Nonequilibrium superconducting thin films with sub-gap and pair-breaking photon illumination

0.48 MB·English

by  Tejas Guruswamy

Nonequilibrium superconducting thin films with sub-gap and pair-breaking photon
illumination
T Guruswamy,∗ D J Goldie, and S Withington
Quantum Sensors Group, Cavendish Laboratory, University of Cambridge,
J J Thomson Avenue, Cambridge, CB3 0HE, UK
(Dated: January 9, 2015)
We calculate nonequilibrium quasiparticle and phonon distributions for a number of widely-
used low transition temperature thin-film superconductors under constant, uniform illumination
by sub-gap probe and pair-breaking signal photons simultaneously. From these distributions we
calculate material-characteristic parameters that allow rapid evaluation of an effective quasiparticle
temperatureusingasimpleanalyticalexpression,forallmaterialsstudied(Mo,Al,Ta,Nb,andNbN)
for all photon energies. We also explore the temperature and energy-dependence of the low-energy
quasiparticle generation efficiency η by pair-breaking signal photons finding η 0.6 in the limit of
≈
thick films at low bath temperatures that is material-independent. Taking the energy distribution of
excess quasiparticles into account, we find η 1 as the bath temperature approaches the transition
5
→
1 temperature in agreement with the assumption of the two-temperature model of the nonequilibrium
0 response that is appropriate in that regime. The behaviour of η with signal frequency scaled by the
2 superconductingenergygapisalsoshowntobematerial-independent,andisinqualitativeagreement
withrecentexperimentalresults. Anenhancementofη inthepresenceofsub-gap(probe)photonsis
n
showntobemostsignificantatsignalfrequenciesnearthesuperconductinggapfrequencyandarises
a
due to multiple photon absorption events that increase the average energy of excess quasiparticles
J
above that in the absence of the probe.
8

PACSnumbers: 74.40.Gh,74.78.-w,29.40.-n,74.25.N-
]
n
o
c I. INTRODUCTION may become important. In this work we concentrate
- on modelling the detection of moderate energy photons
r
p hνsignal < ΩD where these effects are insignificant. The
The energy distribution of the quasiparticle excita-
u photon energies considered are particularly important
tions of a superconductor determines its electrical1 and
s to understand the responsivity, sensitivity and signal-
. thermal transport properties.2 A theoretical understand-
t to-noise of currently-deployed and planned astronomical
a ing of devices including kinetic inductance detectors
instruments performing measurements in the frequency
m (KIDs),3–5 superconducting tunnel junctions,6 supercon-
windowof0.1to10THz. InthecaseofKIDs,thenonequi-
- ducting qubits,7–9 SQUID based parametric amplifiers
libriumstatecreatedbytheinteractionismonitoredusing
d and mixers,10 and quantum capacitance detectors11 re-
n aprobeofenergyhνprobe (cid:28)2∆,whereνprobe istheprobe
quires a description of the quasiparticle system response.
o frequency,soanadditionaldrivetermneedstobeincluded
c Therefore, a method for calculating the excited, possibly in the detailed analysis.
[ nonequilibrium distribution of quasiparticles is central to
An important consideration in any nonequilibrium su-
any device model.
1 perconducting detector operating at low reduced tem-
v Nonequilibrium superconducting detectors rely on pair- peratures T /T (cid:28)1 is that the detected signal is most
b c
1 breaking to create their detected signal. We are partic- influenced by the presence of low-energy quasiparticles
3
ularly interested in thin-film superconductors fabricated because the relaxation of the primary excitation occurs
8
on a dielectric substrate that is cooled to a sufficiently on time-scales that are much shorter than the ultimate
1
low temperature (the bath temperature) T . Typically relaxation time of the low-energy excess. (Here we define
0 b
. Tb ∼ 0.1Tc where Tc is the superconducting transition low-energy quasiparticles to have E <3∆.) We assume
1
temperature. An interacting signal photon of energy the signal photons interact with 100% efficiency i.e. we
0
hν , where h is Planck’s constant and ν the sig- ignoreforexampleanyopticalcouplingefficiencies. Apri-
5 signal signal
1 nal frequency, directly breaks a condensate pair in the maryexcitationwithE =hνsignal−∆relaxestowardsthe
: superconductor provided hνsignal ≥ 2∆ where ∆ is the gapemittingaphonon. Ifthisemittedphononhasenergy
v
superconducting energy gap. Due to the high density of Ω≥2∆, an additional pair may be broken enhancing the
i
X states close to the gap, and for moderate energy photons, number of low-energy excess quasiparticles and thus the
r theinteractioncreatesadistributionofexcitedquasiparti- detected signal. But in a thin-film this emitted phonon
a cleswithpeaks12 atE =∆andE =hν −∆whereE may be lost into the substrate, reducing the number of
signal
is the quasiparticle energy. E is√related to the underlying low-energy excitations. The average number m of low-
Bloch state energy (cid:15) by E = (cid:15)2+∆2. Higher energy energy quasiparticles resulting from the interaction of a
signal photons will directly release an atomic electron via single high-energy photon can be quantified in terms of a
the photoelectric effect. In this case effects associated quasiparticle generation efficiency (or quasiparticle yield)
with localized heating and gap-reduction or “hotspots” η. If the low-energy quasiparticles have average energy
2
(cid:104)E (cid:105) then the efficiency η =m(cid:104)E (cid:105)/hν . Often it is each instance we use measured material properties that
qp qp signal
assumed (cid:104)E (cid:105)=∆.13 characterize films that we deposit by sputtering under
qp
Phonon loss means η ≤1. Most existing work assumes ultra-high-vacuum. Our results are applicable to many
avalueofη ≈0.6forallmaterials,photonfrequenciesand different device designs, as only the phonon escape time
bath temperatures.3 At high temperatures T ≈ Tc our to the substrate τl is device-geometry dependent. The
description of the energy-cascade no longer applies14,15 method assumes that phonon pair-breaking, quasiparticle
becausethermalphononenergiesbecomecomparablewith recombination, and electron-phonon scattering are the
∆,andscatteringandrecombinationoccuroncomparable only significant interaction processes. This assumption is
timescales. Thenatwo-temperaturemodeldescribingthe investigated in section IIB for typical thin-films intended
quasiparticle and phonon systems that assumes η =1 is fordetectorapplications. SectionIIreviewssuperconduct-
most appropriate. Here we show that the full nonequilib- ing parameters for the materials discussed, considers the
rium calculation yields η →1 as T /T →1 in agreement relative contribution from electron-electron scattering for
b c
with the assumption of the two-temperature model. the different materials, describes the numerical method,
and also describes how η is calculated for a pair-breaking
Previous work modelled the effect of incoming energy
at low temperatures with an effective temperature16 or signal. Section III presents an analytical superconductor
as an effective chemical potential17 for the quasiparti- cooling model that can be used to calculate the effec-
tive quasiparticle temperature T∗ for both sub-gap and
cle distribution. To go beyond this analysis, Chang and N
Scalapino18,19 derived a set of coupled nonlinear kinetic pair-breaking photons. Section IV describes detailed nu-
mericalresultsforbothsub-gapandpair-breakingphoton
equationsdescribingtheinteractionsofquasiparticlesand
interactions including the effect of the effect of changes
phononstofindtheirrespectiveenergydistributionsf(E)
in the bath temperature. Section V summarizes the work
and n(Ω). They solved for the nonequilibrium distribu-
done.
tionsresultingfromvariousdriveterms, includingphoton
and phonon injection. A number of investigations have
explored the effect of very high energy photons (x-ray or
II. METHODS
opticalphotons)oninfinitesuperconductors20,21 andthin
films22 calculating a quasiparticle generation efficiency
A. Superconducting parameters
of η ≈ 0.6, ignoring Ω ≥ 2∆ phonon loss. Another ap-
proach13 considered the time-evolution following local
energy deposition into the thin film. Kozorezov et al.23 The superconducting parameters needed for modelling
considered the energy downconversion process after ab- are the characteristic quasiparticle lifetime
sorptionofahighenergyphotonforavarietyofmaterials
Z (0)(cid:126)
and concluded that the materials can be categorized into τ = 1 , (1)
0 2πb(k T )3
three different classes, with the low energy-gap supercon- B c
ductors all having η ≈0.6.
and the characteristic phonon lifetime
We have previously reported24 a numerical approach
that solves the coupled kinetic equations describing the τφ = (cid:126)Nion , (2)
quasiparticleandphonondistributionsf(E)andn(Ω)for 0 4π2N(0) (cid:104)α2(cid:105)∆
bs 0
a superconducting thin-film driven by a sub-gap probe,24
and including the effect of an additional pair-breaking as defined by Kaplan et al.29 N(0) and N(0)bs are the
signal,25 for Al thin-films at low temperatures T ∼0.1T . mass-enhanced and bare single-spin density of states
c
WehavealsoreporteddetailedcalculationsofKIDcharac- at the Fermi energy (cid:15)F respectively, where N(0) =
teristics(qualityfactorsandquasiparticlelifetimesasfunc- Z1(0)N(0)bs, Nion is the ion density, and ∆0 the zero
tionsofreadoutpowerandTb)thatwerecomparedtopre- temperature superconducting energy gap. Z1(0) is the
cisemeasurementsofAlresonatorbehaviourfindinggood electron-phonon renormalization factor Z1(0) = 1+λ,
agreement.26 MorerecentmeasurementsonKIDs27 ofthe where λ = 2(cid:82)∞dΩα2(Ω)F(Ω)/Ω is the dimensionless
0
quasiparticlenumberandrecombinationtime-dependence electron-phonon coupling strength. α2(Ω)F(Ω) is the
on readout power appear to show that they must be ex- Eliashberg function, α2(Ω) is the electron-phonon inter-
plained using nonequilibrium quasiparticle distributions. action matrix element and F(Ω) is the phonon density of
Other recent measurements with a Ta KID28 report a de- states. The material-dependent parameter b is calculated
tector response that has the same qualitative features as withinaDebyemodelsuchthatbΩ2 =α2(Ω)F(Ω)forlow
theenergydependenceofηforAldescribedinGuruswamy phonon energies Ω∼2∆0. The averaged value of the in-
et al.25 Here we apply the method to a number of other teractionmatrixelementis(cid:104)α2(cid:105)=1/3(cid:82)∞dΩα2(Ω)F(Ω).
0
technologically important low temperature superconduc- Values of b and (cid:104)α2(cid:105) for Al, Ta and Nb are already tabu-
tors and discuss the physics involved. We concentrate lated.29 For Mo we use the point-contact measurements
in particular on the refractory elements and extend the of Caro et al.30 while for NbN we use the tunnelling data
earlier analysis to bath temperatures approaching the of Kihlstrom et al.31 and the transport property mea-
superconducting critical temperature T . We study Mo, surements of Semenov et al.32 We note that the precision
c
Al, Ta (in its higher T ∼4.4K form), Nb and NbN. In of our calculations of b are dependent on the accuracy
c
3
within which we can interpret the relevant low-frequency model for phonons29 leads to
parts of the reported α2(Ω)F(Ω) data.

(cid:15)3
3τ (k T )3 if (cid:15)<ΩD
Material Mo Al Ta Nb NbN τ−1 ((cid:15))= 0 B c . (4)
∆0 (µeV) 140 180 700 1470 2560 e−p 3τ (Ωk3DT )3 if (cid:15)≥ΩD
ν (GHz) 68 87 339 711 1240 0 B c
gap
T /K 0.92 1.18 4.6 9.67 16.8
c The electron-electron scattering rate in clean films is
bα(210−(m4meVeV)−2) 21..2682 31..1973 117.3.38 44.60 4.9497 estimated with the Landau-Pomeranchuk formula23
(cid:104) (cid:105)
λ 0.42 0.43 0.69 1.84 1.46
(cid:15)2 r1/2
τ0 (ns) 1310 438 1.78 0.15 0.02 τ−1((cid:15))= s , (5)
τ0φ (ps) 231 260 22.7 4.17 5.98 e−e (cid:126)(cid:15)F 7.96
τφ/τφ 1.04 1.04 0.91 0.94 1.01
ec 0 where r is the radius containing one electron charge
s
divided by the Bohr radius (approximated as 1). For thin
Table I. Characteristic quasiparticle and phonon lifetimes,
and associated parameters. Data from Gladstone et al.33, resistive films the e−e rate can be significantly enhanced
Kaplan et al.29 or Zehnder13 unless otherwise specified. For so that35
Mo and NbN we use measurements of α2(Ω)F(Ω) from Caro e2R (cid:15) (cid:126)π
eotf aτl0φ.3r0eqaunidreKdihfolsrtreonmergeyt aclo.n31serrevsapteiocntivceallyc.uτlaeφcteids tfhroemva(lu8)e. τe−−1e((cid:15),Rsq)= 2π2s(cid:126)q2 ln−1 e2Rsq, (6)
ν =2∆ /h is the gap frequency.
gap 0
where R =ρ/d is the sheet resistance of the thin-film,
sq
ρ is the resistivity and d is the thickness.36 These normal-
The characteristic times calculated are listed in ta- statecalculationssignificantlyoverestimatethee−escat-
ble I. We assume the weak-coupling relationship ∆0 = tering rates in the superconducting state at low tempera-
1.76kBTc for all materials to determine Tc in table I. The tures Tb (cid:28)Tc.37 Figure 1 plots the normal-state scatter-
assumption is reasonable for Mo Al and Ta, but gives a ing rates for a 40 nm thick Mo film and we have used the
slightly higher Tc compared to experiment for Nb (typ- thin-film resistive e−e scattering rate (6). We identify
ically T ∼ 9.3K) and for NbN (T ∼ 15K). However
c c
we do not expect the assumption to significantly affect
the physics of the conclusions drawn; the differences in- 1015
Ω (cid:15)
troduced by strong coupling are reduced because of the D F
shortlifetimesofstateswithenergyE (cid:29)∆.18,29 τφ isthe
ec
value of τφ required for energy conservation calculated 1013
0
from (8) and discussed in more detail in section IIC.
) 1011
1
−
s
( E
1 c
B. Electron-electron vs electron-phonon scattering −
τ 109
Insightintothecontributionfromelectron-electrone−e
sftohcuaentetdelerbciyntrgoconinnestnidheeerrgidineogswtonhf-ecinortneelvareevrsastniot(cid:15)ns(cid:28)cparΩtotecre,isnwsgeisraamstseoussmtT-ece.atsFhiaolyrt 107 ττee−−−11ep
electron-phonon(e−p)inelasticscatterDingdominatesover 105 τe−−1e(Rsq)
−
(e−e) scattering. We previously investigated25 to what
100 101 102 103 104 105
extent this assumption is valid for low-resistivity Al films.
Now we consider the other low-T superconductors using (cid:15)/∆0
c
material parameters typical of those that we measure
experimentallyforthethin-filmswedepositbymagnetron- Figure 1. Energy dependence of normal state scattering rates
sputtering in ultra-high vacuum. in Mo. Electron-phonon rate τe−1p((cid:15)) (solid green), clean
In the normal-state the e−p scattering rate τe−−1p is34 lfiilmmitreesliescttivroenτ-e−el1eec(t(cid:15)r,oRnsqr)a(tdeaτseh−−e1ed((cid:15)r−)ed()d.oCttaeldcublalutieo)n,saanrde ftohrina-
2π (cid:90) (cid:15) 40nm Mo film−with ρ=9.2×10−8Ωm.
τ−1 ((cid:15))= dΩα2(Ω)F(Ω). (3)
e−p (cid:126)
0 the following characteristic energies for each material: (cid:15) ;
F
Ω ; andalow-energycrossoverE , belowwhichthee−e
D c
Assuming α2(Ω) is constant at low energies, a Debye scattering rate exceeds the e−p rate. E /∆ is highest
c
4
Material ρ(10−8Ωm) d(nm) (cid:15)F (eV) ΩD(meV) Ec/∆0 gap ∆(T) was calculated using the BCS gap equation,
Mo 9.2 40 9.32 39.6 6.1 1 (cid:90) ΩD 1−2f(E,T)
Al38 0.8 35 11.6 36.2 1.42 = dE , (7)
(cid:112)
Ta (bcc,epi) 4.1 100 9.5 20.7 0.23 N(0)VBCS ∆(T) E2−∆(T)2
Nb 8.8 100 6.18 23.7 0.14
NbN39,40 250 100 15.6 28.4 0.3 where f(E,T)=(exp(E/k T)+1)−1 is the Fermi-Dirac
B
distribution, and N(0)V is the dimensionless electron-
BCS
TableII.Characteristicmaterialenergiesrelatedtoscattering. phonon coupling constant, calculated from the zero tem-
Ec is the energy at which the electron-phonon scattering perature form of (7) as N(0)VBCS =1/sinh−1(ΩD/∆0).
rate becomes greater than the thin-film resistive electron-
The numerical method converges on a solution that
electronscatteringrate. DatafromGladstoneetal.33,Kaplan
minimizes ξ provided the material parameters satisfy
et al.29 or Kozorezov et al.23 unless otherwise referenced.
The material resistivities and thicknesses used are measured
2πN(0)τφ∆ Ω3
values from thin-films deposited by our group. All data is for 0 0 D =1, (8)
polycrystalinefilmsexceptforTawhereweassumeparameters 9Nionτ0(kBTc)3
typical of the higher-T bcc form.
c
aconsequenceofmatchingthelowenergypartofthemea-
sured Eliashberg function to a Debye-model approxima-
tion that determines b. In our previous work we ensured
for Mo for the material parameters chosen, as shown in
that T satisfied (8),24 but then the T obtained did not
table II. c c
precisely agree with (7) (although the difference is small
During the energy down-conversion, the inelastic scat-
<1%). In this work, since we are also interested in the
tering rate is only relevant above energies E ≥3∆ where
temperature dependence of the response we calculated T
additional pair breaking is possible. We find that E is c
c from (7) and defined an energy-conserving characteristic
below 3∆ for all materials except Mo. Figure 1 shows
phonon lifetime τφ to satisfy (8). Given the difficulty of
that e−p scattering dominates in that case for (cid:15)≈6∆ to ec
interpreting the low-energy α2(Ω)F(Ω) data that we and
104∆ for the modelled Mo film, and e−e scattering may
indeedKaplanet al.29 andothershavenotedweconsider
contribute above and below this energy range. However,
this approach entirely reasonable. The ratios τφ/τφ are
the e−e scattering rate is considerably suppressed in the ec 0
superconducting state at low T/T .37 We conclude that given in table I. Satisfyingly the ratios are within 10%
c
of unity for all materials studied. In our calculation of b
there is negligible contribution from e−e scattering for
for Mo, our error would be of this order given the diffi-
most of these materials, thicknesses, temperatures and
culty of determining the low-energy α2(Ω)F(Ω) from the
signal photon energies. For Mo we note that additional
available data. We note that other estimates of the char-
e−e scattering may increase the quasiparticle generation
acteristic times exist (see for example Parlato et al.42)
efficiency over the results presented here by providing an
although those estimates do not necessarily satisfy the
additional pair-breaking mechanism especially near T .
c
energy-conserving requirement.
In the numerical code the energy-bin distribution for
both the quasiparticles and phonons was also changed
C. Numerical method when calculating bath temperature dependencies. As T
b
is increased, the phonon-bath power flow P(Ω) above
φ−b
We solved the coupled kinetic equations18 describing Ω = 2∆ increasingly resembles the difference between
two thermal distributions, and an increasing fraction of
the quasiparticle and phonon distributions numerically to
power is carried by high energy phonons, as shown in
find the steady-state driven distributions, f(E) n(Ω) for
figure 2. Therefore at high temperatures, the phonon
thequasiparticlesandphononsrespectively,usingNewton-
distributionmustbeaccuratelycalculatedtothesehigher
Raphsoniteration. Drivetermsforpair-breakingandsub-
gapphotons41 wereincludedasnecessaryandthenumeric energies. The energy of the last phonon bin Ωmax signif-
icantly affects the energy conservation of the solutions.
pre-factors of these terms scaled to match the absorbed
Figure 2 (inset) shows that to have a relative power flow
powers, P for sub-gap photons and P for the
probe signal difference ξ < 10−5, required Ω > 15k T for
pair-breaking signal. The numerical method was iterated φ−b max B b
until the solutions conserved energy to within ξ =10−5. Tb/Tc =0.95.
(cid:113) However, the bin width could not be made too large
Thiswasquantifiedasξ = ξ2 +ξ2 ,whereξ is
qp−φ φ−b qp−φ as the energy gap ∆(T), readout photon energy hνp, and
therelativedifferencebetweenthephotonpowerabsorbed signalphotonenergyhν hadtoberoundedtothenearest
s
andpowerflowfromquasiparticlestophonons,andξφ−bis multiple of the bin width, such that the photon-induced
therelativedifferencebetweenthephotonpowerabsorbed peaksinthedistributionsoccurredwithinwell-defineden-
and the power flow from phonons to substrate (held at ergybins. WechosetomodelthephononswithN =2500
a temperature Tb). The numerical method is outlined in uniformly-sized bins with maximum phonon energy Ωmax,
more detail in Goldie and Withington.24 and similarly the quasiparticles with maximum energy
For each material, the temperature dependent energy E =Ω +∆(T). Bin width was therefore variable,
max max
5
4000 tions43 for Γ and Γ results in25
Φ qp
10−2
(N2 −N2 ) 2R
η =(cid:104)E (cid:105) signal probe , (9)
3000 b−10−4 qp Psignal 1+βτl
1) ξφ
− where R and β are the distribution-averaged recombina-
V 10−6
e 2000 tion rate and pair-breaking rates respectively. We define
µ
3− 10 12 14 16 18 (cid:104)Eqp(cid:105) the average energy of the excess low-energy quasi-
m Ω /k T particles where
W max B b
1000
( (cid:82)∞
)φb− (cid:104)Eqp(cid:105)= (cid:82)0∞Eρρ(E(E)()f(f(E(E))−−ff(E(E,,TT))))ddEE. (10)
Ω 0
( 0
P
At low temperatures (cid:104)E (cid:105)≈∆ as usually assumed: tak-
qp
ing account of the detailed energy distributions allows η
1000 to be determined at arbitrary temperatures.
−
0 2 4 6 8 10 12 14
III. SUPERCONDUCTOR COOLING MODEL
Ω (meV)
Weusedthesteady-statedrivendistributionsthatwere
Figure 2. Phonon-bath power flow P(Ω) , at T = 0.1T
φ b b c
(solid red) and T = 0.95T (dashed blue−). Both for NbN, calculated to determine parameters for the thin-film su-
b c
1P6prµoebVe .=I2n×set10o6fWrelmat−iv3e,Ppshigonnalon=-b0a,tτhl/pτ0φow=er1,flaonwddhiνffperroebnec=e pinegrtcoonnd.2u4ctTohricsomoloindgelmisodaenl oauntalliynteidcienxGproelsdsiieonanrdelWatiitnhg-
ξ against the energy of the maximum bin Ω /k T , at T∗ to the power flow P between the quasiparticle and
φ b max B b N
th−e same bath temperatures T = 0.1T (red ) and T = phonon systems (the latter assumed to be the substrate
b c b
×
0.95Tc (blue +). phonons) for a given Tb (cid:28)Tc and material.
1 Σ
depending on material and temperature. The quasiparti- P = s ×
η(P,ν) 1+τ /τ
l pb
cle density of states broadening parameter γ was recalcu-
lated for each simulation, changing for different material  (cid:32)−2∆(T∗)(cid:33) (cid:18)−2∆(T )(cid:19)
and bin width combinations. It was chosen to minimize TN∗ exp k T∗N −Tbexp k T b . (11)
the difference between the quasiparticle number calcu- B N B b
(cid:82)∞
lated as the integral 4N(0) dEρ (E,∆)f(E,T ),
√ ∆ BCS b Σ is a material-dependent constant, τ is the phonon
where ρ (E,∆) = E/ E2−∆2, and the equivalent s pb
BCS pair breaking time, ∼ τφ for T (cid:28) T . T∗ is the effec-
sum over the discretized distribution with the broadened 0 b c N
tive temperature calculated to characterize the driven
density of states calculated with E →E+iγ. In all cases
nonequilibrium quasiparticle distributions so that N =
the approach worked well. qp
4N(0)(cid:82)∞dEρ(E)f(E,T∗). η(P,ν)dependsonthedrive
∆ N
(probeorpair-breakingsignal). GoldieandWithington24
D. Low-energy quasiparticle generation efficiency showed for a sub-gap probe η(P,νprobe) ≡ η2∆ the frac-
for a pair-breaking signal tionofphonon-bathpowerflowcarriedbyexcessphonons
with energy Ω>2∆(T ) that depends on P, ν and
b probe
τ . Here we show that (11) can also be applied for di-
Forasinglepair-breakingphotonwehavedefinedalow- l
rect pair-breaking when η(P,ν ) ≡ η as defined in
energy generation efficiency η = m(cid:104)E (cid:105)/hν . Here signal
qp signal
section IID.
we describe the calculation of η for a constant input
signalpowerP inathin-filmthatmayalsobedriven
signal
by a non-direct pair-breaking constant readout power
IV. RESULTS
P (such as used for KID readout). The absorbed
probe
power creates a number of excess quasiparticles N =
excess
N −N such that the quasiparticle system total A. Effects of a sub-gap probe
signal probe
energy has increased by E =E −E , where
excess signal probe
N and E are the quantities for distributions Here we show and discuss results for modelling of the
signal signal
driven by the signal and probe together. In the steady effect of a sub-gap probe. The data-points (crosses)
state, photons absorbed at a rate Γ create low-energy in figure 3(a) show the calculated T∗ from the full
Φ N
quasiparticles at a rate Γ =mΓ . non-equilibrium solution as a function of P . The
qp Φ probe
Solving a modified set of Rothwarf-Taylor rate equa- solid line shows T∗ evaluated with (11) and from this
N
6
1.8 1 10−9 1
1.6 (a) )0 0.8 (b)
∆
(K)∗N 1.4 η2∆ 0.8 /(Σ0s 10−12 η0 0.6
T 1.2 0.6 P (a)
(b) kB 0.4
1
0.4 10−150 0.2 0.4 0.20 0.2 0.4
10−2 103 108 10−2 103 108
hν /∆ hν /∆
Pprobe (Wm−3) Pprobe (Wm−3) p 0 p 0
1 −0.02
4.2
(ps) 4 2∆ 0.95 η1 −0.04
b η
p 3.8
τ 0.9 0.06
(c) (d) − (c)
3.6
0 0.2 0.4
10−2 103 108 0 10 20 hν /∆
p 0
Pprobe (Wm−3) τl/τ0φ
Figure 4. Parameters for piecewise fits to η for Nb (blue
2∆
Figure 3. (a) Effective quasiparticle temperatures TN∗ as a ×), Ta (green (cid:3)), and Al (red (cid:13)) calculated at a range of
function of Pprobe; full nonequilibrium calculation (blue +), frequencies hνprobe. (a) P0 against scaled frequency. (b) η0
and a fit using (11) (red solid line). (b) Fraction of phonon- (constant section) against scaled frequency. (c) η (slope for
1
bath power flow carried by Ω > 2∆ excess phonons as a linear section) against scaled frequency.
function of P ; full nonequilibrium calculation (blue +),
probe
and piecewise fit (red line) to (12). The dotted vertical line
marks P . (c) Distribution-averaged phonon pair-breaking
0 emphasize the commonality.
time τ . The characteristic phonon lifetime τφ is marked
pb 0
with the dotted line. (d) η as a function of phonon escape
time ratio τ/τφ. The calcu2∆lations are for Nb, with P = Material ∆0 (µeV) Σs (Wm−3K−1) P0 (Wm−3)
l 0 probe
2×103Wm−3, hνprobe =16µeV, and τl/τ0φ =1. MAlo 114800 31..4223·11001100 22..9863·1100−−33
Ta 700 1.17·1014 1.30·101
we also calculate Σ . The calculations are for Nb, Nb 1470 3.83·1015 3.02·103
with P = 2×10s3Wm−3, hν = 16µeV, and NbN 2560 1.29·1016 3.17·104
probe probe · ·
τ /τφ =1. The analytical model is an excellent approx-
l 0
imation to the full nonequilibrium calculation provided Table III. Superconductor cooling model material parameter
Σ , and knee power P at hν = 16µeV and τ/τφ =
thepower-dependenceofη2∆ showninfigure3(b)istaken s 0 probe l 0
1, obtained from fits to results of the full nonequilibrium
into account. The solid line in figure 3(b) is a piecewise
calculation.
fit such that
(cid:40)
η if P ≤P TableIIIsummarizesparametersderivedfromthemod-
0 0
η2∆(P)= (12) ellingtodescribesub-gapphotoninteractionsinallofthe
η ln(P/P )+η if P >P .
1 0 0 0
materials studied. The table shows Σ and knee param-
s
P0 characterizes a “knee” probe power below which η2∆ eter P0 at hνprobe = 16µeV and τl/τ0φ = 1. The values
is constant and η1 characterizes the energy dependence of of Σs shown can also be used for estimates of TN∗ for a
η athigherprobepowers. Botharematerialdependent. direct pair-breaking signal as we will discuss in the next
2∆
Figure 3(c) shows that τ ∼τφ is a good approximation section with an appropriate energy-dependent η.
pb 0
over the range of interest. Figure 3(d) shows that η
2∆
has no significant dependence on the phonon escape time
τ . B. Effects of direct pair-breaking
l
Thedependenceofthe fit parametersP , η , andη on
0 0 1
probe photon energy is shown in figure 4. This includes Here we discuss results for a direct pair-breaking sig-
calculations for Nb (blue ×), Ta (green (cid:3)), and Al (red nal with 2∆ < hν < 10∆. Figure 5 shows how η
signal
(cid:13)). Scalingtheenergyofthephotonby∆ demonstrates varies with signal photon energy. η decreases from unity
0
the monotonic behaviour for all the materials. P shown in the range hν = 2∆ to 4∆, as phonons emitted
0 signal
in figure 4(a) is scaled by the material-dependent Σ to by quasiparticles scattering towards the gap escape the
s
7
1.6 the effect is most-pronounced near hν = 2∆. This
signal
Al results in a higher average excess quasiparticle energy
Ta when the readout power is present alongside the signal
1.4
Nb compared to when only the signal is present, and also
Mo produces the small variation in η between materials. The
1.2 NbN absorbed readout power P and readout frequency
probe
Al, P =0 hν are not scaled by material parameters in this
probe probe
1 figure,unlikethesignal. WhencalculatedwithPprobe =0,
the signal quasiparticle generation efficiency is identical
η
for all materials and is unity at hν =2∆.
signal
0.8
1.4
0.6 0.3 τ/τφ =1
l 0
τ/τφ =5
0.4 1.2 K) 0.25 l 0
(
∗N
T 0.2
0.2
2 4 6 8 10 1
hνs/∆0 100 102 104
η Psignal (Wm−3)
Figure5. Quasiparticlegenerationefficiencyη againstsig- 0.8
power
nal photon energy hν for different materials. Calculated
signal
for Tb = 0.1Tc, Pprobe = 2 103Wm−3, hνprobe = 16µeV,
×
Psignal =2Wm−3, and τl/τ0φ =1. 0.6
2∆(Tb) ∆(Tb)
thin film. Above hνsignal =4∆, the phonons emitted in kBTc kBTc
0.4
scattering can break additional Cooper pairs, increasing
the quasiparticle generation efficiency. The magnitude 0.2 0.4 0.6 0.8 1
of the increase depends on the phonon trapping factor.25 T /T
b c
Measurements of the response of a Ta KID28 at signal
frequencies close to the gap energy show these character-
Figure 6. The quasiparticle generation efficiency η as a func-
istic features. The figure shows that when hν , T
signal b tion of reduced temperature Tb/Tc. The calculation is for for
and τl are scaled by the relevant material parameters, Al, with Pprobe = 0, Psignal = 2Wm−3, hνsignal = 10∆(Tb),
the quasiparticle generation efficiency is the same for all andtwovaluesofthephononescapetime. TheinsetshowsTN∗
materials. Kozorezov et al.23 predicted that the quasi- calculated using the full nonequilibrium model (red ) and
(cid:13)
particle generation efficiency could be categorized into using (11) (blue line) as a function of Psignal with Tb =0.1Tc
three classes of superconductors. The superconductors and τl/τ0φ =1.
included in this work, which fall within either the second
(Ta, Nb, NbN) or third (Al, Mo) class, have a material- Figure 6 shows the variation in η as the bath tempera-
independent η ≈ 0.6. This conclusion differs somewhat ture is changed. As T increases, the scattering between
b
from that of Zehnder.13 We understand this as due to the more numerous thermal quasiparticles and phonons
the current work examining the steady-state response to determinesthestructureofthedrivendistribution,rather
constant incoming power, and scaling all relevant param- than scattering between the excess quasiparticles and
eters by material characteristics, whereas Zehnder’s work phonons. This means the power flow from phonons to
examined the time-dependent response including quasi- bathdoesnothavethesamepeakedstructureandinstead
particle out-diffusion from a localized hot-spot, and chose morecloselyresemblesthedifferencebetweentwothermal
a fixed cutoff time of 10ns to calculate the quasiparticle distributions, as earlier shown in figure 2. This type of
generation efficiency unscaled by material. We emphasize excessquasiparticledistributionmeanstheaverageenergy
that for the low energy photons considered in the present of generated quasiparticles increases without increasing
work localized hot-spot formation (and consequent gap the recombination and pair breaking rates. The figure
suppression) does not occur for the signal powers studied. alsoindicatestemperaturesforwhichk T =2∆(T )and
B b b
Theothereffectshowninfigure5istheenhancementof =∆(T ), above which we expect thermal and nonequilib-
b
η by the readout power due to an increase in the average rium distributions to interact strongly. As T /T →1 we
b c
energy of generated quasiparticles. For hν (cid:39)2∆, it see that η → 1 in agreement with the two-temperature
signal
can be seen that η > 1 which may seem unreasonable. model valid in that regime. We find that T∗ is well-
N
The enhancement is due to multiple photon absorption accounted for by (11), with the value of η shown, for
by the quasiparticles i.e. both signal and probe, and T /T ≤ 0.7. The inset shows T∗ calculated using (11)
b c N
8
for τ /τφ =1 as a function of P with T =0.1T . V. CONCLUSIONS
l 0 signal b c
Wehavedescribedsolutionsofthecoupledkineticequa-
tions that calculate steady-state nonequilibrium quasipar-
0.65 ticle and phonon distributions in a number of techno-
logically important superconducting thin-films driven by
sub-gap and pair-breaking photons. In particular we con-
0.6
siderlowenergysignalphotoninteractionshν ≤10∆,
signal
where localized heating and gap-suppression can be ig-
0.55 nored. Wehavecalculatednumericalparametersforthese
superconductors that can be used in a simple analytical
expression to describe the power-flow between quasiparti-
0.5
clesandphonons. Thisexpressionallowsstraight-forward
η
estimates of the effective quasiparticle temperatures for
0.45
both sub-gap and pair-breaking drives to approximate
the nonequilibrium behaviour without resorting to a full
Al
0.4 numericsolutionofthecoupledkineticequations. Thean-
Ta
alytical expression is shown to give a good account of full
Nb
solutionsofthedetailedequationsforawiderangeofpow-
0.35 Mo
ersandbathtemperatures. Thisisrelevantforpredicting
NbN
the behaviour of thin-film nonequilibrium superconduct-
0.3 ing detectors. We defined a low-energy quasiparticle
0 2 4 6 8 10 12 14
generation efficiency for constant absorbed pair-breaking
τl/τ0φ power and calculated detailed numerical solutions as a
function of photon energies and bath temperature. We
Figure 7. Quasiparticle generation efficiency η for materials have shown that key parameters determining quasipar-
studiedasafunctionofphononescapetimeτl/τ0φ. Calculation ticle generation efficiency are the signal frequency and
usesTb =0.1Tc,Pprobe =0,Psignal =2Wm−3,andhνsignal = the phonon trapping factor. The enhancement of signal
10∆. quasiparticle generation efficiency by absorbed readout
power is demonstrated and explained as multiple-photon
absorption (signal and probe) by the quasiparticles and
Cooper pairs. We estimate this effect can result in an
Figure 7 shows the dependence of η on the phonon increase η of up to 20% at signal photon energies of
trapping factor τl/τpb for Tb = 0.1Tc. As the phonon hνsignal = 2∆. The most sensitive detectors currently
escape time τ /τφ increases, the quasiparticle generation in development may also be able to distinguish between
l 0
efficiency η →0.6 for all materials examined. This con- phonon trapping factors using results presented here. For
clusion is in agreement with earlier Monte Carlo calcula- the low-energy photon interactions studied we also show
tions20,22 for infinite superconductors or where Ω≥2∆ that the quasiparticle generation efficiency is material
phonon loss was ignored. independent.
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