We present a simple experimental scheme for estimating the cryogenic thermal transport properties of thin films using superconducting nanowires. In a parallel array of nanowires, the heat from one nanowire in the normal state changes the local temperature around adjacent nanowires, reducing their switching current. Calibration of this change in switching current as a function of bath temperature provides an estimate of the temperature as a function of displacement from the heater. This provides a method of determining the contribution of substrate heat transport to the cooling time of superconducting nanowire single-photon detectors. Understanding this process is necessary for successful electrothermal modeling of superconducting nanowire systems

Allmaras, J. P.

Beyer, A. D.

Briggs, R. M.

Kozorezov, A. G.

Marsili, F.

Shaw, M. D.

Caltech Authors - Main

Journal of Low Temperature Physics (2018) 193:380–386
https://doi.org/10.1007/s10909-018-2022-0
Thin-Film Thermal Conductivity Measurements Using
Superconducting Nanowires
J. P. Allmaras1,2 · A. G. Kozorezov3 · A. D. Beyer2 · F. Marsili2 ·
R. M. Briggs2 ·M. D. Shaw2
Received: 7 November 2017 / Accepted: 3 July 2018 / Published online: 24 July 2018
© This is a U.S. Government work and not under copyright protection in the US; foreign copyright protection
may apply 2018
Abstract
We present a simple experimental scheme for estimating the cryogenic thermal trans-
port properties of thin films using superconducting nanowires. In a parallel array of
nanowires, the heat from one nanowire in the normal state changes the local temper-
ature around adjacent nanowires, reducing their switching current. Calibration of this
change in switching current as a function of bath temperature provides an estimate of
the temperature as a function of displacement from the heater. This provides a method
of determining the contribution of substrate heat transport to the cooling time of super-
conducting nanowire single-photon detectors. Understanding this process is necessary
for successful electrothermal modeling of superconducting nanowire systems.
Keywords Thermometry · Superconductor · Nanowire
1 Introduction
Superconducting nanowire single-photon detectors (SNSPDs) [1] developed for
near-infrared wavelengths are an exciting detector technology capable of efficiency
exceeding 90% at 1550 nm [2], timing jitter below 5 ps [3], and dark count rates at a
few counts per second [4]. By exploiting these properties, SNSPDs have enabled labo-
ratory experiments of quantum sciences [4, 5], improved quantum key distribution [6],
and demonstrated optical communication from satellites [7]. Despite laboratory and
commercial successes, there are still unresolved questions regarding the fundamental
B J. P. Allmaras
jallmara@caltech.edu
1 Department of Applied Physics and Materials Science, California Institute of Technology,
Pasadena, CA, USA
2 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA
3 Department of Physics, Lancaster University, Lancaster, UK
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Journal of Low Temperature Physics (2018) 193:380–386 381
physics of SNSPDs. The exact details of the photo-detection mechanism are debated in
the literature [8–10], and the precise material properties of common SNSPD materials
are not fully understood.
One area of SNSPD theory that has gained attention over the past decade is
electrothermal behavior. These models attempt to describe the coupled electrical
and thermal behavior of SNSPDs. Electrothermal modeling has been successfully
employed to the NbN material system in order to describe device behavior such as
latching and afterpulsing [11, 12]. While successful in describing the polycrystalline
NbN system, attempts to model the behavior of amorphous WSi embedded in an
amorphous SiO2 dielectric have failed to match experimental measurements. All of
these previous modeling attempts have ignored the effect of substrate heating under
the assumption that the thermal conductivity of the substrate is sufficiently high to
prevent the substrate temperature from changing significantly.
It is commonly known that the thermal transport of thin films differs dramatically
from bulk materials. In the Casimir limit, the mean free path of phonons is limited
by the thickness of the thin film due to surface scattering and can reduce the thermal
conductivity by over order of magnitude compared to the bulk [13]. There is further
experimental evidence that the thermal conductivity of thin SiO2 layers can be reduced
even beyond the Casimir limit [14]. With this in mind, it is necessary to reconsider the
assumption that substrate heating does not play a role in the electrothermal dynamics
of embedded superconducting nanowires.
2 Experimental
2.1 Superconducting Nanowire Thermometry
The development of this thermometry technique is motivated by the desire to better
understand the thermal transport in sputtered dielectric thin films without needing to
drastically alter the SNSPD fabrication workflow. Arrays of WSi nanowires have been
demonstrated where parallel nanowires are fabricated in a co-wound structure [15].
While using normal metal resistors to measure temperature in a noise thermometry or
3ω setup might enable a more accurate measurement, such devices require a different
fabrication workflow and additional laboratory readout electronics.
The superconducting nanowire thermometry technique uses the temperature depen-
dence of superconducting nanowire switching currents to determine the temperature of
a substrate surrounding the nanowire. An array of parallel superconducting nanowires
is fabricated with a single wire acting as the heater while all others act as thermome-
ters. A heater nanowire is biased such that the entire length of the wire is in the normal
state and Joule heating in the wire raises the temperature of the substrate surrounding
the heater as shown in Fig. 1. The switching currents of the thermometer nanowires
are probed under various bias conditions of the heater and correspond to tempera-
tures in the thin film based on the temperature dependence of the switching currents.
Calibration curves of this temperature dependence are obtained for each thermometer
nanowire by using a heater on the cryostat cold plate to increase the temperature of
the device uniformly.
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382 Journal of Low Temperature Physics (2018) 193:380–386
Fig. 1 Superconducting nanowire thermometry scheme. Left shows the cross section of a typical device. A
single nanowire is biased in the normal state, heating the surrounding thin film and changing the switching
currents of the thermometer nanowires. Right shows how the calibrated switching current curve is used
to convert measured switching currents to estimated substrate temperatures surrounding each thermometer
nanowire (Color figure online)
2.2 Device Design
Nanowire arrays were fabricated from 5-nm-thick WSi films sputtered from a com-
pound target on a four-inch Si wafer with 240 nm of thermal oxide. Sixteen parallel
160-nm-wide and 270-µm-long nanowires were patterned with electron beam lithogra-
phy to act as the heater and thermometers. The length ensures a uniform 2D temperature
profile in the active region of the device. After patterning, the nanowires were passi-
vated with 110 nm of sputtered SiO2. Devices were fabricated with 1600- and 400-nm
nanowire pitches to understand the thermal transport at different length scales. Induc-
tors were patterned in series with the active region of the nanowire in order to slow the
reset time of the nanowires and prevent latching, which would reduce the switching
current of the nanowires [16].
2.3 SystemModel
As the motivation for this experiment is to determine an appropriate model for describ-
ing thermal transport in the substrate of SNSPD systems, we use a simple diffusion
and boundary resistance formulation for describing the heat flow in the device. Due to
its large thermal conductivity, the silicon substrate is approximated to have a uniform
temperature TSi which can be elevated above the bath temperature Tbath due to inter-
facial resistance between the silicon die and the gold-plated copper plate. A Cernox
thermometer and resistive heater are used to stabilize the bath temperature at a fixed
value for the measurements. Diffusion in the SiO2 layer is modeled according to the
simple kinetic formula of the thermal conductivity given by Eq. (1).
κSiO2(T ) 
1
3
υl(T )C(T ) (1)
In this form, υ is the phonon mode-averaged sound velocity, l(T ) is the phonon
mean free path, and C(T ) is the heat capacity. The mean free path is split into two
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Journal of Low Temperature Physics (2018) 193:380–386 383
contributions according to l−1(T )  l−1bulk(T )+l−10 where lbulk(T ) describes the phonon
mean free path determined by the same mechanisms that govern the bulk thermal
conductivity and l0 is the temperature-independent mean free path specific to a thin
film. The temperature-dependent bulk mean free path is estimated from literature
values of the bulk thermal conductivity and heat capacity according to Eq. (2).
lbulk(T )  3κBulk(T )
υC(T ) (2)
The interfacial boundary resistance between the SiO2 and Si is modeled according
to a Kapitza boundary with a boundary condition given by Eq. (3).
κSiO2 (T )∇T · nˆ
∣
∣
interface  −R1
(
T r1 − T r1Si
) (3)
The fitting parameters R1 and r1 are used to model the boundary resistance with
the expectation that r1 equals four according to the acoustic mismatch model. The
interface between the silicon and gold-plated copper sample mount is bonded with
GE Varnish, leading to a boundary resistance and silicon temperature with the form
of Eq. (4).
TSi 
(
Ptotal
R2
+ T r2bath
)1/r2
(4)
The fitting parameters R2 and r2 are used to model the boundary resistance and
Ptotal is the total heat dissipated in the heater nanowire. The Kapitza boundary between
the SiO2 and Si is expected to have the same behavior for both the 1600-nm-pitch
and 400-nm-pitch devices. However, because the two separate dies were mounted at
different times on the sample mount, the fitting parameters used to model interfacial
boundary resistance between the silicon and copper plate are expected to change.
The fitting parameters are the SiO2 frequency-independent mean free path, SiO2–Si
Kapitza boundary parameters, and the silicon temperature parameters. Thermal energy
from the heater nanowire is introduced as a time-independent heat source. The thermal
conductivity differential equation and boundary conditions are solved using COMSOL
to find a steady-state temperature distribution.
3 Results
Nanowires were biased with a low noise voltage supply and bias resistor in series with
the nanowire. The current was calculated by dividing voltage across the bias resistor
by the bias resistance. Current–voltage curves of the heater nanowire were used to
extract the total power dissipation at a given bias condition, the power dissipated per
unit length of the nanowire, and the square resistance of the nanowire in the normal
state. Thermometry data were measured for heater powers ranging 1.9–23 nW/µm for
the 1600-nm-pitch device and 0.7–11 nW/µm for the 400-nm-pitch device.
The thermometry results were fitted to the heat transfer model described above to
estimate the thermal conductivity and Kapitza boundary resistance properties of the
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384 Journal of Low Temperature Physics (2018) 193:380–386
Fig. 2 Experimental data and modeling fits. Circles indicate the measured data while the lines are the model
fit. Left shows the 1600-nm-pitch data taken with a bath temperature of 950 mK. There is a reasonable fit
for all data points when using the fit parameters l0 180.2 nm, r1 5, R1 66.1 W/m2K5, r2 4, and
R2 3.70 W/K4. Right shows the 400-nm-pitch data taken at a bath temperature of 1000 mK. The fitting
parameters are l0 96.7 nm, r1 5, R1 66.1 W/m2K5, r2 4, and R2 2.92 W/K4. No set of fitting
parameters is able to match all of the experimental data for the 400-nm-pitch device (Color figure online)
SiO2 films. The best fit to the data required a SiO2–Si Kapitza power r1 of approx-
imately 5 rather than the expected 4 based on the acoustic mismatch model. The
optimized fits with fixed parameters of r1 5, R1 66.1 W/m2K5, and r2 4 are
shown in Fig. 2. The 1600-nm-pitch data model matches the experimental data for all
thermometry data points, while the 400-nm-pitch data cannot be reproduced for the
thermometry data nearest to the heater.
Fitting the model to the two data sets shows that the fixed frequency mean free path
varies significantly between the 400-nm-pitch and 1600-nm-pitch devices. For the
1600-nm-pitch results, the thermal conductivity is approximately 51% of the expected
Casimir limited thermal conductivity based on the total thickness of the SiO2 layer. In
contrast, the 400-nm-pitch results show a thermal conductivity significantly smaller
at 28% of the expected Casimir limit. This result is consistent with the trend of the
literature reported values of the thermal conductivity of SiO2 in thin films [13]. The
drop in thermal conductivity can be understood as decreasing the mean free path of
phonons due to the increased density of nanowires in the film which act as scattering
surfaces for the phonons. This effect dominates any potential increase in conductivity
due to the presence of the metal due to the small thickness fraction of the nanowire
compared to the dielectric and the Kapitza boundary between metal and dielectric.
4 Conclusion
Our nanowire thermometry experiments indicate that the effective thermal conduc-
tivity of the SiO2 thin films used in the optical cavities of SNSPDs is significantly
smaller than the bulk thermal conductivity and even smaller than the Casimir lim-
ited thermal conductivity expected from boundary scattering at the film interfaces.
The effects of substrate heating are currently neglected in electrothermal models of
nanowire systems, but based on these measurements, the substrate temperature can
123
Journal of Low Temperature Physics (2018) 193:380–386 385
increase significantly at the power dissipation levels seen in SNSPD operation. As
an approximate example, if the inductive energy (LI2/2) of a typical WSi single pixel
biased at 8 µA with an electrical reset time constant of 25 ns is dissipated in a rise time
of 2 ns over a 2 µm hotspot length, the corresponding power dissipation is 10 nW/µm.
While the model presented above improves upon the description of the substrate in
an SNSPD, it inadequately describes the energy transfer within the SiO2 thin film
near the heater nanowire, as seen by the deviation from experiment. This may be due
to additional fluctuations which suppress the switching current of the adjacent probe
nanowire. Furthermore, the model does not consider any transient effects present on
the nanosecond timescales relevant to thermal dissipation during a detection event. At
the cost of significant complexity, a model based on the Boltzmann transport equations
[17, 18] may be necessary to resolve these shortcomings. Despite the model’s current
limitations, our experiments suggest that an accurate electrothermal model capable of
predicting the hot spot current and latching behavior of SNSPDs must consider the
effects of substrate heating when wires are embedded in dielectric thin films.
Acknowledgements This work was supported by a NASA Space Technology Research Fellowship (Grant
No. NNX16AM54H). AGK, FM, and MDS acknowledge financial support from DARPA. This work was
carried out at the Jet Propulsion Laboratory under contract with the National Aeronautics and Space Admin-
istration.
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Thin-Film Thermal Conductivity Measurements Using Superconducting Nanowires

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Thin-Film Thermal Conductivity Measurements Using Superconducting Nanowires

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