Single quasiparticle excitation dynamics on a superconducting island

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by  V. F. Maisi

Single quasiparticle excitation dynamics on a superconducting island
V. F. Maisi,1,2, S. V. Lotkhov,3 A. Kemppinen,2 A. Heimes,4 J. T. Muhonen,1,5 and J. P. Pekola1
∗
1Low Temperature Laboratory (OVLL), Aalto University School of Science, P.O. Box 13500, 00076 Aalto, Finland
2Centre for Metrology and Accreditation (MIKES), P.O. Box 9, 02151 Espoo, Finland
3Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany
4Institut fu¨r Theoretische Festko¨rperphysik, Karlsruher Institut fu¨r Technologie,
Wolfgang-Gaede-Str. 1, D-76128 Karlsruhe, Germany
5Centre for Quantum Computation and Communication Technology,
3
School of Electrical Engineering and Telecommunications,
1
University of New South Wales, Sydney NSW 2052, Australia
0
2 We investigate single quasiparticle excitation dynamics on a small superconducting aluminum
island connected to normal metallic leads by tunnel junctions. We find the island to be free of
n
excitationswithinthemeasurementresolutionallowingustodetermineCooperpairbreakingrateto
a
J belessthan3kHz. BytuningtheCoulombenergyoftheislandtohaveanoddnumberofelectrons,
one of them remains unpaired. We detect it by measuring its relaxation rate via tunneling. By
2
injectingelectronswithaperiodicgatevoltage,weprobeelectron-phononinteractionandrelaxation
2
down to a single quasiparticle excitation pair, with a measured recombination rate of 8 kHz. Our
] experiment yields a strong test of BCS-theory in aluminum as the results are consistent with it
n without free parameters.
o
c
-
The quasiparticle excitations describing the micro- is a single electron unable to pair in the condensate and
r
p scopic degrees of freedom in superconductors freeze out hence remaining as an excitation. This parity effect has
u
at low temperatures, provided no energy exceeding the been observed in the past in similar structures [10–12]
s
. superconducting gap ∆ is available. Early experiments but typically with two-electron Andreev tunneling be-
t
a ontheseexcitationswereperformedtypicallyclosetothe ing the main transport process. We focus on devices
m criticaltemperaturewithlargestructuressothatN ,the where Andreev current is suppressed since a high charg-
S
- numberofquasiparticleexcitations,washigh[1–9]. Later ing energy, Ec > ∆, makes tunneling of two electrons
d
on, as the fabrication techniques progressed, it became energetically unfavourable compared to that of a single
n
possible to bring N close to unity to reveal the parity quasiparticle [24–26]. In this case, the transport is dom-
o S
c effect of electrons on a superconducting island [10–14]. inated by single-electron processes allowing simple and
[ In recent years, the tunneling and relaxation dynamics direct probing of the quasiparticle excitations without

of quasiparticles, which we address in this letter, have the interfering multi-electron tunneling.
2
v become a topical subject because of their influence on
For a quantitative descripition of the transport char-
5 practically all superconducting circuits in the low tem-
acteristics, we performed a numerical simulation of the
5 perature limit [15–22].
7 device operation shown in Fig. 1 (c). To describe simul-
2 We study the quasiparticle excitations in a small alu- taneously the charging of the island with electrons and
2. minum island shown in Fig. 1 (a). The island is con- the excitations involvedin superconducting state, we as-
1 nectedvia a thininsulating aluminum oxidelayerto two sign probability P(N,NS) for having N excess electrons
2 normal metallic copper leads to form a single-electron andN quasiparticleexcitationsontheisland. Thetime
S
1 transistor (SET) allowing quasiparticle tunneling. We evolutionof P(N,N ) is described by a master equation
S
:
v bias the SET by a voltage Vb between the source and
i drain and polarize the island by an offset charge n ex-
X g
ar pHreerseseCdginisutnhietsgoaftee-iwslaitnhdacagpaatecitvaonltcaeg.eTVhge≃cuernrge/nCtgI. P˙(N,NS)=NX′,NS′ ΓN′→N,NS′→NSP(N′,NS′), (1)
through the SET is governed by sequential tunneling of
single quasiparticles and it exhibits Coulomb diamonds
whichoverlapeachotherbecauseofthesuperconducting whereP˙(N,N )standsforthetimederivateofP(N,N )
S S
energygap[23,24],observedforourstructureasaregion and ΓN′→N,NS′→NS for the transition rate from state
bounded by the red sawtooths in Fig. 1 (b). P(N ,N ) to P(N,N ). These rates are set by electron
′ S′ S
tunneling between the island and the leads, Cooper pair
Inthesub-gapregime,eV <2∆,ofFig.1(b)thecur-
b breaking and recombination of quasiparticles.
rent should be suppressed if there are no quasiparticle
excitationspresent. Nonetheless,we observeafinite cur- Tunneling rates are calculated by the standard first
rentwhichhasaperiodtwiceaslonginn ascomparedto orderperturbationtheorysothatelectronstunnelinginto
g
the highbiasregion,a unique feature ofa superconduct- the superconducting island to a state with energy E >
ing island due to Cooper pairing of electrons. Its origin ∆ will increase the quasiparticle number and electron
2
of f . This gives the relation
x 10-5 S
(a) (d) 4
V
g Δ 2 NS =√2πD(EF)V ∆kBTSe−∆/kBTS, (3)
Vb N S N I eRI /T-20 wsthaeteres iDn(tEheF)no=rm1a.l4s5ta×te1[02p477]J−an1md−V3tihsetvhoeludmenesoitfythoef
500 nm -4 -1 0 1 island. ForthenormalmetallicleadsweuseFermidistri-
eVb /Δ eR I/Δ butionwithTN =60mK, equalto the base temperature
T 10 0 ofthecryostat. Foradetaileddescriptionofalltransition
4 (b) (c)
-1 rates, see the supplementary information [28].
10
Forthesteady-staterepresentedbyFig.1(c),wesolve
2 10-2 Eq. (1) with P˙(N,N ) = 0 and calculate the current as
S
Δ -3
V /b 0 10 an average of the tunneling rates weighted by the prob-
e
-4 abilities P(N,N ). The parameter values of sample A:
10 S
-2
E = 240 µeV, ∆ = 210 µeV and tunneling resistances
10-5 c
R = 220 kΩ and R = 150 kΩ for the two junctions
T1 T2
-4 10-6 wereusedinthesimulations. Theyweredeterminedfrom
0 1 2 n -1 0 1 measurements in the high bias regime (eV > 2∆) and
g b
| |
hence their values are independent of the sub-gap fea-
FIG. 1: (color online). (a) Scanning electron micrograph of tures.
the sample studied. It is biased with voltage Vb and a gate The simulation of Fig. 1 (c) reproduces the behaviour
offset voltage Vg is applied to a gate electrode. The latter is observed in the experiments. The relaxation rate of a
not shown in the micrograph. The operation is governed by
single quasiparticle excitation via tunneling is expected
few quasiparticle excitations which can be either particle or
to be Γ Γ = (2e2R D(E )V) 1 =
holelike. Aparticlelikeexcitationcanrelaxbytunnelingout qp ≡ N+1→N,1→0 T F −
190 kHz, where we have used the measured dimensions
(filled circle) and a hole by electron tunneling in and filling
the state. (b) Measured source-drain current I as a func- for V = 1.06µm 145nm 25nm. From the fit in the
× ×
tion of bias and gate voltages. (c) Calculated current based sub-gapregimeweobtainΓ =150kHz,consistentwith
qp
on sequential single-electron tunneling model. (d) Measured the prediction. The value ofΓ affects only the value of
qp
current at ng =0 is shown as black dots. Black line is calcu- currentonthelightblueplateauofFig.1(c),nottheac-
lated assuming a quasiparticle generation rate of 3 kHz, and
tual form or size of the terrace. As our simulation based
red one assuming a vanishing generation rate.
onsequentialtunnelingreproducesallthe featuresinthe
sub-gap regime, we conclude that the two-electron peri-
odicity originates from single-electron tunneling and the
number by one,
operation is essentially free of multi-electron tunneling
processes. At odd integer values of n , the characteristic
g
1 ∞ featureofAndreevtunnelingwouldbealinear-in-V cur-
Γ = dEn (E)(1 f (E))f (E+δE), b
NNS−−11→→NN,S e2RT Z∆ S − S N (2) rwehnitchatisloawbsbeinatsivnolotaugredsaatnad. Tahsueblseeaqkuaegnetcdurrorepn[t11in, 2t9h]e,
where RT is the tunneling resistance, nS(E) is the BCS sub-gap region does not vanish even in the zero temper-
density of states, fS/N(E) the occupation probability of aturelimitbutremainsessentiallythesameaspresented
state E in superconductor / normal metal and δE the in Fig. 1. Therefore, all quasiparticle excitations can-
energy gain from charging and biasing [23, 24]. If the not be suppressed at finite bias voltages by lowering the
incoming electron tunnels to the lower branch E < ∆, temperature, if n is close to an odd integer.
− g
it removes an excitation by filling a hole. Similarly, an At even integer values of n we have ideally no cur-
g
outgoing electron from the upper branch removes an ex- rent flow as all electrons are paired. If Cooper pair
citation while if it tunnels out fromthe lowerbranch, an breaking would take place in the island, we would ob-
excitation is generated. tain two quasiparticle excitations in the superconductor.
In the tunneling rates the exact quasiparticle number One of these excitations can then relax by tunneling to
N is accountedforbyanonequilibriumdistributionf . the leads followed by tunneling of a new excitation to
S S
In general this has a complicated form as a function of neutralize the offset charge, similarly as at the odd n
g
energy. However due to the fact that quasiparticles are case described above. This cycle would continue until
injected close to the gap, the resulting tunneling and re- the two quasiparticles recombine to a Cooper pair. As
combination are not sensitive to the functional form of the recombination rate, discussed below, is slower than
f . Oursystemisalsosymmetricwithrespecttothetwo theratesinthecycle,wehaveseveralelectronstunneling
S
branches. Therefore we neglect branch imbalance and throughthedeviceforeachbrokenpair,henceamplifying
paramerizethequasiparticlenumberbyaneffectivelyin- the signal. With this model we obtain an upper bound
creasedtemperature T and a Fermi distribution in case Γ =3 kHz for the pair breaking rate under our
S N N,0 2
→ →
3
experimental conditions based on simulations shown in (a) (b)50
f = 4 kHz
Fig. 1 (d). This generation rate corresponds to energy 4 40 Vb = 120, 200, 280 μV
absorption at 2Γ ∆ = 0.1 aW power on the su-
N N,0 2
→ →
perconducting island.
30
Under constant biasing conditions, there is at most
one quasiparticle present in the subgap regime at low ef/() 2 ef/()20
I I
temperatures. The non-tunneling relaxation on the is-
land is then not possible, since recombinationwould call f = 1 MHz 10
V = 120, 200, 280 μV
for two excitations. Therefore the static case can be de- b
0 0
scribed by pure tunneling without other relaxation pro- 0.0 0.5 1.0 0.0 0.5 1.0
cesses. Tostudyrecombinationoftwoquasiparticlesinto A A
g g
a Cooper pair, we injected intentionally more quasipar- (c)2 (d)
ticlesto theisland. Theinjectionwasdone byaperiodic 10-2
driveofthegatevoltage. Bychangingn ,wechangethe
g
potential of the superconducting island and either pull
qtiuaalsiisplaorwtiecrleedeoxrcictraetaiotenshoinletotytpheeeisxlcaintdatwiohnesnasthpeopteontteina-l Ief/() 1 QQ/SN10-3
is raised and quasiparticles tunnel out. The number of
injected quasiparticles and the number of quasiparticles f = 1 MHz
on the island can then be determined from the resulting Vb = 120, 200, 280 μV 10-4
0
current curves with the help of simulations. In the ex- 0.0 0.5 1.0 0.20 0.25 0.30
periment we approach two different limits which we will A T /T
g S C
discuss in the following: When the pumping frequency
is high, N is large. Such situation can be described by
S
a thermal model, i.e., by an increased time-independent FIG. 2: (color online). (a)-(b) Measured current of Sample
effective temperature of the superconducting island. In A against gate voltage amplitude Ag at f =4 and 1000 kHz
the opposite limit of low frequency, N is small. Then at bias voltage values Vb = 120,200,280 µV shown as blue,
S
redandgreen dotsrespectively. Black linesshowsimulations
the thermal model fails and we have to account for the
assuminganelevatedtemperatureonthesuperconductingis-
exact time dependent number of quasiparticles.
land. (c) Similar measurement for Sample B at f = 1 MHz.
In Fig. 2 (a) we show the measured current for three
(d) The electron-phonon heat flux in the superconducting
differentvaluesofbiasvoltageVb. Thegatedriveissinu- state normalized by that in the normal state extracted from
soidal around n = 1/2 with amplitude A expressed in the measurements (black circles). Temperature is expressed
g g
units of e/Cg and frequency f = 1 MHz corresponding withrespecttothecriticaltemperatureTC =∆/1.76kB. The
tofastpumping. Withoutaccumulationofquasiparticles theoretical result of Eq. (4) is shown by the black line. Solid
and dotted red lines show the recombination and scattering
tothe island,the currentwouldshowquantizedplateaus
partofEq.(4)correspondinly. Theopengreysymbolisfrom
withspacingef,similartotheSINISturnstile[23]. How-
Sample B.
ever, as the island has a surplus of quasiparticles, i.e. it
is heated up, the current is substantially higher. We
use now the thermal model where the heat injection to ΣV(T5 T5) and T is the electron temperature [32].
the island by electron tunneling is balanced by electron- WeNex−pecPt the theNrmal model to be a good approach
phonon interaction. The heat flux into the phonon bath if N 1. With the high frequency and large ampli-
S
is given by tude d≫rive in Fig. 2 (a), we have N 10 quasiparticles
S
∼
Q×˙enpS(=E)n24SζΣ((E5V)k+B5 ǫR)0∞(cid:16)1dǫ−ǫ3E((∆En2+(ǫǫ,)T(cid:17)S()fS−(En()ǫ−,TfPS)()ER−∞+∞ǫd))E, afpdr=eesqe4unakttHfeozrf,osArhgothw∼ens1ei,ndsFuagitgag..e2stI(ifbnt)gh,tethhfaertetqhthueeremntchayelrmimsoaldolewmleforaedidleslatioss
(4) N approachesunity. Asafurtherproofoftheoverheat-
S
whereΣisthematerialconstantforelectron-phononcou- ing, we repeated the high frequency measurement using
pling, ζ(z) the Riemann zeta function and n(ǫ,T) = sampleBwithmeasuredparametervaluesE =620µeV,
c
(exp(ǫ/(k T)) 1) 1 the Bose-Einstein distribution of ∆ = 270 µeV, R = 1800 kΩ, R = 960 kΩ and
B − T1 T2
−
the phonons at temperature T . See supplementary in- V = 800 nm 60 nm 15 nm. The result is shown in
P
× ×
formationfor derivation. The same result is obtained by Fig2(c). Again,thesimulations(blacklines)areableto
kinetic Boltzmann equation calculations [30, 31]. The reproduceallnon-trivialfeaturesofthemeasuredcurves.
simulations based on the thermal model are shown as Asasummaryofthethermalmodelfits,werepeatedthe
black lines in Fig. 2 (a). The electron-phonon coupling measurement of Fig. 2 (a) at different frequencies and
constant Σ = 1.8 109 WK 5m 3, used in simula- determined by numerical simulations the temperature of
− −
tions, was measured· in the normal state, where Q˙ = the superconducting island and the heat injected into it
ep
4
50 electron-phonon relaxation is disregarded.
With the simulations based on instantaneous quasi-
40
20
particle number we can reproduce the experimental fea-
30 tures precisely with no free parameters in the calcula-
tion. At the lowest frequency, f = 4 kHz, we have only
20 10 one quasiparticle present for most of the time. Hence
the curves are not sensitive to the recombination. As
10
f = 4 kHz f = 10 kHz frequency is increased, the simulations without electron-
ef) 0 0 phonon relaxation deviate from the experimental data.
I/( For f = 10 kHz we probe the recombination rate of a
4
single qp pair only, Γ = 8 kHz. We checked
N N,2 0
→ →
10 this by artificially changing the recombination rates for
N >2, without any significant difference in the curves.
S
2 At higher frequencies, the recombination for N >2 be-
S
comes significant as well. The results of the two mod-
f = 100 kHz f = 1 MHz els approacheach other and the thermal model becomes
0 0 valid.
0.0 0.5 1.0 0.0 0.5 1.0
In summary, a small superconducting island at low
A
g temperatures has allowed us to study the dynamics of
single electronic excitations and their relaxation. Under
quiescent conditions we found a vanishing Cooper pair
FIG. 3: (color online). Red circles show the measured cur-
rent for f = 4 −1000 kHz at V = 280 µV. Black lines breaking rate within the measurement resolution: based
b
are simulations based on Eq. (1). Recombination rates are on the measurementnoise we obtained an upper limit of
takentomatchtheheatfluxintheregimewherethethermal 3 kHz for this rate. On the other hand, by periodically
modelapplies(Eq.(4)). Dottedbluelinesarecalculatedwith pumpingelectrons,wecontrollablyincreasedthenumber
vanishingelectron-phononrelaxationrateandsolidgraylines
of quasiparticles and were able to measure the recombi-
with thethermal model.
nationratesboth inthe largequasiparticlenumber limit
and for a single quasiparticle pair: Γ = 8 kHz.
N N,2 0
→ →
basedonthemeasuredcurrent. Theresultsareshownin The recombination rates are in quantitative agreement
Fig. 2 (d) as black circles. The results match well with withtherelaxationmeasuredathighertemperaturesand
the expected electron-phonon coupling of the supercon- in the normal state.
ductor, Eq. (4), presented as the solid black line. We thank D. V. Averin, G. Scho¨n, F. W. J. Hekking,
For studying the deviation from the thermal model, D. Golubev, T. Heikkila¨ and A. Zorin for discussions.
we measured the characteristics at four frequencies f = The work has been supported partially by LTQ (project
4,10,100and1000kHz atV =280µV, showninFig.3. no. 250280)CoEgrant,theEuropeanCommunity’sSev-
b
The thermal model is presented now as solid gray lines. enthFrameworkProgrammeunderGrantAgreementNo.
A more adequate description of the data at low frequen- 238345 (GEOMDISS) and the National Doctoral Pro-
cies is obtained by simulations based on Eq. (1), where gramme in Nanoscience (NGS-NANO). We acknowledge
we keep track on the number of excitations during the the provisionof facilities and technical support by Aalto
operationandtakeintoaccounttherecombinationrates. University at Micronova Nanofabrication Centre.
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scattering which can, respectively, be expressed as
QQ˙˙rsecc == V3ζπΣ(V5T)ΣkS5B5e−(cid:0)k∆B/kTBST∆S.4+ 47(kBTS)2∆3(cid:1)e−2∆/kBTS, [1∗] AEl.ecFt.roGn.icWadydatrte,ssV:.viMlle..mDamisiti@riemvi,keWs..fiS. Moore and F.
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W. Sheard, Phys.Rev.Lett. 16, 1166 (1966).
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Single quasiparticle excitation dynamics on a superconducting island
-
Supplementary information
V. F. Maisi,1,2, S. V. Lotkhov,3 A. Kemppinen,2 A. Heimes,4 J. T. Muhonen,1,5 and J. P. Pekola1
∗
1Low Temperature Laboratory (OVLL), Aalto University School of Science, P.O. Box 13500, 00076 Aalto, Finland
2Centre for Metrology and Accreditation (MIKES), P.O. Box 9, 02151 Espoo, Finland
3Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany
3 4Institut fu¨r Theoretische Festko¨rperphysik, Karlsruher Institut fu¨r Technologie,
1 Wolfgang-Gaede-Str. 1, D-76128 Karlsruhe, Germany
0 5Centre for Quantum Computation and Communication Technology,
2 School of Electrical Engineering and Telecommunications,
University of New South Wales, Sydney NSW 2052, Australia
n
a
J
2 MASTER EQUATION
2 Wemodelthesuperconductingislandofthemainarticlebywritingdownamasterequationbyassigningaprobability
P(N,N ) for having N excess electrons and N quasiparticles on the island. The time derivate of P(N,N ) is then
S S S
]
n
co P˙(N,NS)=N′,N′ ΓN′→N,NS′→NSP(N′,NS′), (1)
- XS
r
up where ΓN′→N,NS′→NS is a transition rate from state (N′,NS′) to (N,NS). For N′ = N,NS′ = NS we insert a rate
s ΓN→N,NS→NS = − N′,NS′ ΓN→N′6=N,NS→NS′6=NS, which is a sum of all rates out from the state (N,NS). Next we
. considertheratesarisingfromelectrontunnelingbetweennormalmetallicleadsandtheislandandtherecombination
t P
a and pair breaking in the superconductor.
m
- ELECTRON TUNNELING
d
Withinfirstorder,thetunnelingratesaregivenbyFermigoldenrulesimilarlyasinRefs.[1]. Nowasweconsideralso
n
explicitly N , we need to split the rates so that an electron tunneling into the upper branch of the superconducting
o S
c island to energy E > ∆, increases the excitation number NS by one and an electron tunneling into lower branch,
[ E < ∆, decrease it by one as shown in Fig. 1. Likewise, an electron tunneling out from the upper branch decreases
−
N and an electron tunneling out from the lower branch increases it by one. Hence we obtain rates
2 S
v
5 Γ = 1 ∞dE n (E)(1 f (E,N ))f (E+δE (N))
5  N−1→N,NS−1→NS j=1,2e2RT,j Z∆ S − S S N j
7 X
arXiv:1212.2 where R is theΓΓΓtNNNu−++nn111→→→elNNNin,,,gNNNSSSre+−+s111i→→→stNNNanSSSce===ofjjjjXXX===u111n,,,222cteeeio222RRRn111TTTj,,,,jjjnZZZ−−∆(−−∞∞∞E∆∆d),ddEEEthnennSSSB(E((CEE)Sf))f(Sd1S(eE(−nEs,fiN,tSNyS(SE)o()f,1(sN1−ta−Stf)e)fNsfN,(NE(fE(E+(+E+δ,δENδEjE(jNj()N(,N)o)))c)),c))upation prob(2a)-
T,j S S S
bilityforstateE ontheislandifwehaveN quasiparticles. andf (E)occupationprobabilityinthenormalmetallic
S N
lead at energy E. δE (N) the energy cost from biasing and charging energy [1]. We assume f to be a Fermi dis-
j N
tribution with temperature equal to the bath temperature T =60 mK. For the superconductor we need to have f
N S
suchthatthe number ofexcitationsequalsN . Forthe tunneling rates,aswellasfor the electron-phononinteraction
S
discussed below, the functional form of the distribution is irrelevant as long as the excitations are close to the gap
edges E ∆, where we inject them. We also neglect the branch imbalance,i.e. assume no chemicalpotential shift,
≈±
∗Electronicaddress: ville.maisi@mikes.fi
2
N I S I N
(a) (b) (c) (d)
Figure 1: (color online). The four different tunneling processes changing N and NS. (a) Electron tunneling out of the island
and removing an excitation. (b) Electron tunneling into the island creating and excitation. (c) Electron tunneling into the
island and removing a hole excitation. (d) Electron tunneling out and creating a hole excitation.
δµ = 0, because our drive is symmetrical with respect to the two branches. Hence we use a Fermi distribution with
temperature T . The relation between N and T is then
S S S
NS =2D(EF)V ∞dE E fE √2πD(EF)V ∆kBTSe−∆/kBTS, (3)
√E2 ∆2 ≈
Z∆ − p
where D(E ) is the normal state density of states at Fermi level and V the volume of the island. The tunneling
F
current is obtained then as
I =e Γ Γ P(N,N ). (4)
N→N+1,NS→NS±1− N→N−1,NS→NS±1 S
N,XNS,±(cid:16) (cid:17)
Including higher order tunneling to Eqs. (1) and (4) is straightforward. For example Andreev current gives rise to
terms Γ . These rates are calculated similarly as in Refs. [2–4] since quasiparticle number N is not
changingN.±I2n→clNu,dNinSg→ANSndreevtunnelingtermsintothecalculationsinmaintext,doesnotchangetheresultsasES >∆.
c
For E < ∆, those terms contribute because Andreev tunneling thresholds are exceeded before the single-electron
c
tunneling thresholds [2–4].
ELECTRON-PHONON INTERACTION
To obtain the recombination rates for Eq. (1) and Eq. (4) of the main article we use Hamiltonian
H =H +H +H . (5)
s p ep
Here H +H is the non-perturbated part and H the perturbation from electron-phonon interaction. The BCS
s p ep
Hamiltonian of the superconductor is
Hs = Ekγk†σγkσ, (6)
kσ
X
with Ek = ǫ2k+|∆k|2. The diagonalizing fermionic operators γkσ are connected to the electron operators ck′σ′ by
c†kσ =vk∗σγp(kσ)+ukσγk†σ, where we use notation
−
v if σ = u if σ =
vkσ = kv if σ =↑ ukσ = uk if σ =↑ . (7)
k k
(cid:26)− − ↓ (cid:26) − ↓
The coefficients satisfy u 2+ v 2 =1 and ∆ v /u =E ǫ leading to
| k| | k| ∗k k k k− k
1 ǫ
v 2 =1 u 2 = 1 k . (8)
k k
| | −| | 2 − E
(cid:18) k(cid:19)
Similarly, we have for the phonons
H = ~ω b b . (9)
p q †q q
q
X
3
The coupling of these two systems has the form
Hep =ν ωq1/2(c†kck qbq+c†kck+qb†q). (10)
−
k,q
X
as in Ref. [6]. The operator of heat flux to the phonons is
i i
H˙p = ~[H,Hp]= ~[Hep,Hp]=iν ωq3/2(c†kck−qbq−c†kck+qb†q). (11)
k,q
X
By using the Kubo formula in the interaction picture we obtain
i t
H˙ = H˙ dt H˙ (t),H (t)
p p 0− ~ ′ p ep ′ 0
D E D1 E tZ−∞ Dh iE
= ~ ωq2 dt′
(
kXk′σσq′ Z−∞ (12)
c†kσ(t)ck−qσ(t)bq(t), c†k′σ′(t′)ck′+qσ′(t′)b†q(t′) 0−
Dh iE
c†kσ(t)ck+qσ(t)b†q(t), c†k′σ′(t′)ck′ qσ′(t′)bq(t′) .
− 0)
Dh iE
Next we use bq(t)=e−iωqtbq and c†kσ(t)=vk∗σγ (kσ)(t)+ukσγk†σ(t) with γkσ(t)=γkσe−iEkt/~. As an example, let us
consider the first non-zero term in Eq. (12). W−e obtain
eiωq(t′−t)ei(−Ekt+Ek−qt−Ek′t′+Ek′+qt′)/~vk∗σvk−qσvk∗′σ′vk′+qσ′ [γ−(kσ)γ−†(k−qσ)bq,γ−(k′σ′)γ−†(k′+qσ′)b†q] 0
= eiωq(t′−t)ei(Ek−q−Ek)(t−t′)/~|vkσ|2|vk−qσ|2 [γ−(kσ)γ−†(k−qσ)bqD,γ−(k−qσ)γ−†(kσ)b†q] 0 E (13)
= eiωq(t′−t)ei(Ek−q−Ek)(t−t′)/~ vkσ 2 vk qσ 2D(1 fk)fk q(nq+1) fk(1 fk qE)nq ,
| | | − | − − − − −
n o
wherewehavepairedtheoperatorsask =k +q. Forsomeoftheterms,thereexiststwodifferentpairingpossibilities.
′
From the second commutator of Eq. (12) we find similarly a term
e−iωq(t′−t)e−i(Ek−q−Ek)(t−t′)/~ vkσ 2 vk qσ 2 (1 fk)fk q(nq+1) fk(1 fk q)nq , (14)
| | | − | − − − − −
n o
where the dummy summationindex has beenchangedfromk to k. Now we cancombine these two terms anddo the
′
time integration to obtain energy conservation rules as delta functions. By combining all terms similarly, assuming
∆ =∆eiφ, where ∆ is real, and changing the summing to integration we obtain from Eq. (12)
k
Q˙ =πν2N(E )D(q) ∞ dǫ d3q ω2
ep F k q
Z−∞ Z n
1 ∆2 (1 f )f (n +1) f (1 f )n δ(E E +~ω )+
− EkEk−q − k k−q q − k − k−q q k− k−q q
(cid:16)1+ ∆2 (cid:17)(cid:16)(1 f )(1 f )(n +1) f f n (cid:17)δ(E +E +~ω )+ (15)
EkEk−q − k − k−q q − k k−q q k k−q q
(cid:16)1+ ∆2 (cid:17)(cid:16)f f (n +1) (1 f )(1 f )n (cid:17)δ( E E +~ω )+
EkEk−q k k−q q − − k − k−q q − k− k−q q
(cid:16)1 ∆2 (cid:17)(cid:16)f (1 f )(n +1) (1 f )f n (cid:17)δ( E +E +~ω ) ,
− EkEk−q k − k−q q − − k k−q q − k k−q q
(cid:16) (cid:17)(cid:16) (cid:17) o
where N(E ) is the density of states of the normal state and D(q) = V the density of states of the phonons. V
F (2π)3
is the volume of the system. Now, d3q = 2π ∞dqq2 1 d(cos(θ)) where θ is the angle between k and q. Further
0 1
dmeoltrae fǫukn=cti~o22nmks2,ovǫekr−qco=s(θ~)2,(2kwm−eq)n2e=edRǫk~2mkFqcosR(θ) and ωR−q = clq, where cl is the speed of sound. For integrating the
1
dEk q − Ek q m m
(cid:12)d(cos(−θ))(cid:12) = ǫk−−q ~2kFq =nS(Ek−q) ~2kFq, (16)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
4
where we have identified the density of states of the superconductor n (E)= E = E . Therefore we obtain
S ǫ √E2 ∆2
−
Q˙ = ν2V N(E )c2 ∞ dǫ ∞dqq3 m n (E )
ep 2π F l Z−∞ kZ0 ~2kF S k−q n
1 ∆2 (1 f )f (n +1) f (1 f )n
− EkEk−q − k k−q q − k − k−q q Ek−q=Ek+~ωq>0
+++ (cid:16)(cid:16)(cid:16)(cid:16)111+−+ EEEkkk∆∆∆EEE222kkk−−−qqq(cid:17)(cid:17)(cid:17)(cid:17)(cid:16)(cid:16)(cid:16)(cid:16)ff(kk1f(−k1−−fqk(f)nk(q−1+q−)1(fn)kq−−+q()11(n)−q−f+k(1)1()−1−−fkf)fkkff−kk−−q)qqnnnqqq(cid:17)(cid:17)(cid:17)(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)EE(cid:12)(cid:12)(cid:12)Ekkk−−−qqq===−E−kEE−kk−~+ω~~qωω>qq>>000o. (17)
(cid:12)
Now, we change the integration variables to E = ǫ2 +∆2 and ǫ=~ω = ~c q(cid:12). Hence, we have dǫ =n (E )dE
k k q l k S k k
and dq =(~c ) 1dǫ. We can also use notation E =E and E =E . Then, equation (17) yields
l − k ′ k q
p −
Q˙ = mν2VN(EF) ∞dE ∞dǫǫ3n (E)n (E )
ep π~6k c2 S S ′
F l Z0 Z0 n
1 ∆2 (1 f(E))f(E )(n(ǫ)+1) f(E)(1 f(E ))n(ǫ)
− EE′ − ′ − − ′ E′=E+ǫ>0
+ (cid:16)1+ E∆E2′(cid:17)(cid:16)(1−f(E))(1−f(E′))(n(ǫ)+1)−f(E)f(E′)n(ǫ) (cid:17)E(cid:12)(cid:12)(cid:12)′= E ǫ>0 (18)
+ (cid:16)1+ E∆E2′(cid:17)(cid:16)f(E)f(E′)(n(ǫ)+1)−(1−f(E))(1−f(E′))n(ǫ)(cid:17)(cid:12)(cid:12)(cid:12)E′=−E+−ǫ>0
+ (cid:16)1− E∆E2′(cid:17)(cid:16)f(E)(1−f(E′))(n(ǫ)+1)−(1−f(E))f(E′)n(ǫ)(cid:17)(cid:12)(cid:12)(cid:12)E′=−E ǫ>0 .
(cid:16) (cid:17)(cid:16) (cid:17)(cid:12) − o
Finally we can change integration variable E and the dummy variable E as follows: f(cid:12)or first term: no changes, for
′ (cid:12)
second term: change of sign for E , for third term: change of sign for E and for the fourth term: change of sign for
′
E and E . Then, we see that both E and E cover both positive and negative values and we obtain
′ ′
Q˙ = mν2VN(EF) ∞ dE ∞dǫǫ3n (E)n (E+ǫ)
ep π~6k c3 S S ×
F l Z−∞ Z0 (19)
1 ∆2 (1 f(E))f(E+ǫ)(n(ǫ)+1) f(E)(1 f(E+ǫ))n(ǫ) .
− E(E+ǫ) − − −
(cid:16) (cid:17)(cid:16) (cid:17)
With some algebra, we can simplify Eq. (17) to
Q˙ = ΣV ∞dǫ ǫ3(n(ǫ,T ) n(ǫ,T )) ∞ dE n (E)n (E+ǫ) 1 ∆2 (f(E) f(E+ǫ)). (20)
ep 24ζ(5)k5 S − P S S − E(E+ǫ) −
B Z0 Z−∞ (cid:18) (cid:19)
Here we have identified the electron-phonon coupling constant Σ = 24ζ(5)kB5mν2N(EF) by taking the limit ∆ 0,
π~6kFc3l →
which yields the normal-state result Q˙ =ΣV(T5 T5). One arrives to Eq. (20) also by using kinetic equations for
ep S − P
the electronic excitations under electron-phonon interaction [7].
For obtaining the recombination and scattering rates, we identify in Eq. (18) the first and last term to correspond
to scattering and the two middle ones to recombination and pair breaking. Since the heat rate is set dominantly
by the quasiparticle occupation near the gap, we can again parametrize the number of quasiparticles by an effective
temperature T and the Fermi distribution f = (exp(E/k T )+1) 1 without any loss off generality. For T
S E B S − P
≪
T ∆/k , satisfied in the experiments, we obtain the recombination and scattering heat fluxes as
S B
≪
Q˙rec(TS) ≈ 2π~bD(EF)V ∞ dξ ∞ dξ′(E+E′)3 1+ E∆E2 fEfE′
Z−∞ Z−∞ (cid:20) ′(cid:21)
πVΣ 7
≈ 3ζ(5)k5 (kBTS∆4+ 4(kBTS)2∆3)e−2∆/kBTS,
B
Q˙ (T ) πbD(E )V ∞ dξ ∞ dξ θ(E E )(E E )3 1 ∆2 f
sc S ≈ ~ F ′ − ′ − ′ − EE E
Z−∞ Z−∞ (cid:20) ′(cid:21)
≈ VΣTS5e−∆/kBTS.
InsteadofparametrisationwithrespecttoT ,wecouldalsoparametrizetheeffectivetemperatureT bythequasipar-
S S
ticle number N with Eq. (3) and ask for the rate Γ entering the master equation for P(N,N ). This
S N→N,NS+2→NS S
5
accounts for the transition of N +2 to N quasiparticles, whereas the number of excess electrons N is unchanged.
S S
As the quasiparticles lie close to the gap the energy loss during this process is 2∆ and therefore the rate is given by
Q˙ (N +2)
rec S
Γ = . (21)
N→N,NS+2→NS 2∆
In these terms the heat balance then is fully described by the master equation for P(N,N ).
S
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