Electrothermal feedback in superconducting nanowire
single-photon detectors
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Kerman, Andrew J. et al. “Electrothermal feedback in
superconducting nanowire single-photon detectors.” Physical
Review B 79.10 (2009): 100509. © 2009 The American Physical
Society.
As Published
http://dx.doi.org/10.1103/PhysRevB.79.100509
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Final published version
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Wed May 25 18:18:03 EDT 2016
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http://hdl.handle.net/1721.1/51752
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RAPID COMMUNICATIONS
PHYSICAL REVIEW B 79, 100509共R兲 共2009兲
Electrothermal feedback in superconducting nanowire single-photon detectors
Andrew J. Kerman,1 Joel K. W. Yang,2 Richard J. Molnar,1 Eric A. Dauler,1,2 and Karl K. Berggren2
1Lincoln
Laboratory, Massachusetts Institute of Technology, Lexington, Massachusetts 02420, USA
Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
共Received 5 December 2008; revised manuscript received 30 January 2009; published 26 March 2009兲
2Research
We investigate the role of electrothermal feedback in the operation of superconducting nanowire singlephoton detectors 共SNSPDs兲. It is found that the desired mode of operation for SNSPDs is only achieved if this
feedback is unstable, which happens naturally through the slow electrical response associated with their
relatively large kinetic inductance. If this response is sped up in an effort to increase the device count rate, the
electrothermal feedback becomes stable and results in an effect known as latching, where the device is locked
in a resistive state and can no longer detect photons. We present a set of experiments which elucidate this effect
and a simple model which quantitatively explains the results.
DOI: 10.1103/PhysRevB.79.100509
PACS number共s兲: 85.25.Oj, 74.78.⫺w, 85.60.Gz
1098-0121/2009/79共10兲/100509共4兲
comes fast enough to counterbalance the Joule heating before it runs away, resulting in a stable resistive domain,
known as a self-heating hotspot.9,10
In a standard treatment of these hotspots,9 solutions to a
one-dimensional heat equation are found in which a normalsuperconducting 共NS兲 boundary propagates at constant velocity vNS for fixed device current Id.9,11 This results in a
solution of the form
vNS = v0
␣共Id/Ic兲2 − 2
1
2
⬇ 共I2d − Iss
兲,
冑␣共Id/Ic兲 − 1 ␥
共1兲
2
where v0 ⬅ 冑Acs␬h / c is a characteristic velocity 共Acs is the
wire’s cross-sectional area, ␬ is its thermal conductivity, and
c and h are the heat capacity and heat transfer coefficient to
the substrate, per unit length, respectively兲, Ic is the critical
current, and ␣ ⬅ ␳nI2c / h共Tc − T0兲 is known as the Stekly parameter, which characterizes the ratio of Joule heating to
Rn
IL
Id
Detector
e)
d)
f)
RL
I0
L
b)
c)
RS
bias tee
Rn
Id
L
RL
I0
RDC
IL
time [ns]
Iswitch/IC
Detector
a)
scaled
detection efficiency
Superconducting nanowire single-photon detectors
共SNSPDs兲 combine high speed, high detection efficiency
共DE兲 over a wide range of wavelengths, and low dark
counts.1–4 Of particular importance is their high singlephoton timing resolution of ⬃30 ps,4 which permits extremely high data rates in photon-counting communications
applications.5,6 Full use of this electrical bandwidth is limited, however, by the fact that the maximum count rates of
these devices are much smaller 共a few hundred MHz for
10 ␮m2 active area and decreasing as the area is increased2兲,
limited by their large kinetic inductance and the input impedance of the readout circuit.2,7 To increase the count rate,
therefore, one must either reduce the kinetic inductance 共by
using a smaller active area or different materials or substrates兲 or increase the load impedance.7 However, either of
these approaches causes the wire to “latch” into a stable resistive state where it no longer detects photons.8 This effect
arises when negative electrothermal feedback, which in normal operation allows the device to reset itself, is made fast
enough that it becomes stable. We present experiments which
probe the stability of this feedback, and we develop a model
which quantitatively explains our observations.
The operation of an SNSPD is illustrated in Fig. 1共a兲. A
nanowire 共typically ⬃100 nm wide and 5nm thick兲 is biased
with a dc current I0 near its critical current Ic. The nanowire
has kinetic inductance L and is read out using a load impedance RL 共typically a 50⍀ transmission line兲. When a photon
is absorbed, a short 共⬍100 nm long兲 normal domain is
nucleated, giving the wire a resistance Rn共t兲. This results in
Joule heating which causes the normal domain 共and consequently, Rn兲 to expand in time exponentially. The expansion
is counteracted by negative electrothermal feedback from the
load RL, which forms a current divider with Rn, and diverts a
current IL into the load 共so that the current in the nanowire is
reduced to Id ⬅ I0 − IL兲, reducing the heating. However, in a
correctly functioning device, this feedback is unstable: the
inductive time constant is long enough so that before IL becomes appreciable, Joule heating has already increased Rn,
so that Rn Ⰷ RL. The current Id then drops nearly to zero,
turning off the heating and allowing the nanowire to quickly
cool down and return to the superconducting state, after
which Id recovers with a time constant ␶e ⬅ L / RL.2 If one
attempts to shorten ␶e too much, the negative feedback be-
RS [Ω
Ω]
FIG. 1. 共Color online兲 Speedup and latching of nanowire detectors with increased load impedance. 共a兲 electrical model of detector
operation. A hotspot is nucleated by absorption of a photon, producing a resistance Rn in series with the wire’s kinetic inductance L. 共b兲
Experimental circuit, including series resistor RS, bias tee, and impedance of current source Rdc. 共c兲 and 共d兲 Averaged pulse shapes for
RS = 0 , 250⍀ 共L ⬃ 50 nH兲; dashed lines are predictions with no free
parameters. 共e兲 Iswitch vs RS. As RS is increased, Iswitch decreases,
becoming less than Ic. 共f兲 DE at I0 = 0.975Iswitch vs RS 共open
squares兲. Also shown 共crosses兲 are the expected DEs assuming that
latching affects DE simply by limiting I0 共obtained from DE vs I0 at
RS = 0兲.
100509-1
©2009 The American Physical Society
RAPID COMMUNICATIONS
PHYSICAL REVIEW B 79, 100509共R兲 共2009兲
KERMAN et al.
a)
Detector
S
G
b)
L
Rp
G
RP
Rn
G
Id
S
Au
RP
v
I0
L
Rn
G
RL
RL
RL=333Ω
Ω
Ic
e)
34Ω
Ω
Iss
f)
d)
323Ω
Ω
Ilatch
323Ω
Ω
34Ω
Ω
34Ω
Ω
Rn/RL
Voltage [mV]
33Ω
Ω
Id [µ
µA]
c)
Bias current I0 [µ
µA]
FIG. 2. 共Color online兲 Hotspot stability measurements. 共a兲 Device schematic; two ground-signal-ground probes perform a highimpedance three-point measurement of Rn, with RL determined by
probe position. 共b兲 Equivalent electrical circuit. 共c兲 and 共d兲 Example
V-I curves, with L = 60 nH and L = 605 nH, respectively. 共e兲 and 共f兲
Inferred Id and Rn / RL for the data shown in 共d兲. In 共e兲 the relaxation
oscillations in the region Ic ⬍ I0 ⬍ Ilatch are shown schematically.
Dashed lines show 共e兲 Id = Ic and Id = Iss. 共f兲 Rss = RL共I0 / Iss − 1兲.
conduction cooling in the normal state9 共␳n is the normal
resistance of the wire per unit length and T0 and Tc are the
substrate and critical temperatures兲. Equation 共1兲 is valid
when T0 Ⰶ Tc, and the approximate equality holds
when 兩Id − Iss兩 Ⰶ Iss with ␥ ⬅ 共Tc − T0兲共c / ␳n兲冑h / ␬Acs and
2
⬅ 2h共Tc − T0兲 / ␳n. The physical meaning of Eq. 共1兲 is clear:
Iss
the NS boundary is stationary only if the local power density
共⬀I2d兲 is equal to a fixed value; if it is greater, the hotspot will
expand 共vNS ⬎ 0兲, if less it will contract 共vNS ⬍ 0兲.
We can use Eq. 共1兲 to describe the electrothermal circuit
in Fig. 1共a兲, by combining it with the circuit equation
IdRn + LdId / dt = RL共I0 − Id兲 共where dRn / dt = 2␳nvNS兲. To
determine when the device will latch, we analyze the
stability of the resulting second-order nonlinear system for
small deviations from its steady-state solution 关Id → Iss , Rn
→ RL共I0 / Iss − 1兲 ⬅ Rss兴 to obtain a damping coefficient ␨
I
= 4I0ss 冑␶th / ␶e, where ␶th ⬅ RL / 2␳nv0 is a thermal time constant.
This can be re-expressed in terms of Rtot ⬅ RL + Rss thus: ␨
= 41 冑␶th,tot / ␶e,tot 共␶e,tot ⬅ L / Rtot and ␶th,tot ⬅ Rtot / 2␳nv0兲, which
clearly shows that the stability is determined by a ratio of
electrical and thermal time constants.
In normal device operation, where the damping ␨ is small,
the feedback cannot stabilize the hotspot during the initial
photoresponse, as described above. However, as I0 is increased, ␨ increases, making the hotspot more stable 共this
occurs because Rss ⬀ I0 and larger Rss gives a shorter inductive time constant ␶e,tot兲. Eventually, at a bias current
I0 = Ilatch the device latches. For a correctly functioning device, Ilatch ⬎ Ic, so that latching does not affect its operation.
However, if ␶e is decreased, Ilatch decreases, and eventually it
becomes less than Ic. This prevents the device from being
biased near Ic, resulting in a drastic reduction in
performance.12
Devices used in this work were fabricated from
⬃5-nm-thick NbN films, deposited on R-plane sapphire substrates in a UHV dc magnetron sputtering system 共base pressure ⬍10−10 mbar兲. Film deposition was performed at
a wafer temperature of ⬃800 ° C and a pressure of
⬃10−8 mbar.13 Aligned photolithography and liftoff were
used to pattern ⬃100-nm-thick Ti films for on-chip resistors8
and Ti:Au contact pads. Patterning of the NbN was then
performed with e-beam lithography.3 Devices were tested in
a cryogenic probing station at 2 K as described in Refs.
2 and 3.
Figures 1共c兲–1共f兲 show data for a set of 共3 ␮m
⫻ 3.3 ␮m area兲 devices having various resistors RS in series
with the 50⍀ readout line8 关Fig. 1共b兲兴, so that RL = 50⍀ + RS.
For RS = 0, these devices had similar performance to those in
Ref. 3. Panels 共c兲 and 共d兲 show averaged pulse shapes for
devices with RS = 0 , 250⍀, respectively. Clearly, the reset
time can be reduced; however, this comes at a price. Panels
共e兲 and 共f兲 show, for devices with different RS, the current
Iswitch ⬅ min共Ic , Ilatch兲 above which each device no longer detects photons and the measured DE at I0 = 0.975Iswitch. The
data show that as RS is increased, Iswitch decreases far below
IC 共due to reduction in Ilatch兲, resulting in a significantly reduced DE.14
To investigate the latched state, we fabricated devices designed to probe the stability of self-heating hotspots as a
function of I0, L, and RL. Each device consisted of three
sections in series, as shown in Fig. 2共a兲: a 3-␮m-long 100nm-wide nanowire where the hotspot was nucleated, a wider
共200 nm兲 meandered section acting as an inductance, and a
series of nine contact pads interspersed with Ti-film resistors.
Also shown are the two electrical probes, which result in the
circuit of Fig. 2共b兲: a high-impedance 共R p = 20 k⍀兲 threepoint measurement of Rss. We varied RL by moving the
probes along the line of contact pads and L by testing different devices 共with different L兲. We tested 66 devices on three
chips and selected from these only unconstricted12 nanowires
with nearly identical linewidths 共Ic ⬇ 22– 24 ␮A兲 and with
RL = 20⍀ – 1000⍀ and L = 6 – 600 nH.
For each L and RL, we acquired a dc V-I curve like those
shown in Figs. 2共c兲 and 2共d兲, sweeping I0 downward starting
from high values where the hotspot was stable.15 These data
can be converted to Id and Rn, as shown in Figs. 2共e兲 and 2共f兲
关for the data of Fig. 2共d兲兴. From data of this kind, Ilatch can be
identified by the sudden jumps in Id: for I0 ⬎ Ilatch, Id is fixed
共at Iss兲, independent of I0 and RL, as predicted by Eq. 共1兲.16
For the largest values of RL, Id never reaches Ic 关shown by a
horizontal dashed line in Fig. 2共e兲兴 because once it latches
Id → Iss ⬍ Ic. As RL is decreased, Ilatch increases as expected,
until another feature appears when Ilatch ⬎ Ic. In this region
共Ic ⬍ I0 ⬍ Ilatch兲 the nanowire can neither superconduct nor
latch and instead undergoes relaxation oscillations,9,17 as indicated in the figure, producing a periodic pulse train with a
frequency that increases as I0 is increased.14 The average
resistance 共from the dc V-I curve兲 increases with this frequency, producing the observed continuous decrease in Id
until Ilatch is reached.
The data in Fig. 3 show the measured Ilatch as RL and L are
varied, plotted in dimensionless form as 2␶e / ␶th共⬀L / RL2 兲 vs
共Ilatch / Iss兲2, which can be thought of as defining the boundary
between stable and unstable hotspots. Our simple model de-
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ELECTROTHERMAL FEEDBACK IN SUPERCONDUCTING…
d⌬T ␥␳n␶a d2l
=
− h⌬T.
dt
2 dt2
c
L=605 nH
217nH
stable
unstable
L=6.0 nH
6.0nH
FIG. 3. 共Color online兲 Summary of hotspot stability results.
Data are shown from three different chips 共indicated by different
colors兲. Circles, squares, and triangles are data for L = 6 – 12, 15–60,
and 120–600 nH, respectively. In the ␶e Ⰷ ␶a limit 共where NS
domain-wall motion dominates the feedback兲, the data approach the
dashed line, which is the prediction based on Eq. 共1兲. The solid
curves are obtained from Eq. 共8兲 with a phase margin of 30°; each
curve corresponds to a fixed L in the set 共6,15,30,60,600兲 nH and
spans the range of RL in the data. The dotted lines extend these
predictions over a wider range of RL. For ␶e Ⰶ ␶th the NS domain
walls are effectively fixed and the temperature feedback dominates.
In this regime the feedback is always unstable when Rn ⱗ RL 共or
equivalently, I0 ⱗ 2Iss兲, as shown in the inset; the minimum stable
Rn are all above the dashed-dotted line 共Rn = RL兲.
scribed above predicts a line of slope 1 共indicated by the
dashed line兲. The data do approach this line, although only in
the ␶e Ⰷ ␶th limit. This is consistent with the assumption of
constant 共or slowly varying兲 Id under which Eq. 共1兲 was derived. As ␶e / ␶th is decreased, the data trend downward, away
from this line, and Ilatch / Iss becomes almost independent of
␶e / ␶th 共all data approach the same vertical asymptote兲; this
implies a minimum Ilatch / Iss, or equivalently, a minimum
Rn / RL, below which the hotspot is always unstable. This is
shown in the inset: the measured minimum stable Rn is always greater than ⬃RL, over a range of L values from 6 to
217 nH 共shown by solid symbols—solid lines are guides for
the eyes兲.
This behavior can be explained in terms of a time scale ␶a
over which the temperature profile of the hotspot stabilizes
into the quasisteady-state form which yields Eq. 共1兲. For
power-density variations faster than this, the NS boundaries
do not have time to start moving, resulting instead in a temperature deviation ⌬T. Since the NS boundary occurs at T
⬇ Tc, where ␳n is strongly temperature dependent 共defined by
d␳ / dT ⬅ ␤ ⬎ 0兲, this changes Rn, giving a second parallel
electrothermal feedback path which dominates for frequencies ␻ Ⰷ ␶−1
a . We can describe this by replacing Eq. 共1兲 with
冉
冊
d2l dl
␥␳n
2
␶a 2 +
= I2d␳共⌬T兲 − Iss
␳n ,
dt
2
dt
共2兲
共3兲
Here, l is the hotspot length, ␳共⌬T兲 is the resistance per unit
length 关with ␳共0兲 ⬅ ␳n兴, and Rn = ␳共⌬T兲l. In Eq. 共2兲, ␶a is the
characteristic time over which 2vNS = dl / dt adapts to changes
in power density: for slow time scales dt Ⰷ ␶a, we have
␶ad2l / dt2 Ⰶ dl / dt and Eq. 共2兲 reduces to Eq. 共1兲 共with
⌬T = 0兲. For faster time scales, ␶ad2l / dt2 becomes appreciable and acts as a source term for temperature deviations in
Eq. 共3兲. When dt Ⰶ ␶a, ␶ad2l / dt2 Ⰷ dl / dt and Eqs. 共2兲 and 共3兲
2
␳n − h⌬T.18
can be combined to give c · d⌬T / dt ⬇ I2d␳共⌬T兲 − Iss
In this limit, if RL ⱗ Rn the bias circuit including RL begins to
look like a current source, which then results in positive
feedback: a current change produces a temperature and resistance change of the same sign. Therefore, the hotspot is always unstable when Rn ⱗ RL.
Expressing Eqs. 共2兲 and 共3兲 in dimensionless units
共i ⬅ Id / I0, r ⬅ Rn / RL, ␭ ⬅ l␤Tc / RL, and ␪ ⬅ T / Tc兲 and expanding to first order in small deviations 共␦i , ␦r , ␦␭ , ␦␪兲 from
steady state, we obtain
␦i⬘ = − 共i0␦i + i−1
0 ␦r兲,
共4兲
␦r = ␩共i0 − 1兲␦␪ + ␩−1␦␭,
共5兲
␶a
␶
␦␭⬙ + ␦␭⬘ = 2␩2 e 共␦␪ + 2i0␩−1␦i兲,
␶e
␶th
共6兲
␦␪⬘ =
␶
⌰␶th␶a
␦␭⬙ − e ␦␪ .
␩ ␶ e␶ c
␶c
共7兲
Here, the prime denotes differentiation with respect to
t / ␶e, i0 ⬅ I0 / Iss, ⌰ ⬅ 共Tc − T0兲 / Tc, ␩ ⬅ ␤Tc / ␳n characterizes
the resistive transition slope, and ␶c ⬅ c / h is a cooling
time constant. When ␶e Ⰷ ␶th , ␶a, the system reduces to
␦i⬙ + i0␦i⬘ − 4␶e / ␶th ⬇ 0, which has damping coefficient
␨ = i0共4冑␶e / ␶th兲−1, as above. In the opposite limit, where
␶e Ⰶ ␶th , ␶a, we obtain ␦i⬙ + i0␦i⬘ + 共2␩⌰␶e / ␶c兲共i0 − 2兲 ⬇ 0. In
agreement with our argument above, the oscillation frequency becomes negative for Rn ⬍ RL 共I0 ⬍ 2Iss兲.
We characterize the stability of the system of Eqs. 共4兲–共7兲
using its “open loop” gain Aol: we assume a small oscillatory
perturbation by replacing ␦r in Eq. 共4兲 with ⌬re j␻t and responses 共␦i , ␦␪ , ␦␭ , ␦r兲e j␻t. Solving for Aol ⬅ ␦r / ⌬r, we obtain
␶
Aol =
共
␶
␶
共
兲
.
␻
兲关2j␻␩⌰ ␶␶ − 共1 + j␻ ␶␶ 兲共1 + j␻ ␶␶ 兲兴
4 ␶the 1 + j␻ ␶e − 4␩⌰␻2共i0 − 1兲 ␶ae
j ␻i0 1 +
j
i0
c
a
c
a
e
e
e
共8兲
The stability of the system can then be quantified by the
phase margin ␲ + arg关Aol共␻0兲兴, where ␻0 is the unity gain
共兩Aol兩 = 1兲 frequency. In the extreme case, when the phase
margin is zero 共arg关Aol共␻0兲兴 = −␲兲, the feedback is positive.
The solid lines in Fig. 3 show our best fit to the data. Note
that although the stability is determined only by ␶e / ␶th and i0
in the two extreme limits 共not visible in the figure兲, in the
100509-3
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KERMAN et al.
intermediate region of interest here this is not the case, so
several curves are shown. Each solid curve segment corresponds to a single L, over the range of RL tested; the dotted
lines continue these curves for a wider range of RL. The data
are grouped into three inductance ranges: 6–12, 15–60, and
120–600 nH, indicated by circles, squares, and triangles, respectively. We used fixed values ⌰ = 0.8 and ␩ = 6.5, which
are based on independent measurements, and fitted ␶a
= 1.9 ns and ␶c = 0.47 ns to all data.19 Separate values of
␳nv0 were fitted to data from each of the three chips, differing at most by a factor of ⬃2. These fitted values were
␳nv0 ⬃ 1 ⫻ 1011 ⍀ / s; since ␳n ⬃ 3 ⫻ 109 ⍀ / m, this gives v0
⬃ 30 m / s, which is a reasonable value.
A natural question to ask in light of this analysis is
whether it suggests a method for speeding up these devices.
The most obvious way would be to increase the heat transfer
1 G.
Gol’tsman, O. Minaeva, A. Korneev, M. Tarkhov, I.
Rubtsova, A. Divochiy, I. Milostnaya, G. Chulkova, N. Kaurova, B. Voronov, D. Pan, J. Kitaygorsky, A. Cross, A. Pearlman, I. Komissarov, W. Slysz, M. Wegrzecki, P. Grabiec, and R.
Sobolewski, IEEE Trans. Appl. Supercond. 17, 246 共2007兲; J. A.
Stern and W. H. Farr, ibid. 17, 306 共2007兲; F. Marsili, D. Bitauld, A. Gaggero, R. Leoni, F. Mattioli, S. Hold, M. Benkahoul,
F. Levy, and A. Fiore, European Conference on Lasers and
Electro-Optics 2007, p. 816; S. N. Dorenbos, E. M. Reiger, U.
Perinetti, V. Zwiller, T. Zijlstra, and T. M. Klapwijk, Appl. Phys.
Lett. 93, 131101 共2008兲; A. D. Semenov, P. Haas, B. Günther,
H.-W. Hübers, K. Il’in, M. Siegel, A. Kirste, J. Beyer, D. Drung,
T. Schurig, and A. Smirnov, Supercond. Sci. Technol. 20, 919
共2007兲; S. Miki, M. Fujiwara, M. Sasaki, B. Baek, A. J. Miller,
R. H. Hadfield, S. W. Nam, and Z. Wang, Appl. Phys. Lett. 92,
061116 共2008兲.
2 A. J. Kerman, E. A. Dauler, W. E. Keicher, J. K. W. Yang, K. K.
Berggren, G. N. Gol’tsman, and B. M. Voronov, Appl. Phys.
Lett. 88, 111116 共2006兲.
3
K. M. Rosfjord, J. K. W. Yang, E. A. Dauler, A. J. Kerman, V.
Anant, B. M. Voronov, G. N. Gol’tsman, and K. K. Berggren,
Opt. Express 14, 527 共2006兲.
4 E. A. Dauler, A. J. Kerman, B. S. Robinson, J. K. W. Yang, B.
Voronov, G. Gol’tsman, S. A. Hamilton, and K. K. Berggren, J.
Mod. Opt. 56, 364 共2009兲.
5
D. Rosenberg, S. W. Nam, P. A. Hiskett, C. G. Peterson, R. J.
Hughes, J. E. Nordholt, A. E. Lita, and A. J. Miller, Appl. Phys.
Lett. 88, 021108 共2006兲; H. Takesue, S. W. Nam, Q. Zhang, R.
H. Hadfield, T. Honjo, K. Tamaki, and Y. Yamamoto, Nat. Photonics 1, 343 共2007兲.
6
B. S. Robinson, A. J. Kerman, E. A. Dauler, D. M. Boroson, S.
A. Hamilton, J. K. W. Yang, V. Anant, and K. K. Berggren, Proc.
SPIE 6709, 67090Z 共2007兲.
7 A potential alternative has recently been demonstrated by A.
Korneev, A. Divochiy, M. Tarkhov, O. Minaeva, V. Seleznev, N.
Kaurova, B. Voronov, O. Okunev, G. Chulkova, I. Milostnaya,
K. Smirnov, and G. Gol’tsman, J. Phys.: Conf. Ser. 97, 012307
coefficient h, which increases both Iss and v0, moving the
wire further into the unstable region and allowing its speed
to be increased further without latching. However, at present
it is unknown how much h can be increased before the DE
begins to suffer. At some point, the photon-generated hotspot
will disappear too quickly for the wire to respond in the
desired fashion. In any case, experiments like those described here will be a useful measurement tool in future work
for understanding the impact of changes in the material
and/or substrate on the thermal coupling and electrothermal
feedback.
We acknowledge helpful discussions with Sae Woo Nam,
Aaron Miller, and Enectalí Figueroa-Feliciano. This work is
sponsored by the United States Air Force under Contract No.
FA8721-05-C-0002.
共2008兲, although latching may affect this as well.
J. K. W. Yang, A. J. Kerman, E. A. Dauler, V. Anant, K. M.
Rosfjord, and K. K. Berggren, IEEE Trans. Appl. Supercond.
17, 581 共2007兲.
9 A. Vl. Gurevich and R. G. Mints, Rev. Mod. Phys. 59, 941
共1987兲, and references therein.
10 Since the negative feedback opposes the fast joule heating that
produces the rising edge of the output pulse, it may also affect
the timing jitter of this edge.
11 This description is further simplified by the fact that near the NS
boundary all material properties can be approximated by their
values at Tc.
12
A. J. Kerman, E. A. Dauler, J. K. W. Yang, K. M. Rosfjord, V.
Anant, K. K. Berggren, G. N. Gol’tsman, and B. M. Voronov,
Appl. Phys. Lett. 90, 101110 共2007兲.
13 R. J. Molnar, E. A. Dauler, A. J. Kerman, and K. K. Berggren
共unpublished兲.
14
For the circuit in Fig. 1共b兲, if Ilatch ⬎ Ic, the detector oscillates
关see Fig. 2共e兲 and associated discussion兴 for the time constant of
the bias tee, after which it senses the larger Rdc = 5 k⍀ and
latches. The absence of this burst of pulses is a signature for
Ilatch ⬍ Ic.
15 The results were almost identical when sweeping I upward,
0
since the dark counts of the device allow it to lock into the
latched state if it is stable.
16
From the observed Iss ⬇ 5 ␮A and ␳n ⬃ 3 ⫻ 109 ⍀ m−1, Eq. 共1兲
gives h ⬃ 5 ⫻ 10−3 W m−1 K−1; this gives ␣ ⬃ 30 共TC ⬇ 10 K,
T0 = 2 K, and IC = 22 ␮A兲.
17 R. H. Hadfield, A. J. Miller, S. W. Nam, R. L. Kautz, and R. E.
Schwall, Appl. Phys. Lett. 87, 203505 共2005兲.
18
This equation is identical to that governing a transition-edge sensor under the action of negative electrothermal feedback; see,
e.g., K. D. Irwin, G. C. Hilton, D. A. Wollman, and J. M. Martinis, J. Appl. Phys. 83, 3978 共1998兲.
19
From the fitted ␶c = 0.47 ns and the h inferred above 共Ref. 16兲,
we obtain c = 2.2⫻ 10−12 J m−1 K−1. Reference 8 gives cel = 1.2
⫻ 10−12 and cph = 4.9⫻ 10−12 J m−1 K−1.
8
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