Fano fluctuations in superconducting nanowire single-photon detectors

A. G.Kozorezov1, C. Lambert1, F. Marsili2, M. J. Stevens3, V. B.

Verma3, J.P.Allmaras2, M. D. Shaw2, R. P. Mirin3, Sae Woo Nam3

1Department of Physics, Lancaster University, Lancaster, UK,2 Jet Propulsion Laboratory, California Institute of Technology,

4800 Oak Grove Dr., Pasadena, California 91109, USA,3National Institute of Standards and Technology, 325 Broadway, Boulder, CO 80305, USA

(Dated: February 9, 2017)

Because of their universal nature, Fano fluctuations are expected to influence the response ofsuperconducting nanowire single-photon detectors (SNSPDs). We predict that photon countingrate (PCR) as a function of bias current (IB) in SNSPDs is described by an integral over a trans-verse coordinate-dependent complementary error function. The latter describes smearing of localresponses due to Fano fluctuations of the amount of energy deposited into electronic system. Thefinite width, σ, of the PCR vs IB arises from fluctuations in the energy partition between quasipar-ticles and phonons during the energy down-conversion cascade. In narrow-nanowire SNSPDs thelocal responses are uniform, and the effect of Fano-fluctuations on σ is dominant. In wide-nanowireSNSPDs with strong coordinate dependence of local responses due to vortex-antivortex unbind-ing and vortex entry from edges, Fano-fluctuations smear singularities imprinted by vorticity on thetransition part of PCR curve. We demonstrate good agreement between theory and experiments fora series of bath temperatures and photon energies in narrow-wire WSi SNSPDs. The time-resolvedhotspot relaxation curves predicted by Fano fluctuations match the Lorentzian shapes observed inexperiments over the whole range of bias currents investigated except for their tails.

I. INTRODUCTION

The conversion of light into detectable excitationsconstitutes the key process in photodetection. Energyflow and relaxation pathways are the central topic whichis equally interesting in many areas from traditionalscintillators1 or new graphene-based materials2,3. Theefficiency of detection depends on competition betweendifferent energy flow pathways.

Fano fluctuations describe variations in the numberof charge carriers generated in a single-particle or single-photon sensor. Upon impact with a particle or absorp-tion of a photon, energy is deposited in the sensor. Thisenergy is partitioned between charged and neutral ele-mentary excitations, for example between electrons andphonons. Fano fluctuations are caused by the branchingprocesses and result in variations in the fraction of energydeposited in each system. Fano fluctuations are knownto determine the theoretical limit of spectral resolutionof many types of spectrometers, and are a limiting factorin the noise characteristics of CCDs and CMOS imagesensors4,5, as well as superconductor sensors such as su-perconducting tunnel junctions and microwave kinetic in-ductance detectors6,7. Fano fluctuations may also be sig-nificant in sensors lacking an energy gap in the spectrumof elementary excitations, for example in superconduct-ing transition edge microcalorimeters grown on a solidsubstrate8–10 and magnetic microcalorimeters11,12. Therelevant parameter of the detector material is the Fanofactor, which quantifies the branching variance: a smallerfactor indicates better resolving power.

To date, the role of Fano fluctuations in SNSPDshas not been discussed in the literature, likely becauseit was viewed as irrelevant. It has generally been as-

sumed that an ideal SNSPD should exhibit sharp spec-tral and current thresholds for photodetection, charac-terized by a step function in detection efficiency whenplotted as a function of bias current13. The broadeningof this step function into the sigmoidal shape observedin experiments has been attributed to inhomogeneitiesin nanowire width or thickness, or to variations in theposition of the photon absorption site affecting vortexentry at the edge of the wire competing with unbindingof vortex-antivortex pairs away from the edge14–22.

Here, we present a study of the influence of Fanofluctuations on the current and spectral dependence ofthe detection efficiency of SNSPDs. We found that asigmoidal shape is expected even in the absence of theseinhomogeneities. We show that the energy deposited intothe electronic system by monochromatic photons fluctu-ates about the mean value due to the partition betweenquasiparticles (QP) and phonons. This occurs during en-ergy down-conversion, with a variance given by the Fanofactor. We show that photon counting rate vs. bias cur-rent in SNSPDs in general is described by an integralover transverse coordinate-dependent complementary er-ror function with a width σ determined by variance ofFano fluctuations.

II. FANO FLUCTUATIONS IN SNSPDS

A. Energy down-conversion cascade

We start by describing the first moments followingthe absorption of a photon in the SNSPD. We adapt thepicture of energy down-conversion developed in our earlywork23 for a thin, disordered metal film. Fig.1 schemat-

2

FIG. 1: Schematic picture of photoelectron-hole energy down-conversion cascade in a metal

FIG. 2: Electron-electron, τee, electron-phonon, τep, andphonon-electron, τpe, scattering-out times vs excitation en-ergy in WSi and NbN.

ically illustrates the evolving down-conversion cascade.At t = 0, a photon is absorbed and an electron-holepair is generated. The sum of electron and hole en-ergies is the energy of a photon, Eλ. On average halfthis energy is given to the electron and half to the hole.The energy and time scales in Fig.1 are given to illus-trate an absorption of optical photon. E∗1 is the thresh-old energy separating the higher-energy interval, whereelectron-electron scattering with large momentum trans-fer dominates inelastic relaxation of the electron(hole)due to electron-phonon interaction. For excitations ofenergy ε < E∗1 the electron-phonon interactions totallydominate both the phonon-electron and electron-electroninteractions until the mean energies of interacting elec-trons and phonons reach a lower threshold, Ω1, belowwhich electron-electron and phonon-electron interactionsin the disordered film dominate. Usually Ω1 ΩD,where ΩD is the Debye energy. Fig.2 shows scattering-out relaxation times as a function of quasiparticle en-ergy calculated from Fermi energy over the range of en-ergies up to Debye energy. Electron-electron scatteringtime as a function of energy was calculated from Alt-shuler and Aronov’s formula for a disordered 2D nor-mal metal24, while electron-phonon and phonon-electronscattering times were taken from Chang25. It is impor-tant to note that the expressions for electron-phonon

and phonon-electron times were taken neglecting vertexrenormalisation by strong disorder, which is valid for therange of energies εl/hc ≥ 1, where l is the elastic meanfree path for electrons and c - is the sound velocity26,27.This inequality holds true for higher energies, while atlower energies, close to Ω1, the role of disorder becomesimportant. The characteristic time τ0 is taken for WSiand NbN to be 5-10 and 0.3-0.6 ns respectively, whichfalls into the expected range for the clean limit, consis-tent with the magnitude of the Debye and critical tem-peratures if the median value of these is taken as theeffective coupling strength28. As follows from Fig.2, inboth NbN and WSi the electron-phonon interaction dom-inates the electron-electron scattering rate over a signif-icant spectral range. This means that the loss of highfrequency non-equilibrium (athermal) phonons from thethin film, with thickness comparable to the phonon meanfree paths, may be significant.

Using the estimate for electron-electron scattering-out time for large momentum transfers for ε > ΩD, weobtain that τee(ε) > τep(ε > ΩD) = τs for ε < E1 withE1 close to 1 eV23. The energy loss for electron/holeof the initial pair due to electron-electron interactionswith electrons of the Fermi distribution at equilibrium isslower. It can be roughly estimated from the expressionfor τoutee (ε)24

1

τoutee

(ε) ∝∫ ε

0

dω

∫ ω

0

dε′∫ ∞

0

dqq2Wq1

qVF

ε ∝∫ ε

0

dωω

∫ ω

0

dε′∫ ∞

0

dqq2Wq1

qVF

ε = −2

3

ε

τoutee (ε)(1)

where ω and q are the energy and momentum transferat the collision, ε′ is the energy of a particle the electroncollides with, Wq is the matrix element describing the in-teraction, VF is Fermi velocity. Solving the last equationyields

ε(t)

ε0=

[1 +

4

3

t

τoutee (ε0)

]−1/2

(2)

where ε0 = ε(0). The energy of an electron excitedat ε as follows from (2) halves at 9/4τoutee (ε). Theloss of half of the energy due to sequential emissionof phonons takes ∼ (ε/2ΩD)τs. Both rates are equal

at E∗1 , E∗1 = E1

(9

2

ΩDE1

)1/3

. For Ω1 < ε < E∗1 the

electron-phonon scattering is the dominant mechanismof electron/hole energy relaxation. With E1 ∼ 1 eV andΩD ∼ 30 meV we obtain for typical materials E∗1 ∼ 500meV. When the energy of initial electron/hole (or sec-ondary excitations) is close to E∗1 their subsequent cool-ing proceeds mostly via sequential emission of phonons.Since the characteristic values for τs are tens of fs, unlessthe materials have a small Debye energy, cooling from500 meV down to ΩD will last a fraction of a picosecond.

3

By the end of it, which completes before the lifespan ofthe first emitted phonons, most of the photon energy hasbeen transferred to high energy (Debye) phonons. It isconvenient to consider this highly non-equilibrium stateas the natural initial condition for the subsequent evolu-tion of interacting quasiparticles and phonons. In Fig.1this condition is called a phonon bubble.

B. Modelling Fano-fluctuation

We consider a simple model of photon detection30,31.Following the absorption of a photon of energy Eλ, a frac-tion of the energy, E < Eλ, is deposited into electronicexcitations. During the energy down-conversion cascade,a hotspot is created23. For a narrow and thin wire weassumed that the hotspot spans the width of the wire,W , occupying volume VHS = WLHSd, with LHS ≥ Wbeing its length along the wire and d is the wire thick-ness. In this situation LHS can be determined in tomog-raphy experiments30. We assume that after completionof the cascade the quasiparticle distribution is thermal-ized owing to a strong electron-electron interaction in adisordered nanowire. Therefore, we assumed that afterthe absorption of a photon the QP temperature instantlyincreased from bath temperature, Tb, to excitation tem-perature, Tex. The hotspot excitation temperature is de-termined from the thermal balance

EHS(IB , Tex, B)− EHS(IB , Tb, B) =

∫ Tex

Tb

dT ′

C(T ′, IB , B) = EQPHS (IB , Tex, B)− EQPHS (IB , Tb, B) +

2N(0)VHSkB

∫ ∞−∞

dξ

∫ T−1b

T−1ex

dβ1

exp(βε) + 1

∂ε

∂β= E (3)

where EHS(IB , T, B) is the internal energy of thehotspot, N(0) is the normal state density of states at

the Fermi level, ε =√ξ2 + ∆2(IB , T, B) is the quasipar-

ticle energy, ∆ is the order parameter in current carryingnanowire, B is the external magnetic field and E the ac-tual amount of energy deposited in the electronic system.

EQPHS (IB , T, B) = 4N(0)VHS

∫ ∞0

dεερ(ε, IB , T, B)

exp (ε/T ) + 1(4)

Here ρ(ε, T, IB , B) is the density of states within thehotspot in units of N(0). The dependencies on bias cur-rent IB , temperature T and magnetic field B originatefrom the pair-breaking energy and the order parameterbeing functions of T , IB and B. The last term in theexpression (3) originates from the dependence of QP dis-persion relations in the hotspot on temperature.

In an ideal SNSPD there is a count event every timethe energy E exceeds the threshold E∗, determined from

EHS(IB , Tb, B) + E∗ = EHS(IB , Ts, B) (5)

where Ts is the temperature at which the hotspot under-goes transition from the superconducting to the normal

state. This temperature depends on bias current and ex-ternal magnetic field and can be found for a narrow wireusing the solution of the Usadel equation for density ofstates32 and the dependence of the order parameter oncurrent, temperature and magnetic field30,31.

In wider wires the characteristic shape of a hotspotis more complicated. If the photon absorption site isnot close to the edge of the wire, then the hotspot iscylindrical. Its radius, RHS , is smaller than its width,W > RHS d, the hotspot does not span the wirewidth, and its creation results in a current density redis-tribution. The latter depends on the lateral coordinate ofthe absorption site, r (2-dimensional geometry). Finally,in very thick wires the hotspot may be spherical (or closeto semi-spherical for shallow absorption), with the radiusRHS mind,W. The geometry of current flow in thiswide and thick nanowire becomes 3-dimensional.

In wide thin nanowires, the change in the current flowfacilitates vortex entry at the edges of the nanowire or un-binding of vortex-antivortex pairs. In this situation, Fanofluctuations result in supercurrent density fluctuationsand their description becomes more complicated. Thefluctuating order parameter, current density and temper-ature in the hotspot become dependent on the coordinateof the absorption site and are connected through a morecomplicated relation. The coordinate-dependent detec-tion current, Idet(y,E, Tb, B, λ) is introduced through theappropriate simulation15,18, and the implicit conditionfor the E∗ threshold to trigger the detection click can bewritten as

IB = Idet(y, T∗, Tb, B, λ) = Idet(y,E

∗, Tb, B, λ) (6)

where T ∗ = T (E∗) is the threshold temperature of thehotspot corresponding to energy deposition E∗.

The normalised probability distribution P (E) de-scribing energy deposition into the electronic system isGaussian,

∫∞−∞ dEP (E) = 1,

P (E) =1√2πσ

exp

[− (E − E)2

2σ2

](7)

The distribution is centered around a mean value E =χEλ, where χ < 1, χ = E/Eλ is the photon yield, whichwe defined as the ratio of the energy deposited in thehotspot after the absorption of the photon (E) to thephoton energy (Eλ). The full width at half maximum of

the distribution P (E) is 2√

2 ln 2σ, where σ is the vari-ance. Since a count occurs when E ≥ E∗, simple inte-gration yields the normalised PCR for a narrow wire inthe form

PCRNW =

∫ ∞E∗

dEP (E) =1

2erfc

(E∗ − E√

2σ

)=

1

2erfc

[E(IB , Ts, B)− E(IB , Tb, B)− E√

2σ

](8)

For a wide wire the photon count occurs if the bias cur-rent exceeds the minimum of detection current, which

4

is a function of the coordinate y across the wire anddepends on hotspot temperature (deposited energy),bath temperature, magnetic field, photon wavelength,Idet (y, T (E), Tb, B, λ) = Idet (y,E, Tb, B, λ). In this sit-uation

PCRWW =∫∞

0dEP (E)

1

W

∫ W/2

−W/2dyΘ [IB−

Idet (y,E, Tb, B, λ)] =∫∞

0dEP (E)w (E, Tb, B, λ) =

1

W

∫ W/2

−W/2dy erfc

[IB − Idet (y, χEλ, Tb, B, λ)√

2σI ′det (y, χEλ, Tb, B, λ)

](9)

where Θ(x) is the Heaviside function, w (T (E), Tb, B, λ)is the fraction of the wire width where generationof a vortex results in formation of a normal do-main across the wire, and I ′det (y, χEλ, Tb, B, λ) =∂I ′det (y,E, Tb, B, λ) /∂E|E=χEλ In wide nanowires, theexpression (9) must be further convolved if the light po-larization results in a spatial profile of the absorptionsites.

The variance is determined from two statistically in-dependent processes as σ2 = σ2

1 + σ22 . Following photon

absorption, a rapid process of energy down-conversionis initiated, engaging numerous electronic and phononexcitations. Phonons can be divided into two groups.Non-pair-breaking phonons with energy hΩ smaller thantwice the order parameter 2∆ are decoupled from thecondensate and can only be reabsorbed by excited QPs.In thin films we neglect their reabsorption, assuming there-absorption time to be considerably longer than theirescape into a substrate or thermalization time. In con-trast, higher energy phonons can break Cooper pairs andexchange energy with the electronic system. In a bulk su-perconductor, if the distance from the absorption site toescape interface far exceeds the phonon mean free path,none of the pair-braking phonons escapes into the ther-mal bath before QPs thermalise. By the end of down-conversion the energy of a photon splits between QPs andnon-pair-breaking phonons. σ1 describes statistical fluc-tuations of E originating from fluctuations of numbers ofpair-breaking phonons under the assumption that noneof the pair-breaking phonons from the down-conversioncascade escapes. It can be written σ2

1 = FεE, where 2εis the mean energy which is necessary to spend in orderto generate one pair of QPs. In superconducting filmswith thickness comparable to the mean free path of pair-breaking phonons, some of the highly energetic (ather-mal) phonons will escape. Thus, instead of E we mustwrite E′ = (1 − χa)E, where we introduce 0 < χa < 1to account for the average fraction of athermal phononsescaping from the nanowire prior to random partitionof energy between QPs and non-pair-breaking phonons,which are de-coupled from condensate. The Fano factorin most superconductors is F ' 0.26,7.

The energy loss from the film due to escaping ather-mal phonons is a random process, depending on meanfree paths of phonons and transmission characteristics

into the substrate. Therefore, there appears the secondindependent contribution to the variance, σ2. If all ather-mal phonons are re-absorbed in the film, which is the casefor thick films illuminated from the top, then χa = 0 andσ2 = 0 while σ2

1 = FεE becomes the known Fano fluc-tuations variance. σ2 originates from fluctuations in thetransmission of athermal phonons to the substrate9,10:σ2

2 = J(Eλ)εEλ, we call this contribution phonon down-conversion noise. This contribution to Fano noise hasalso been expressed in conventional form33 with the fac-tor J(Eλ). The accurate determination of Fano-factorJ(Eλ) is not an easy problem. It involves evaluation ofstatistical fluctuations of loss of successive generations ofathermal phonons, which are part of evolving distributionof the down-conversion process. We may, however, de-rive the lower limit to J(Eλ). While lower-energy longer-living phonons from later generations account for the ma-jority of energy loss, the opposite is true for fluctuationsof energy loss. With multiplication of phonon numbersand decrease of their mean energy, the relative fluctua-tion of energy loss from phonons from later-generationsdecreases. We estimate the contribution to J(Eλ) fromphonons of first generation (phonon bubble) designat-ing it J1(Eλ) and ignore contributions from subsequentathermal phonon generations arriving in the lower limit,J1(Eλ) < J(Eλ).

In a thin film, photons are absorbed homogeneouslythrough the depth of the film, and, similarly, phononsof the first generation (phonon bubble) are also ho-mogeneously generated while the energetic photoelec-tron(hole) performs a random walk in the film. Thus,taking the limit d → 0 and m = 0 we simplify the ex-pression for J1(Eλ), which was derived in earlier work33

and obtain

J1 = 2ΩDε

lpe,Dd

∫ 1

cos(θc)

dξξη(ξ)

1

4

(lpe,Dd

)4

−

∫ d/lpe,Dξ

0

dxx3

[e−x +

1

2η(ξ) sin2(θc/2)

(1− e−2x

)](10)

Here θc is the angle of total internal reflection, so thatphonons that impinge the escape interface at larger an-gles stay inside the plane parallel film until they undergoscattering-assisted conversion and move at smaller anglesθ < θc, η is the phonon transmission coefficient throughthe interface with the substrate for incidence below thecritical angle, and lpe,D is the mean free path of Debyephonons with respect to absorption by electrons. Thisresult has the following meaning. The variance of thenumber of phonons emitted into the critical cone fromany energy interval follows a binomial distribution. Thecontributions from phonons of different energies are sta-tistically independent. Therefore, total variance is thesum (integral) of contributions from individual groups ofphonons of the flat distribution of phonons of the firstgeneration. Individual factors in the expression (10) re-flect : ΩD - the dominant phonon energy in the phonon

5

bubble,lpe,Dd - probability of survival until reaching the

escape interface, the inner integral - averaging phononcontributions over their distribution, accounting for theirprobabilities to reach the interface depending on energyand angle of incidence within the critical cone. Note thata binomial distribution is close to a normal distributionin expression (7) under the assumption of large phononnumbers in each energy interval. Thus, integration overthe spectrum works well for phonons of later generations.For first-generation phonons it works for the dominantphonon group with energies close to Debye energy, andthe second integral in (10) is an approximation of thedominant phonon contribution.

The total variance due to Fano noise can be writ-ten as σ2 = FeffεEλ, where the effective Fano-factor isFeff = F (1 − χa) + J . Our estimate of J relies on us-ing the simplest model of one phonon-mode with a lineardispersion relation (Debye) approximation. For phononsof the first few generations we have εl/hc ≥ 1 , and ourestimate of J1 is not affected by disorder. Given theseapproximations this estimate provides only a rough indi-cation of the magnitude of the effect.

In the next section we compare our simulation withexperiments on WSi SNSPDs. Because of the largeacoustic mismatch between WSi (W3Si) and the SiO2

substrate, even for normal incidence we have χ ≤ 0.5.Taking χ = 0.5 for WSi on amoprphous SiO2 substratewe obtain 1.0 ≤ J ≤ 1.5 for 5 ≤ τ0 ≤ 10 ns. For NbNon SiO2, a similar calculation yields 0.8 ≤ J ≤ 1.0 for0.5 ≤ τ0 ≤ 1.0 ns. With these estimates we conclude thatphonon down-conversion noise even due to phonons of thefirst generation dominates Fano fluctuations, σ2 σ1.This is not surprising because the fluctuations of thenumber of phonons falling within the escape cone and thefluctuations of ultimate number of quasiparticles are bothproportional to the square root of the respective meannumbers. Fluctuations for athermal phonons are strong,partly because of their large energy and hence smallernumbers, but more importantly because only a signif-icantly smaller fraction of all athermal phonons (thosewithin the escape cone) contribute to fluctuations in thecorresponding energy loss. The subsequent phonon gen-erations also contribute to fluctuations increasing σ2, al-though each subsequent contribution becomes smaller,because of the twofold increase of the number of phononswith every generation. Thus, even if we overestimatedJ1, there is still an extra contribution due to lower en-ergy phonons, so that most probably σ2 σ1. Becausewe cannot provide credible estimates for contributions tooverall variance from later phonon generations we willuse Feff as a fitting parameter. The likely range of vari-ation of Feff must be consistent with our estimate forJ1.

C. The effect of Fano-fluctuations on the shape ofPCR curves in narrow, homogeneous SNSPDs

The shape of PCR curve vs bias current at fixedvalues of external parameters T, λ, B is affected by thebias dependence of the energy terms in the numeratorof the argument of the complimentary error function inequation (8), i.e E(IB , Ts, B) − E(IB , Tb, B) − E. Alsoin the current-carrying superconductor both variances σ1

and σ2 depend on IB . σ1 depends on the order param-eter at critical point ∆s, while σ2 due to contributionsof phonons of later generations may weakly depend onthe threshold energy Ω1, because both ∆s and Ω1 de-pend on IB . We defined the cutoff current of the SNSPDat fixed wavelength, bath temperature, and magneticfield, Ico(λ, Tb, B), as the inflection point of the PCRvs IB curve and used the approximation E(IB , Ts, B) −E(IB , Tb, B) − E ≈ α−1(IB , λ, Tb, B)(IB − Ico(λ, Tb, B))to obtain

PCR =1

2erfc

[IB − Ico(λ, Tb, B)√

2σ(IB , Tb, B)α(IB , λ, Tb, B)

](11)

This is the modification of the equation (9) for the caseof a narrow wire with a detection current, that is notdependent on y-coordinate across the wire. As seen from(11), PCR as a function of bias current differs from theideal complimentary error function because of the cur-rent dependence of α and σ in the denominator. If thewidth of the PCR curve is small relative to de-paringcurrent, we may keep only the first term in series ex-pansion of E(IB , Ts, B) − E(IB , Tb, B) − E. This cor-responds to α being independent of bias current. Simi-larly, if the width of the PCR curve is small relative tothe de-paring current, the dependence of σ on bias cur-rent can also be neglected. Therefore, deviations of PCRfrom an ideal complimentary error function reflect eitherthe specific features of Fano-fluctuations in the current-carrying superconducting nanowire (through σ) or strongnon-linearity of the SNSPD response (through α).

The down-conversion in a strongly disorderedcurrent-carrying nanowire is different from typical super-conducting sensors with no significant disorder, such assuperconducting tunnel junctions and many types of mi-crowave kinetic inductance detectors. In these detectors,the temperature of the hotspot does not rise close to thecritical temperature TC even for photon energies as highas a few keV. The transport current is small comparedto the de-pairing current and the order parameter staysclose to its zero current-temperature value. Therefore,the energy required to generate a quasiparticle is con-stant during the downconversion process, ε = 1.75∆(0) .By contrast, in an SNSPD at the edge of the transition tothe normal state, the temperature of the hotspot reachesthe critical value Ts, while the order parameter remainsnonzero at the level ∆(IB , Ts), but is significantly smallerthan its zero temperature value for a given bias cur-rent. As a result, some of the non-pair-breaking phononswith energies below 2∆(t) that were emitted shortly after

6

photon absorption regain their pair-breaking capabilitylater, when their energy starts exceeding the decreasing2∆(t). This happens if down-conversion and thermalisa-tion times are faster than phonon escape time from thefilm. In this asymptotic limit ε = 1.75∆(Ts), and it de-creases with bias current, while Ts increases30,31. Thecorresponding σ1 also decreases, resulting in a steeperrise of the PCR vs IB . The dependence of the secondcontribution σ2 on IB is weaker, reflecting the fact thatσ2 is the contribution from the initial generations of ex-citations when the order parameter is much less affected.Therefore, to experimentally observe the bias dependenceof the variance σ, σ1 and σ2 must be comparable in mag-nitude. This situation is easier to fulfil in larger gap NbNthan in WSi.

The other factor, α, depends on bias cur-rent considerably stronger, because it is de-termined by a highly non-linear function ofbias current E(IB , Ts, B) − E(IB , Tb, B) − E,

α−1(IB , λ, Tb, B) =∂ (E(IB , Ts, B)− E(IB , Tb, B))

∂IB

∣∣∣χEλ(neglecting potential dependence of E of IB). Itcan be analysed only numerically. With the factorσ(IB , Tb, B)α(IB , λ, Tb, B) being independent of biascurrent, or if estimating it at inflection points ofexperimental curves is a good approximation, thenthe predicted PCR curves with any other parameters(Tb, λ, B) changing must exhibit parallel shifts alongIB axis. With a stronger non-linear dependence of thefactor σ(IB , Tb, B)α(IB , λ, Tb, B) on IB , the predictedPCR curves will exhibit changes of slopes of the risingparts with onsets at different bias currents. Moreover,in general σ(IB , Tb, B)α(IB , λ, Tb, B) is non-monotonicin the transition range. The four speculative patternsof PCR vs IB can therefore be predicted as shown inFig.3. While the pattern in Fig.3a is not uncommon, thepattern resembling Fig.3d was observed experimentallyin narrow, W=30 nm, NbN SNSPDs29.

D. The effect of Fano-fluctuations on the shape ofPCR curves in wide SNSPDs

In wide SNSPDs, the sensor response depends on they-coordinate of the absorption site across the nanowire.In a spatially homogeneous nanowire this dependence willaffect the shape of the PCR curves in the transitionregion, reflecting the curvature of the predicted depen-dence of detection current vs coordinate y. Two shapesof Idet(y,E, Tb, λ) are discussed in the literature: bell-shaped15 and w-shaped18,37. Fig.4a shows schematicallythese Idet profiles. For illustration we will take the w-shaped profile depicted in Fig.4a with averaged detectioncurrent at one half and a difference between maximumand minimum detection currents of ∼0.1 of the criticaldepairing current. We obtain the PCR-curve shown inFig.4b. Fano-fluctuations will smear this curve. How-ever, in order to evaluate this smearing, apart from the

FIG. 3: Predicted patterns of PCR vs IB

FIG. 4: a. Schematic profiles of Idet vs transverse coordinatey: top curve - bell shaped, bottom - w-shaped. b. PCR forw-shaped profile of Idet, best fit error function shape -red anddifference between the two shapes -cyan

variance σ we must evaluate the sensitivity of Idet(y) rel-ative to fluctuations of the deposited energy, the deriva-tive in the denominator in (9). This depends on the ma-terial properties and detection model. Instead, shownsuperimposed is the closest fit error function curve anddifference between the two curves. As is seen, the dif-ference between the two curves can be resolved exper-imentally allowing us to study the co-ordinate depen-dent response, on the background of either normally dis-tributed static spatial inhomogeneities or dynamic Fano-fluctuations. Similarly, the distinguishing features of thePCR curve for the bell-shaped profile of detection cur-rent can also be resolved experimentally.

Observation of smooth error function-like sigmoidalshapes in wide-wire SNSPDs will indicate a much weakerco-ordinate dependence than shown in Fig.4a. Such aweak co-ordinate dependence on its own cannot be re-sponsible for the observable width of the transition re-gion, δIB , Idet,max − Idet,min δIB . In this case theshapes of PCR curves of wide-wire SNSPDs will closelyresemble PCR curves of narrow-wire SNSPDs. Spatialingomogeneity connected to vortex initiation is totally

7

FIG. 5: Photon count rate(PCR) for a WSi SNSPD operatedin the single-photon detection regime as a function of bathtemperature a) from 250 mK to 2 K with the increment 250mK and wavelength b) λ=1200, 1350, 1450, 1550 and 1650nm. Solid curves-theoretical simulation, solid circles - exper-iment. Dashed lines indicate the slopes of the outer PCRcurves

smeared by fluctuations, which also determine the widthof the transition region, δIB . Moreover, hot belt andhot spot models37 will further merge with appropriatereplacements Ts ↔ T ∗, Ico ↔ Idet and α(IB , λ, Tb, B)↔I ′det (y, χEλ, Tb, B, λ).

III. COMPARISON WITH EXPERIMENT

A. PCR vs IB curves

For comparison with experiment, we have chosen thePCR vs IB curves of WSi SNSPDs measured at differentexcitation wavelengths and bath temperatures30 shownin Fig.5. To simulate these data we used the hotspot dy-namics model developed in31, modified to include Fanofluctuations. In the absence of Fano fluctuations and spa-tial inhomogeneities this model predicts step-like PCRcurves.

Fig.5a shows the experimental and fitted PCR vs IBcurves at different temperatures. The fitting parameterswere the same as in Marsili et al30 except for a higherphoton yield χ = 0.32, allowing the switching current tobe ∼ 60% of the de-pairing current. In the absence ofmeasurements of the de-pairing current, its value is anadditional fitting parameter, allowing a better match be-tween theory and experiment. For evaluation of σ2 weused J = 1.3. The calculated values of J1 for WSi ona-SiO2 range between 0.5 to 1.5 (for η = 0.5) and 0.33 to1.0 (for η = 0.3) taking τ0 in the interval 2 to 10 ns. lpe,Dvaries in the limits from 0.8 nm for τ0 =2 ns to 4 nm forτ0 =10 ns. Note that there is no information in the litera-

ture about the strength of elastic scattering of phonons inamorphous WSi film. For comparison, in a 6.9 nm-thickamorphous SiO2 film, the phonon mean free path defin-ing heat conductivity at T > 50 K is comparable to filmthickness34. The qualitative agreement between theoryand experiment in Fig.5a is good. The simulated curvesfor the whole range of bath temperatures from 0.25 to2 K are close to experimental results, simulations alsoreproduce the steeper slopes of the PCR curves with in-creased bath temperature. This feature is clearly seen inthe experimental data and is an extra consistency checkof our kinetic model.

Also consistent with experimental data is the groupof simulated PCR curves for different photon wave-lengths at fixed bath temperature, which have almostidentical slopes. We used the same set of fitting parame-ters as in Fig.5a, fixing the bath temperature and allow-ing λ to change. Steeper slopes at smaller bias currents(higher Ts in fixed-λ experiment) originate in the nonlin-ear E(IB , Ts, B) − E(IB , Tb, B) dependence. The samenonlinearities are effective for similar range of currentsdetermining the shape of PCR curves also for the sec-ond experiment, i.e. PCR vs bias current at fixed bathtemperature and variable λ. However, in the second ex-periment, shown in Fig. 5b, the effect of nonlinearity isnearly balanced by the square root dependence of vari-ance σ on photon energy. As a result the correspondingslopes remain almost unchanged, as expected from themodel.

B. Pclick vs tD curves

We have shown that Fano fluctuations play an im-portant role in smearing the detection threshold. Thisconclusion holds regardless of the ultimate detectionmechanism, which primarily affects the magnitude of thethreshold energy E∗ and correspondingly the effectiveTs. Here we discuss the role of Fano fluctuations in de-termining the hotspot relaxation dynamics. As reportedin Ref.30, we coupled optical pulse pairs separated by avariable delay tD to an SNSPD operating in the two pho-ton regime. In the two photon detection regime, a pho-toresponse pulse can be efficiently triggered only if twophotons generate two overlapping hotspots. As shown inFig.6, we measured the probability of a response pulse,or click (Pclick), as a function of tD for different bias cur-rents. The Pclick vs tD curves have Lorentzian shape andbecome wider if the bias current is increased. We definedthe hotspot relaxation time as the half width at half max-imum of each Lorentzian curve. This profile is evidencedby the shape of the two-photon PCR(tD) as a functionof time delay between two single-photon pulses30.

In an idealized model with a sharp threshold forsingle-photon detection, Pclick vs tD curves will havea rectangular shape with a width determined by thehotspot relaxation time31. According to our model, Fanofluctuations smear the sides of these rectangular curves,

8

FIG. 6: PCR as a function of time delay tD for a series ofbias currents: (a) experiment, (b) theory.

transforming them to bell-shape curves. Neglecting dif-fusive expansion31, we let the first pulse at t = 0 de-posit an energy E with probability P (E), which createsa hotspot with initial temperature Tex = T (E, Tb, IB).Subsequently the hotspot relaxes, and its temperaturefollows the functional dependence T (E, Tb, IB , t). Thelatter can be found as a solution of the kinetic equation31.At an instance of time t = tD , the hotspot energy, EHS ,thus decreases to EHS(T (E), Tb, IB , tD). Then the nor-malized two-photon Pclick becomes

PCR(tD) =

∫ ∞0

dEP (E)

∫ ∞E∗−E(T (E),Tb,IB ,tD)

dE′P (E′) =

1

2

∫ ∞0

dEP (E)erfc

[E(IB , Ts)− E(T (E), Tb, IB , tD)− E√

2σ

](12)

Fig.6b shows the simulated shapes of Pclick(tD) for sev-eral different values of bias current using the following pa-rameters, J = 1.3, χ = 0.32, Isw/Idep=0.68 and τ0 = 5.0ns.

As expected, Fano fluctuations play a domi-nant role in shaping the photoresponse Pclick(tD),also providing further evidence that in amorphousWSi nanowires hotspot relaxation proceeds via self-recombination. While the simulated curves in Fig.6can be better matched to the experimentally observedLorentzians over the hotspot lifetime defining range oftime delays, their tails differ from experiment. The re-sults of such a match are shown in Fig.7. The simulatedtails fall between that of Lorentzian and Gaussian. It ispossible that diffusion makes most of its contribution at

FIG. 7: Fitting the simulated PCR(tD) for series of bias cur-rents by the Lorentzians.

the tails of PCR(tD), causing reduced self-recombinationat the periphery of the hotspot. Simulations in Fig.6 lookreasonably close to experimental curves. They were doneat parameter values close to our of earlier work30,31 as-suming switching current 32% less than de-paring currentand correspondingly more efficient energy deposition pa-rameter. In general larger difference between the switch-ing and de-paring currents requires larger factors χ tomatch theory and experiment. Physically that meansthat achieving the higher critical temperature in a wirewith bias current being a smaller fraction of the de-paringcurrent requires more deposited energy. Also, decreasingIsw/Idep and increasing χ results in a weaker dependenceof hotspot relaxation time (half width at half maximum)on bias current as evidenced by a weaker dependence ofhotspot relaxation time on IB at lower currents. Thisdependence becomes weaker also for larger values of τ0.Allowing Isw/Idep to be free fitting parameter we mayshift simulated PCR curves along the bias current axisachieving better fit to experiment. These simulationsmust be considered together with best fits in Fig.5 whichrequired the same set of fitting parameters as for simu-lations in Fig.6 except for Isw/Idep=0.6. Achieving goodfit with the same material parameters and fixed values forIsw/Idep, σ and χ is very challenging. We could have fit-ted each separate experiment better allowing some flex-ibility of each individual parameter. The fact that wecan semi-quantitatively fit all the different experiments,allowing 12% variation of a single Isw/Idep fitting pa-rameter and keeping the rest the same, shows both thevalidity of the model and the accuracy of the main as-sumptions.

Simulation of PCR(tD) for two-photon detectionwith variable time delay requires the two additional pa-rameters, τ0, determining the time scale, and cutoff tem-perature, determining the level of cooling of hotspot be-low which it cannot be detected at arrival of the second

9

photon. Both parameters are irrelevant for single photondetection. It is important to note that for a given photonwavelength the ratio of cutoff to critical temperature Tsis not an entirely independent parameter. It depends onthe ratio of bias- to de-paring currents. However, thisdependence is highly non-linear and its effect on simu-lations differs from the linear scaling of the bias relativeto de-pairing current, which is sufficient for fitting singlephoton counting rates. In two-photon experiments withvariable time delay, both linear re-scaling of the bias rel-ative to depairing current and nonlinearity of ratio of Tcoto Ts are important, and hence credibility of both setsof simulations depends on assumed applicability of BCSmodel. Finally, the role of diffusion, which has not beendiscussed so far is also important. The simple model30,31,although acknowledging its potential role, ignored it withsome support from experimental data.

C. Discussion

Our simulations of single-photon PCR-curves andtwo-photon pump-and-probe experiments with variabletime delay were based on a narrow nanowire hot-beltmodel. The analogous hot-belt model was claimed to beirrelevant for both NbN and WSi SNSPDs with nanowirewidth exceeding 150 nm37. Our experiments and simu-lations therefore allow testing of the hot-belt model pre-dictions for a wide range of experiments and comparisonto predictions of the hotspot model where we can iden-tify expected differences. The hot belt model is expectedto work for the two-photon detection experiment. It wasdemonstrated to produce qualitatively similar but quan-titatively strongly different results to the hotspot model(except for SNSPDs in external magnetic fields in a cer-tain range of bias currents). In contrast, the agreementwith both experiments that comes as the result of ouranalysis is much closer than what could have been in-ferred on assumption that hot-belt and hotspot modelswork for two-photon and one-photon experiments respec-tively.

The validity criterion for the narrow-wire model isτth ≥ τD, where τth is the characteristic thermalisationtime, controlling suppression of the gap within the ini-tial volume, and τD is characteristic diffusion time acrossthe nanowire37. We do not use the coherence length asthe radius R0 of phonon bubble. This scale is not rele-vant for the initial state, because the radius of phononbubble must be exactly the same for a normal metal.

We chose R0 =√

4Dtd + l2pe,D as the more appropriate

spacial scale. Here td stands for the descent time for pho-toelectron(hole) from the level E∗1 to Fermi energy. Themeaning of R0 is the length of the random walk, that theprimary photoelectron and hole perform while disposingtheir excess energy to phonons, creating a phonon bub-ble. The second term under the square root accountsfor extra volume expansion due to phonons of the bub-

ble moving on average a distance of their mean free pathprior to being re-absorbed by electrons. For WSi usingd = 5 nm, D = 0.75cm2/s30, τ0 = 10 ns, ΩD = 34 meV39,

td = τ02E∗1ΩD

(TcΩD

)3

, τpe,D =τ0γ

TcΩD

and γ=89 we obtain

for volume of phonon bubble V0 = πR20d=2044 nm3 ex-

ceeding initial volume used for the estimate of τth inref.37. Correspondingly, the energy density is smaller re-sulting in lower temperature of of electrons and phononsin the hotspot Te = Tph = 4.1Tc and τth=8.7 ps for ab-sorption of a 1550 nm photon as estimated from energyconservation. There is currently no consensus in the lit-erature regarding the magnitude of τ0 with recent mea-surements of magnetoresistance38 yielding τ0=1.9 ns. Wehave chosen τ0=10 ns for two reasons. It fits the measure-ments of electron-phonon relaxation time39 over the lowtemperature range. It also better fits the expected mag-nitude of τ0 inferred from scaling according to Ω2

D/Tc3

law, which must work during the formation of phononbubble when effects of disorder on electron-phonon in-teractions are not important. Phonon escape from theSNSPD film and diffusive expansion during thermalisa-tion both reduce the energy density within the volumefilled with non-equilibrium excitations resulting in fur-ther slowdown of thermalization. A more important pro-cess is likely to be diffusive expansion, which may resultin a substantial increase of τth. Indeed, during the firstpicosecond evolving hotspot with the initial volume of2044 nm3 expands to fill the volume of 4700 nm3. Re-ducing energy density within the evolving hotspot re-sults in the temperature of thermalised quasiparticles andphonons of 3.1Tc and τth > 19.0 ps (>9.5 ps if to assumeτ0=5 ns). The representative value for expansion time ofhotspot across the width of the wire is the diffusion timefrom the centre of the strip, τD ∼ W 2/16D ' 14.1 ps.The account of both phonon loss and diffusion during thestage of thermalisation must be done within the refinedmodel. In the absence of such a model and in view of sig-nificant uncertainty of material parameters the questionof validity of one or the other model remains open.

The experimental data in Figs.5a and b have smoothsigmoidal shapes close to predicted error functions. In-deed, there is no evidence of coordinate-dependent re-sponse in the transition region. With any of the bell orw-shaped coordinate-dependent responses, one expectsthe specific change of curvature at the inflection pointfrom concave below, IB < Ico, to convex above cutoffcurrent, IB > Ico as shown in Fig.4. In contrast, for anarrow wire, the predicted error function shape is convexon the left and concave on the right of the inflection point,Ico. In order to check whether the experimental curvesin Fig.5 can be approximated by error functions in Fig.8we show the results of matching the curves from Fig.5 to

error functions, PCR =A

2erfc

(Ico − IB

∆I

), where A and

∆I are fitting parameters. No clear signs of coordinate-dependent response are seen in Fig.8.

In a recent paper on MoSi SNSPDs, the PCR curves

10

FIG. 8: Fitting PCR vs IB for λ = 1200, 1350, 1450, 1550and 1650 nm by error functions.

were measured and fitted with error functions for a wave-length range of 750 to 2050 nm. It was found that atlow photon energies the fit agrees very well with thedata. However, at high energies, the shape of the curvesstarts to deviate from simple error function40. The au-thors hypothesize that these deviations might indicatea coordinate-dependent response of the SNSPD. The ob-served deviations, however, are not what can be expectedfrom coordinate-dependent response. Firstly, the curva-tures of PCR-shapes on both sides of inflection pointsremain consistent with fluctuation-induced homogeneoussensor response. Secondly, the experimental curve showssmaller counts both in the lower bias tail and on ap-proach to saturation level. This is not indicative of aninhomogenious response, for which on approach to Imaxdetphoton counting rate must exhibit a discontinuity in itsderivative. It is possible that the observed deviationsare connected with non-linearity of SNSPD response asgiven by the function α in expression (11). This non-linearity is of the same origin as the observed substantialnon-linearity of current-energy relation in MoSi40. Theseresults may be also explained within the hotspot modelprovided that detection current is only weakly dependenton coordinates of absorption site so that predictions ofhotspot model for the shape of PCR vs IB will be nodifferent and both models further merge as explained inSection II.

The effects of an external magnetic field on PCRvs bias current17,22,35,36 deserve special attention, sincethey can give insight into the detection mechanisms inSNSPDs22,35. Features due to Fano fluctuations werenot accounted for in any of the previous work. There areseveral ways that external magnetic fields can combinewith Fano fluctuations and thus cause the PCR curvesto change shape. In an external perpendicular magneticfield, the threshold E∗ depends on the field magnitude,as seen in (5) and (8). Both the critical current andthe de-pairing energy depend on magnetic field, with theformer being dominant for weak magnetic fields. WithIc(B) decreasing, while B increases, the obvious effect in

weak fields is the shift of PCR curves towards lower cur-rents, as observed in22,35,36. The magnitude of the shiftsdepends on the exact functional dependence of Ic(B).

Vodolazov et al22,35 observed shape transformationsof the PCR curves with varying magnetic field and pho-ton wavelength above the crossover current. Their resultswere interpreted on the basis of a model assuming vortextrapping by compact hotspots having a radius depend-ing on the photon wavelength. The SNSPD response ina weak magnetic field was suggested as the definitive ex-periment for identification of the detection mechanism37.Within our model with uniform current density, thereis no such a crossover. Nonetheless, the shifts may as-sume a more complicated pattern due to the interplay be-tween the critical current, the de-pairing energy, and thecomplicated nonlinear E(IB , Ts, B)−E(IB , Tb, B) depen-dence. At shorter photon wavelengths, the PCR curvesshift towards lower bias, where the contribution of mag-netic field term to depairing energy increases relative tothe supercurrent term. Significant non-linearity of theenergy-current relation was recently reported over thespectral interval 750-2050 nm40. A full understandingof the detection mechanisms in SNSPDs will require amore detailed study of PCR shapes through the transi-tion range at the low current tail, near inflection point(s)and close to saturation together with thorough study ofthe role of magnetic field affecting non-linearities of re-sponse and Fano effect.

IV. CONCLUSIONS

In summary, we have shown that Fano fluctuationsplay a fundamental role in superconducting nanowiresingle-photon detectors. They are an essential factorin determining the exact shape of both single-photonphotoresponse and time-delayed, two-photon photore-sponse. The special features of PCR(IB) curves, suchas slope transformation, positions of inflection point(s)versus wavelength, bath temperature and magnetic fieldreveal a wealth of sensor physics and will do a significantservice for unambiguous determination of the detectionmechanism(s).

V. ACKNOWLEDGEMENTS

AGK and CL acknowledge financial support fromthe Engineering and Physical Sciences Research Coun-cil. AGK, FM and MDS acknowledge financial supportfrom DARPA, J.P.A. was supported by a NASA SpaceTechnology Research Fellowship. AGK is grateful toD.Vodolazov for numerous discussions.

11

1 R. T. Williams, J. Q. Grim, Qi Li, K. B. Ucer, G. A. Bizarri and A. Burger, Scintillation Detectors of Radiation: Excitationsat High Densities and Strong Gradients in ”Excitonic and Photonic Processes in Materials” ed. Jai Singh and Richard T.Williams, Springer Series in Materials Science 203 (2015)

2 K. J. Tielrooij, J. C. W. Song, S. A. Jensen, A. Centeno, A. Pesquera, A. Zurutuza Elorza, M. Bonn, L. S. Levitov and F.H. L. Koppens, Photoexcitation cascade and multiple hot-carrier generation in graphene, Nature Physics 9, 248252 (2013)

3 Justin C. W. Song, Leonid S. Levitov, Energy Flows in Graphene: Hot Carrier Dynamics and Cooling, J. Phys.: Condens.Matter 27, 164201 (2015)

4 J. Janesick, T. Elliott, R. Bredthauer, C. Chandler, B. Burke, ”Fano fluctuations-Limited CCDs”, Proc. SPIE, 0982, 70(1988).

5 G.F. Knoll, Radiation Detection and Measurement, fourth ed., Wiley and Sons, 2010.6 M.Kurakado, Possibility of high resolution detectors using superconducting tunnel junctions, Nucl. Instr. Meth. 196,

275(1982)7 N.Rando, A.Peacock, A. van Dordrecht et al,The properties of niobium superconducting tunneling junctions as X-ray

detectors, Nucl. Instr. Meth A 313, 173 (1992).8 B. Cabrera, R. M. Clarke, P. Colling, A. J. Miller, S. Nam, and R. W. Romani, Detection of single infrared, optical, and

ultraviolet photons using superconducting transition edge sensors Appl. Phys. Lett. 73, 735 (1998).9 D.D.E. Martin, P. Verhoeve, A. Peacock, A.G. Kozorezov, J.K. Wigmore, H. Rogalla, R. Venn, Resolution limitation due

to phonon losses in superconducting tunnel junctions, Appl.Phys. Lett. 88, 123510 (2006).10 38. Kozorezov, A. G. Wigmore, J. K. Martin, D. Verhoeve, P.; Peacock, A., Resolution limitation in superconducting

transition edge photon detectors due to downconversion phonon noise, Applied Physics Letters, 89, 223510 (2006).11 A.G. Kozorezov, C.J.Lambert, S.R. Bandler, M.A. Balvin, S.E. Busch, P.N. Nagler, J-P. Porst, S.J. Smith, T.R. Stevenson,

J.E. Sadleir, Athermal energy loss from x-rays deposited in thin superconducting films on solid substrates, Phys. Rev. B 87,104504 (2013).

12 A.E. Lita, A.J. Miller, S. Nam, Energy Collection Efficiency of Tungsten Transition-Edge Sensors in the Near-Infrared, JLow Temp Phys, 151, 125 (2008).

13 A. D. Semenov, et al., ”Quantum detection by current carrying superconducting film” Physica C 351, 349 (2001).14 A. Semenov, et al., ”Spectral cut-off in the efficiency of the resistive state formation caused by absorption of a single-photon

in current-carrying superconducting nano-strips” Eur. Phys. J. B 47, 495 (2005).15 A.Engel,J.Lonsky,X.Zhang, Detection Mechanism in SNSPD: Numerical Results of a Conceptually Simple, Yet Powerful

Detection Model, IEEE Ttans. Appl, Superconductivity, 25, 2200407 (2015)16 L. N. Bulaevskii, M. J. Graf, C. D. Batista, and V. G. Kogan,Vortex-induced dissipation in narrow current-biased thin-film

superconducting strips, Phys. Rev. B 83, 144526 (2011).17 L. N. Bulaevskii, M. J. Graf, and V. G. Kogan, Vortex-assisted photon counts and their magnetic field dependence in

single-photon superconducting detectors, Phys. Rev. B 85, 014505 (2012).18 D.Yu. Vodolazov, Current dependence of the red boundary of superconducting single-photon detectors in the modified

hot-spot model, Phys. Rev. B 90, 054515 (2014)19 A. N. Zotova and D. Yu. Vodolazov, Intrinsic detection efficiency of superconducting nanowire single photon detector in the

modified hot spot model, Supercond. Sci. Technol. 27 125001 (2014)20 J. J. Renema, R. Gaudio, Q. Wang, Z. Zhou, A. Gaggero, F. Mattioli, R. Leoni, D. Sahin, M. J. A. de Dood, A. Fiore, and

M. P. van Exter, Experimental Test of Theories of the Detection Mechanism in a Nanowire Superconducting Single PhotonDetector, PRL, 112, 117604 (2014)

21 A.Engel, J.J.Renema, K.Ilin, A. Semenov, Detection mechanism of superconducting nanowire single-photon detectors, Su-percond. Sci. Technol. 28, 114003 (2015).

22 D. Yu.Vodolazov, Yu. P.Korneeva, A. V.Semenov, A. A.Korneev, G. N.Goltsman, Vortex-assisted mechanism of photoncounting in a superconducting nanowire single-photon detector revealed by external magnetic field, Physical Review B92,104503(2015)

23 A.G. Kozorezov, A.F. Volkov, J.K. Wigmore, A. Peacock, A. Poelaert, R. den Hartog, Quasiparticle-phonon downconversionin nonequilibrium superconductors, Phys. Rev. B61, 11807 (2000).

24 B.L.Altshuler and A.G.Aronov in ”Electron-electron Interactions in Disordered systems”, ed. by A.L.Efros and M.Pollak,Elsevier Science Publishers B.V., 1985

25 J-J Chang in ”Nonequilibrium Superconductivity” ed. by D.N.Langenberg and A.I.Larkin, Elsevier Acince Publishers B.V.1986

26 A.Schmid, Electron-phonon interaction in an impure metal, Z. Phys. 259, 421 (1973)27 M. Yu. Reizer and A. V. Sergeev, The effect of the electron-phonon interaction of the conductivity of impure metals , Sov.

Phys. JETP 65, 1291 (1987)28 S.B. Kaplan et al, Quasiparticle and phonon lifetimes in superconductors, Phys.Rev. B 14, 4854 (1976)29 F. Marsili, F. Bellei, F. Najafi, A. Dane, E. Dauler, R. Molnar, and K. Berggren, Efficient Single Photon Detection from

500 nm to 5 m Wavelength, Nano Lett., 12, 4799 (2012)30 F. Marsili, M. J. Stevens, A. Kozorezov, V. B. Verma, Colin Lambert, J. A. Stern, R. D. Horansky, S. Dyer, S. Duff,

D. P. Pappas, A. E. Lita, M. D. Shaw, R. P. Mirin, and S. W. Nam, Hotspot relaxation dynamics in a current-carryingsuperconductor, Phys. Rev. B 93, 094518 (2016)

12

31 A.G.Kozorezov, C. Lambert, F. Marsili, M. J. Stevens, V. B. Verma, Colin Lambert, J. A. Stern, R. Horansky, S. Dyer,M. D. Shaw, R. P. Mirin, and S. W. Nam, Quasiparticle recombination in hotspots in superconducting current-carryingnanowires, Physical Review B92, 064504 (2015)

32 A. Anthore, H. Pothier, and D. Esteve, Density of States in a Superconductor Carrying a Supercurrent, PRL 90, 127001(2003)

33 A.G. Kozorezov, J.K.Wigmore, D. Martin, P. Verhoeve, A. Peacock, Electron energy down-conversion in thin superconduct-ing films, Phys. Rev.B 75, 094513 (2007).

34 E. T. Swartz, R. O. Pohl, Thermal boundary resistance, Reviews of Modern Physics, 61, 605 (1989)35 A. A. Korneev, Y. P. Korneeva, M. Y. Mikhailov, Y. P. Pershin, A. V. Semenov, D. Y. Vodolazov, A. V. Divochiy, Y. B.

Vakhtomin, K. V. Smirnov, A. G. Sivakov, A. Y. Devizenko, and G. N. Goltsman, Characterization of MoSi SuperconductingSingle-Photon Detectors in the Magnetic Field, IEEE Transactions on Applied Superconductivity, 25, 6975120 (2015).

36 J. J. Renema, R. J. Rengelink, I. Komen, Q. Wang, R. Gaudio, K. P. M. opt Hoog, Z. Zhou, D. Sahin, A. Fiore, P. Kes,J. Aarts, M. P. van Exter, M. J. A. de Dood, and E. F. C. Driessen, The effect of magnetic field on the intrinsic detectionefficiency of superconducting single-photon detectors, Appl. Phys. Lett. 106, 092602 (2015).

37 D.Vodolazov, Theory of single photon detection by ’dirty’ current-carrying superconducting strip based on the kineticequation approach, arXiv:1611.06060 (2016)

38 X.Zhang,A.Engel,Q.Wang,A.Schilling,A.Semenov,M.Sidorova,H.-W.Hbers,I.Charaev,K.Il’in, and M.SIegel, Characteristicsof superconducting tungsten silicide WxS−1−x for single photon detection, Phys.Rev.B 94, 174509 (2016)

39 M.Sidorova, A.Semenov, A.Korneev, G.Chulkova, Yu.Korneeva, M.Mikhailov, A.Devizenko, A.Kozorezov, G.Goltsman,Electron-phonon relaxation time in ultrathin tungsten silicon film, arXiv:1607.07321

40 Misael Caloz, Boris Korzh, Nuala Timoney, Markus Weiss, Stefano Gariglio, Richard J. Warburton, Christian Schonenberger,Jelmer Renema, Hugo Zbinden, and Felix Bussi‘eres, Optically probing the detection mechanism in a molybdenum silicidesuperconducting nanowire single-photon detector, arXiv:1611.08238