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Virtanen, P.; Giazotto, F.Fluctuation of heat current in Josephson junctions

Published in:AIP ADVANCES

DOI:10.1063/1.4914077

Published: 01/01/2015

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Please cite the original version:Virtanen, P., & Giazotto, F. (2015). Fluctuation of heat current in Josephson junctions. AIP ADVANCES, 5(2), 1-10. [027140]. https://doi.org/10.1063/1.4914077

Fluctuation of heat current in Josephson junctionsP. Virtanen, and F. Giazotto

Citation: AIP Advances 5, 027140 (2015);View online: https://doi.org/10.1063/1.4914077View Table of Contents: http://aip.scitation.org/toc/adv/5/2Published by the American Institute of Physics

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AIP ADVANCES 5, 027140 (2015)

Fluctuation of heat current in Josephson junctionsP. Virtanen1,a and F. Giazotto21Low Temperature Laboratory, Department of Applied Physics, Aalto University,P.O. Box 15100, FI-00076 AALTO, Finland2NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56127 Pisa, Italy

(Received 19 January 2015; accepted 20 February 2015; published online 27 February 2015)

We discuss the statistics of heat current between two superconductors at differenttemperatures connected by a generic weak link. As the electronic heat in supercon-ductors is carried by Bogoliubov quasiparticles, the heat transport fluctuations followthe Levitov–Lesovik relation. We identify the energy-dependent quasiparticle trans-mission probabilities and discuss the resulting probability density and fluctuationrelations of the heat current. We consider multichannel junctions, and find that heattransport in diffusive junctions is unique in that its statistics is independent of thephase difference between the superconductors. C 2015 Author(s). All article content,except where otherwise noted, is licensed under a Creative Commons Attribution 3.0Unported License. [http://dx.doi.org/10.1063/1.4914077]

I. INTRODUCTION

Heat transport through junctions between superconductors is significantly affected by super-conductivity.1 In tunnel Josephson junctions, superconducting phase coherence manifests as acomponent of the thermal conductance that oscillates with the phase difference between the super-conductors,2–5 a prediction which was confirmed by recent experiments.6,7 In general, both the signand the magnitude of the oscillations depend on the transparency of the junction in question.4

Studies of heat transport in Josephson junctions have largely concentrated on the ensembleaverage value of the heat currents. However, the heat transferred in a given time is in generalnot constant, but has fluctuations.8,9 When mesoscopic systems are considered, fluctuations in heattransport can lead to fluctuations in other quantities — such as the energy stored on a small metalisland — and eventually, in more accessible observables, such as charge current10–12 or temperaturemeasured by a generic temperature probe.13 In addition to the theoretical question on how thecoherent physics of phase differences in superconducting order parameters manifest in statisticalproperties of heat transport, questions on fluctuations can also be of interest in systems that utilizemesoscopic superconductors in a nonequilibrium settings for example for radiation detection.14

In this work, we find the statistics of temperature-driven quasiparticle heat transport in aJosephson junction of arbitrary transparency, as illustrated in Fig. 1. Previously, the energy transportstatistics has been discussed in the tunneling limit,15 and we recover these results as a specialcase. The energy transport separates to elementary quasiparticle transport events. In agreementwith the Levitov–Lesovik formula,16,17 the elementary quasiparticle transmission probabilities aregoverned by the total transmission eigenvalues Tn±(E) [Eq. (13)] of the Bogoliubov–de Gennesscattering problem of the interface. We compute the heat current noise and discuss its parameterdependence in single-channel junctions. We derive results corresponding to dirty-interface18 anddiffusive19 multichannel junctions. The heat statistics in the diffusive limit turns out to have nophase dependence, except in the presence of spectral broadening.

aElectronic mail: pauli.virtanen@aalto.fi

2158-3226/2015/5(2)/027140/10 5, 027140-1 ©Author(s) 2015

027140-2 P. Virtanen and F. Giazotto AIP Advances 5, 027140 (2015)

FIG. 1. (a) Heat current IE flows from superconductor L to another superconductor R coupled to it, driven by a differencein the temperatures TL > TR. The heat current is modulated by the phase difference ϕ between the order parameters ∆L, ∆Rof the superconductors. The Josephson junction connecting the two is described by the set of (spin-independent) transmissioneigenvalues {τn} of the normal-state scattering matrix of the interface. (b) The excitations that carry heat in superconductorsare Bogoliubov quasiparticles, whose density of states N (E) is illustrated. Heat current and its fluctuation statistics is fullydetermined by their energy-dependent transmission probabilities Tn±(E). Each normal-state quantum channel (τn) splits intoinequivalent particle-hole transmission channels (±) due to superconductivity.

II. MODEL

We consider heat transport between two superconducting terminals that are connected by ageneric contact described by the transmission eigenvalues {τj}. The terminals are assumed to lieat different temperatures. We make use of the Keldysh-Nambu Green function formulation fortransport in superconducting structures,20–22 and the quasiclassical boundary condition descriptionof a weak link between bulk superconductors in the diffusive limit.23,24 As in Refs. 2–5, we assumethe superconducting order parameter phase difference ϕ across the junction is kept constant by asuitable biasing circuit, similarly as in the previous experiments.6,7

At equilibrium, electrons inside a superconducting terminal at temperature T with supercon-ducting gap ∆ are described by the quasiclassical equilibrium Green function g(E),

g = *,

gr (gr − ga)h0

0 gr+-, (1)

gr =

*......,

EE2 − |∆|2

∆E2 − |∆|2

− ∆∗E2 − |∆|2 − E

E2 − |∆|2

+//////-

, (2)

in Keldysh(ˇ)–Nambu(ˆ) space. Here, ga = −τ3(gr)†τ3, where τ3 is the third spin matrix in theNambu space. Temperature enters in the equilibrium distribution function h0 = tanh E

2T . To modelinelastic effects in the superconductors, we introduce a phenomenological Dynes parameter byreplacing E → E ± iΓ/2 in gr/a, where Γ is a relaxation rate modeling possible microscopic mecha-nisms.25,26

The statistics of heat transferred in time t0 in a two-point measurement protocol can bedescribed via the generating functions8,9,27

Wα(u, t0) = Tr[eiuHαU(t0)e−iuHαρ(0)U(t0)†] , (3)

which correspond to changes in the internal energy of terminals α = L,R. Here, Hα are the BCSHamiltonians of the superconducting terminals, u is a scalar counting field, and U(t) the timeevolution operator. How to calculate this generating function using a Keldysh approach is discussedin Refs. 27–30. Main features of the approach can be seen by first differentiating Eq. (3) to obtain,

∂uWα(u, t0) = i⟨Hα(t0) − Hα(0)⟩uW (u, t0) , (4)

where ⟨X⟩u,α = Tr[XU+,αρ(0)U†−,α]/Tr[U+,αρ(0)U†−,α] is an expectation value computed with modi-fied time evolution operators U±,α = e±iuHα/2Ue∓iuHα/2 including the counting field with differingsigns on Keldysh branches during time [0, t0]. As noted in Ref. 27, this corresponds to time shiftson observables corresponding to lead α, similarly to charge counting where the counting field also

027140-3 P. Virtanen and F. Giazotto AIP Advances 5, 027140 (2015)

shifts the conjugate variable of the measured observable. In the Fourier transformed Green functionrepresentation25 above, such time shifts on the Keldysh contour are represented by27

gα(E,u) = eiuEσ1/2gα(E)e−iuEσ1/2 . (5)

The expectation value in Eq. (4) can be evaluated via the quasiclassical boundary conditionapproach,23,24 and integrating in u produces the action of superconducting contacts27–30

ln WR(u, t) = 12

n

Tr ln1 +

τn4([gL, gR(u)]+ − 2) + C , (6)

where C is a normalization constant, and Tr includes energy integration in addition to Keldysh-Nambu matrix trace. Note that the counting field enters as time shifts in a reservoir correlationfunction,27 similarly as in the master equation formulation of Ref. 9. Here and below, we fix α = Rand describe the statistics of energy transferred into the right terminal.

III. GENERATING FUNCTION

An important difference in energy transport compared to charge statistics follows from the factthat an Andreev reflection does not transfer energy. In the present formulation, this is visible in thefact that (Γ → 0+)

g(E,u) = gr(E) ⊗

1 , |E | < ∆ ,eiuEσ1

2 *,

1 2h0

0 −1+-

e−iuEσ1

2 , |E | > ∆ . (7)

There is no energy transfer at sub-gap energies, where there are no quasiparticles, assuming nobroadening in the spectrum of the superconductors.

The generating function can be found by direct substitutions into Eq. (6). It is however useful tomake use of g2

L/R = 1 and rewrite

1 +τn4([gL, gR(u)]+ − 2) = [qn + gLgR(u)][qn + gR(u)gL]

(1 + qn)2 , (8)

where qn = −1 + 2/τn + 2√

1 − τn/τn are the eigenvalues of the hermitian square of the correspond-ing transfer matrix,28,31 so that

ln WR(u) = t0

∞

−∞

dE2π

n

ln det[qn + gLgR(u)] + C ′ , (9)

where C ′ is a normalization constant.The product structure g = gr ⊗ V of Eq. (7) implies that the Keldysh and Nambu components

can be diagonalized separately. This yields

ln WR(u) = 2t0

∞

max(∆L,∆R)dE2π

n

αβ=±1

lnµα + qnλβ

1 + qnλβ, (10)

where t0 is the measurement time, and {λ,1/λ} and {µ,1/µ} are the eigenvalues of grLgrR and

VLVR(u), respectively.Solving the eigenvalue problems, Eq. (10) can be rewritten in the form of a characteristic

function of a multinomial distribution:

ln WR(u) = 2t0

∞

0

dE2π

n

β=±1

lnk

eikuEpk,n, β(E) , (11)

027140-4 P. Virtanen and F. Giazotto AIP Advances 5, 027140 (2015)

with the event probabilities

pk,n, β(E) =

Tnβ(E)[1 − fR(E)] fL(E) , k = +1 ,Tnβ(E)[1 − fL(E)] fR(E) , k = −1 ,1 − p+1,n, β(E) − p−1,n, β(E) , k = 0 ,0 , otherwise .

(12)

Here, fL/R are the electron Fermi distribution functions in the left and right terminals, and

Tnβ(E) = θ(E2 − |∆L |2)θ(E2 − |∆R|2) 4λβqn(1 + λβqn)2 , (13a)

λ = exp arccoshE2 − |∆L∥∆R| cos ϕ

E2 − |∆L |2

E2 − |∆R|2. (13b)

Here, ϕ is the phase difference between the order parameters of the two superconductors. Thegenerating function describes events where a quasiparticle at energy E > ∆ attempts to move fromthe left to the right (k = +1) or from the right to the left (k = −1). The probability that such atransmission event succeeds depends both on the transparency of the transmission channel (qn, τn),and on superconductivity (λ).

The generating function is non-periodic in u, and the corresponding probability density, whichis obtained as a Fourier transform of WR(u),8,9 describes a continuous distribution.27 This is incontrast to charge transport, where the fact that transferred electronic charge is quantized to multi-ples of e is reflected as a periodicity in the counting field, S(χ) = S(χ + 2π), also in the su-perconducting case.29 Note however that the calculation for statistics of charge current betweensuperconductors leads to “event probabilities” not satisfying 0 ≤ p ≤ 1, attributed to a problem ininterpreting the supercurrent in terms of classical transport events.29 This problem does not arise inthe results discussed here, as the supercurrent flow does not contribute directly to energy transfer.

The probabilities Tn,± are the transmission eigenvalues of a Bogoliubov–de Gennes (BdG)scattering problem. The corresponding scattering matrix (for ∆L = ∆R) can be found in Ref. 32.Direct evaluation of the transmission eigenvalues using the results there gives

eig tt† = {Tn,±} , (14)

which coincides with Eq. (13) above. A similar connection can be made to results in N/S junc-tions,33 (∆L = 0, ∆R = ∆ > 0), after identifying the barrier reflectivity in Ref. 33 as Z = (q − 1)/

4q. That the energy statistics is related to these scattering matrices stems from the fact that thecounting statistics of Bogoliubov quasiparticles, which carry all of the electronic energy current,must follow the Levitov–Lesovik result.16,17

Superconductivity has a significant impact on the transmission eigenvalues. In the normal state,they are simply Tnβ = 4qn/(1 + qn)2 = τn independent of the particle-hole channel index β = ±1. Inthe superconducting state, they deviate significantly from this, as illustrated in Fig. 2. In particular,there is a transmission resonance in one of the channels:

Tn,−(Eres) = 1 , Eres = ±∆

1 +τn

1 − τnsin2ϕ

2. (15)

The above result applies for ∆L = ∆R, but the resonance appears also for ∆L , ∆R. This has a largeeffect especially for junctions whose normal-state transparency is small, in which a large part of thetotal heat current is carried by the resonance.4

A. Including broadening of density of states

We can extend the above results to include broadening of the density of states in the super-conductors, by adding a Dynes parameter Γ > 0. Also in this case, the final generating functionobtains the form (11). As the factorization (7) does not apply, the expressions for the transmissioneigenvalues Tnβ(E) need to be extracted from the determinant in Eq. (9).

027140-5 P. Virtanen and F. Giazotto AIP Advances 5, 027140 (2015)

FIG. 2. Transmission eigenvalues Tn,±(E) calculated for τn = 0.7 and ∆L =∆R =∆. Location of the resonance (15) isindicated with a dashed line. At ϕ = 0, Tn,±=τn.

Straightforward calculation yields for Tn,±(E) the indirect expressions

Tn,+Tn,− =16q2

ncos2(Im θL) cos2(Im θR)�1 + q2

n + 2qn cosh θL cosh θR − 2qn sinh θL sinh θR cos ϕ�2 , (16a)

1Tn,++

1Tn,−= 1 + tan(Im θL) tan(Im θR) cos ϕ (16b)

+(q2

n + 1)[cosh(Re θL) cosh(Re θR) − sinh(Re θL) sinh(Re θR) cos ϕ]2qn cos(Im θL) cos(Im θR) ,

where θL/R = arctanh ∆L/R

E+i(Γ/2) . The above formulas reduce to Eq. (13) for Γ → 0+. For Γ > 0, theresults cannot however be written in terms of a single λ as in Eq. (13).

The eigenvalues Tn,±(E) are plotted for Γ > 0 in Fig. 3. In general, a finite Γ induces a smallnumber of states inside the gaps of the superconductors, so that the junction is transparent for heattransport also at sub-gap energies. Similarly as for above-gap transport, the transmission throughthe two channels ± can differ significantly. Moreover, we can observe that sharp sub-gap resonancesappear in one of the two channels — these are associated with the Andreev bound states and aremost prominent in transparent junctions. Moreover, we note that the phase dependence of sub-gapheat transport in high-transparency channels (left panel) can be substantial, whereas it is not asprominent at low transparencies (right panel).

FIG. 3. Transmission eigenvalues Tn,±(E) at ϕ = π (solid) and ϕ = 0 (dotted) for Γ= 0.1∆, and τn = 0.7 (left panel) orτn = 0.01 (right panel). Here, ∆R = 2∆L.

027140-6 P. Virtanen and F. Giazotto AIP Advances 5, 027140 (2015)

IV. HEAT FLOW STATISTICS

Let us now compute the first moments of the heat statistics. Direct differentiation of Eq. (11)with respect to u yields the average heat current,

IE =n

∞

−∞

dE2π

E[Tn,+(E) + Tn,−(E)][ fL(E) − fR(E)] , (17)

±

Tn± =2τn(E2 − ∆2)[E2 − ∆2cos2ϕ

2 + rn∆2sin2ϕ2 ]

(E2 − ∆2(1 − τnsin2ϕ2 ))2

, (18)

where rn = 1 − τn. This result coincides with that found in Ref. 4. Note, however, that the coeffi-cientsDee,Dhe defined in Ref. 4 do not coincide with Tn±, even though the sums do.

Taking the second derivative, we find the heat current zero-frequency noise

SE =n

β=±1

∞

−∞

dE2π

E2Tnβ

fL(1 − fR) + fR(1 − fL) (19)

−Tnβ( fL − fR)2.

In the tunneling limit, this result coincides with that found in Ref. 15. The behavior of the heatcurrent noise away from equilibrium is shown in Fig. 4, for a fixed value of TR. These resultstake into account the temperature dependences ∆L(TL), ∆R(TR) of the gap on the two sides of thejunctions, as a difference ∆L , ∆R reduces contributions from the resonant transmission. The noisedecreases as the temperature TL decreases, and saturates at TL < TR to a constant value that dependson TR. The result becomes independent of the phase ϕ at TL > Tc, when the left terminal switches tothe normal state. We emphasize the difference of SE between the opaque (left panel) and transparentjunction limit (right panel). In particular, in the former case, the heat current noise is minimized forϕ = 0 whereas in the latter junction SE is minimized for ϕ = π. This behavior resembles the one ofthe thermal conductance of a temperature-biased Josephson weak-link, as predicted in Refs. 4 and5. Moreover, when normalized by conductance, the heat noise is smaller for transparent junctions,due to the last negative term in Eq. (19), which is second order in Tnβ.

At equilibrium (TL = TR = T), the heat current noise obeys the fluctuation relation SE = 2GthT2

that connects it to the thermal conductance Gth = dIE /dTR |TL=TR. This relation can be used to

define an effective noise temperature T∗(TL,TR,{τn}, ϕ) such that SE = 2T2∗Gth(T∗), which can be

useful for characterizing the behavior of the noise, as much of the variation with phase and trans-parency is contained in the thermal conductance. Such a temperature plot is shown in Fig. 5.The result is fairly insensitive to the values of τn and ϕ, and the results fall nearly on the same

FIG. 4. Heat current noise SE as a function of temperature TL, for different values of ϕ and τn (single channel), normalizedto its equilibrium normal-state value at T =Tc. The temperature TR =Tc/2 is kept fixed, and Γ= 0. Here, Tc is the BCScritical temperature, and we take the temperature dependence of the energy gaps ∆L(TL), ∆R(TR) into account via the BCSrelation, taking ∆L(0)=∆R(0).

027140-7 P. Virtanen and F. Giazotto AIP Advances 5, 027140 (2015)

FIG. 5. Temperature T∗(TL,TR) at which SE(TL,TR)= 2T 2∗Gth(T∗). TR sweeps from 0.1Tc to Tc in steps of 0.1Tc, for

Γ= 0. Shown for a single-channel system with τ = 0.1, ϕ = π (solid) and τ = 0.9, ϕ = π/4 (dashed).

curves for different values of these parameters. At low temperatures, the result converges towardsT∗ = max(TL,TR).

More generally, the generating function obeys the fluctuation relation9

WR(u) = WR(−u − iT−1L + iT−1

R ) . (20)

For the probability distribution of transferring energy ε into terminal R in time t0 this implies

PR(ε, t0)PR(−ε, t0) = eε/TRe−ε/TL , t0 → ∞ . (21)

This fluctuation relation is independent of the channel transmissions Tnβ, and therefore does notcontain information about superconductivity.

A. Diffusive junctions

It is possible to average the above generating functions over known distributions of trans-mission eigenvalues {τn}, to obtain results for certain types of multichannel junctions. Let usin particular consider the universal31,34 transmission eigenvalue distribution of a short diffusivejunction,19

τn

=

1

0dτ ρ(τ) , ρ(τ) = 1

τ√

1 − τ. (22)

Changing the integration variable to q for β = + and q−1 for β = − in Eq. (11):

ln WR(u) = 2t0

∞

0

dE2π

∞

0

dqqτ(q)1 − τ(q)ρ(τ(q)) (23)

× ln[1 + 4λq(1 + λq)2 F(E,u)] ,

where F(E,u) = (eiuE − 1) fL(1 − fR) + (e−iuE − 1) fR(1 − fL). The diffusive distribution (22) can-cels the τ(q) dependent prefactors. The integral can then be evaluated:

ln WR(u) = 2t0

∞

max(∆L,∆R)dE2π

4 arcsinh2

F(E,u) . (24)

Note that the result does not depend on λ. The heat transport statistics of a diffusive junction isindependent of the phase difference between the superconductors. Moreover, the only difference tothe normal-state result is the presence of a gap max(|∆L |, |∆R|) in the energy integration.

The absence of phase oscillations arises as coherent contributions from different channelscancel each other. The sign of the phase oscillation is different for large and small τn, and diffusive

027140-8 P. Virtanen and F. Giazotto AIP Advances 5, 027140 (2015)

FIG. 6. Heat transparency of a short diffusive junction (left panel) and tunnel junction channel τ = 0.01 (right panel), forinelastic Dynes parameter Γ= 0.05∆. The diffusive-limit phase-independent result for Γ→ 0+ is shown as a dotted line in theleft panel.

junctions contain the exact balance of low and high-transparency channels necessary for the sum tocancel. The diffusive limit distribution is unique in the sense that log qn are uniformly distributed,31

which is crucial for the above result.Moreover, it should be noted that the phase dependence reappears in the diffusive limit if

inelastic scattering is present (Γ > 0). The technical reason for this is that it is only in the limitΓ → 0 that superconductivity appears solely as a scale factor λ in the square transfer matrixeigenvalues q. We can illustrate the reappearance of phase oscillations via the normalized valueM(E) = n,±Tn,±(E)/[2n τn] that can be understood as a transparency for the heat current (17).The result is shown in Fig. 6. At sub-gap energies, the emergence of the diffusive junction minigapof size Eg = |∆∥ cos ϕ

2 | is evident. Moreover, as pointed out above for single transparent channels,in contrast with low-transparency tunnel junctions, the relative sub-gap change in heat conductivitycan be several orders of magnitude in transparent junctions (left panel).

B. Dirty interfaces

A second universal, potentially experimentally interesting eigenvalue distribution is the “dirtyinterface” distribution18,35

ρ(τ) = g

π

1

τ3/2√

1 − τ, g =

n

τn . (25)

FIG. 7. Heat transparency M (E)=n,±Tn,±(E)/[2nτn] for the dirty interface transmission distribution. Shown forΓ= 0.05∆ (solid) and Γ→ 0+ (dashed).

027140-9 P. Virtanen and F. Giazotto AIP Advances 5, 027140 (2015)

Experimentally, this has been found to match results existing in high-transparency tunnel junc-tions.36

The generating function can be evaluated also in this case, starting from Eq. (23):

ln WR(u) = 2t0

∞

max(∆L,∆R)dE2π

4√

2gsinh2 z(E,u)4

(26)

×

1 +E2 − ∆L∆R cos ϕE2 − ∆2

L

E2 − ∆2

R

,

where z(E,u) = arccosh[1 + 2F(E,u)]. The associated heat transparency entering the heat current isillustrated in Fig. 7. We can note that the above-gap transport resembles that in tunnel junctions, andthe sub-gap part that of diffusive junctions. However, in contrast to the tunnel junction result,3,4 theenergy integral in Eq. (26) is convergent, and does not require cutoffs in the above-gap resonance.

V. DISCUSSION

Electronic transport of heat in Josephson junctions is facilitated by transfer of quasiparti-cles from one side to the other. Andreev reflections transfer no heat, and so only quasiparticlescontribute. Consequently, the heat transport statistics follows directly from the counting statisticsof these excitations. Exactly as in normal-state junctions,16,17 this is determined by the transmissioneigenvalues of an appropriate scattering problem. In the superconducting state, this statistics isdescribed by Bogoliubov–de Gennes transmission eigenvalues. We find their analytical expressions,Eqs. (13) and (16).

We considered both above-gap and sub-gap transport of heat, the latter by using a toy modelfor the broadening of the density of states in the superconductors. The general picture that emergesis that sub-gap heat transport is significantly more sensitive to the phase difference in transparentjunctions than in tunnel junctions. Interestingly, in diffusive junctions it is in fact only the sub-gaptransport that has any phase dependence at all.

The fluctuation statistics of heat transport can be measured using similar approaches as previ-ously used for studying the heat currents.6,7 For example, one can make the heat capacity ofone of the terminals small, and then observe the fluctuation of its total energy via tempera-ture measurements using established experimental techniques.14 Setups utilizing several metalislands10,13 equipped with fast NIS electron thermometry37 could also be a viable approach forprobing the statistics experimentally.

In summary, we consider heat transport driven by a temperature difference across Josephsonjunctions of varying transparency. We obtain the generating function of fluctuation statistics inclosed form. In addition to describing the fluctuations in heat transport, the results provide a wayto understand the average heat current in terms of elementary transmission events. Open questionsremain in taking into account the effects of voltage, temperature, and order parameter fluctua-tions in the superconductors, and modeling inelastic effects starting from microscopic rather thanphenomenological models.

ACKNOWLEDGMENTS

P.V. acknowledges the Academy of Finland for financial support. The work of F.G. has beenpartially funded by the European Research Council under the European Union’s Seventh Frame-work Programme (FP7/2007-2013)/ERC grant agreement No. 615187-COMANCHE, and by theMarie Curie Initial Training Action (ITN) Q-NET 264034.1 K. Maki and A. Griffin, Phys. Rev. Lett. 15, 921 (1965).2 G. D. Guttman, B. Nathanson, E. Ben-Jacob, and D. J. Bergman, Phys. Rev. B 55, 3849 (1997).3 G. D. Guttman, E. Ben-Jacob, and D. J. Bergman, Phys. Rev. B 57, 2717 (1998).4 E. Zhao, T. Löfwander, and J. A. Sauls, Phys. Rev. Lett. 91, 077003 (2003).5 E. Zhao, T. Löfwander, and J. A. Sauls, Phys. Rev. B 69, 134503 (2004).6 F. Giazotto and M. Martinez-Perez, Nature 492, 401 (2012).

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