Electron cooling by phonons in superconducting proximity structures
EElectron cooling by phonons in superconducting proximity structures
Danilo Nikoli´c, Denis M. Basko, and Wolfgang Belzig Fachbereich Physik, Universit¨at Konstanz, D-78467, Germany Universit´e Grenoble Alpes and CNRS, LPMMC, 25 rue des Martyrs, 38042 Grenoble, France (Dated: November 10, 2020)We investigate the electron-phonon cooling power in disordered electronic systems with a specialfocus on mesoscopic superconducting proximity structures. Employing the quasiclassical KeldyshGreen’s function method, we obtain a general expression for the cooling power perturbative in theelectron-phonon coupling, but valid for arbitrary electronic systems out of equilibrium. We apply ourtheory to several disordered electronic systems valid for an arbitrary relation between the thermalphonon wavelength and the electronic mean free path due to impurity scattering. Besides recoveringthe known results for bulk normal metals and BCS superconductors, we consider two experimentallyrelevant geometries of superconductor-normal metal proximity contacts. Both structures feature asignificantly suppressed cooling power at low temperatures related to the existence of a minigap inthe quasiparticle spectrum. This improved isolation low cooling feature in combination with thehigh tunability makes such structures highly promising candidates for quantum calorimetry.
I. INTRODUCTION
In experiments on quantum thermodynamics it is im-portant to understand the ultimate limits of thermal en-ergy transfer in nanoscale systems. A prime candidatefor ultra-low temperature detectors of single heat quantaare so-called proximity thermometers, that consist of nor-mal metals in contact with a superconductor offering agreat variability by structuring. An important limitingfactor in heat control is the unavoidable coupling of elec-tronic systems out of equilibrium to the phonon bath.A general description for proximity thermometers is stillmissing and we close that gap in this article.At low temperatures the electron-phonon couplingplays an important role in description of heat removalfrom hot electrons [1]. Besides the theoretical impor-tance, understanding this effect has a practical mean-ing in quantum calorimetry [2–4]. Particularly, fluctu-ations of the electron-phonon cooling power, related tothe electron-phonon thermal conductance by a Nyquist-like relation [4, 5], provide a fundamental limitation forthe minimum portion of energy which can be detectedby its electron heating effect. From the theoretical pointof view, the problem of electron cooling by phonons isintimately related to that of ultrasound attenuation byelectrons, since both problems are concerned with energyexchange between electrons and phonons.Numerous experiments on normal metals have shownthat the power (typically, per unit volume) transferredfrom hot electrons at temperature T e to cold phononsat T ph can be written as Q ( T e , T ph ) = Q ( T e ) − Q ( T ph ),where Q ( T ) ∝ T p . The well-known result p = 5 forclean normal metals has been proven experimentally [6–8] in agreement with theory [8, 9]. Electron scatteringon impurities modifies the power p . Due to the so-calledPippard ineffectiveness condition [10], disordered metalswith fully screened Coulomb interaction have a power p = 6 at low temperatures, so that the cooling power isweaker than in the clean case [11–15], as has been veri-fied experimentally [16, 17]. The crossover between the T and T behaviors occurs at a temperature when thethermal phonon wavelength λ ph is of the order of theelectronic mean free path (cid:96) due to the impurity scatter-ing.The energy exchange between electrons and phononshas also been studied in bulk BCS superconductors [13,18–20] and superconducting proximity structures [21].The presence of a gap in the quasiparticle spectrum leadsto a significant suppression of the cooling power at lowtemperatures and makes these systems advantageous forquantum calorimetry applications. In Ref. [21], the au-thors studied the influence of the proximity effect on thecooling power by solving the kinetic equation with theelectron-phonon collision integral in the clean limit (i. e., (cid:96) (cid:29) λ ph ).In this Article, we calculate the energy current betweenelectrons and phonons, kept at temperatures T e and T ph ,respectively, in superconducting proximity structures foran arbitrary relation between (cid:96) and λ ph . We only as-sume (cid:96) (i) to be small compared to the superconduct-ing coherence length and to the typical size of the struc-ture, and (ii) to be large compared to the Fermi wave-length, so that the proximity effect can be describedby the quasiclassical diffusive Usadel equation. Underthese conditions, the energy exchange between electronsand phonons is local on the scale (cid:96) . As a result, thespatial dependence of the cooling power is disentangledfrom its dependence on the phonon momentum. Interst-ingly, the latter dependence is the same as for a normalmetal [10, 12, 18].The Article is organized as follows. In Sec. II we firstspecify the electron-phonon interaction in the co-movingframe of reference, and present a very general expressionfor the cooling power in terms of the electronic stress re-sponse function. This expression is perturbative in theelectron-phonon interaction, but valid for an arbitraryout-of-equilibrium electronic system. Then we show howthis stress response function can be found in a prox-imitized superconducting structure using the quasiclas-sical Keldysh Green’s formalism. The central result of a r X i v : . [ c ond - m a t . s up r- c on ] N ov this chapter is the cooling power expression mentionedabove. In Sec. III we illustrate this approach by apply-ing it to several electronic structures. Besides recover-ing the known results for a bulk normal metal and abulk BCS superconductor, we consider two geometriesof mesoscopic superconductor-normal metal proximitystructures: a normal metal tunnel-coupled to a bulk su-perconductor and a bilayer of a normal metal in contactwith a superconductor. We find a strong suppression ofthe cooling power at low temperatures that is related tothe formation of a minigap in the spectrum. Finally, inSec. IV we summarize our work and give concluding re-marks. II. GENERAL FRAMEWORKA. Electron-phonon interaction and cooling power
Since we are going to describe several structures withnormal and superconducting parts, we do not specify theelectronic Hamiltonian here. We assume the electronsto be in the diffusive limit because of impurity scat-tering, and the Coulomb interaction is assumed to bevery strong. The electrons are described by the usualfermionic field operators ˆ ψ † ( r ) and ˆ ψ ( r ). We omit thespin indices for compactness (the spin multiplicity willgive an additional factor of 2 in the final result). Even-tually, we will only need the electronic quasiclassicalKeldysh Green’s functions, built from these electronicoperators, and satisfying the Eilenberger equation in thepresence of impurities.The acoustic phonons are described via the lattice dis-placement field ˆ u ( r ), giving the displacement of an atominitially located at the point r . In the standard quanti-zation procedure for lattice vibrations, the displacementoperator takes the following form:ˆ u ( r ) = (cid:88) q λ e q λ (cid:115) ρ L ω q λ (cid:16) ˆ b q λ + ˆ b †− q λ (cid:17) e i qr , (1)where ˆ b q λ (ˆ b † q λ ) is the annihilation (creation) operatorof a phonon with momentum q and polarization char-acterized by a unit polarization vector e q λ , ω q λ is thephonon frequency which in general depends on the mo-mentum and polarization, ρ is the mass density of thematerial and L is the sample volume. For the polariza-tion λ = l, t , t e q ,l = q /q , while two transverse vectors e q ,t , e q ,t are chosen so that e q ,t · q = e q ,t · q = 0and e q ,t · e q ,t = 0. The dispersion relation is as-sumed to be the usual relation for acoustic phonons, ω q ,l = c l q , ω q ,t = ω q ,t = c t q , where c l/t is the lon-gitudinal/transverse speed of sound in the material, re-spectively. Due to isotropy, we assume the two transversemodes to have the same velocity, i.e., c t = c t = c t . The lattice Hamiltonian isˆ H ph = (cid:88) q λ ω q λ ˆ b † q λ ˆ b q λ . (2)As discussed in Refs. [12, 13, 18–20], the electron-phonon interaction in disordered systems is most con-veniently described in a co-moving reference frame, i. e.,attached to the oscillating ions of the crystal lattice, sincethe impurities oscillate together with the lattice. Theelectron-phonon interaction Hamiltonian is assumed tohave the form ˆ H e − ph = (cid:90) d r ˆ σ ij ( r ) ˆ u ij ( r ) , (3)where ˆ σ ij and ˆ u ij are the stress and strain tensors, re-spectively,ˆ σ ij ( r ) = 14 m (cid:18) ∂∂r i − ∂∂r (cid:48) i (cid:19)(cid:32) ∂∂r j − ∂∂r (cid:48) j (cid:33) ˆ ψ † ( r ) ˆ ψ ( r (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) r = r (cid:48) + δ ij p F m ˆ ψ † ( r ) ˆ ψ ( r ) , (4)ˆ u ij ( r ) = 12 (cid:18) ∂ ˆ u i ∂r j + ∂ ˆ u j ∂r i (cid:19) , (5)and the summation over the repeated Cartesian indices i, j = x, y, z is implied. The Coulomb interaction is as-sumed to be very strong, so that electronic charge densityfluctuations are completely suppressed. Formally thisis described by dressing the electron-phonon vertex bythe Coulomb interaction in the random phase approx-imation [12, 13, 19, 20] and results in the subtractionfrom the first term in ˆ σ ij ( r ) of its projection on the elec-tron density. Here, p F is the electron Fermi momen-tum defined via the average of p over the Fermi surface, p F = ( (cid:104) p (cid:105) F ) / , and m the free electron mass.With the electron-phonon interaction at hand we candefine the operator for the total energy current flowinginto the phonons due to the electron-phonon interactionin the whole sample:˙ˆ H ph = i [ ˆ H e − ph , ˆ H ph ] = i (cid:90) d r [ˆ u ij ( r ) , ˆ H ph ] ˆ σ ij ( r ) . (6)We are interested in the cooling power when the phononsare in thermal equilibrium at temperature T ph . Since theenergy current (6) is linear in the phonon operators, ithas zero average over any density matrix of the directproduct form, ˆ ρ e ⊗ e − ˆ H ph /T ph , with an arbitrary electronicdensity matrix ˆ ρ e . To obtain a non-zero value to leadingorder in the electron-phonon coupling, one has to perturbsuch a phonon state to the first order in ˆ H e − ph . Thisamounts to calculating the linear response of d ˆ H ph /dt to the perturbation ˆ H e − ph , which can be done using theKubo formula: P = (cid:42) d ˆ H ph dt (cid:43) = − i t (cid:90) −∞ dt (cid:48) (cid:104) [ ˙ˆ H ph ( t ) , ˆ H e − ph ( t (cid:48) )] (cid:105) , (7)where all the operators are represented in the interactionpicture and the average is taken over the non-interactingdensity matrix ˆ ρ e ⊗ e − ˆ H ph /T ph .Since we are dealing with a non-equilibrium situation,it is natural to use the Keldysh Green’s function formal-ism. Expanding the commutator in the Kubo formulaand identifying various Keldysh Green’s function compo-nents [22] (see Appendix A), we end up with the followingexpression for the cooling power: P = 14 (cid:90) d r d r (cid:48) ∞ (cid:90) −∞ dω π ω × (cid:26) D Kijkl ( r , r (cid:48) , ω ) (cid:2) Π Rklij ( r (cid:48) , r , ω ) − Π Aklij ( r (cid:48) , r , ω ) (cid:3) − (cid:2) D Rijkl ( r , r (cid:48) , ω ) − D Aijkl ( r , r (cid:48) , ω ) (cid:3) Π Kklij ( r (cid:48) , r , ω ) (cid:27) , (8)where Π R,A,Kijkl ( r , r (cid:48) , ω ) and D R,A,Kijkl ( r , r (cid:48) , ω ) are the re-tarded, advanced and Keldysh components of Green’sfunctions built from the bosonic operators (4) and (5),respectively. For phonons in thermal equilibrium at tem-perature T ph , D Kijkl ( ω ) = (cid:2) D Rijkl ( ω ) − D Aijkl ( ω ) (cid:3) coth ω T ph . (9) Being mainly interested in various proximity structures,we can make further assumptions about the spatial de-pendence. Assuming the phonons to be unaffected bythe proximity effect, we take them spatially homoge-neous, D ijkl ( r , r (cid:48) , ω ) = D ijkl ( r − r (cid:48) , ω ). On the otherhand, the situation for the electronic polarization opera-tor, Π ijkl ( r , r (cid:48) , ω ) can be rather complicated, due to theproximity effect. Still, we assume the electrons to bein the quasiclassical regime, so we pass to the Wigner(mixed) representation where the spatial dependence ofΠ ijkl ( r , r (cid:48) , ω ) is decomposed into the center of mass, R = ( r + r (cid:48) ) /
2, and the relative coordinate component, x = r − r (cid:48) . Fourier transform is, therefore, performed inthe following way D ijkl ( r , r (cid:48) , ω ) = (cid:90) d q (2 π ) D ijkl ( q , ω ) e i q ( r − r (cid:48) ) , (10)Π ijkl (cid:16) R + x , R − x , ω (cid:17) = (cid:90) d q (2 π ) Π ijkl ( R , q , ω ) e i qx . (11)In leading order of the quasiclassical approximation, thespatial convolution becomes a simple product in theWigner representation. Then the total cooling power canbe written as a volume integral, P = (cid:82) d R Q ( R ), wherethe position-dependent power per volume is given by Q ( R ) = (cid:90) d q (2 π ) ∞ (cid:90) −∞ dω π ω (cid:2) D Rijkl ( q , ω ) − D Aijkl ( q , ω ) (cid:3) (cid:26)(cid:2) Π Rklij ( R , q , ω ) − Π Aklij ( R , q , ω ) (cid:3) coth ω T ph − Π Kklij ( R , q , ω ) (cid:27) . (12)This formula is very general and applicable to an arbi-trary electronic system out of equilibrium. CombiningEqs. (5) and (1), we obtain the phonon spectral functionexplicitly: D Rijkl ( q , ω ) − D Aijkl ( q , ω ) = (cid:88) λ T q λij T q λkl D R − Aλ ( q , ω ) , (13a) D R − Aλ ( q , ω ) = iπ qρ c λ [ δ ( ω + c λ q ) − δ ( ω − c λ q )] , (13b) T q ,lij ≡ q i q j q , T q ,tκij ≡ e q ,tκi q j + e q ,tκj q i q ( κ = 1 , . (13c) B. Electronic polarization operator
Electrons in the proximitized superconductor are de-scribed by the quasiclassical Green’s function ˇ g ( t, t (cid:48) ; r , n )with n and r being, respectively, the unit vector whichindicates the direction of momentum and the center of mass coordinate [23, 24]. The Green’s function has 2 × g = (cid:18) ˆ g R ˆ g K ˆ g Z ˆ g A (cid:19) , (14)where each component is itself a 2 × (cid:90) ˇ g ( t, t (cid:48)(cid:48) ; r , n ) ˇ g ( t (cid:48)(cid:48) , t (cid:48) ; r , n ) dt (cid:48)(cid:48) = ˇ1 × δ ( t − t (cid:48) ) , (15)and satisfying the Eilenberger equation [25]: (cid:20) ˆ τ ∂ t + ˆ τ ∆ + i ˆ τ ˇ V + (cid:104) ˇ g (cid:105) n τ , ˇ g (cid:21) + v F n · ∇ ˇ g = 0 , (16)where ˆ τ i are the Pauli matrices in the Gor’kov-Nambuspace and (cid:104) . . . (cid:105) n denotes the average over the direc-tions n . The electron-impurity scattering time τ and theFermi velocity v F define the mean free path (cid:96) = v F τ .The commutator in Eq. (16) includes the convolutionover time. Thus, the time derivative ∂ t , the supercon-ducting gap ∆, and the perturbation ˇ V should be un-derstood as integral operators in the time variables withkernels δ (cid:48) ( t − t (cid:48) ), ∆( r ) δ ( t − t (cid:48) ), and ˇ V ( r , t, n ) δ ( t − t (cid:48) ),respectively.In Keldysh space, ∂ t and ∆ are proportional to the2 × V c , in which case the left-lower corner of theGreen’s function ˆ g Z = 0. However, the goal of this sub-section is to evaluate the three components (retarded,advanced, and Keldysh) of the electronic polarization op-erator, Π R,A,Kijkl ( r , r (cid:48) , ω ), needed in Eq. (12) for the coolingpower. In a disordered superconductor, the calculationof the polarization operator involves summation of lad-der diagram series, which is rather cumbersome [13]; inaddition, here we are interested in a proximity system,lacking translational invariance. A more convenient way,equivalent in the quasiclassical limit p F (cid:96) (cid:29)
1, is to calcu-late the response of the electronic stress σ ij to an appliedexternal classical strain u c ij using the Eilenberger equa-tion [26]. Indeed, the average stress tensor (4) in termsof the quasiclassical Keldysh-Green’s functions reads σ ij ( r , t ) = πN p F m (cid:10) ( n i n j − δ ij /
3) Tr ˆ g K ( t, t ; r , n ) (cid:11) n , (17) where N is the normal density of states at the Fermilevel per spin projection. If the Green’s function is foundto the first order in the perturbing stress, the result de-termines the retarded component of the polarization op-erator, Π Rijkl , since the latter coincides with the Kubosusceptibility (up to the sign). To find the advancedKeldysh components of the polarization operator, onehas to include the quantum component u q ij of the strain.Thus, one has to consider the perturbation ˇ V with thefollowing structure:ˇ V ( t, t (cid:48) ; r , n ) = − p F m ( n i n j − δ ij / δ ( t − t (cid:48) ) e − iωt × (cid:18) u c ij ( r ) u q ij ( r ) u q ij ( r ) u c ij ( r ) (cid:19) , (18)and calculate the response of ˇ g to the first order in thisperturbation from Eq. (16). The three components of theelectronic polarization operator can then be determinedas [26]Π Rij,kl ( r , r (cid:48) , ω ) = − πN (0)2 p F m (cid:28)(cid:18) n i n j − δ ij (cid:19) Tr δ ˆ g K ( t, t ; r , n ) e iωt δu cl kl ( r (cid:48) ) (cid:29) n , (19a)Π Aij,kl ( r , r (cid:48) , ω ) = − πN (0)2 p F m (cid:28)(cid:18) n i n j − δ ij (cid:19) Tr δ (ˆ g R + ˆ g A )( t, t ; r , n ) e iωt δu q kl ( r (cid:48) ) (cid:29) n , (19b)Π Kij,kl ( r , r (cid:48) , ω ) = − πN (0)2 p F m (cid:28)(cid:18) n i n j − δ ij (cid:19) Tr δ (ˆ g K + ˆ g Z )( t, t ; r , n ) e iωt δu q kl ( r (cid:48) ) (cid:29) n . (19c)Note that in the presence of the quantum component u q ij ,the lower left corner of Eq. (14), ˆ g Z (cid:54) = 0.We assume to be in the dirty limit, ωτ (cid:28)
1, ∆ τ (cid:28) V = 0) has the followingangular structure [27]:ˇ g ( t, t (cid:48) ; r , n ) = (cid:90) d(cid:15) π e − i(cid:15) ( t − t (cid:48) ) (cid:2) ˇ g ( (cid:15) ; r ) −− v F τ n · ˇ g ( (cid:15) ; r ) ∇ ˇ g ( (cid:15) ; r ) + O (( (cid:96) ∇ g ) ) (cid:3) . (20)The isotropic part ˇ g ( (cid:15) ; r ) satisfies the Usadel equa-tion [27] which should be solved in each specific geometryof the proximitized system. In this subsection, we will as-sume that ˇ g ( (cid:15) ; r ) is known. Constraint (15) for ˇ g impliesˇ g ( (cid:15) ; r ) ˇ g ( (cid:15) ; r ) = ˇ1 × + O (( (cid:96) ∇ g ) ) . (21)Usually, in the dirty limit it is sufficient to work withthe Usadel equation without invoking the full Eilenberger equation (16) at all. It is the angular structure of the per-turbation, n i n j − δ ij /
3, proportional to the second spher-ical harmonics, that obliges us to work with Eq. (16).Let us first assume that the perturbation is a smoothfunction of space and time. The linear in ˇ V correction, δ ˇ g , to the leading order in τ ∆, τ ∂ t ˇ V , (cid:96) ∇ ˇ V , satisfies[ˇ g , δ ˇ g − (cid:104) δ ˇ g (cid:105) n ] = − iτ [ ˇ V , ˇ g ] , (22)ˇ g δ ˇ g + δ ˇ g ˇ g = 0 , (23)obtained by linearizing Eqs. (16) and (15), respectively.Angular averaging gives (cid:104) δ ˇ g (cid:105) n = 0 because (cid:104) ˇ V (cid:105) n = 0.Adding up (22) and (23), multiplying by ˇ g on the left,and using Eq. (21), we obtain the response, local in spaceon the scale (cid:96) : δ ˇ g = iτ (cid:0) ˇ V − ˇ g ˇ V ˇ g (cid:1) [1 + O ( τ ∆ , (cid:96) ∇ )] . (24)Let us now consider u ij ( r ) ∝ e i qr − iωt assuming ωτ (cid:28) τ (cid:28)
1, but not q(cid:96) (cid:28)
1. As just seen, the spatial scaleof the nonlocality in the response is (cid:96) , while ˇ g , foundfrom the Usadel equation, depends on r on a longer scale.Then, to find the response at q ∼ /(cid:96) , one can neglectthe r dependence of ˇ g and seek the correction δ ˇ g in theform δ ˇ g ( r , n ) = δ ˇ g ( n ) e i qr . The linearized Eilenbergerequation becomes[ˇ g , δ ˇ g − (cid:104) δ ˇ g (cid:105) n ] + 2 i(cid:96) ( qn ) δ ˇ g = 2 iτ [ˇ g , ˇ V ] , (25)and the correction again satisfies Eq. (23). Multiplyingit with 1 and ( qn ), averaging over the angles, and us-ing (cid:104) ˇ V (cid:105) n = 0, (cid:104) n ˇ V (cid:105) n = 0, we obtain (cid:104) ( qn ) δ ˇ g (cid:105) n = 0, (cid:104) ( qn ) δ ˇ g (cid:105) n = 0. Adding up Eq. (23) with its angularaverage subtracted, we obtainˇ g δ ˇ g + i(cid:96) ( qn ) δ ˇ g = ˇ g (cid:104) δ ˇ g (cid:105) n + iτ ˇ g ˇ V − iτ ˇ V ˇ g , (26) i(cid:96) ( qn )ˇ g δ ˇ g + δ ˇ g = (cid:104) δ ˇ g (cid:105) n + iτ ˇ V − iτ ˇ g ˇ V ˇ g , (27)where the second equation is obtained from the first bymultiplying by ˇ g . This gives δ ˇ g = iτ − i(cid:96) ( qn )ˇ g (cid:96) ( qn ) (cid:18) (cid:104) δ ˇ g (cid:105) n iτ + ˇ V − ˇ g ˇ V ˇ g (cid:19) , (28) (cid:104) δ ˇ g (cid:105) n iτ = (cid:28) (cid:96) ( qn ) (cid:96) ( qn ) (cid:29) − n (cid:28) ˇ V − ˇ g ˇ V ˇ g (cid:96) ( qn ) (cid:29) n . (29) Bearing in mind the structure of the strain-strain spectralfunction (13), it is convenient to define three componentsof the polarization operator:Π λ ( q , ω ) = T q λij T q λkl Π ijkl ( q , ω ) , λ = l, t , t , (30)which can be found from Eq. (28) for the following per-turbations:ˇ V λ ( t, t (cid:48) ; r , n ) = δ ( t − t (cid:48) ) e i qr − iωt Φ λ ( n ) (cid:18) u c λ u q λ u q λ u c λ (cid:19) , (31)Φ l ( n ) = ( qn ) q − , Φ t ,t ( n ) = ( qn )( e t ,t n ) q . (32)As a result,Π R/Aλ ( q , ω ) = 2 N (cid:18) p F m (cid:19) Y λ (0) + iτ Y λ ( q(cid:96) ) ∞ (cid:90) −∞ d(cid:15) Tr (cid:110) ˆ g R/A ( (cid:15) + ) ˆ g K ( (cid:15) − ) + ˆ g K ( (cid:15) + ) ˆ g A/R ( (cid:15) − ) (cid:111) , (33a)Π Kλ ( q , ω ) = 2 N (cid:18) p F m (cid:19) iτ Y λ ( q(cid:96) ) ∞ (cid:90) −∞ d(cid:15) Tr (cid:8) ˆ g K ( (cid:15) + ) ˆ g K ( (cid:15) − ) − ˆ g R − A ( (cid:15) + ) ˆ g R − A ( (cid:15) − ) (cid:9) , (33b)where λ = l, t , t (cid:15) ± = (cid:15) ± ω/ g R − A = ˆ g R − ˆ g A .The term with Y λ (0) is not captured by the quasiclassi-cal theory and is inserted by noting that the response at ω = 0 is determined by the Fermi sea, and thus is (i) in-sensitive both to disorder and to superconductivity, and(ii) is local on the spatial scale of the Fermi wavelength,so it can be evaluated for a clean Fermi gas at q = 0 [26].The factors Y λ ( q(cid:96) ) coming from angular averages (seeAppendix B) are given by Y l ( ξ ) = − ξ − (1 + ξ /
3) arctan ξ ξ ( ξ − arctan ξ ) , (34a) Y t ,t ( ξ ) = ξ (1 + 2 ξ / − (1 + ξ ) arctan ξ ξ . (34b)Remarkably, Eqs. (33) have a separable form: the depen-dence on q and λ is factorized from the rest which con-tains the frequency and coordinate dependence and allinformation about the superconductivity and the prox-imity effect. The q , λ dependence is entirely containedin the factors Y λ ( q(cid:96) ), and is the same as calculated for a normal metal [12]. C. Final expression for the cooling power
Having obtained the expressions (33) for the polariza-tion operator, we can rewrite Eq. (12) for the coolingpower per unit volume as follows: Q = (cid:88) λ (cid:90) d q (2 π ) (cid:90) ∞−∞ dω π ω D R − Aλ ( q , ω ) × N (cid:18) p F m (cid:19) (cid:18) iτ (cid:19) Y λ ( q(cid:96) ) F ( R , ω ) . (35)Here F ( R , ω ) denotes the factor F = (cid:90) ∞−∞ d(cid:15) ×× Tr (cid:26) (cid:20) ˆ g R − A ( (cid:15) + ) coth ω T ph − ˆ g K ( (cid:15) + ) (cid:21) ˆ g K ( (cid:15) − ) − (cid:20) ˆ g K ( (cid:15) + ) coth ω T ph − ˆ g R − A ( (cid:15) + ) (cid:21) ˆ g R − A ( (cid:15) − ) (cid:27) , (36)whose frequency dependence comes from (cid:15) ± = (cid:15) ± ω/ g ( R , (cid:15) ± ) whose spatial argument is omitted inEq. (36) for brevity. Since the q dependence of the elec-tronic spectral function is solely contained in Y λ ( q(cid:96) ), theintegration over q is straightforwardly performed by re-solving the δ functions in Eq. (13b), which yields Q = N τ πρ (cid:18) p F m (cid:19) ∞ (cid:90) dω π (cid:88) λ ω c λ Y λ ( ω(cid:96)/c λ ) F ( R , ω ) . (37)This very general formula is the main result of this pa-per and it is applicable to variety of electronic systemsincluding superconducting proximity structures. If theelectron-electron relaxation is sufficiently fast we can as-sume the electrons to be in thermal equilibrium at tem-perature T e [28], so that ˆ g K ( (cid:15) ) = ˆ g R − A ( (cid:15) ) tanh[ (cid:15)/ (2 T e )],then Eq. (37) further simplifies adopting the form Q ( T e , T ph ) = N τ πρ (cid:18) p F m (cid:19) (cid:90) ∞ dω π (cid:88) λ ω c λ Y λ ( ω(cid:96)/c λ ) × (cid:18) coth ω T e − coth ω T ph (cid:19) I ( R , ω ) , (38)where I ( R , ω ) = 2 (cid:90) ∞−∞ d(cid:15) [ n F ( (cid:15) − ) − n F ( (cid:15) + )] ×× Tr (cid:8) ˆ g R − A ( R , (cid:15) + ) ˆ g R − A ( R , (cid:15) − ) (cid:9) , (39)and n F ( (cid:15) ) = [exp( (cid:15)/T e ) + 1] − is the Fermi distribution.The whole information about the electronic properties ofthe system is contained in the function I ( R , ω ). Essen-tially, our task from now on is to calculate it for varioussystems. Another important quantity we are interestedin is the thermal conductance per unit volume: K ( T ) = ∂Q ( T e , T ) ∂T e (cid:12)(cid:12)(cid:12)(cid:12) T e = T . (40)Eqs. (38) and (40) yield an expression for the thermalconductance of the same form as Eq. (38), but with areplacementcoth ω T e − coth ω T ph → ω T sinh[ ω/ (2 T )] . (41)In the following section we shall make use of the devel-oped formalism to calculate the electron-phonon coolingpower in various electronic systems. III. APPLICATION TO SPECIFICSTRUCTURESA. Bulk normal metal
In a bulk normal metal, the retarded and advancedcomponent of ˇ g are just ˆ g R/A = ± ˆ τ , so the function I ( ω ) from Eq. (39) simply reads I ( ω ) = 16 ω . Pluggingit into Eq. (38), we obtain the cooling power per unitvolume in the form Q ( T e ) − Q ( T ph ) with Q ( T ) given by: Q ( T ) = 2 N τπ ρ (cid:18) p F m (cid:19) (cid:88) λ ∞ (cid:90) dω ω c λ Y λ ( ω(cid:96)/c λ ) e ω/T − . (42)Since the functions Y λ ( ξ ) [see Eq. (34)] are rather com-plicated, the integral should be calculated numerically.Still, it simplifies in two limiting cases.At low temperatures, T (cid:28) c λ /(cid:96) , we employ Eqs. (B3a)and (B3c) for Y λ ( ξ ), and arrive at the well-known T dependence [12, 13], Q ( T ) = 32 π N c t τρ (cid:96) (cid:18) p F m (cid:19) (cid:18) c t c l (cid:19) T T ∗ , (43)where the crossover temperature T ∗ ≡ c t /(cid:96) . At hightemperatures, T (cid:29) c λ /(cid:96) , we use Eqs. (B3b) and (B3d)for Y λ ( ξ ) we end up with the following expression for the c t /c l = 0.5 l o n g - w a v e l e n g t h l i m i t s h o r t - w a v e l e n g t h l i m i t K / K ∗ T/T ∗ −9 −6 −3 T/T ∗ = 1.0c t /c l K / K ∗ FIG. 1. (Color online) Electron-phonon thermal conductanceper unit volume in a bulk normal metal as a function of tem-perature T for c l /c t = 0 .
5. The dotted and dashed linescorrespond to the high- and low- T limits, respectively. Theinset shows the same quantity as a function of c t /c l for thetemperature T /T ∗ = 1 . cooling power: Q ( T ) = N c t τρ (cid:96) (cid:18) p F m (cid:19) (cid:18) ζ (5)3 π c t c l T T ∗ + 4 π T T ∗ (cid:19) , (44)where ζ ( x ) is the Riemann zeta function. In the cleanlimit, τ, (cid:96) → ∞ with (cid:96)/τ = v F , the second term ∝ T vanishes, while the first one gives the standard Q ( T ) ∝ T result for clean metals [8, 9].The thermal conductance per unit volume, K ( T ) = dQ ( T ) /dT , is shown in Fig. 1. As can be seen fromEq. (42), K ( T ) has a natural unit K ∗ = 2 N τπ ρ (cid:96) (cid:18) p F m (cid:19) , (45)so that K ( T ) /K ∗ is a dimensionless function of two di-mensionless parameters T /T ∗ ≡ T (cid:96)/c t and c t /c l (notethat c t /c l < / √ K/K ∗ as a function of T /T ∗ for c t /c l = 0 . T limits areindicated by the dashed and dotted lines, respectively.Their validity depends on c t /c l , but roughly speakingthe two limits are reached for T /T ∗ < . T /T ∗ > c t /c l ratio at temperature T /T ∗ = 1 .
0. Estimation ofthe crossover temperature in copper with (cid:96) = 10 nm is T ∗ ≈ . c l = 4 . c t = 2.3 km/s. B. Bulk BCS superconductor
In a superconductor, the retarded and advanced com-ponents of the quasiclassical Keldysh Green’s functionˆ g R/A ( (cid:15) ) can be parameterized in terms of the normal, g ,and the anomalous, f, f † , Green’s functions (we omit thesuperscripts R, A for compactness):ˆ g ( (cid:15) ) = (cid:18) g ( (cid:15) ) f ( (cid:15) ) f † ( (cid:15) ) − g ( (cid:15) ) (cid:19) , g ( (cid:15) ) + f ( (cid:15) ) f † ( (cid:15) ) = 1 . (46)Assuming the superconducting gap to be real, ∆ = ∆ ∗ ,we also have f = f † . Substituting the parametriza-tion (46) into Eq. (39) and employing the relation ˆ g R = − ˆ τ (ˆ g A ) † ˆ τ [30], we obtain I ( ω ) = 16 ∞ (cid:90) −∞ d(cid:15) [ n F ( (cid:15) − , T ) − n F ( (cid:15) + , T )] × [Re g R ( (cid:15) + ) Re g R ( (cid:15) − ) − Im f R ( (cid:15) + ) Im f R ( (cid:15) − )] . (47)We note that this expression is rather general and willbe applied to the bulk homogeneous superconductor im-mediately below, as well as to other proximity structuresin the following subsections. We also note that since the Green’s functions depend on the electronic temperature T e via the superconducting gap, the cooling power canno longer be represented in the form Q ( T e ) − Q ( T ph ). Inthe following, we will focus on the thermal conductanceper unit volume, Eq. (40), which depends only on onetemperature. It is given by K ( T ) K ∗ = (cid:90) ∞ ω I ( ω ) dω T sinh [ ω/ (2 T )] × (cid:20) Y l ( ω(cid:96)/c l )( c l /(cid:96) ) + 2 Y t ( ω(cid:96)/c t )( c t /(cid:96) ) (cid:21) . (48)In a bulk homogeneous superconductor, the quasiclas-sical Green’s functions are given by g ( (cid:15) ) = − i(cid:15) √ ∆ − (cid:15) , f ( (cid:15) ) = f † ( (cid:15) ) = ∆ √ ∆ − (cid:15) . (49)The retarded/advanced Green’s function is obtained bythe substitution (cid:15) → (cid:15) ± iη . The broadening parameter η can be taken to be infinitesimal, or finite, describing levelbroadening due to some relaxation processes [31, 32].The temperature dependence of the superconducting gapis assumed to be [33]∆( T ) = ∆ tanh (cid:16) . (cid:112) T c /T − (cid:17) , (50)where ∆ is the superconducting gap at zero temperatureand T c is the critical temperature.Plugging the Green’s functions (49) into Eq. (47), weobtain I ( ω ) of a bulk BCS superconductor, I ( ω ) = 16 (cid:90) ∞−∞ d(cid:15) [ n F ( (cid:15) − , T ) − n F ( (cid:15) + , T )] × θ ( | (cid:15) + | − ∆) θ ( | (cid:15) − | − ∆) (cid:113) (cid:15) − ∆ (cid:113) (cid:15) − − ∆ × (cid:0) | (cid:15) + (cid:15) − | − ∆ sign (cid:15) + (cid:15) − (cid:1) (51)[ θ ( x ) is the Heaviside step function], illustrated inFig. 2(a) for different temperatures. At T = T c , thenormal state dependence I ( ω ) = 16 ω is recovered. Themain difference between the normal and superconduct-ing cases is the presence of a gap in I ( ω ) at ω < T /T c = 0 . T (cid:28) ∆, I (0 < ω < (cid:114) πω ∆ Tω + 2∆ (cid:16) − e − ω/T (cid:17) e − ∆ /T , (52a) I ( ω = 2∆ + 0 + ) = 16 π ∆ . (52b)The thermal conductance per unit volume K ( T ), be-sides T /T ∗ and c l /c t , now depends on another dimension-less parameter T ∗ / ∆ . For aluminum with the electronic / Δ K / K ∗ T/T c I (a) ω/Δ T/T c = 0.1T/T c = 0.5T/T c = 0.7T/T c = 0.8T/T c = 0.9T/T c = 1.0 (b) c t /c l = 0.5N state, T ∗ /Δ = 0.5T ∗ /Δ = 0.5T ∗ /Δ = 1.0T ∗ /Δ = 2.0T ∗ /Δ = 5.0 −18 −15 −12 −9 −6 −3 FIG. 2. (Color online) (a) The I ( ω ) function of a bulk BCSsuperconductor for various temperatures. At T = T c thenormal state result I ( ω ) = 16 ω is recovered. (b) Electron-phonon thermal conductance per unit volume of a bulk super-conductor as a function of temperature T for c t /c l = 0 . T ∗ / ∆ . The dashed blue line correspondsto the normal case with T ∗ / ∆ = 0 . mean free path of (cid:96) = 10 nm, T ∗ / ∆ ≈ .
1. The val-ues for the longitudinal and transverse speed of soundare taken c l = 6 . c t = 3 . K ( T ) in Fig. 2(b) for c t /c l = 0 . T ∗ / ∆ . Increase of T ∗ / ∆ almost does not change the shape of the curvesjust shifting them along the vertical axis. At low tem-peratures, the cooling power in a BCS superconductoris exponentially suppressed in comparison to the normalstate [see the dashed blue line in Fig. 2(b)] due to thepresence of the superconducting gap: K ( T (cid:28) ∆) K ∗ = 693 π ζ (cid:18) (cid:19) (cid:18) c t c l (cid:19) T T ∗ e − ∆ /T . (53) This difference diminishes at higher temperatures, andfinally at T = T c the normal case is recovered. C. Thin SIN contact
Let us now consider a simple SN proximity structureconsisting of a small island of normal metal coupled toa massive superconducting electrode via a weak tunnelcontact. Then we can neglect suppression of supercon-ductivity by the inverse proximity effect in the supercon-ductor, and focus on the proximity effect in the normalpart. In the zero-dimensional limit it can be describedby the quantum circuit theory [34]. The quasiclassicalGreen’s function of the normal metal ˆ g R/AN satisfies thezero-dimensional analog of the Usadel equation [34–37]: − i(cid:15) (cid:104) ˆ τ , ˆ g R/AN ( (cid:15) ) (cid:105) + Γ (cid:104) ˆ g R/AS ( (cid:15) ) , ˆ g R/AN ( (cid:15) ) (cid:105) = 0 , (54)where ˆ g R/AS ( (cid:15) ) is the solution for a homogeneous BCS su-perconductor given in Eq. (49) and the first commutatordenotes the so-called leakage of coherence [see the insetin Fig. 3]. Γ is half of the rate of electron escape fromthe island into the bulk electrode, related to the tunnelcontact conductance G via G = 4 e N V Γ, where V isthe island volume, so that 1 / ( N V ) is the electronic or-bital mean level spacing in the island. Γ must be smallcompared to the island Thouless energy E Th , defined asthe inverse time needed for an electron to cross the is-land, in order for the island to be in the zero-dimensionallimit; at the same time, we need Γ (cid:29) / ( N V ) for the T/T c = 0.01 N / N ε/Δ Γ/Δ = 1.0Γ/Δ = 0.5Γ/Δ = 0.3Γ/Δ = 0.2Γ/Δ = 0.1 S G T N leakage FIG. 3. (Color online) The density of state N ( (cid:15) ) in a thinnormal metal coupled to a massive superconducting lead viaa tunnel contact for several values of Γ / ∆ , and temperature T /T c = 0 .
01. The Dynes broadening parameter is η/ ∆ =10 − . The inset shows a schematic view of the structure. Coulomb blockade effects to be negligible [40]. Eq. (54)can also describe a planar structure when both V and G are proportional to the contact area, while Γ is inde-pendent of the area; in this geometry the normal layerthickness d should be larger than the mean free path butsmall enough so that the time needed for an electron totravel the distance d is smaller than 1 / Γ (see the nextsubsection for more details).The solution of Eq. (54) readsˆ g N ( (cid:15) ) = a ˆ τ + b ˆ τ √ a + b , (55) a ≡ − i(cid:15) (cid:18) √ ∆ − (cid:15) (cid:19) , b ≡ Γ∆ √ ∆ − (cid:15) , and the retarded and advanced functions are obtained bysubstituting (cid:15) → (cid:15) ± iη , as in the previous subsection.Since Eq. (55) has the same structure as for a bulk su-perconductor, Eq. (49), we expect the presence of a mini-gap in the quasiparticle spectrum [38, 39]. This featureis clearly seen in Fig. 3 that shows the density of states(DOS) per unit volume, N ( (cid:15) ) = N Re g RN ( (cid:15) ) for the tem- I ω/Δ / Δ (a) T/T c = 0.01 Γ/Δ = 1.0Γ/Δ = 0.5Γ/Δ = 0.3Γ/Δ = 0.2Γ/Δ = 0.1 (b) Γ/Δ = 0.2T/T c = 0.01T/T c = 0.1T/T c = 0.3T/T c = 0.7T/T c = 1.0 FIG. 4. (Color online) I ( ω ) function in a thin normal metalcoupled to a massive superconducting lead for (a) differentvalues of Γ / ∆ , and temperature T /T c = 0 .
01, (b) differenttemperatures and Γ / ∆ = 0 .
2. The broadening parameter η/ ∆ = 10 − . perature T /T c = 0 .
01 and several values of Γ / ∆ , witha finite Dynes broadening parameter η/ ∆ = 10 − . Theminigap is narrower than the bulk gap ∆ and for smallΓ (cid:28) ∆ it is determined by Γ (see the black line in Fig. 3that corresponds to Γ / ∆ = 0 . I ( ω ) from Eq. (47) is plotted in Fig. 4(a) for the tem-perature T /T c = 0 .
01 and several values of Γ. Similarlyto the bulk superconductor case, it has a gap determinedby the minigap in the island DOS, strongly dependenton Γ [see Fig. 4(a)]. For Γ (cid:28) ∆ , the gap is approxi-mately 2Γ [the black line in Fig. 4(a)]. Since we are atlow temperature, the gap is empty. The same functionfor various temperatures and the Γ / ∆ = 0 . T = T c .Plugging this I ( ω ) into Eq. (48), one arrives at thethermal conductance K ( T ) per unit volume, shown inFig. 5(a) for various values of Γ / ∆ , T ∗ / ∆ = 1, and T/T c K / K ∗ (a) T ∗ /Δ = 1.0c t /c l = 0.5N state Γ/Δ = 0.1Γ/Δ = 0.2Γ/Δ = 0.5Γ/Δ = 1.0BCS state −15 −12 −9 −6 −3 c t /c l = 0.5 (b) Γ/Δ = 0.2 T ∗ /Δ = 0.5T ∗ /Δ = 1.0T ∗ /Δ = 2.0T ∗ /Δ = 5.0 −9 −6 −3 FIG. 5. (Color online) Electron-phonon thermal conductanceper unit volume as a function of temperature T in a thinnormal metal coupled to a massive superconducting lead for c t /c l = 0 . / ∆ and T ∗ / ∆ = 1 . T ∗ / ∆ and Γ / ∆ = 0 .
2. The dashedviolet and blue lines correspond to the cases of a bulk normalmetal and a bulk superconductor, respectively, at T ∗ / ∆ =1 . c t /c l = 0 .
5. All curves lie between those for the bulksuperconductor (the dashed blue line) and the normalstate (the dashed violet line) and K ( T ) is suppressed to-wards larger Γ. This clearly follows from the fact thatthe minigap in the DOS grows with Γ, always remainingsmaller then ∆ (Fig. 3). With increasing temperaturesall curves tend towards the normal state which is recov-ered at T = T c . For small Γ, e.g. Γ / ∆ = 0 .
1, theminigap is very narrow and this case shows a similar be-havior like the normal metal even at low temperatures,
T /T c ∼ .
07 [the black line in Fig. 5(b)]. On the otherhand, for Γ / ∆ = 1 . T ∗ / ∆ , we plot K ( T ) in Fig. 5(b)for various values of the α parameter and Γ / ∆ = 0 . c t /c l = 0 .
5. Similarly to the bulk superconductor, in-crease of T ∗ / ∆ does not change the shape of the curves,but just shifts them downwards. D. Thin SN bilayer
Finally, let us consider a thin SN bilayer in the dirtylimit ∆ τ (cid:28)
1, shown schematically in Fig. 6 and de-scribed in the corresponding caption. The difference be-tween this geometry and the structure studied in the pre-vious subsection is twofold: (i) both the normal metaland the superconductor are thin, and (ii) the contact isnot considered in the tunneling limit, i. e. the interfaceconductance per unit area G is a measure of imperfectionof the SN interface which would be perfectly transpar-ent in the ideal case with G → ∞ . Both these featureslead to the inverse proximity effect in the superconduc-tor that has now to be treated on equal footing with theproximity effect in the normal layer. SN 𝑦𝑧 𝜎 ! 𝜎 " −𝑑 ! 𝑑 " FIG. 6. (Color online) A schema of a thin S N bilayer consist-ing of a normal metal (orange) of a thickness d N coupled to asuperconductor (blue) of a thickness d S . The nonideal SN in-terface is characterized by the electric conductance G per unitarea, whereas σ N/S denotes the normal state conductivity ofthe N / S layer material.
The Green’s functions for this system were found inRef. [42]. Assuming the system to be homogeneous in theplane ( x, y dimensions), we arrive at an one-dimensionalproblem along the transverse ( z ) direction. Parametriz-ing the Green’s function ˆ g by the proximity angle θ such that ˆ g = ˆ τ cos θ + ˆ τ sin θ , we can write the Usadel equa-tion in each of the two materials as [24] D d θdz = − i(cid:15) sin θ − ∆ cos θ, (56)where D = v F (cid:96)/ z <
0. In principle ∆ has to bedetermined selfconsistently for a given geometry, whichwe neglect here for simplicity. Eq. (56) should be supple-mented by the boundary conditions. At the SN interface, z = 0, we have [41] σ S dθdz (cid:12)(cid:12)(cid:12)(cid:12) z =0 − = σ N dθdz (cid:12)(cid:12)(cid:12)(cid:12) z =0 + = G sin[ θ (0 + ) − θ (0 − )] , (57)where σ N/S = 2 e N ,N/S D N/S is the normal state con-ductivity of the normal metal/superconductor. At thefree surfaces, z = − d S , z = d N (Fig. 6), there is nocurrent flow and the boundary conditions are simply( dθ/dz ) | z = − d S = ( dθ/dz ) | z = d N = 0. ξ/σ = 0.1ξ/σ = 0.5ξ/σ = 1.0d N /ξ = d N /ξ = 0.2T/T c = 0.01 N / N ε/Δ GGG
FIG. 7. (Color online) Local DOS in the normal (solid lines)and superconducting (dotted lines) parts of a thin SN bilayerfor various G , the equal thicknesses of the layers d N = d S =0 . ξ , and the temperature T /T c = 0 . Assuming the system to be thinner than the supercon-ducting coherence length, d S + d N (cid:28) ξ ≡ (cid:112) D/ ∆, wecan seek the solution in the form θ ( z <
0) = θ S + θ (cid:48)(cid:48) S z + d S ) + . . . , (58a) θ ( z >
0) = θ N + θ (cid:48)(cid:48) N z − d N ) + . . . , (58b)where the second term is small compared to the mainone by a factor ∼ d S,N /ξ , and subsequent terms are1even smaller. Then boundary conditions (57) lead to thefollowing system of nonlinear equations: G e N S d S sin( θ S − θ N ) = i(cid:15) sin θ S + ∆ cos θ S , (59a) G e N N d N sin( θ S − θ N ) = − i(cid:15) sin θ N . (59b)The retarded and advanced solutions are obtained byshifting (cid:15) → (cid:15) ± iη . Note that Eq. (59b) has exactly thesame form as Eq. (54), with Γ given by the coefficienton the left-hand side of Eq. (59b). When the coefficienton the left-hand side of Eq. (59a) is small compared to∆, that is ( G ξ/σ S )( ξ/d S ) (cid:28)
1, then θ S is close to itsbulk value, and we recover the results of the previoussubsection. When Γ (cid:28) ∆, the relevant energy scale inthe normal metal is (cid:15) ∼ Γ, so the length scale controllingthe expansion in Eq. (58b) is (cid:112) D/ Γ, and the condition d N (cid:28) (cid:112) D/ Γ is equivalent to Γ (cid:28) E Th = D/d N . Inthe opposite limit of a thick superconductor, d S (cid:29) ξ , thecorrection to θ S is small and the results of the previoussubsection are recovered when G ξ/σ S (cid:28) N ( (cid:15), z ) = N Re cos θ RN ( (cid:15), z ) [in fact, the z dependenceis weak in the regime of the expansion (58)]. We take D N = D S = D , σ N = σ S = σ , T = 0 . T c , d N = d S = 0 . ξ , and plot in Fig. 7 N ( (cid:15), d N ) (solid lines) and N ( (cid:15), − d S ) (dotted lines) as a function of energy for var-ious G . The main feature as in all gapped systems is theminigap which is smaller than ∆ and shrinking as G de-creases. The spectrum in both N and S layer is smeared,due the inverse proximity effect, taken into account hereand neglected in the previous subsection. More detailson these results can be found in Ref. [42].Having found the Green’s functions, we calculate I ( z, ω ) from Eq. (47). We plot I ( d N , ω ) (solid lines)and I ( − d S , ω ) (dotted lines) for different transparencies[controlled by G ] of the SN interface [Fig. 8(a)] and differ-ent temperatures [Fig. 8(b)], other parameters being thesame as in Fig. 7. As before, I ( d N , ω ) exhibits a gap thatstrongly depends on G . The edge of the gap is not sharpdue to the smeared spectrum previously shown in Fig. 7.Increasing temperature leads to shrinking and filling ofthe gap until T = T c , where we arrive at the normal statein both layers. Plugging I ( z, ω ) into Eq. (48), we obtainthe electron-phonon thermal conductance K ( T ) in thenormal metal and the superconductor per unit area ofthe structure. In Fig. 9(a) we plot K ( T ) on the nor-mal side for various transparencies of the SN interfacefor c t /c l = 0 . T ∗ / ∆ = 1 . G the effect is strongersince the minigap is getting larger. In Fig. 9(b) wepresent K ( T ) on the superconducting side for the sameparameters as in Fig. 9(a). As in all superconductingstructures, at T = T c , all curves converge to the normalstate one. Since the minigap is always smaller than ∆,the low-temperature suppression of K ( T ) is weaker thanin the bulk superconductor case [the dashed orange line / Δ ω/Δ I ξ/σ = 0.5ξ/σ = 0.1 d N /ξ = d S /ξ = 0.2 (a) T/T c = 0.01ξ/σ = 1.0 GGG ξ/σ = 0.1d N /ξ = d S /ξ = 0.2 (b) G T/T c = 0.01T/T c = 0.7T/T c = 1.0 FIG. 8. (Color online) I ( d N , ω ) (solid lines) and I ( − d S , ω )(dotted lines) (a) for different G and T /T c = 0 .
01, (b) forvarious temperatures and G ξ/σ = 0 .
1. Other parameters arethe same as in Fig. 7. in Fig. 9(a,b)] but, depending on G , much stronger thanin the normal case [the blue line in Fig. 9(a,b)]. Fig. 9(c)shows K ( T ) on the normal (solid lines) and the super-conducting side (dotted lines) of a thin SN bilayer forvarious values of T ∗ / ∆ and the conductance of the SNinterface G ξ/σ = 0 .
1. As in the previous subsections in-creasing T ∗ / ∆ just shifts the curves downwards. Onenotes that the effect is stronger in the S region due tothe inverse proximity effect visible in Fig. 7. IV. CONCLUSIONS
We have studied electron cooling by phonons in super-conducting proximity structures. Using the quasiclassi-cal approximation and perturbation theory in electron-phonon coupling, we obtained a rather general formulafor the cooling power and the thermal conductance,Eq. (37), that is applicable to an arbitrary electronicsystem, even nonequilibrium. We focused on situationswhen electrons and phonons are in equilibrium amongthemselves, but have different temperatures. In the sim-2 K / K ∗ T/T c N state ξ/σ = 0.1 ξ/σ = 0.5 ξ/σ = 1.0BCS state
GGG T ∗ /Δ = 1.0c t / c l = 0.5d N /ξ = d S /ξ = 0.2(a) N layer −12 −9 −6 −3 N state ξ/σ = 1.0 ξ/σ = 0.5 ξ/σ = 0.1BCS state
GGG T ∗ /Δ = 1.0c t / c l = 0.5d N /ξ = d S /ξ = 0.2(b) S layer −12 −9 −6 −3 G ξ/σ = 0.1c t / c l = 0.5d N /ξ = d S /ξ = 0.2(c) T ∗ /Δ = 0.5T ∗ /Δ = 1.0T ∗ /Δ = 2.0 −6 −3 FIG. 9. (Color online) Electron-phonon thermal conductance per unit area (a) on the normal and (b) superconducting side ofa thin SN bilayer as a function of temperature T for various G , c t /c l = 0 .
5, and T ∗ / ∆ = 1 .
0. The dashed blue and orangelines correspond to the cases of a bulk normal metal and a bulk superconductor. Panel (c) shows K ( T ) per unit area on thenormal (solid lines) and the superconducting side (dotted lines) of a thin SN bilayer for various values of T ∗ / ∆ , c t /c l = 0 . G ξ/σ = 0 .
1. Other parameters are the same as in Fig. 7. ple cases of a bulk normal metal and a bulk BCS super-conductor we recovered the previously known results.Subsequently, we illustrated our theory on two sim-ple geometries of a superconductor-normal metal contact.Due to the presence of a proximity minigap, these het-erostructures exhibit a strong suppression of the coolingpower at low temperatures which makes them suitablecandidates for making quantum thermal detectors. Ourtheory can serve as a tool for optimizing the structurein order to improve the detector sensitivity, which couldserve as a benchmark for future experiments.
ACKNOWLEDGMENTS
We thank Jukka Pekola and Bayan Karimi for numer-ous useful discussions. D.N. thanks Universit´e GrenobleAlpes and CNRS for hospitality during his visit. Thiswork was funded through the European Union’s Horizon2020 research and innovation programme under MarieSklodowska-Curie actions (Grant No. 766025).
Appendix A: Keldysh Green’s functions
In the derivation of Eq. (8) based on the Kubo for-mula we make use of the contour-ordered Green’s func-tion which can be constructed for an arbitrary set ofbosonic fields ϕ α ( t ) (the index α incorporating the spa- tial coordinates and all other indices) as˘ D αβ ( t, t (cid:48) ) = (cid:18) D αβ ( t, t (cid:48) ) D <αβ ( t, t (cid:48) ) D >αβ ( t, t (cid:48) ) ˜ D αβ ( t, t (cid:48) ) (cid:19) , (A1a) iD αβ ( t, t (cid:48) ) = (cid:104)T ϕ α (1) ϕ β (1 (cid:48) ) (cid:105) , (A1b) iD <αβ ( t, t (cid:48) ) = (cid:104) ϕ β (1 (cid:48) ) ϕ α (1) (cid:105) , (A1c) iD >αβ ( t, t (cid:48) ) = (cid:104) ϕ α (1) ϕ β (1 (cid:48) ) (cid:105) , (A1d) i ˜ D αβ ( t, t (cid:48) ) = (cid:104) ˜ T ϕ α (1) ϕ β (1 (cid:48) ) (cid:105) , (A1e)where T ( ˜ T ) denotes chronological (antichronological)time ordering. These functions are not independentand by performing the Larkin-Ovchinnikov rotation [43],ˇ D → ˇ L ˇ τ ˇ D ˇ L † with ˇ L = (1 − i ˇ τ ) / √ τ being thesecond Pauli matrix, we pass to the so-called Keldyshspace obtainingˇ D αβ ( t, t (cid:48) ) = (cid:18) D Rαβ ( t, t (cid:48) ) D Kαβ ( t, t (cid:48) )0 D Aαβ ( t, t (cid:48) ) (cid:19) . (A2)The newly introduced functions satisfy the following re-lations [ θ ( t ) being the Heaviside step function]: D Rαβ ( t, t (cid:48) )= θ ( t − t (cid:48) ) (cid:104) D >αβ ( t, t (cid:48) ) − D <αβ ( t, t (cid:48) ) (cid:105) , (A3) D Aαβ ( t, t (cid:48) )= θ ( t (cid:48) − t ) (cid:104) D <αβ ( t, t (cid:48) ) − D >αβ ( t, t (cid:48) ) (cid:105) , (A4) D Kαβ ( t, t (cid:48) )= D >αβ ( t, t (cid:48) ) + D <αβ ( t, t (cid:48) ) , (A5)which are used to derive Eq. (8) from Eq. (7). Appendix B: Calculation of the Y λ ( q(cid:96) ) factors The factors Y λ ( q(cid:96) ) coming from angular averages thatare given by3 Y λ ( q(cid:96) ) = (cid:28) Φ λ ( n )1 + (cid:96) ( qn ) (cid:29) n (cid:18) − (cid:28)
11 + (cid:96) ( qn ) (cid:29) n (cid:19) − + (cid:28) Φ λ ( n )1 + (cid:96) ( qn ) (cid:29) n , (B1)where (cid:104) . . . (cid:105) n denotes averaging over the directions of theFermi velocity and Φ λ ( n ) are given in Eq. (32). Evalua-tion of the angular averages gives (cid:28)
11 + (cid:96) ( qn ) (cid:29) n = arctan q(cid:96)q(cid:96) , (cid:28) ( qn ) /q − /
31 + (cid:96) ( qn ) (cid:29) n = q(cid:96) − (1 + q (cid:96) /
3) arctan q(cid:96)q (cid:96) , (cid:28) [( qn ) /q − / (cid:96) ( qn ) (cid:29) n = (1 + q (cid:96) / ( q(cid:96) ) arctan q(cid:96) −− q (cid:96) / q(cid:96) ) , (cid:28) ( e tκ n )( qn ) /q (cid:96) ( qn ) (cid:29) n = 0 , Y l ( q(cid:96) ) = − q(cid:96) − (1 + q (cid:96) /
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