# Current-Crowding-Free Superconducting Nanowire Single-Photon Detectors via
Local Helium Ion Irradiation

Stefan Strohauer<EMAIL_ADDRESS>Walter Schottky Institute, Technical
University of Munich, 85748 Garching, Germany  TUM School of Natural Sciences,
Technical University of Munich, 85748 Garching, Germany  Fabian Wietschorke
Christian Schmid  Walter Schottky Institute, Technical University of Munich,
85748 Garching, Germany  TUM School of Computation, Information and
Technology, Technical University of Munich, 80333 Munich, Germany  Stefanie
Grotowski  Walter Schottky Institute, Technical University of Munich, 85748
Garching, Germany  TUM School of Natural Sciences, Technical University of
Munich, 85748 Garching, Germany  Lucio Zugliani  Walter Schottky Institute,
Technical University of Munich, 85748 Garching, Germany  TUM School of
Computation, Information and Technology, Technical University of Munich, 80333
Munich, Germany  Björn Jonas  Walter Schottky Institute, Technical University
of Munich, 85748 Garching, Germany  TUM School of Computation, Information and
Technology, Technical University of Munich, 80333 Munich, Germany  Kai Müller
Walter Schottky Institute, Technical University of Munich, 85748 Garching,
Germany  TUM School of Computation, Information and Technology, Technical
University of Munich, 80333 Munich, Germany  Munich Center for Quantum Science
and Technology (MCQST), 80799 Munich, Germany  Jonathan J. Finley
<EMAIL_ADDRESS>Walter Schottky Institute, Technical University of Munich,
85748 Garching, Germany  TUM School of Natural Sciences, Technical University
of Munich, 85748 Garching, Germany  Munich Center for Quantum Science and
Technology (MCQST), 80799 Munich, Germany

(July 18, 2024)

# Current-Crowding-Free Superconducting Nanowire Single-Photon Detectors

Stefan Strohauer<EMAIL_ADDRESS>Walter Schottky Institute, Technical
University of Munich, 85748 Garching, Germany  TUM School of Natural Sciences,
Technical University of Munich, 85748 Garching, Germany  Fabian Wietschorke
Christian Schmid  Walter Schottky Institute, Technical University of Munich,
85748 Garching, Germany  TUM School of Computation, Information and
Technology, Technical University of Munich, 80333 Munich, Germany  Stefanie
Grotowski  Walter Schottky Institute, Technical University of Munich, 85748
Garching, Germany  TUM School of Natural Sciences, Technical University of
Munich, 85748 Garching, Germany  Lucio Zugliani  Walter Schottky Institute,
Technical University of Munich, 85748 Garching, Germany  TUM School of
Computation, Information and Technology, Technical University of Munich, 80333
Munich, Germany  Björn Jonas  Walter Schottky Institute, Technical University
of Munich, 85748 Garching, Germany  TUM School of Computation, Information and
Technology, Technical University of Munich, 80333 Munich, Germany  Kai Müller
Walter Schottky Institute, Technical University of Munich, 85748 Garching,
Germany  TUM School of Computation, Information and Technology, Technical
University of Munich, 80333 Munich, Germany  Munich Center for Quantum Science
and Technology (MCQST), 80799 Munich, Germany  Jonathan J. Finley
<EMAIL_ADDRESS>Walter Schottky Institute, Technical University of Munich,
85748 Garching, Germany  TUM School of Natural Sciences, Technical University
of Munich, 85748 Garching, Germany  Munich Center for Quantum Science and
Technology (MCQST), 80799 Munich, Germany

(July 18, 2024)

###### Abstract

Detecting single photons is essential for applications such as dark matter
detection, quantum science and technology, and biomedical imaging.
Superconducting nanowire single-photon detectors (SNSPDs) excel in this task
due to their near-unity detection efficiency, sub-Hz dark count rates, and
picosecond timing jitter. However, a local increase of current density
(current crowding) in the bends of meander-shaped SNSPDs limits these
performance metrics. By locally irradiating the straight segments of SNSPDs
with helium ions while leaving the bends unirradiated, we realize current-
crowding-free SNSPDs with simultaneously enhanced sensitivity: after
irradiation with 800, locally irradiated SNSPDs showed a relative saturation
plateau width of 37 while fully irradiated SNSPDs reached only 10. This larger
relative plateau width allows operation at lower relative bias currents,
thereby reducing the dark count rate while still detecting single photons
efficiently. We achieve an internal detection efficiency of 94 for a
wavelength of 780 with a dark count rate of 7 near the onset of saturating
detection efficiency.

current crowding, superconducting thin film, radiation damage, He ion
irradiation,

Superconducting nanowire single-photon detectors (SNSPDs) [1] are widely used
across fields such as quantum science and technology [2, 3, 4, 5, 6, 7, 8, 9],
astronomy [10], optical communication [11, 12], biology [13, 14], and medicine
[15]. Their ability to detect single photons with near-unity efficiency [16,
17], sub- dark count rate [18], picosecond timing jitter [19], and nanosecond
reset time [20] makes them ideally suited for demanding applications such as
photonic quantum computing [21, 22, 23, 24, 25, 26, 27], particle and dark
matter detection [28, 29, 30, 31], or infrared fluorescence microscopy for in
vivo deep brain imaging [13, 14]. One strategy to further enhance detection
efficiency, dark count rate, and timing jitter of SNSPDs is to reduce the
effect of current crowding in their bends. Current crowding locally increases
the current density in the bends, limiting the maximum applicable bias current
through the detector, and thus limiting also the maximum achievable detection
efficiency, dark count rate, and timing jitter [32, 33]. In literature,
methods to reduce the effect of current crowding consist of optimized bend
geometries [32, 34, 35, 36, 33], or increased superconductor thickness in the
bends [37, 38]. The use of optimized bend geometries is either limited to
relatively low-fill factor SNSPDs, which have lower absorption and detection
efficiency, or requires a spiral or special meander design with significant
wire length overhead, which is incompatible with dense SNSPD arrays and limits
the timing properties of the detector. In contrast, variable thickness SNSPDs
successfully reduce current crowding also for highly efficient and compact
large fill factor SNSPDs. In this work, we introduce a novel method to obtain
not only current-crowding-free SNSPDs of arbitrary geometry and fill factor
but also to simultaneously enhance their sensitivity.

##  I Current crowding at low temperatures

Current crowding describes a non-uniform distribution of current density in an
electrical conductor and has its origin for example in variations in material
properties or device geometry. In the context of SNSPDs, current crowding
leads to a local increase of the supercurrent density in the 180 bends and at
kinks or discontinuities of the SNSPD. Once this current density exceeds the
critical value for the superconductor, the SNSPD switches to the normal
conducting state. This maximum current defines the switching current
$I_{\mathrm{sw}}$. In other words, current crowding limits $I_{\mathrm{sw}}$
of the SNSPD.

It is worthwhile to note that the effect of current crowding is most
pronounced at low temperatures ($<0.7\,T_{\mathrm{c}}$), while at temperatures
close to $T_{\mathrm{c}}$ the switching current practically coincides for
straight and bent wires due to the divergence of the coherence length at
temperatures close to the critical temperature [39, 40]. To quantify the
effect of current crowding for our devices, we measured the temperature
dependence of the switching current for SNSPDs (in meander form with 180
bends) and straight wires (without any bend), both consisting of 250 wide and
8 thick NbTiN wires. As shown for two representative devices in Figure 1, the
switching current increases and the effect of current crowding becomes more
prominent towards low temperatures, reaching a saturating (relative)
difference between $I_{\mathrm{sw}}$ of the straight wire and the SNSPD at
temperatures below 1.6 (1). In fact, at temperatures below 1, the switching
current of NbTiN meander SNSPDs with a fill factor of 71 and 250 wide wires is
reduced to only 60 of $I_{\mathrm{sw}}$ of corresponding straight wires.

At the same time, a high switching current, as close as possible to the limit
set by the depairing current density, is desirable because it allows larger
plateaus of saturating detection efficiency [41, 19]. This makes operation
with high detection efficiency at simultaneously low dark count rates
possible. Moreover, high switching currents allow high bias currents through
the SNSPDs that yield high and easily detectable voltage pulses and low timing
jitter [38]. Since high switching currents are reached for low temperatures
where current crowding limits the maximum applicable bias current, a method to
avoid current crowding would be highly advantageous. Moreover, current-
crowding-free SNSPDs are expected to exhibit fewer dark counts since they
originate primarily from bends or constrictions with strong current crowding
[34, 42, 43, 37, 44]. Compared to methods presented in literature that solely
mitigate current crowding [32, 34, 35, 36, 33, 37, 38], the following section
introduces a solution to obtain current-crowding-free SNSPDs of arbitrary
geometry that simultaneously exhibit enhanced sensitivity.

Figure 1: Switching current versus temperature for an SNSPD and a straight
wire. Current crowding causes a reduced switching current for the SNSPD over
the straight wire, especially towards low temperatures.

##  II Design and concept of current-crowding-free SNSPDs with enhanced
sensitivity

To demonstrate current-crowding-free SNSPDs of enhanced sensitivity, we
combine the two effects of reduced switching current and enhanced sensitivity
of SNSPDs after helium (He) ion irradiation [45, 46]. As discussed in the
previous section, the 180 bends are the main contribution to current crowding
in a meander-type SNSPD, thus limiting its switching current. By locally
irradiating only the straight segments of the SNSPD while leaving its bends
unirradiated, we reduce the critical current density of the straight segments
while making them simultaneously more sensitive to single photons. Once the He
ion fluence is large enough, the critical current of the straight segments
with homogeneous current density will be lower than the critical current of
the bends. Then, one can expect the overall critical current (or switching
current) of the SNSPD to be given by the critical current of the straight
segments and not being limited by current crowding anymore. To investigate
this hypothesis, three types of devices as shown in Figure 2a were fabricated
on the same chip: SNSPDs that were _locally_ (only their straight segments) or
_fully_ irradiated, as well as _straight wires_ without any bend. They were
fabricated from an 8 thick NbTiN film on a Si substrate with 130 thermally
grown SiO2. The nominal design of the detectors consists of 250 wide nanowires
in meander form with 100 gaps (fill factor 71) and covers an active area of
[product-units = repeat]20 x 20. This high fill factor, along with the
nominally rectangular design of the bends between neighboring nanowires, was
chosen to have a pronounced effect of current crowding. The reference straight
wires were fabricated on the same chip, also with a wire width of 250. All
devices were characterized prior to irradiation by measuring their switching
current distributions and mean switching currents at a temperature of 1. After
characterization of the unirradiated devices, a He ion microscope (Zeiss Orion
Nanofab) with an acceleration voltage of 30 was used to irradiate them with He
ions. An irradiation area of [product-units = repeat]25 x 15 was centered on
the [product-units = repeat]20 x 20 detectors to homogeneously expose the
straight segments of the nanowires while leaving the bends unirradiated. In
this way, locally irradiated SNSPDs as shown in Figures 2a and 2b were
obtained. For the reference straight wires, an irradiation area of [product-
units = repeat]5 x 15 was chosen. Moreover, two detectors were fully
irradiated ([product-units = repeat]25 x 25 irradiation area). After
irradiation, the devices were characterized again, and for several devices the
process of irradiation and subsequent measurement was repeated multiple times.

Figure 2: Current-crowding-free SNSPDs. a, Local irradiation of the straight
detector segments is used to obtain current-crowding-free SNSPDs. Fully
irradiated SNSPDs and straight wires were fabricated as reference devices. b,
Scanning electron microscope image of an SNSPD after local irradiation. The
irradiation area is indicated by the white dashed rectangle. c, d, Switching
current versus He ion fluence for _locally_ as well as _fully_ irradiated
SNSPDs and straight wires for the low fluence regime (c) and for the full
measured range up a fluence of 1400 (d).

##  III Reducing current crowding by local He ion irradiation

As shown in Figures 2c and 2d, the mean switching current of locally
irradiated SNSPDs stays constant for doses up to 90+-20, while it continuously
decreases for fully irradiated SNSPDs and straight wires. As anticipated,
locally irradiated SNSPDs exhibit a constant $I_{\mathrm{sw}}$ until the He
ion fluence reaches a level that reduces $I_{\mathrm{sw}}$ of the straight
wires to comparable values. Beyond this point, $I_{\mathrm{sw}}$ of locally
irradiated SNSPDs and straight wires follow a similar curve. The slightly
smaller $I_{\mathrm{sw}}$ of locally irradiated SNSPDs compared to that of the
straight wires can be attributed to the higher likelihood of imperfections in
one of the many straight segments of an SNSPD, unlike in a single straight
wire.

Figure 3 shows the temperature dependence of the maximum switching current for
differently irradiated SNSPDs and straight wires, determined from the
underlying switching current distributions [47, 40]. The critical temperature
decreases continuously with increasing He ion fluence in accordance with
literature [46, 45] and does not depend on device type or irradiation type.
For decreasing temperature, the $I_{\mathrm{sw}}(T)$ curves start to differ
and show a smaller slope for fully irradiated SNSPDs compared to that of
locally irradiated SNSPDs and irradiated straight wires. Moreover, for similar
He ion fluences and low temperatures, $I_{\mathrm{sw}}$ is much smaller for
fully irradiated SNSPDs than for locally irradiated SNSPDs or straight wires.
For example, at temperatures below 1 and for a He ion fluence of 300 it is
only 30 instead of 40. While for a He ion fluence of 60 the switching current
of the straight wire is still higher than that of the locally irradiated
SNSPD, they almost coincide for the two fluences 110 and 130. Those
observations support the hypothesis that for small He ion fluences
$I_{\mathrm{sw}}$ of the locally irradiated SNSPDs is not reduced, while it
starts to coincide with that of the straight wires for fluences above 90+-20.
In this case, the irradiation induced reduction of the switching current in
the straight segments is the limiting factor for $I_{\mathrm{sw}}$ instead of
current crowding in the bends. The stronger the effect of current crowding in
the unirradiated device, the higher the He ion fluence required for local
irradiation such that current crowding no longer plays a role. Since current
crowding limits the switching current of SNSPDs more as the fill factor
increases, this method is particularly useful for high fill factor or micro-
scale SNSPDs, especially if the detector requires He ion irradiation anyway to
become single-photon sensitive [48, 49].

Figure 3: Switching current $I_{\mathrm{sw}}$ of straight wires (s. w.),
locally irradiated SNSPDs (local), and fully irradiated SNSPDs (full) versus
temperature. The He ion fluence used for irradiation is given by the numbers
in front of the device type in units of .

##  IV Enhanced sensitivity of current-crowding-free SNSPDs

Helium ion irradiation enhances the sensitivity of SNSPDs. However, since we
do not irradiate the whole SNSPD but intentionally leave the bends
unirradiated and, consequently, do not have the irradiation induced decrease
of switching current, we expect an enhanced performance due to two effects:
(1) the He ion irradiation induced higher sensitivity, and (2) the higher
available bias currents compared to fully irradiated SNSPDs of the same He ion
fluence. Figure 4a shows a comparison of the normalized count rates between
two devices, one irradiated locally and one irradiated fully, both with the
same He ion fluences and for photons of 780 wavelength. The locally irradiated
SNSPD clearly shows higher switching currents and larger saturation plateaus
of the count rates, for which the internal detection efficiency is unity.
Moreover, since the dark count rate at a certain bias current $I_{\mathrm{b}}$
mainly depends exponentially on the ratio $I_{\mathrm{b}}/I_{\mathrm{sw}}$,
the dark count rate of locally irradiated SNSPDs is lower than that of fully
irradiated SNSPDs at similar absolute bias currents or normalized count rates.
Since our SNSPDs exhibit very low dark count rates, we measured it over an
extended period of time for selected devices. As shown in Figure 4a for the
device locally irradiated with 400, we reach dark count rates of 7 and 0.55 at
internal detection efficiencies of 94 and 31, respectively. Also the two
regimes of intrinsic dark counts and dark counts originating from black body
radiation are clearly visible: the first is given by the strongly current
dependent contribution in the high current regime, while the second is the
weaker current dependent contribution at lower currents [50]. The black body
radiation induced dark count rate could be further suppressed by using cold
band-pass filters [51].

In Figure 4b we present the relative count rate saturation plateau width,
defined as

$\sigma_{\mathrm{rel}}=\frac{I_{\mathrm{c}}-I_{\mathrm{sat}}}{I_{\mathrm{c}}}\;,$
(1)

with the critical current $I_{\mathrm{c}}$, and $I_{\mathrm{sat}}$
representing the current where the saturation plateau begins (where the
normalized count rate reaches 0.95). The critical current is defined as the
smallest bias current where the SNSPD shows non-zero resistance. Since we use
a shunt resistor for the CR measurements to prevent latching of the detectors,
the SNSPDs transition to the relaxation oscillation regime at $I_{\mathrm{c}}$
before switching to the latching state [52]. Most importantly, the relative
saturation plateau width is higher for locally irradiated SNSPDs compared to
fully irradiated SNSPDs of the same He ion fluence. Furthermore, the relative
saturation plateau width increases strongest as a function of He ion fluence
for values smaller than 200 and, as elaborated in the supplementary, it peaks
between 600 and 1000 before it decreases again for higher fluences.

To assess the performance of the device locally irradiated with 800 for
optical communication in the C-band, we measured its normalized count rate for
photons with a wavelength of 1550 and compare it in Figure 4c with its
performance for 780 photons. For both wavelengths the detector shows
saturating internal detection efficiency, even for 1550 where the onset of
photon detection is shifted to higher bias currents due to the lower
sensitivity to longer wavelength photons. Since it is the same device, the
dark count rate is identical for both measurements.

Moreover, we found that the detection pulse height (relevant for assessing the
requirements of the readout electronics) of locally irradiated SNSPDs is
higher than that of fully irradiated SNSPDs of the same fluence, while the
pulse decay time (related to the detector’s dead time) is similar. These
quantities are further discussed in the supplementary.

Figure 4: Detection performance of SNSPDs after local or full irradiation with
He ions. a, Normalized count rate (solid symbols) and dark count rate (open
symbols) versus bias current after local (orange) or full (blue) irradiation
with He ions. The He ion fluence used for irradiation is shown as annotations
at the corresponding count rate curves. b, Relative saturation plateau width
versus He ion fluence after local or full irradiation, calculated according to
Equation 1. The solid lines serve as a guide to the eye. c, Comparison of the
normalized count rate between photons of 780 and 1550 wavelength for a
detector locally irradiated with 800. All shown dark count rate data was taken
such that the integration time per data point was at least 10 times higher
than the inverse of the measured dark count rate.

##  V Conclusions

Current crowding strongly limits the maximum bias current, sensitivity, and
dark count rate of SNSPDs at low temperatures. We demonstrated a method to
obtain current-crowding-free SNSPDs by irradiating the straight segments of
SNSPDs locally with He ions while leaving their bends unirradiated. Up to a He
ion fluence of 90+-20 the critical current is not reduced, meaning the photon
sensitivity is enhanced without sacrificing any of the SNSPD’s critical
current. Above this threshold, the straight segments of the locally irradiated
SNSPDs show a critical current lower than that of the bends such that current
crowding in the bends does not limit the overall critical current anymore.

The boost in performance for locally irradiated SNSPDs is twofold: (1) the
sensivitiy of the SNSPD is enhanced due to He ion irradiation, and (2) the
sensitivity is higher than for fully irradiated SNSPDs due to the higher
available bias currents. Using this approach, we achieved a relative
saturation plateau width of 37 for a locally irradiated SNSPD compared to 10
for a fully irradiated SNSPD, both irradiated with 800. This larger relative
plateau width means that the SNSPD can be operated at lower relative bias
currents with lower dark count rates and still detect single photons
efficiently (e. g., 7 dark count rate at an internal detection efficiency of
94). Therefore, this method is particularly beneficial for applications that
require high detection efficiency combined with low dark count rate, such as
quantum key distribution, brain imaging, and dark matter detection.
Furthermore, it is ideally suited for high fill factor or micro-scale SNSPDs
that are prone to current crowding, especially if the detectors require He ion
irradiation anyway to become single-photon sensitive [48]. By combining site-
selective [46] with local He ion irradiation, one can realize highly and
homogeneously performing SNSPD arrays. Moreover, local He ion irradiation can
be further used to study the effect of current crowding in different regions
of superconducting devices, including SNSPDs and the influence of current
crowding in their bends onto properties such as dark count rate, detection
efficiency, and switching current.

###### Acknowledgements.

We gratefully acknowledge support from the German Federal Ministry of
Education and Research (BMBF) via the funding program “Photonics Research
Germany” (projects MOQUA (13N14846) and MARQUAND (BN106022)) and the funding
program “Quantum technologies – from basic research to market” (projects
PhotonQ (13N15760), SPINNING (13N16214), QPIS (16K1SQ033), QPIC-1 (13N15855)
and SEQT (13N15982)), as well as from the German Research Foundation (DFG)
under Germany’s Excellence Strategy EXC-2111 (390814868) with the projects
PQET (INST 95/1654-1) and MQCL (INST 95/1720-1). This research is part of the
“Munich Quantum Valley”, which is supported by the Bavarian state government
with funds from the “Hightech Agenda Bayern Plus”.

## Methods

###  V.1 Fabrication of NbTiN SNSPDs

To fabricate SNSPDs and straight wires, we deposited an 8 thick NbTiN film
using DC reactive magnetron sputtering onto a Si substrate with a 130 thick
thermally grown SiO2 layer. The superconductor thickness was controlled by
measuring the sputtering rate and choosing the sputtering time
correspondingly. Subsequently, we patterned the SNSPDs and straight wires with
electron beam lithography and reactive ion etching, followed by contact pad
fabrication using optical lithograpy and gold evaporation [53]. The detector
design consists of 250 wide nanowires in meander form with 100 gaps (fill
factor 71) and covers an active area of [product-units = repeat]20 x 20. To
have a pronounced effect of current crowding, a nominally rectangular design
of the bends between neighboring nanowires of the SNSPDs was chosen. Similar
to the detectors, the straight wires were designed to be 250 wide and
fabricated on the same chip to ensure best comparability. After
characterization of the unirradiated devices, we used a He ion microscope
(Zeiss Orion Nanofab) for irradiation with He ions with an acceleration
voltage of 30. For the locally irradiated devices we chose an irradiation area
of [product-units = repeat]25 x 15, centered on the [product-units = repeat]20
x 20 detectors. Two detectors were fully irradiated with an irradiation area
of [product-units = repeat]25 x 25. For the reference straight wires, an
irradiation area of [product-units = repeat]5 x 15 was chosen. After
irradiation, we characterized the devices again and repeated the process of
irradiation and subsequent measurements for several devices multiple times.

###  V.2 Low-temperature measurements

All detectors were pre-characterized in a cryogenic probe station (Janis) at
4.5. Subsequently, devices exhibiting a similar switching current of 42+-2 at
4.5 were selected for wire bonding and further characterization at
temperatures down to 150 in a closed cycle cryostat with adiabatic
demagnetization refrigeration (kiutra GmbH). Further characterization
consisted of measuring the switching current of each device 10 – 250 times to
determine a switching current distribution and its mean switching current at a
temperature of 1. The switching current measurements were performed by ramping
the bias voltage applied at a 50 resistor in series to the respective device
and determining the current at which the total resistance increases by more
than 1. Temperature dependent switching current measurements were performed at
temperature ramp rates of 0.075/min, ensuring that during one bias voltage
sweep the temperature change was less than 0.1.

## References

  * Gol’tsman  _et al._ [2001] G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, Picosecond superconducting single-photon optical detector, Applied Physics Letters 79, 705 (2001).
  * Takesue  _et al._ [2007] H. Takesue, S. W. Nam, Q. Zhang, R. H. Hadfield, T. Honjo, K. Tamaki, and Y. Yamamoto, Quantum key distribution over a 40-dB channel loss using superconducting single-photon detectors, Nature Photonics 1, 343 (2007).
  * Chen  _et al._ [2022] J.-P. Chen, C. Zhang, Y. Liu, C. Jiang, D.-F. Zhao, W.-J. Zhang, F.-X. Chen, H. Li, L.-X. You, Z. Wang, Y. Chen, X.-B. Wang, Q. Zhang, and J.-W. Pan, Quantum Key Distribution over 658 km Fiber with Distributed Vibration Sensing, Physical Review Letters 128, 180502 (2022).
  * Liu  _et al._ [2023] Y. Liu, W.-J. Zhang, C. Jiang, J.-P. Chen, C. Zhang, W.-X. Pan, D. Ma, H. Dong, J.-M. Xiong, C.-J. Zhang, H. Li, R.-C. Wang, J. Wu, T.-Y. Chen, L. You, X.-B. Wang, Q. Zhang, and J.-W. Pan, Experimental Twin-Field Quantum Key Distribution over 1000 km Fiber Distance, Physical Review Letters 130, 210801 (2023).
  * Bussières  _et al._ [2014] F. Bussières, C. Clausen, A. Tiranov, B. Korzh, V. B. Verma, S. W. Nam, F. Marsili, A. Ferrier, P. Goldner, H. Herrmann, C. Silberhorn, W. Sohler, M. Afzelius, and N. Gisin, Quantum teleportation from a telecom-wavelength photon to a solid-state quantum memory, Nature Photonics 8, 775 (2014).
  * Takesue  _et al._ [2015] H. Takesue, S. D. Dyer, M. J. Stevens, V. Verma, R. P. Mirin, and S. W. Nam, Quantum teleportation over 100 km of fiber using highly efficient superconducting nanowire single-photon detectors, Optica 2, 832 (2015).
  * Shibata  _et al._ [2014] H. Shibata, T. Honjo, and K. Shimizu, Quantum key distribution over a 72 dB channel loss using ultralow dark count superconducting single-photon detectors, Optics Letters 39, 5078 (2014).
  * Valivarthi  _et al._ [2016] R. Valivarthi, M. G. Puigibert, Q. Zhou, G. H. Aguilar, V. B. Verma, F. Marsili, M. D. Shaw, S. W. Nam, D. Oblak, and W. Tittel, Quantum teleportation across a metropolitan fibre network, Nature Photonics 10, 676 (2016).
  * Natarajan  _et al._ [2012] C. M. Natarajan, M. G. Tanner, and R. H. Hadfield, Superconducting nanowire single-photon detectors: Physics and applications, Superconductor Science and Technology 25, 063001 (2012).
  * Wollman  _et al._ [2021] E. E. Wollman, V. B. Verma, A. B. Walter, J. Chiles, B. Korzh, J. P. Allmaras, Y. Zhai, A. E. Lita, A. N. McCaughan, E. Schmidt, S. Frasca, R. P. Mirin, S. W. Nam, and M. D. Shaw, Recent advances in superconducting nanowire single-photon detector technology for exoplanet transit spectroscopy in the mid-infrared, Journal of Astronomical Telescopes, Instruments, and Systems 7, 011004 (2021).
  * Grein  _et al._ [2015] M. E. Grein, A. J. Kerman, E. A. Dauler, M. M. Willis, B. Romkey, R. J. Molnar, B. S. Robinson, D. V. Murphy, and D. M. Boroson, An optical receiver for the Lunar Laser Communication Demonstration based on photon-counting superconducting nanowires, Advanced Photon Counting Techniques IX 9492, 949208 (2015).
  * Biswas  _et al._ [2017] A. Biswas, M. Srinivasan, R. Rogalin, S. Piazzolla, J. Liu, B. Schratz, A. Wong, E. Alerstam, M. Wright, W. T. Roberts, J. Kovalik, G. Ortiz, A. Na-Nakornpanom, M. Shaw, C. Okino, K. Andrews, M. Peng, D. Orozco, and W. Klipstein, Status of NASA’s deep space optical communication technology demonstration, in  _2017 IEEE International Conference on Space Optical Systems and Applications (ICSOS)_ (IEEE, Naha, 2017) pp. 23–27.
  * Xia  _et al._ [2021] F. Xia, M. Gevers, A. Fognini, A. T. Mok, B. Li, N. Akbari, I. E. Zadeh, J. Qin-Dregely, and C. Xu, Short-Wave Infrared Confocal Fluorescence Imaging of Deep Mouse Brain with a Superconducting Nanowire Single-Photon Detector, ACS Photonics 8, 2800 (2021).
  * Tamimi  _et al._ [2023] A. Tamimi, M. Caldarola, S. Hambura, J. C. Boffi, N. Noordzij, J. W. N. Los, A. Guardiani, H. Kooiman, L. Wang, C. Kieser, F. Braun, A. Fognini, and R. Prevedel, Deep mouse brain two-photon near-infrared fluorescence imaging using a superconducting nanowire single-photon detector array (2023).
  * Ozana  _et al._ [2021] N. Ozana, A. I. Zavriyev, D. Mazumder, M. B. Robinson, K. Kaya, M. H. Blackwell, S. A. Carp, and M. A. Franceschini, Superconducting nanowire single-photon sensing of cerebral blood flow, Neurophotonics 8, 035006 (2021).
  * Marsili  _et al._ [2012] F. Marsili, F. Bellei, F. Najafi, A. E. Dane, E. A. Dauler, R. J. Molnar, and K. K. Berggren, Efficient Single Photon Detection from 500nm to 5µm Wavelength, Nano Letters 12, 4799 (2012).
  * Korneev  _et al._ [2012] A. Korneev, Yu. Korneeva, I. Florya, B. Voronov, and G. Goltsman, NbN Nanowire Superconducting Single-Photon Detector for Mid-Infrared, Physics Procedia 36, 72 (2012).
  * Shibata  _et al._ [2015] H. Shibata, K. Shimizu, H. Takesue, and Y. Tokura, Ultimate low system dark-count rate for superconducting nanowire single-photon detector, Optics Letters 40, 3428 (2015).
  * Korzh  _et al._ [2020] B. Korzh, Q. Y. Zhao, J. P. Allmaras, S. Frasca, T. M. Autry, E. A. Bersin, A. D. Beyer, R. M. Briggs, B. Bumble, M. Colangelo, G. M. Crouch, A. E. Dane, T. Gerrits, A. E. Lita, F. Marsili, G. Moody, C. Peña, E. Ramirez, J. D. Rezac, N. Sinclair, M. J. Stevens, A. E. Velasco, V. B. Verma, E. E. Wollman, S. Xie, D. Zhu, P. D. Hale, M. Spiropulu, K. L. Silverman, R. P. Mirin, S. W. Nam, A. G. Kozorezov, M. D. Shaw, and K. K. Berggren, Demonstration of sub-3 ps temporal resolution with a superconducting nanowire single-photon detector, Nature Photonics 14, 250 (2020).
  * Cherednichenko  _et al._ [2021] S. Cherednichenko, N. Acharya, E. Novoselov, and V. Drakinskiy, Low kinetic inductance superconducting MgB 2 nanowires with a 130 ps relaxation time for single-photon detection applications, Superconductor Science and Technology 34, 044001 (2021).
  * Slussarenko and Pryde [2019] S. Slussarenko and G. J. Pryde, Photonic quantum information processing: A concise review, Applied Physics Reviews 6, 041303 (2019).
  * Gyger  _et al._ [2021] S. Gyger, J. Zichi, L. Schweickert, A. W. Elshaari, S. Steinhauer, S. F. Covre Da Silva, A. Rastelli, V. Zwiller, K. D. Jöns, and C. Errando-Herranz, Reconfigurable photonics with on-chip single-photon detectors, Nature Communications 12, 1408 (2021).
  * Ferrari  _et al._ [2018] S. Ferrari, C. Schuck, and W. Pernice, Waveguide-integrated superconducting nanowire single-photon detectors, Nanophotonics 7, 1725 (2018).
  * Sprengers  _et al._ [2011] J. P. Sprengers, A. Gaggero, D. Sahin, S. Jahanmirinejad, G. Frucci, F. Mattioli, R. Leoni, J. Beetz, M. Lermer, M. Kamp, S. Höfling, R. Sanjines, and A. Fiore, Waveguide superconducting single-photon detectors for integrated quantum photonic circuits, Applied Physics Letters 99, 181110 (2011).
  * Reithmaier  _et al._ [2013] G. Reithmaier, S. Lichtmannecker, T. Reichert, P. Hasch, K. Müller, M. Bichler, R. Gross, and J. J. Finley, On-chip time resolved detection of quantum dot emission using integrated superconducting single photon detectors, Scientific Reports 3, 1901 (2013).
  * Reithmaier  _et al._ [2015] G. Reithmaier, M. Kaniber, F. Flassig, S. Lichtmannecker, K. Müller, A. Andrejew, J. Vučković, R. Gross, and J. J. Finley, On-Chip Generation, Routing, and Detection of Resonance Fluorescence, Nano Letters 15, 5208 (2015).
  * Majety  _et al._ [2023] S. Majety, S. Strohauer, P. Saha, F. Wietschorke, J. J. Finley, K. Müller, and M. Radulaski, Triangular quantum photonic devices with integrated detectors in silicon carbide, Materials for Quantum Technology 3, 015004 (2023).
  * Polakovic  _et al._ [2020] T. Polakovic, W. Armstrong, G. Karapetrov, Z. E. Meziani, and V. Novosad, Unconventional applications of superconducting nanowire single photon detectors, Nanomaterials 10, 1198 (2020).
  * Shigefuji  _et al._ [2023] M. Shigefuji, A. Osada, M. Yabuno, S. Miki, H. Terai, and A. Noguchi, Efficient low-energy single-electron detection using a large-area superconducting microstrip (2023), arXiv:2301.11212 [cond-mat, physics:physics, physics:quant-ph] .
  * Hochberg  _et al._ [2019] Y. Hochberg, I. Charaev, S.-W. Nam, V. Verma, M. Colangelo, and K. K. Berggren, Detecting Sub-GeV Dark Matter with Superconducting Nanowires, Physical Review Letters 123, 151802 (2019).
  * Chiles  _et al._ [2022] J. Chiles, I. Charaev, R. Lasenby, M. Baryakhtar, J. Huang, A. Roshko, G. Burton, M. Colangelo, K. Van Tilburg, A. Arvanitaki, S. W. Nam, and K. K. Berggren, New Constraints on Dark Photon Dark Matter with Superconducting Nanowire Detectors in an Optical Haloscope, Physical Review Letters 128, 231802 (2022).
  * Clem and Berggren [2011] J. R. Clem and K. K. Berggren, Geometry-dependent critical currents in superconducting nanocircuits, Physical Review B 84, 10.1103/PhysRevB.84.174510 (2011).
  * Jönsson  _et al._ [2022] M. Jönsson, R. Vedin, S. Gyger, J. A. Sutton, S. Steinhauer, V. Zwiller, M. Wallin, and J. Lidmar, Current Crowding in Nanoscale Superconductors within the Ginzburg-Landau Model, Physical Review Applied 17, 064046 (2022).
  * Akhlaghi  _et al._ [2012] M. K. Akhlaghi, H. Atikian, A. Eftekharian, M. Loncar, and A. H. Majedi, Reduced dark counts in optimized geometries for superconducting nanowire single photon detectors, Optics Express 20, 23610 (2012).
  * Henrich  _et al._ [2013] D. Henrich, L. Rehm, S. Dorner, M. Hofherr, K. Il’in, A. Semenov, and M. Siegel, Detection Efficiency of a Spiral-Nanowire Superconducting Single-Photon Detector, IEEE Transactions on Applied Superconductivity 23, 2200405 (2013).
  * Charaev  _et al._ [2017] I. Charaev, A. Semenov, S. Doerner, G. Gomard, K. Ilin, and M. Siegel, Current dependence of the hot-spot response spectrum of superconducting single-photon detectors with different layouts, Superconductor Science and Technology 30, 025016 (2017).
  * Baghdadi  _et al._ [2021] R. Baghdadi, E. Schmidt, S. Jahani, I. Charaev, M. G. W. Müller, M. Colangelo, D. Zhu, K. Ilin, A. D. Semenov, Z. Jacob, M. Siegel, and K. K. Berggren, Enhancing the performance of superconducting nanowire-based detectors with high-filling factor by using variable thickness, Superconductor Science and Technology 34, 035010 (2021).
  * Xiong  _et al._ [2022] J.-M. Xiong, W.-J. Zhang, G.-Z. Xu, L.-X. You, X.-Y. Zhang, L. Zhang, C.-J. Zhang, D.-H. Fan, Y.-Z. Wang, H. Li, and Z. Wang, Reducing current crowding in meander superconducting strip single-photon detectors by thickening bends, Superconductor Science and Technology 35, 055015 (2022).
  * Henrich  _et al._ [2012] D. Henrich, P. Reichensperger, M. Hofherr, J. M. Meckbach, K. Il’in, M. Siegel, A. Semenov, A. Zotova, and D. Yu. Vodolazov, Geometry-induced reduction of the critical current in superconducting nanowires, Physical Review B 86, 144504 (2012).
  * Hortensius  _et al._ [2012] H. L. Hortensius, E. F. C. Driessen, T. M. Klapwijk, K. K. Berggren, and J. R. Clem, Critical-current reduction in thin superconducting wires due to current crowding, Applied Physics Letters 100, 182602 (2012).
  * Frasca  _et al._ [2019] S. Frasca, B. Korzh, M. Colangelo, D. Zhu, A. E. Lita, J. P. Allmaras, E. E. Wollman, V. B. Verma, A. E. Dane, E. Ramirez, A. D. Beyer, S. W. Nam, A. G. Kozorezov, M. D. Shaw, and K. K. Berggren, Determining the depairing current in superconducting nanowire single-photon detectors, Physical Review B 100, 054520 (2019).
  * Semenov  _et al._ [2015] A. Semenov, I. Charaev, R. Lusche, K. Ilin, M. Siegel, H.-W. Hübers, N. Bralović, K. Dopf, and D. Y. Vodolazov, Asymmetry in the effect of magnetic field on photon detection and dark counts in bended nanostrips, Physical Review B 92, 174518 (2015).
  * Zhang  _et al._ [2014] L. Zhang, L. You, D. Liu, W. Zhang, L. Zhang, X. Liu, J. Wu, Y. He, C. Lv, Z. Wang, and X. Xie, Characterization of superconducting nanowire single-photon detector with artificial constrictions, AIP Advances 4, 067114 (2014).
  * Zhang  _et al._ [2022] X. Zhang, X. Zhang, J. Huang, C. Yang, L. You, X. Liu, P. Hu, Y. Xiao, W. Zhang, Y. Wang, L. Li, Z. Wang, and H. Li, Geometric origin of intrinsic dark counts in superconducting nanowire single-photon detectors, Superconductivity 1, 100006 (2022).
  * Zhang  _et al._ [2019] W. Zhang, Q. Jia, L. You, X. Ou, H. Huang, L. Zhang, H. Li, Z. Wang, and X. Xie, Saturating Intrinsic Detection Efficiency of Superconducting Nanowire Single-Photon Detectors via Defect Engineering, Physical Review Applied 12, 044040 (2019).
  * Strohauer  _et al._ [2023] S. Strohauer, F. Wietschorke, L. Zugliani, R. Flaschmann, C. Schmid, S. Grotowski, M. Müller, B. Jonas, M. Althammer, R. Gross, K. Müller, and J. J. Finley, Site-Selective Enhancement of Superconducting Nanowire Single-Photon Detectors via Local Helium Ion Irradiation, Advanced Quantum Technologies 6, 2300139 (2023).
  * Brenner  _et al._ [2012] M. W. Brenner, D. Roy, N. Shah, and A. Bezryadin, Dynamics of superconducting nanowires shunted with an external resistor, Physical Review B 85, 224507 (2012).
  * Charaev  _et al._ [2024] I. Charaev, E. K. Batson, S. Cherednichenko, K. Reidy, V. Drakinskiy, Y. Yu, S. Lara-Avila, J. D. Thomsen, M. Colangelo, F. Incalza, K. Ilin, A. Schilling, and K. K. Berggren, Single-photon detection using large-scale high-temperature MgB2 sensors at 20 K, Nature Communications 15, 3973 (2024).
  * Wang  _et al._ [2024] Y.-Z. Wang, W.-J. Zhang, X.-Y. Zhang, G.-Z. Xu, J.-M. Xiong, Z.-G. Chen, Y.-Y. Hong, X.-Y. Liu, P.-S. Yuan, L. Wu, Z. Wang, and L.-X. You, Free-space coupled, large-active-area superconducting microstrip single-photon detector for photon-counting time-of-flight imaging, Applied Optics 63, 3130 (2024).
  * Kahl  _et al._ [2015] O. Kahl, S. Ferrari, V. Kovalyuk, G. N. Goltsman, A. Korneev, and W. H. P. Pernice, Waveguide integrated superconducting single-photon detectors with high internal quantum efficiency at telecom wavelengths, Scientific Reports 5, 10941 (2015).
  * Shibata  _et al._ [2013] H. Shibata, K. Shimizu, H. Takesue, and Y. Tokura, Superconducting Nanowire Single-Photon Detector with Ultralow Dark Count Rate Using Cold Optical Filters, Applied Physics Express 6, 072801 (2013).
  * Kerman  _et al._ [2013] A. J. Kerman, D. Rosenberg, R. J. Molnar, and E. A. Dauler, Readout of superconducting nanowire single-photon detectors at high count rates, Journal of Applied Physics 113, 144511 (2013).
  * Flaschmann  _et al._ [2023] R. Flaschmann, L. Zugliani, C. Schmid, S. Spedicato, S. Strohauer, F. Wietschorke, F. Flassig, J. J. Finley, and K. Müller, The dependence of timing jitter of superconducting nanowire single-photon detectors on the multi-layer sample design and slew rate, Nanoscale 15, 1086 (2023).