal con-
ductance
G
(
T
) =
∂
̇
Q
a
→
b
∂T
a
∣
∣
∣
∣
∣
T
a
=
T
=
h
′
(
T
a
) =
nAT
n
−
1
,
(3)
where the last equality holds for any power law.
Throughout this manuscript, we will use exponent
n
to
refer to the heat flow power law [Eq. (2)]. Consequently,
the exponent for the thermal conductance will be
n
−
1.
Accurate experimental investigations of the above re-
lations require independent determination of the three
quantities
̇
Q
a
→
b
,
T
a
, and
T
b
. In some cases, one can sub-
stitute 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 dif-
ferences
|
T
b
−
T
c
|
, or if a non-zero direct thermal link be-
tween
b
and
c
exists, and
̇
Q
b
→
c
= 0 is known to sufficient
accuracy.
We will discuss how supercurrent thermometry allows
one to approach the ideal thermal measurement in prac-
tice with a relatively simple measurement setup. The
electrical connections for a single-body electron ther-
mometry experiment are illustrated in Fig. 2. The metal-
lization pattern on the chip, corresponding to the ”Sam-
ple” 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, the largest
resistance on the chip is the heater wire,
R
heat
≈
10 Ω.
For direct electron heating and thermometry, a galvanic
connection between the heater and thermometer is re-
quired. The electrical resistance
R
i
of this connection is
less than 1 Ω. In the following, we show that the amount
of current flowing through this connection is negligible.
Due to the low impedance of the sample, it is straight-
forward to realize an accurate low-frequency current
bias using external voltage sources and large bias resis-
tors. Operating the voltage sources in such a manner
that
V
heat
(
±
)
=
±
V
heat
ensures that the heating cur-
rent
I
heat
=
V
heat
/R
bias,heat
flows through the resistor
R
heat
. To the 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
=
V
probe
/R
bias,probe
, and
only a small fraction of it flows through
R
i
or
R
heat
.
More quantitatively, the relative 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: If a fraction
α
of
I
heat
were to flow
through
R
i
and the JJ element, one would observe a
switching current
I
c
±
αI
heat
that depends on the po-
larity of the heating current. We find
|
α
|
<
1%. Con-
secutively, the maximum leakage current encountered in
the experiment is less than 2 nA, contributing to a maxi-
mum of 4
×
10
−
18
W of unaccounted heating. To further
4
null the effect of the heating current leakage, and any
other unaccounted 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 volt-
age preamplifier. The heating power in the target elec-
tron gas can be then evaluated as
P
=
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
=
R
heat
I
2
heat
instead. Fi-
nally, we note that our use of leads made of only su-
perconducting materials ensures that all on-chip dissipa-
tion takes place in the normal metal reservoirs, and that
electronic heat conductance along the leads is suppressed
to a negligible 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 us-
ing e-beam lithography, e-beam evaporation, and lift-off
processes. In the normal-state film, the superconduct-
ing character of the Ti adhesion layer can be expected
to be completely suppressed by the inverse proximity ef-
fect. The surface of the Au film was cleaned with in-situ
Ar ion milling before the superconducting contacts were
formed. To create the mechanically suspended struc-
tures, the SiO
2
film was first patterned using e-beam
lithography and Ar ion milling. Finally, the SiO
2
plat-
form and the metallic structures were released by remov-
ing the underlying Si with an isotropic XeF
2
etch.
III.
THERMAL MEASUREMENTS
A.
Electron-phonon coupling in bulk
Early work on measurement of current-induced dise-
quilibrium between electrons and phonons was carried
out using local noise thermometry
22
. We realize a simi-
lar non-suspended device geometry with heater and ther-
mometer connections [Fig. 3(a), only one half of the
device was tested].
We find
n
= 5
.
07 (
σ
= 0
.
03);
A
= 2
.
23
×
10
−
8
WK
−
n
(
σ
rel
= 0
.
01) [Fig. 3(b)]. The
fitted exponent is very close to the value
n
= 5 cor-
responding to a clean-limit 3D metal
23
. We note that
the calculation presented in Ref. 24 predicts a slightly
20
40
60
80
T (mK)
10
-13
10
-12
10
-11
G (W/K)
10
0
10
2
P
heater
(fW)
25
40
60
80
100
120
T (mK)
10 μm
5 μm
(a)
(b)
(d)
(c)
n = 5.07
± 0.05
n - 1
= 1.95
... 2.05
FIG. 3. Verification experiments.
Electron-phonon coupling
on bulk substrate:
(a) SEM image of a device nominally iden-
tical 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 conduc-
tance datasets. Data from the small membrane [device in
panel (c), magenta markers], and the large membrane [device
in Fig. 4(b), red and blue markers corresponding to two in-
dependent measurements]. Thermal conductance obtained as
the local numerical derivative of the measured heating char-
acteristic. Lines are power-law fits to the data with a back-
ground heating offset.
higher temperature-dependent exponent for a thin metal-
lic film on a bulk dielectric. The nominal volume of
the gold island is
V
= 57
μ
m
2
×
50 nm = 2
.
9
μ
m
3
.
The fitted
A
yields an electron-phonon coupling constant
Σ =
A/V
= 7
.
7
×
10
9
WK
−
n
m
−
3
. We have neglected the
effect of the Ti adhesion 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. 25, which employed
a much thinner and more disordered film (
d
= 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
= 2
μ
m and length
l
= 7
.
5
μ
m. The
geometry factor is 4
w/l
= 1
.
07. We find
n
= 2
.
98 (
σ
=
0
.
03);
A
= 2
.
46
×
10
−
10
WK
−
n
(
σ
rel
= 0
.
02) [magenta
data in Fig. 3(d)].
The larger membrane [Fig.4 (b)] has four support legs
5
with a nominal width of 4
.
5
μ
m and length 10
.
25
μ
m. We
find
n
= 3
.
05 (
σ
= 0
.
02);
A
= 2
.
62
×
10
−
10
WK
−
n
(
σ
rel
=
0
.
01) using the Right heater/Left thermometer combina-
tion [red data in Fig 3(d)], and
n
= 3
.
01 (
σ
= 0
.
02);
A
= 2
.
37
×
10
−
10
WK
−
n
(
σ
rel
= 0
.
01) using the Left
heater/Right thermometer combination [blue data in
Fig 3(d)].
The fitted exponents are very close to the integer value
n
= 3 (corresponding to
κ
∝
T
2
). This would indicate
that the phonons are quantized in the thickness direc-
tion, but not in the lateral direction. The prefactor
A
is appears to be independent of the geometry factor, in-
dicating that phonon scattering in the support legs is
weak. Phononic thermal conductance in suspended and
patterned structures with a similar geometry have been
analyzed theoretically in detail in Refs. 26 and 27.
C.
Electron-phonon coupling in a thin suspended system
To study the electron-phonon coupling on a suspended
membrane, 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 com-
pleted device. The upward buckling of the platform is
caused by stresses in SiO
2
layer from the film growth pro-
cess (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)-(g). The temperature calibration curves of the
two supercurrent thermometers are shown in Figs. 4(h).
The critical currents after suspending the membrane
are a factor 3–4 smaller than in the sample with bulk
dielectrics. We suspect this is caused by degradation of
the SNS weak link due to the additional processing steps
and ageing of the sample. While the principle of super-
current 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 postulate to follow a power law
̇
Q
=
A
(
T
n
el
−
T
n
ph
)
−
P
0
,
(4)
where the excess heating term
P
0
accounts for the back-
ground heat loads that are beyond experimental control
and are responsible for saturation of the thermometer
signal at low bath temperatures and low heating pow-
ers. Details of the thermometer calibration and data
analysis are presented in Appendix B. The dual-heater,
dual-thermometer configuration allows us to indepen-
dently measure the quantities
̇
Q
,
T
el
, and
T
ph
appear-
ing in the above equation. By swapping the roles of
the left and right halves of the nominally symmetric de-
sign, we obtain two independent measurements of the
power law. We find
n
= 4
.
55 (
σ
= 0
.
01);
A
= 3
.
78
×
10
−
9
WK
−
n
(
σ
rel
= 0
.
003) using Right heater data [Red
10 μm
Heater L
Heater R
SNS R
SNS L
Left
electron
gas
Joule
heating
Joule
heating
Right
electron
gas
Membrane phonons
Bulk phonons
(a)
(c)
(b)
-10
0
10
A
-10
0
10
A
-4
0
4
V
-10
0
10
A
-50
0
50
V
-4
0
4
V
-50
0
50
V
-10
0
10
A
Left
Right
20
40
60
80
T
mc
(mK)
2.5
3
3.5
I
sw
(
A)
(h)
(d)
(e)
(f)
(g)
I
sw
(μA)
T
mc
(mK)
20
40
60
80
FIG. 4. Experiment for studing electron-phonon coupling on
suspended membrane. (a) Annotated 3D rendering of sam-
ple. (b) SEM image of the measured device. (c) Thermal
block diagram of the experiment. (d)-(f) Base temperature
I-V characteristics of the thermometers and heating elements
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 devi-
ation (2
σ
) evaluated from 30 measurements per temperature
point.
markers in Figs. 5(a), (b)], and
n
= 4
.
55 (
σ
= 0
.
01);
A
=
3
.
75
×
10
−
9
WK
−
n
(
σ
rel
= 0
.
003) using Left heater data
[Blue markers in Figs. 5(a), (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 stud-
ies by Karvonen and Maasilta
13,14
, where they obtained
n
≈
4
.
5 at temperatures up to 0
.
4 K employing quasi-
particle thermometry on a SiN membrane. The fitted
value for the excess heating term is 0
.
9
×
10
−
16
W and
−
2
.
1
×
10
−
16
W for the Right and Left heater data re-
spectively. A negative
P
0
indicates that the background
heating was higher in the phonon temperature measure-
ment configuration compared to the electron temperature
measurement.
The theoretical work that most closely models our
experimental system is that by Anghel and cowork-
ers
12,28,29
, wherein they consider a sheet consisting of
6
10
-18
10
-16
10
-14
P (W)
20
30
40
50
60
70
80
T (mK)
10
-18
10
-16
10
-14
Applied heating (W)
10
-18
10
-17
10
-16
10
-15
10
-14
A (T
el
n
- T
ph
n
) - P
0
(W)
(a)
(b)
Electron Thermometry
Heat LEFT
Heat RIGHT
Phonon Thermometry
Heat LEFT
Heat RIGHT
Electron-Phonon
power law
Heat LEFT
(n = 4.55 ± 0.02)
Heat RIGHT
(n = 4.55 ± 0.02)
FIG. 5. Electron-phonon power law. (a) The observed heat-
ing characteristic at base temperature for all four heater and
thermometer combinations. (c) Comparison of the applied
heating power and the calculated electron-power heat flow
using the power law in Eq. (4) with best-fit parameters.
finite-thickness metal and dielectric layers. In Ref. 12
in particular, they extend their analysis to cover the
range of temperatures where the phonon dimensional-
ity 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
≈
~
c
t
/
(2
k
B
d
), where
c
t
is the transversal (shear wave)
speed of sound, and
d
is the total thickness of the mem-
brane. We estimate
T
C
for our device as
T
C
≈
~
2
k
B
(
d
SiO2
c
t,
SiO2
+
d
Ti
c
t,
Ti
+
d
Au
c
t,
Au
)
−
1
= 30 mK
.
Hence, our experiment covers the range
T/T
C
=
0
.
7
...
2
.
7. In the numerical calculation of Ref. 12, the
local power-law exponent
∂
ln
̇
Q
∂
ln
T
el
is found to lie between
4.5 and 4.7 for
T/T
C
= 2
...
10, constituting the plateau
region.
The quality of our experimental data does not per-
mit a reliable extraction of local exponents. However,
the global exponent given by an unweighted fit is dom-
inated by the data at higher temperatures, and there-
fore measures primarily the plateau exponent. To ver-
ify this, we have performed a separate power law fit
using only the data where
T
el
≥
2
T
C
= 59 mK, which
yields
n
= 4
.
56 (
σ
= 0
.
02) using Right heater data, and
n
= 4
.
51 (
σ
= 0
.
02) using Left heater data.
In conclusion, we have used an SNS critical current
thermometer to study electron-phonon coupling in a mi-
cromechanical 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 spectrum. In
particular, we have observed an exponent 4
.
51
≤
n
≤
4
.
56 in the suspended membrane in the plateau tem-
perature region corresponding to the phononic 2D-to-3D
crossover, in agreement with a recent theoretical predic-
tion.
Acknowledgments
We thank D.-V. Anghel for im-
portant discussions during the preparation of the
manuscript. This material is based upon work supported
by, or in part by, the U. S. Army Research Labora-
tory and the U. S. Army Research Office under con-
tracts W911NF-13-1-0390 and W911NF-18-1-0028, and
Academy of Finland under contract 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 litera-
ture value for thermally grown SiO
2
is 2
×
10
10
dyn cm
−
2
compressive
30
, and for evaporated metals one finds
2
...
9
×
10
9
dyn cm
−
2
tensile
31
. The stress forces be-
come evident in the deformation of the structure after
the underlying Si substrate is removed. Ignoring the
effect of the metal films, and using
E
= 70 GPa as
the Young’s modulus of SiO
2
, we predict an engineering
strain
=
−
0
.
029. The observed distortion is well repro-
duced by a finite-element model where the non-suspended
surroundings of the membrane are rigidly compressed by
that amount. The result of the finite-element analysis is
shown in Fig. 6.
7
FIG. 6. Simulated deformation of the approximate large
membrane geometry under isotropic strain
=
−
0
.
029 ap-
plied to the support frame. The lateral size of the mesh is
approximately 2
μ
m. To be compared with Fig. 4(b).
At the cryogenic temperatures of the experiment, the
stress and strain will be different due to thermal con-
traction. We did not possess the capability to directly
measure the deformation of the membrane in the cryo-
stat. Instead, we note the literature values of the to-
tal linear thermal contraction (
l
0 K
−
l
300 K
)
/l
300 K
of the
constituent materials
32
:
−
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 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 er-
rors that arise in the measurements of thermal prop-
erties of nanostructures performed with secondary ther-
mometers. By definition, a primary thermometer is one
that does not need calibration against other thermome-
ters. Primary thermometers are based on a physical
measurands with an
a priori
known temperature depen-
dence. A secondary thermometer can be constructed
from any temperature-dependent measurand by calibrat-
ing 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
) depen-
dence from physical properties of the weak link. In prac-
tice, however, one cannot independently and accurately
determine all factors affecting the measured critical cur-
rent. Instead, one calibrates the empirical
I
c
against the
cryostat thermometer, which is assumed to be traceable
to primary temperature standards. Many practical ther-
mometers 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
susceptible 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 bolometer-like electronic systems,
such as ours, a common source of background heating
is electronic noise that originates from hotter tempera-
ture stages of the refrigerator, or from room temperature
electronics. Although filtering and shielding of the sam-
ple and experimental wiring helps, the heating typically
cannot be completely eliminated.
The question we wish to address here is whether sys-
tematic errors can be avoided in a measurement that em-
ploys a non-linear secondary thermometer, and is subject
to non-zero 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 experi-
ment under steady state heating as
P
ext
+
P
0
=
h
(
T
e
)
−
h
(
T
b
)
(B1)
M
=
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
) = Σ
V T
5
. The
thermometer measurand
M
is a function of
T
e
only.
In the first step of the experiment, one performs a cal-
ibration of
M
(
T
e
) by setting
P
ext
= 0 and sweeping
T
b
.
In all physically relevant scenarios,
h
(
T
) grows at least
quadratically (superlinear growth is sufficient for the ar-
gument), and hence
T
e
tends to
T
b
at high temperatures
even in the presence of finite
P
0
. We establish a
temper-
ature calibration
by associating the value
M
(
T
e
) of the
measurand with the temperature
T
b
.
Continuing the experiment, one obtains readings of the
secondary electron thermometer by measuring
M
and in-
verting 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
) =
h
(
T
e
)
−
P
0
.
(B3)
8
To finalize our analysis, we rewrite the heat balance law
Eq. (B1) in terms of
T
meas
instead of
T
e
. We find
P
ext
=
h
(
T
meas
)
−
h
(
T
b
)
.
(B4)
In what appears to be a fortuitous coincidence, the back-
ground 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
+
P
0
,e
1
=
h
e
1
(
T
e
1
)
−
h
e
1
(
T
p
)
(B5)
P
0
,e
2
=
h
e
2
(
T
e
2
)
−
h
e
2
(
T
p
)
(B6)
P
0
,p
= [
h
e
1
(
T
p
)
−
h
e
1
(
T
e
1
)] +
(B7)
[
h
e
2
(
T
p
)
−
h
e
2
(
T
e
2
)] +
h
p
(
T
p
)
−
h
p
(
T
b
)
.
M
1
=
M
1
(
T
e
1
)
.
(B8)
M
2
=
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
) =
A
i
T
n
i
. The experiment does not al-
low direct phonon heating or phonon thermometry. The
two mesoscopic thermometers
M
1
and
M
2
sense the local
electronic 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 demon-
strate 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 temperature
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 exper-
iment consists of the triplets (
P
ext
,T
meas,e
1
,T
meas,e
2
).
The preceding single-body analysis is directly applica-
ble 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
+
P
0
,tot
=
h
p
(
T
p
)
−
h
p
(
T
b
)
,
(B10)
where we have introduced
P
0
,tot
=
P
0
,e
1
+
P
0
,e
2
+
P
0
,P
.
Importantly, Eq. (B10) is of the same form as Eq. (B1).
Noting that
M
2
is determined uniquely by
T
p
(since
P
0
,e
2
is fixed), it fulfills the condition for a non-linear sec-
ondary thermometer for
T
p
. This completes the anal-
ogy, implying that a power law
h
p
(
T
) can be determined
exactly by fitting a model
P
ext
=
A
p
(
T
meas,e
2
)
n
p
+
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
=
A
e
1
{
[(
T
meas,e
1
)
n
p
+
C
]
n
e
/n
p
−
[(
T
meas,e
2
)
n
p
+
C
]
n
e
/n
p
}
,
(B12)
where the free parameter
C
stands for
P
0
,tot
/A
p
. The
phonon heat law exponent
n
p
can be determined inde-
pendently in the manner described above, and is there-
fore considered a known value.
3.
Experimental data analysis
Experimental data with a single thermal body is ana-
lyzed by fitting it with a model
P
ext
=
A
(
T
meas
)
n
+
B.
(B13)
In light of the preceding analysis, assuming the true
h
(
T
)
is a 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 in-
stabilities when used to fit experimental data. Instead,
we use a simpler phenomenological model
P
ext
=
A
[(
T
meas,
1
)
n
−
(
T
meas,
2
)
n
] +
B,
(B14)
and verify the validity of the fit by observing small resid-
uals.
In conclusion, our analytical studies show that thermal
power laws can be determined from mesoscopic experi-
ments using secondary thermometers even in the presence
of unknown background heating offsets. We have verified
the above analytical conclusions with numerical simu-
lations. However, experimental statistical noise can in-
troduce systematic biases when parameters of non-linear
models are fitted.
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