Phase-coherent thermoelectricity and non-equilibrium Josephson current in Andreev interferometers
PPhase-coherent thermoelectricity and non-equilibrium Josephson current in Andreevinterferometers
Mikhail S. Kalenkov
I.E. Tamm Department of Theoretical Physics, P.N. Lebedev Physical Institute, 119991 Moscow, Russia
Andrei D. Zaikin
Institute for Quantum Materials and Technologies,Karlsruhe Institute of Technology (KIT), 76021 Karlsruhe, Germany andNational Research University Higher School of Economics, 101000 Moscow, Russia
We develop a detailed theory describing a non-trivial interplay between non-equilibrium effectsand long-range quantum coherence in superconducting hybrid nanostructures exposed to a temper-ature gradient. We establish a direct relation between thermoelectric and Josephson effects in suchstructures and demonstrate that at temperatures exceeding the Thouless energy of our device bothphase-coherent thermoelectric signal and the supercurrent may be strongly enhanced due to non-equilibrium low energy quasiparticles propagating across the system without any significant phaserelaxation. By applying a temperature gradient one can drive the system into a well pronounced π -junction state, thereby creating novel opportunities for applications of Andreev interferometers. I. INTRODUCTION
It is well known that Cooper pairs can penetrate deepinto a normal metal attached to a superconductor. Asa result of this proximity effect, normal metals may alsoacquire superconducting properties [1–3]. At sufficientlylow temperatures T such macroscopic quantum coher-ence of electrons in normal metals is limited either bythermal fluctuations or by electron-electron interactions[4, 5]. Accordingly, proximity induced superconductingcoherence extends into a normal metal at a typical lengthequal to the shortest of two different length scales, so-called thermal length L T ∼ (cid:112) D/T (here and below D is the electron diffusion coefficient) and Cooper pair de-phasing length L ϕ that remains temperature independent[4–6] at low enough T .As a consequence of this proximity effect, macroscopicphase coherence can be established in the structure thatconsists of two superconductors ( S ) connected via a nor-mal metal ( N ) layer (with normal state resistance R n )forming the so-called SN S junction. Hence, a non-vanishing supercurrent I J may flow across such junc-tions which depends periodically on the superconductingphase difference χ between two superconductors. Pro-vided the normal metal layer remains shorter that L ϕ electron-electron interactions inside it can be neglected.Then the magnitude of dc Josephson current I J in such SN S junctions is controlled by the thermal length L T reaching appreciable values I J > ∼ E Th / ( eR n ) in the lowtemperature limit [7–9] and dropping down to exponen-tially small values I J ∝ e − √ πT/E Th at temperatures ex-ceeding an effective Thouless energy E Th of the N -layer.In somewhat more complicated geometries, such as,e.g., that of an SN S transistor [10], one can also controlboth the magnitude and the sign of I J by applying anexternal voltage and driving the quasiparticle distribu-tion function out of equilibrium [10–13]. Very recently itwas demonstrated [14] that yet another efficient way to control the Josephson current is to expose the junctionto a temperature gradient. This situation can be real-ized in structures analogous to those considered in Refs.[10–13], e.g., in a system composed of two superconduct-ing and two normal terminals interconnected by normalmetallic wires forming a cross. Following [14] below wewill denote this structure as X - junction .Similarly to the case of voltage biased X -junctions [10–13], by applying a temperature gradient one also drivesthe quasiparticle distribution function out of equilibrium.At the same there also exists an important differencebetween these two situations. Namely, in the voltage-biased case the Josephson current I J remains exponen-tially small in the high temperature limit T (cid:29) E Th ,whereas exposing an X -junction to a thermal gradientmay yield substantial supercurrent stimulation in thistemperature range [14]. Thus, applying a temperaturegradient, one can effectively support long-range phasecoherence in the SN S -type of structures at high enoughtemperatures where the equilibrium Josephson currentalready becomes negligible.Exposing the system to a temperature gradient onealso generates electric currents and/or voltages inside thesample. This is the essence of the so-called thermoelec-tric effect in superconducting structures [15]. The mag-nitude of this effect becomes large as soon as electron-hole symmetry in a superconductor is violated in someway [16]. On top of that, at low enough temperaturesthermoelectric signals are phase-coherent, thus depend-ing periodically on the phase of a superconducting con-densate. In hybrid superconducting structures involv-ing normal metals such phase-dependent thermoelectric-ity yields a variety of interesting and non-trivial effectswhich were extensively studied both experimentally [17–21] and theoretically [22–28]. In addition, thermoelectriceffects in superconductors give rise to a number of ap-plications ranging from refrigeration and thermometry[29] to phase-coherent caloritronics [30] and thermal log- a r X i v : . [ c ond - m a t . s up r- c on ] F e b ics [31] aiming to transmit information in the form ofenergy.It turns out that thermoelectric and Josephson effectsin the presence of a temperature gradient are intimatelyrelated to each other [14]. For instance, in the case of X -junctions with two normal terminals kept at differenttemperatures T and T (both exceeding the Thouless en-ergy E Th ) a phase-coherent thermoelectric voltage signal V generated at these terminals is related to the Joseph-son current as [14] eV ( χ ) ∼ eI J ( χ ) R n ∼ E | /T − /T | . (1)This result demonstrates that the thermoelectric voltage V does not decay exponentially even in the high temper-ature limit T , (cid:29) E Th having exactly the same tempera-ture dependence as I J . Such non-exponential dependenceof both V and I J on temperature is due to the presence ofnon-equilibrium low energy quasiparticles suffering littledephasing while propagating through normal wires con-necting two superconducting terminals.One can also demonstrate [14] that a non-vanishingthermoelectric signal V may only occur in asymmetric X -junctions. The same observation holds for the non-equilibrium contribution to I J and, furthermore, a ratherstrong degree of the junction asymmetry is required forthis contribution to reach appreciable values [32]. It ap-pears, however, that the latter observations are of nogeneral validity being merely specific to the X -junctiongeometry.Here we will develop a general microscopic theory ofboth thermoelectric and Josephson effects in supercon-ducting hybrid structures exposed to a temperature gra-dient. Specifically, we will address the four-terminal hy-brid structures – frequently called Andreev interferome-ters – which geometry differs from that of an X -junction[14, 32]. An example of such structure is displayed inFig. 1We will work out a new approach to the descriptionof the phase-coherent transport in quasi-one-dimensionalconductors interconnected between each other and at-tached to bulk external normal and superconducting ter-minals. For this purpose we reformulate the standardquasiclassical theory of superconductivity [3] in the spiritof Nazarov’s circuit theory [33, 34] extending the latterwith the emphasis put on quasi-one-dimensional metallicconductors.With the aid of this approach we will set up adetailed theory describing a non-trivial interplay be-tween proximity-induced quantum coherence and non-equilibrium effects in Andreev interferometers exposedto a temperature gradient. In particular, we will ad-dress both thermoelectric and Josephson effects whichdemonstrate a number of interesting new features whichcan be directly observed in modern experiments. Wewill also emphasize a close relation between these twonon-equilibrium effects in the presence of a temperaturegradient. The structure of our paper is as follows. In Sec. II,we introduce our model system and describe the qua-siclassical formalism serving as a basis for our furtherconsiderations. Section III is devoted to extending thecircuit theory and adopting it to superconducting hybridstructures under consideration. In Sec. IV, we presenta detailed analysis of both thermoelectric and Joseph-son effects in four-terminal Andreev interferometers un-der the influence of a temperature gradient. In Sec. V,we briefly discuss the results and formulate our main con-clusions. As usually, technical details of our calculationare relegated to the Appendixes. II. THE MODEL AND QUASICLASSICALEQUATIONS
Below we will consider a metallic heterostructure whichconsists of two superconducting and two normal termi-nals interconnected by five quasi-one-dimensional normalwires (of lengths L p , L S , , L N , and cross sections A p , A S , , A N , ) as it is shown in Fig. 1. We will assumethat two superconducting electrodes S and S are de-scribed by the order parameter values | ∆ | exp( iχ , ) andare kept at temperature T and the same electrostatic po-tential which – without loss of generality – can be setequal to zero. Two normal electrodes N and N are,in turn, kept at different temperatures T and T . Pro-vided both normal terminals are disconnected from anyexternal circuit no electric current can flow into or out ofthese terminals, i.e. the conditions I N = 0 , I N = 0 (2)should apply.In the presence of a temperature gradient quasiparticledistribution function inside normal wires is driven out ofequilibrium. As a result, electrostatic potentials of twonormal terminals N and N do not anymore equal tozero due to the thermoelectric effect. These two thermo-electric voltages – respectively V and V – induced bythe temperature gradient T − T will be evaluated be-low along with the supercurrent I S flowing between twosuperconducting terminals S and S .Electron transport in metallic heterostructures can beconveniently described by means of quasiclassical Usadelequations [3] for 4 × ⊗ Nambu spaceˇ G = (cid:18) ˆ G R ˆ G K G A (cid:19) . (3)These equations read iD ∇ (cid:0) ˇ G ∇ ˇ G (cid:1) = (cid:2) ˇΩ , ˇ G (cid:3) , ˇ G ˇ G = 1 , (4)ˇΩ = (cid:18) ˆΩ 00 ˆΩ (cid:19) , ˆΩ = (cid:18) ε + eV ∆ − ∆ ∗ − ε + eV (cid:19) . Here and below D is the diffusion coefficient, ε labels thequasiparticle energy and ∆ is the superconducting order ˆ I N ˆ I N ˆ I S ˆ I S ˆ I p L N L N L S L S L p p p N T N T S | ∆ | e iχ S | ∆ | e iχ FIG. 1. Four terminal Andreev interferometer under consid-eration. parameter. The current density j is related to the matrixˇ G (3) in a standard manner as j = − σ e (cid:90) dε Sp(ˆ τ ˇ G ∇ ˇ G ) K , (5)where σ is the normal state Drude conductivity and ˆ τ i are Pauli matrices in the Nambu space.The Keldysh component ˆ G K of the Green function ma-trix (3) can be expressed via retarded ( ˆ G R ) and advanced( ˆ G A ) components of this matrix in the formˆ G K = ˆ G R ˆ h − ˆ h ˆ G A , (6)where ˆ h = h L + ˆ τ h T is the matrix distribution functionconveniently parameterized by two different distributionfunctions h L and h T . In normal conductors the latterfunctions obey diffusion-like equations D ∇ (cid:2) D T ∇ h T + Y∇ h L + j ε h L (cid:3) = 0 , (7) D ∇ (cid:2) D L ∇ h L − Y∇ h T + j ε h T (cid:3) = 0 , (8)where D T,L and Y denote dimensionless kinetic coeffi-cients and j ε represents the spectral current D T = 14 Sp(1 − ˆ G R ˆ τ ˆ G A ˆ τ ) = ν + 14 | F R + F A | , (9) D L = 14 Sp(1 − ˆ G R ˆ G A ) = ν − | F R − F A | , (10) Y = 14 Sp( ˆ G R ˆ τ ˆ G A ) = − (cid:16) | F R | − | ˜ F R | (cid:17) , (11) j ε = 14 Sp (cid:16) ˆ τ (cid:104) ˆ G R ∇ ˆ G R − ˆ G A ∇ ˆ G A (cid:105)(cid:17) = 12 Re (cid:16) F R ∇ ˜ F R − ˜ F R ∇ F R (cid:17) , (12) ν = Re G R is the local density of states and G R,A , F R,A and ˜ F R,A are components of retarded and advanced Green functionsˆ G R,A = (cid:18) G R,A F R,A ˜ F R,A − G R,A (cid:19) . (13)Note that the kinetic coefficient Y specifically accountsfor the presence of electron-hole asymmetry in our sys-tem.In the vicinity of the interfaces between normal wiresand bulk metallic terminals the Green functions maychange abruptly, in which case quasiclassical equations(4) cannot be applied and should be supplemented byproper boundary conditions. In the case of diffusive con-ductors considered here these boundary conditions read[34] A σ − ˇ G − ∇ ˇ G − = A σ + ˇ G + ∇ ˇ G + = e π (cid:88) n T n [ ˇ G − , ˇ G + ]4 + T n ( { ˇ G − , ˇ G + } − , (14)where ˇ G − and ˇ G + are the Green-Keldysh matrices re-spectively at the left and the right sides of the interface,the sum runs over all conducting channels of the inter-face and T n is the transmission of the n -th conductingchannel.In the next section we will work out a circuit theorythat will allow us to establish a formal solution of theabove equations for the system displayed in Fig. 1 and,more generally, for an arbitrary network of quasi-one-dimensional normal wires. III. CIRCUIT THEORY
Our approach is to a certain extent similar to Nazarov’scircuit theory which was originally formulated for low en-ergy transport [33] and subsequently generalized to ar-bitrary energies [34]. This circuit theory, being quitegeneral, can in principle be employed for conductors ofarbitrary dimensionality. Still, in the case of spatially ex-tended low dimensional structures it sometimes remainsrather complicated for practical calculations. Below wewill focus our attention on quasi-one-dimensional conduc-tors and reformulate the quasiclassical theory of super-conductivity in a form that appears more suitable bothfor quantitative calculations and for qualitative analysis.
A. Extended conductors
Let us rewrite Eqs. (7), (8) in the matrix form D ∇ (cid:104) ˆ D ∇ ˆ H + ˆ τ j ε ˆ H (cid:105) = 0 , (15)where we definedˆ H = (cid:18) h T h L (cid:19) , ˆ D = (cid:18) D T Y−Y D L (cid:19) . (16) x ˜ x ˆ H ( x ) ˆ H (˜ x )ˆ I ˆ I ˆ I ˆ I k a) b) FIG. 2. (a) A segment of a quasi-one-dimensional normal wireand (b) several normal wires connected to each other in thenode.
In the case of quasi-one-dimensional conductors Eq. (15)yields ˆ D ˆ H (cid:48) + ˆ τ j ε ˆ H = − e ˆ I/ ( A σ ) . (17)where A is the wire cross section andˆ I = (cid:18) I T I L (cid:19) = − A σ e (cid:18) Sp( ˇ G ˇ G (cid:48) ˆ τ ) K Sp( ˇ G ˇ G (cid:48) ) K (cid:19) (18)is the matrix current which remains conserved along thenormal wire segment. The total electric current I flow-ing across the wire is linked to the I T -component of thematrix current by means of the following relation I = 12 (cid:90) I T dε. (19)It will be convenient for us to introduce the matrixevolution operator ˆ U which obeys the equationˆ D ˆ U (cid:48) ( x, ˜ x, ε ) + ˆ τ j ε ˆ U ( x, ˜ x, ε ) = 0 (20)combined with the initial conditionˆ U ( x, x, ε ) = 1 , (21)where x , ˜ x are two coordinates along the normal wire.With the aid of this evolution operator we can resolveEq. (17) and establish the relation between the matrixcurrent ˆ I and the matrix distribution function ˆ H at thepoints x and ˜ x , cf. Fig. 2a. It readsˆ G x, ˜ x ˆ H ( x ) + ˆ G ˜ x,x ˆ H (˜ x ) = − e ˆ I, (22)where ˆ G x, ˜ x is 2 × G x, ˜ x = σj ε A ˆ τ (cid:104) − ˆ U ( x, ˜ x, ε ) (cid:105) − . (23)It obeys the following relationsˆ G x, ˜ x + ˆ τ ˆ G T ˜ x,x ˆ τ = 0 , ˆ G x, ˜ x + ˆ G ˜ x,x = ˆ G j , (24)where we defined ˆ G j = ˆ τ G j and G j = σj ε A .Equation (22) represents an important result. Its formclosely resembles that of the standard Kirchhoff law fornormal electric circuits where the matrix distribution function ˆ H plays the role analogous to that of the volt-age. This observation enables one to operate with bothmatrix distribution functions ˆ H ( x ) and conductance ma-trices ˆ G x, ˜ x employing the standard rules of electric en-gineering. For instance, in Appendix A we demonstratethat the conductance matrices of several elements con-nected either in series or in parallel can be replaced by asingle equivalent conductance matrix in the same way asit is routinely done for conductances in normal electriccircuits. Furthermore, in Appendix B we also establisha direct matrix analogue of the standard transformationbetween Y-shaped and ∆-shaped electric circuits.Finally, we verify that in the absence of superconduc-tivity (i.e. provided all terminals in our structure remainnormal) the conductance matrix ˆ G x, ˜ x reduces to a trivialform ˆ G x, ˜ x = σ A x − ˜ x ˆ1 . (25) B. Boundary conditions
Let us now specify the boundary conditions at thepoints where the wire is connected to another wire and/orto a bulk metallic terminal.
1. Intersection nodes
At the nodes where several normal wires cross eachother (see Fig. 2b) the sum of the matrix currents flowinginto the node equals to zero (cid:88) ˆ I k = 0 . (26)The same applies for the spectral currents (cid:88) A k j ε,k = 0 . (27)The matrix distribution function ˆ H ( x ) remains continu-ous across the node taking the same value for all wiresat their end points connected to the same node.
2. Interface barriers
Turning now to inter-metallic interfaces we introducethe distribution functions h T,L − and h T,L + on both sides ofthe interface and rewrite Eq. (6) asˆ G K − = ˆ G R − ˆ h − − ˆ h − ˆ G A − , ˆ h − = h L − + ˆ τ h T − , (28)ˆ G K + = ˆ G R + ˆ h + − ˆ h + ˆ G A + , ˆ h + = h L + + ˆ τ h T + . (29)With the aid of these equations we can identically trans-form the Keldysh component of the boundary conditions(14) to the following matrix Kirchhoff-like equationˆ G − + ˆ H − + ˆ G + − ˆ H + = − e ˆ I, (30)where ˆ G + − = (cid:18) g T g Y + A σj ε / − g Y + A σj ε / g L (cid:19) , (31)ˆ G − + = − (cid:18) g T g Y − A σj ε / − g Y − A σj ε / g L (cid:19) (32)are the interface matrix conductances,ˆ H ± = (cid:18) h T ± h L ± (cid:19) (33) define the matrix distribution functions on both sides ofthe barrier and j ε = e π A σ (cid:88) n (cid:34) T n Sp(ˆ τ [ ˆ G R − , ˆ G R + ])4 + T n ( { ˆ G R − , ˆ G R + } − − T n Sp(ˆ τ [ ˆ G A − , ˆ G A + ])4 + T n ( { ˆ G A − , ˆ G A + } − (cid:35) , (34) g T ( ε ) = e π (cid:88) n T n Sp (cid:104) ( ˆ G R + − ˆ τ ˆ G A + ˆ τ )( ˆ G R − − ˆ τ ˆ G A − ˆ τ )(4 + T n ( ˆ G R − ˆ G R + + ˆ τ ˆ G A − ˆ G A + ˆ τ − (cid:105)(cid:16) T n ( { ˆ G R − , ˆ G R + } − (cid:17) (cid:16) T n ( { ˆ G A − , ˆ G A + } − (cid:17) , (35) g L ( ε ) = e π (cid:88) n T n Sp (cid:104) ( ˆ G R + − ˆ G A + )( ˆ G R − − ˆ G A − )(4 + T n ( ˆ G R − ˆ G R + + ˆ G A − ˆ G A + − (cid:105)(cid:16) T n ( { ˆ G R − , ˆ G R + } − (cid:17) (cid:16) T n ( { ˆ G A − , ˆ G A + } − (cid:17) , (36) g Y ( ε ) = − e π (cid:88) n T n Sp (cid:104)
8( ˆ G R − ˆ G A + ˆ τ + ˆ G R + ˆ G A − ˆ τ ) + 4 T n ( ˆ G R + − ˆ G R − )( ˆ G A + − ˆ G A − )ˆ τ + T n [ ˆ G R − , ˆ G R + ][ ˆ G A − , ˆ G A + ]ˆ τ (cid:105)(cid:16) T n ( { ˆ G R − , ˆ G R + } − (cid:17) (cid:16) T n ( { ˆ G A − , ˆ G A + } − (cid:17) . (37)In the tunneling limit T n (cid:28) j ε and the interface con-ductances g L,T, Y in Eqs. (30)-(32) reduce to j ε = G N A σ Sp(ˆ τ [ ˆ G R − , ˆ G R + ] − ˆ τ [ ˆ G A − , ˆ G A + ]) , (38) g T ( ε ) = G N G R + − ˆ τ ˆ G A + ˆ τ )( ˆ G R − − ˆ τ ˆ G A − ˆ τ ) , (39) g L ( ε ) = G N G R + − ˆ G A + )( ˆ G R − − ˆ G A − ) , (40) g Y ( ε ) = − G N G R − ˆ G A + ˆ τ + ˆ G R + ˆ G A − ˆ τ ) . (41)where G N = ( e /π ) (cid:80) n T n is the normal state interfaceconductance.We observe that the matrix relation between the dis-tribution functions at the opposite sides of the interfacebarrier (30) has the same structure as Eq. (22) we de-rived for a quasi-one-dimensional wire. Hence, the circuittheory developed here allows to treat both diffusive wiresand interface barriers on equal footing, thereby greatlysimplifying the whole consideration. With the aid of theabove equations it already becomes straightforward toevaluate the quasiparticle distribution functions every-where inside our system, as it will be demonstrated be-low. x ˜ x ˆ IS N
FIG. 3. A normal wire attached to a superconducting termi-nal.
3. Subgap electron transport
Owing to the absence of quasiparticle states in bothterminals S and S at subgap energies the matrix con-ductance of an SN interface acquires a particularly sim-ple structure at such energies. Let ˇ G − be the bulk Greenfunction for a superconducting terminal. Employing thecondition ˆ G R − = ˆ G A − applicable at subgap energies we ob-serve from Eqs. (34), (36) and (37) that g L -component ofthe matrix conductance vanishes identically, while its g Y -component is linked to the spectral current j ε by meansof a simple formula j ε = 2 A σ g Y . (42)Making use of the above relations we conclude that thatat subgap energies the matrix conductance for the SNinterface can be parameterized by the two spectral con-ductances g T and G j , i.e.ˆ G − + = (cid:18) − g T G j (cid:19) , ˆ G + − = (cid:18) g T G j (cid:19) . (43)Considering now a normal wire connected to the super-conducting terminal via some barrier (Fig. 3) and em-ploying the Kirchhoff rule for series resistances derived inAppendix A one can evaluate the effective subgap matrixconductance for a complex resistor consisting of both theinterface barrier and the attached normal wire. After asimple calculation we obtainˆ G ,x = (cid:18) − G G j (cid:19) , ˆ G x, = (cid:18) G G j (cid:19) , (44)where G ( ε ) is a spectral parameter characterizing boththe interface and the normal wire. Note that the abovestructure of the subgap matrix conductance is by nomeans accidental being consistent with the fundamentalobservation stating that the heat spectral current εI L vanishes identically at subgap energies.For fully transparent interfaces the boundary condi-tions (14) reduce to a simple continuity condition forthe Green functions across the interface which readsˇ G − = ˇ G + . In this case the functions Y and D L vanishat the SN interface and, hence, the matrix ˆ D becomessingular at this interface. Hence, special care should betaken while evaluating the subgap matrix conductance ofthe normal wire attached directly to the superconductingelectrode. In Appendix C we demonstrate that the ma-trix conductance structure in Eq. (44) remains preservedalso in this particular case. IV. THERMOELECTRIC AND JOSEPHSONEFFECTS
Let us now employ the above circuit theory formal-ism in order to describe phase-coherent thermoelectric anJosephson effects in Andreev interferometers displayed inFig. 1.As we already indicated above, keeping the normal ter-minals N and N at different temperatures T and T one drives the quasiparticle distribution functions insideour structure out of equilibrium which in general makesthe whole problem rather difficult to deal with. Somesimplifications can be achieved provided we assume thatthe order parameter value | ∆ | in both superconductingterminals strongly exceeds both temperatures T , andthermoelectric voltages eV , as well as the characteristicThouless energy of our structure E Th = D/L (where L = L p + L S + L S ) along with temperature T of bothsuperconducting terminals. Except pointed out other-wise, below we will adopt this assumption which allows to disregard the effect of overgap quasiparticles and focusour attention only on subgap electron transport.Further drastic simplifications are achieved if we makeuse of Eq. (22) which allows us to write down the rela-tions between the matrix currents and the matrix distri-bution functions for each of the five wires in our structure.We obtain (cid:18) G S G j (cid:19) ˆ H p = − e ˆ I S , (45) (cid:18) G S − G j (cid:19) ˆ H p = − e ˆ I S , (46)ˆ G N ( ˆ H N − ˆ H p ) = e ˆ I N , (47)ˆ G N ( ˆ H N − ˆ H p ) = e ˆ I N , (48)ˆ G p p ˆ H p + ˆ G p p ˆ H p = − e ˆ I p , (49)where ˆ H p and ˆ H p are the matrix distribution functionsat the nodes p and p and the spectral matrix conduc-tances ˆ G N , , ˆ G p p and ˆ G p p can be expressed in theform ˆ G N , = (cid:32) G TN , G Y N , − G TN , G LN , (cid:33) (50)ˆ G p p = − (cid:18) G Tp G Y p − G j / − G Y p − G j / G Lp (cid:19) , (51)ˆ G p p = (cid:18) G Tp G Y p + G j / − G Y p + G j / G Lp (cid:19) . (52)The distribution functions ˆ H N , inside the normal ter-minals read ˆ H N , = (cid:18) h TN , h LN , (cid:19) , (53) h TN , = 12 (cid:20) tanh ε + eV , T , − tanh ε − eV , T , (cid:21) , (54) h LN , = 12 (cid:20) tanh ε + eV , T , + tanh ε − eV , T , (cid:21) . (55)In addition, it is necessary to employ the continuityconditions (26) for the matrix current at the crossingpoints p and p , i.e.ˆ I S + ˆ I N = ˆ I p , ˆ I S + ˆ I N + ˆ I p = 0 . (56)Making use of the above system of linear equations onecan evaluate the matrix distribution functions ˆ H p andˆ H p as well as the matrix currents depending on the dis-tribution functions h T,LN , in the normal terminals. Thenone can derive the general expressions for thermoelectricvoltages V , V along with the supercurrent I S as func-tions of the phase difference χ = χ − χ between thesuperconducting electrodes, temperatures T and T andother relevant parameters. A. Symmetric structures
Let us focus our attention on a special case of sym-metric interferometers in which case the whole analysisbecomes simpler due to the presence of extra symmetryconditions. Setting L N = L N ≡ L N , L S = L S ≡ L S , A N = A N ≡ A N , and A S = A S ≡ A S we gain ex-tra relations between spectral conductances for differentwire segments: G S = G S = G S , (57) G T,LN , = G T,LN , (58) G Y N = − G Y N = G Y N , G Y p = 0 . (59)With the aid of these relation it becomes possible to de-couple the equations for the combinations of the distri-bution functions ˆ H p ± ˆ τ ˆ H p and write (cid:18) G S + G TN G Y N + G j − G Y N − G j G LN + 2 G Lp (cid:19) ( ˆ H p + ˆ τ ˆ H p )= ˆ G N ( ˆ H N + ˆ τ ˆ H N ) , (60) and (cid:18) G S + G TN + 2 G Tp G Y N − G Y N G LN (cid:19) ( ˆ H p − ˆ τ ˆ H p )= ˆ G N ( ˆ H N − ˆ τ ˆ H N ) . (61)These equations can easily be resolved providing the ex-pressions for ˆ H p and ˆ H p in terms of the terminal dis-tribution functions ˆ H N and ˆ H N .The conditions (2) yield (cid:90) ( G S + 2 G Tp )[ G LN G TN + ( G Y N ) ] G LN ( G TN + G S + 2 G Tp ) + ( G Y N ) ( h TN − h TN ) dε = 0 , (62)and (cid:90) G TN (2 G S G Lp + G j ) + G S [ G TN G LN + ( G Y N ) ]( G S + G TN )(2 G Lp + G LN ) + ( G j + G Y N ) ( h TN + h TN ) dε + (cid:90) G Y N ( G S G Lp + G j ) + G j [ G TN G LN + ( G Y N ) ]( G S + G TN )(2 G Lp + G LN ) + ( G j + G Y N ) ( h LN − h LN ) dε = 0 . (63)The general expression for the supercurrent I S can be derived from Eq. (45) together with matrix distribution functionˆ H p evaluated from Eqs. (60) and (61). We obtain eI S = − (cid:90) (cid:40) G S [ G TN G LN + ( G Y N ) ] − G j G Y N ( G S + 2 G Tp ) G LN ( G TN + G S + 2 G Tp ) + ( G Y N ) ( h TN − h TN ) + G j ( h LN + h LN ) (cid:41) dε. (64)Let us emphasize that Eqs. (62)-(64) involve no approx-imations and represent a full solution of the problem ofthe subgap electron transport in symmetric Andreev in-terferometers displayed in Fig. 1. Equations (62)-(63) al-low to evaluate the thermoelectric voltages V , inducedin the normal terminals, while Eq. (64) determines theJosephson current flowing between the superconductingterminals S and S in the presence of a temperaturegradient applied to normal ones N and N .Below we will specifically address the limits of suffi-ciently high and low temperatures and explicitly resolveEqs. (62)-(64) in these two limits. B. High temperature limit
We first consider the high temperature limit T , (cid:29) E Th . It is easy to observe that in this limit energies | ε | ∼ T , provide the main contribution to the integralin Eq. (62). In this case the spectral conductances G Y N are exponentially small and, hence, can be disregarded,the conductances G Tp and G TN in Eq. (62) can simplybe replaced by their normal state values, respectively G np and G nN , and, finally, the conductance G S can be takenin the form [3] G S (cid:39) G nS (cid:32) αL S (cid:115) D | ε | (cid:33) , α (cid:39) . . (65)where G nS is the corresponding normal state conduc-tance. Performing all these manipulations in Eq. (62),combining it with Eqs. (54), introducing the voltages V = ( V + V ) / δV = V − V and making use of theinequality | δV | (cid:28) | V | that holds in the high temperaturelimit considered here, one arrives the following relationbetween V and δV : δVV = − α I G nN G nS ( G nN + G nS + 2 G np )( G nS + 2 G np ) × LL S (cid:114) E Th (cid:18) √ T − √ T (cid:19) , (66)where I = ∞ (cid:90) dx √ x cosh x = 2(1 − / √ √ π ζ (3 / ≈ . . (67) In order to evaluate the thermoelectric voltage V itis necessary to employ Eq. (63) combined with Eqs.(54) and (55). In the leading order in the parameter E Th /T , (cid:28) h TN + h TN in Eq. (63) by its normal state value.Then after simple algebra we obtain V = 12 G nN G nS (2 G np + G nN ) [ I J ( T , χ ) − I J ( T , χ )] + (cid:18) T − T (cid:19) (cid:90) [ K Y ( ε, χ ) + K j ( ε, χ )] εdε, (68) K Y ( ε, χ ) = − G nS + G nN eG nS G nN G Y N (2 G S G Lp + G j )( G S + G TN )(2 G Lp + G LN ) + ( G j + G Y N ) , (69) K j ( ε, χ ) = − G j eG nS (cid:34) G TN G LN + ( G Y N ) ( G S + G TN )(2 G Lp + G LN ) + ( G j + G Y N ) G nS + G nN G nN − G nN (2 G np + G nN ) (cid:35) , (70)where I J ( T, χ ) is the equilibrium Josephson current inour structure at temperature T . Note that the first termin Eq. (68) proportional to the difference of the Joseph-son currents I J ( T , χ ) − I J ( T , χ ) agrees with that previ-ously derived by Virtanen and Heikkil¨a [23] whereas thelast term ∝ /T − /T does not coincide with the cor-responding extra contribution to V found in that work.We observe that the kernel K Y (69) is fully determined by electron-hole asymmetry in the spectrum, and it van-ishes identically provided this asymmetry is absent. Incontrast, the kernel K j (70) has a mixed origin and re-mains non-zero even if electron-hole symmetry would berestored.Now let us specify the expression for the Josephsoncurrent I S in the presence of a temperature gradient T − T . In the limit T , (cid:29) E Th Eq. (64) reduces to I S = 12 [ I J ( T , χ ) + I J ( T , χ )] − e (cid:90) G S [ G TN G LN + ( G Y N ) ] − G j G Y N ( G S + 2 G Tp ) G LN ( G TN + G S + 2 G Tp ) + ( G Y N ) ( h TN − h TN ) dε. (71)The last term in Eq. (71) can be significantly simplifiedmaking use of Eq. (62). With this in mind after somealgebraic manipulations from Eq. (71) we obtain I S = 12 [ I J ( T , χ ) + I J ( T , χ )] + G np δV. (72)Equations (66), (68) and (72) provide the expressionsboth for thermoelectric voltages V and δV and for theJosephson current I S which are formally exact in the hightemperature limit T , (cid:29) E Th . They also illustrate anintimate relation between thermoelectric and Josephsoneffects in the presence of a temperature gradient.In the above expressions one can identify the two types of terms, quasi-equilibrium and non-equilibriumones. Quasi-equilibrium terms contain the combinations I J ( T ) ± I J ( T ) which decay exponentially at tempera-tures exceeding E Th . Thus, at sufficiently high values of T , these quasi-equilibrium terms can be safely neglectedand the expressions for V , δV and I S will be dominatedby non-equilibrium ones which – according to Eqs. (66),(68) and (72) – yield slower (power law) temperature de-pendencies, i.e. V ∝ T − T (73)and δV ∝ I S ∝ (cid:18) T − T (cid:19) (cid:18) √ T − √ T (cid:19) . (74)We also note that in the limit L p → V and δV vanish identically together withthe non-equilibrium contribution to the supercurrent I S .These observations fully agree with the results derivedearlier for the case of symmetric X -junctions [14].In order to specify the quasi-equilibrium contributionsto both thermoelectric voltages and the Josephson cur-rent it suffices to simply evaluate the equilibrium super-current I J at a given temperature T (cid:29) E Th . This taskcan easily be accomplished (see Appendix D) with theresult I J ( T, χ ) = 1283 + 2 √ E Th eL A S A p σ ( A S + A N + A p ) sin χ × (cid:18) πTE Th (cid:19) / e − √ πT/E Th , (75)which reduces to the standard expression for the super-current in SNS junctions [7, 8] provided we set A p = A S and A N → | ε | ∼ E Th only a numerical solution is possible.In order to proceed we first evaluate the non-equilibriumterms in Eqs. (66), (68) and (72) approximately employ-ing analytic results for the spectral conductances derivedat higher energies | ε | > E Th and then combine these ap-proximate results with a numerically exact calculation.Notice that in the interesting for us high energy limit | ε | (cid:29) E Th the kernels (69) and (70) can be significantlysimplified. In this limit we can neglect higher powers of G j and G Y conductances and replace G T,LN,p by its normalstate values. Moreover, G S can be replaced by G nS in K Y kernel whereas in K j kernel it is necessary to keepcorrection term for G S . As a result we obtain K Y (cid:39) − G Y N e (2 G np + G nN ) G np G nN , (76) K j (cid:39) − G nN G j ( G nS − G S )8 eG nS (2 G np + G nN )( G nS + G nN ) . (77)Here the expression for G S is defined in Eq. (65), whereasthe conductances G j and G Y N can easily be recovered em-ploying the solution of the Usadel equations at higherenergies | ε | (cid:29) E Th worked out in Appendix D, cf. Eqs.(D13) and (D14). Substituting Eqs. (65), (D13), (D14)into Eqs. (76) and (77), combining the latter two equa-tions with Eq. (68) and formally extending the integral -0.015-0.012-0.009-0.006-0.003 0 0.003 0.006 20 30 40 50 60 T /E Th e V / E T h . numericsanalyticsEqs. (67)-(69)Eqs. (67), (75), (76) FIG. 4. Symmetric part of the thermoelectric voltage V as afunction of temperature T . Solid line corresponds to a nu-merically exact solution of the Usadel equation, short dashedline indicates our analytic result in Eq. (78), dotted and longdashed lines represent Eq. (68) combined with exact and ap-proximate expressions for K Y and K j defined respectively inEqs. (69), (70) and in Eqs. (76), (77). The chosen parametervalues T = 40 E Th , χ = π/ L N = L S = L p = L/ A S = A N = A p remain the same for all curves. over ε to all energies we arrive at the following result forthe symmetric part of the induced thermoelectric voltage V = 12 G nN G nS (2 G np + G nN ) [ I J ( T , χ ) − I J ( T , χ )]+ 323 + 2 √ E sin χe (2 G np + G nN ) (cid:18) T − T (cid:19) σ A S A p ( A S + A N + A p ) × (cid:40) A p ( A S + A N + A p ) L (3 L − L p ) L N ( L + L p ) − αG nN L S ( G nS + G nN ) (cid:41) , (78)where I J ( T, χ ) is defined in Eq. (D1). The asymmetricpart of the thermoelectric voltage δV and the supercur-rent I S are then defined respectively by Eqs. (66) and(72) combined with Eq. (78).From the above results we observe that at high tem-peratures T , (cid:29) E Th symmetric Andreev interferome-ters exhibit purely sinusoidal dependence of both ther-moelectric voltages V and δV as well as the supercurrent I S on the Josephson phase χ , i.e. V ∝ δV ∝ I S ∝ sin χ .Provided the non-equilibrium contribution G np δV in Eq.(72) exceeds the quasi-equilibrium one, the sign of thesupercurrent I S is negative for any positive value of thecombination in the curly brackets in Eq. (78). Hence, inthis case the system is driven into a π -junction state, cf.also Eq. (74).In order to verify the accuracy of the employed sim-ple approximations we also evaluated the thermoelectricvoltages V and δV numerically by directly solving the Us-adel equations. The corresponding results are displayedin Figs. 4 and 5 as functions of temperature for one of0 -0.00012-0.0001-8×10 -5 -6×10 -5 -4×10 -5 -2×10 -5 T /E Th e δ V / E T h . numericsanalytics FIG. 5. Asymmetric part of the thermoelectric voltage δV as a function of T . Solid line corresponds to a numericallyexact solution of the Usadel equation, dashed line indicatesour analytic result defined in Eq. (66) combined with Eq.(78). The parameter values are the same as in Fig. 4. the two normal terminals together with our analytic esti-mates for V and δV . For both these quantities we observea remarkably good agreement between numerically exactresults and those defined by our Eqs. (78) and (66).Note that the first and the second terms in curly brack-ets in Eq. (78) originate respectively from K Y - and K j -terms. We verified that our simple approximation (76)for K Y remains sufficiently accurate in a wide range ofparameters of our structure. At the same time, the ap-proximation (77) for K j may sometimes become less ac-curate since the high energy expansion in Eq. (65) is notsupposed to work well at lower energies | ε | < ∼ E Th . C. Lower temperatures
Let us now briefly address the opposite low tempera-ture limit T , (cid:28) E Th . In this limit the integrals in Eqs.(62)-(63) are restricted to energy intervals where all spec-tral conductances behave as smooth functions of energyand with a good accuracy can be expanded in Taylor se-ries near ε = 0. Below we will also make use of the factthat diagonal elements of the matrix conductances areeven functions of ε , whereas their off-diagonal elementsare odd functions of energy.At temperatures well below the Thouless energy E Th the combination of spectral conductances in front of theterm ( h TN − h TN ) under the energy integral in Eq. (62)can be simplified and replaced by( G nS + 2 G np ) G nN ( G nN + G nS + 2 G np ) [1 + c ( ε/E Th ) ] , c ∼ . (79)Then Eq. (62) yields δVV = − π c T − T E , (80) i.e. for symmetric interferometers the inequality | δV | (cid:28)| V | holds also in the low temperature limit T , (cid:28) E Th .As before, the symmetric part of the thermoelectricvoltage V can be derived from Eq. (63). Making useof the fact that the spectral conductances G S and G TN,p evaluated in the zero energy limit exactly coincide withtheir normal state values due to the so-called reentranceeffect [3, 36, 37] we obtain V = π
12 ( G Y N ) (cid:48) G nS G Lp (0) + G (cid:48) j G nN G LN (0) eG nS G nN [2 G Lp (0) + G LN (0)] ( T − T ) . (81)Here and below G Lp,N (0) define the spectral conductancesin the zero energy limit while ( G Y N ) (cid:48) and G (cid:48) j represent thederivatives of the corresponding spectral conductanceswith respect to energy at ε = 0. For the supercurrent I S we get I S = 12 [ I J ( T , χ ) + I J ( T , χ )] + π V T − T E ˜ G, (82)where V and ˜ G are defined respectively in Eqs. (81) and(D15). Having in mind that at low T the temperature de-pendence of the equilibrium supercurrent is determinedby the expression I J ( T, χ ) (cid:39) I J (0 , χ ) + π G (cid:48) j T , G (cid:48) j < , (83)we conclude that in this limit the leading T -dependentcorrection to the supercurrent behaves as ∼ − ( T + T ) /E and originates from the quasi-equilibrium con-tribution to I S . At the same time, the non-equilibriumterms produce only a subleading correction ∼ ( T − T ) /E which can be safely neglected in the low tem-perature limit.Qualitatively the same conclusion holds also in thelimit T (cid:28) E Th (cid:28) T in which case I S is again dominatedby the quasi-equilibrium contribution I S (cid:39) I J ( T , χ ) / E Th (cid:28) | ∆ | .It is easy to demonstrate that the latter condition canactually be relaxed while Eqs. (80), (81) and (82) willremain applicable also at E Th > ∼ | ∆ | (though still for T , , T (cid:28) | ∆ | ). In this case one should be somewhatmore cautious since the contribution of quasiparticleswith overgap energies should also be taken into account.Fortunately, however, at | ε | (cid:29) T , , T the spectral cur-rent I T depends neither on temperature nor on thermo-electric voltages V , , i.e. eI T = − G j sgn ε . Combiningthis expression for I T with Eq. (64) we again arrive atEq. (D15).In Fig. 6 we compare Eqs. (81) and (80) for the ther-moelectric voltages V and δV as functions of T withnumerically exact results for these quantities obtaineddirectly from the Usadel equation. We observe that at1 ⋅ -3 ⋅ -3 ⋅ -3
0 0.1 0.2 0.3 02 ⋅ -4 ⋅ -4 ⋅ -4 T /E Th e V / E T h . e δ V / E T h numericsanalytics FIG. 6. Thermoelectric voltages V and δV as functions oftemperature T for T = T = 0 . E Th , χ = π/ | ∆ | = E Th , L N = L S = L p = L/
3, and A S = A N = A p . Solid lines cor-respond to numerically exact solutions of the Usadel equationwhile dashed lines display analytic results for V (81) and δV (80), where the prefactors are also evaluated numerically. low temperatures our analytic results indeed remain veryaccurate and start deviating from numerical curves at T > ∼ . E Th . D. Asymmetric structures
Our general approach can also be applied in or-der to describe both thermoelectric and non-equilibriumJosephson effects in asymmetric Andreev interferometerspresented in Fig. 1. The resulting expressions for V , V and I S – which provide a complete solution of our prob-lem in this general case – turn out to be rather lengthyand cumbersome. For that reason both the final resultsand their derivation are relegated to Appendix E. Forsymmetric structures these general results reduce to thecorresponding expressions found above.The behavior of asymmetric Andreev interferometersis in many respects qualitatively similar to that of asym-metric X -junctions [14, 32]. At high temperatures T , (cid:29) E Th coherent thermoelectric voltages V and V gener-ated at two normal metallic electrodes N and N in thepresence of a temperature gradient are defined in Eqs.(E14) and (E15). Similarly to the symmetric case, forasymmetric structures both thermoelectric voltages V , depend periodically on the phase difference χ and canbe represented as a sum of a quasi-equilibrium contribu-tion ∝ I J ( χ, T ) − I J ( χ, T ) and a non-equilibrium onebeing proportional to 1 /T − /T and dominating theresult at higher temperatures. In contrast to symmetricstructures, here the voltage difference δV = V − V isnot anymore small and can be of the same order as thevoltage V = ( V + V ) / I S in the limit T , (cid:29) E Th is given by Eq. (E18) which also consists of quasi-equilibrium terms (defined as a sum of I J ( χ, T ) and I J ( χ, T ) ”weighted” by certain combinations of nor-mal state wire conductances) and non-equilibrium onesdecaying as ∝ /T − /T . At high enough tem-peratures this non-equilibrium contribution exceeds thequasi-equilibrium one, thereby signaling about the pos-sibility of a π -junction state. For strongly asymmetricstructures and high temperatures the absolute value of I S can be rather large and may even reach the same or-der of magnitude as the equilibrium Josephson currentat T →
0, cf. [32].At low temperatures T , (cid:28) E Th both quasi-equilibrium and non-equilibrium contributions to V and V are defined in Eqs. (E20), (E21) and have the sametemperature dependence ∝ T − T . The Josephson cur-rent I S is expressed by Eq. (E22). Combining this resultwith Eq. (83) we obtain I S (cid:39) I J (0 , χ ) minus small termsproportional to T and T associated with the temper-ature correction to the Josephson current (83) plus anextra term ∝ T − T of a non-equilibrium origin. Thelatter term turns out to be parametrically larger thanthat for symmetric structures. V. CONCLUSIONS
We worked out a detailed theory describing a non-trivial interplay between proximity-induced quantum co-herence and non-equilibrium effects in Andreev interfer-ometers exposed to an arbitrary temperature gradient.We elaborated a circuit theory applying it to the anal-ysis of quantum coherent effects in the network of inter-connected diffusive quasi-one-dimensional normal wiresattached to external normal and superconducting termi-nals. We formulated transparent rules of the circuit the-ory resembling the standard Kirchhoff rules well knownin the circuit electrodynamics. Our theory allows to ex-plicitly derive the solution for the kinetic Usadel equa-tion in terms of the spectral conductances. One of thekey advantages of our approach is that it enables one tounambiguously identify different contributions responsi-ble for a variety of physical phenomena involved in ourproblem.We demonstrated that the thermoelectric voltage re-sponse V to an externally imposed temperature gradi-ent T − T depends periodically on the superconductingphase difference χ and is determined by the two groupsof terms originating from different physical mechanisms.One of them is the quasi-equilibrium contribution pro-portional to the difference of the equilibrium Josephsoncurrents I J ( χ, T ) − I J ( χ, T ). This contribution exactlycoincides with that identified previously [23]. It plays asignificant role as long as at least one of the two tem-peratures – T or T – remains below an effective Thou-less energy E Th of our device. At the same time, thisquasi-equilibrium contribution decays exponentially withincreasing temperature and eventually becomes vanish-ingly small in the limit T , (cid:29) E Th .The thermoelectric response, however, does not vanish2in this limit because of another contribution of essentiallynon-equilibrium origin. This non-equilibrium contribu-tion involves different terms both related and unrelatedto particle-hole asymmetry generated by the temperaturegradient. As a result, with increasing temperature thethermoelectric voltage signal decays only as a power law eV ∼ E | /T − /T | and dominates the system behav-ior at high enough temperatures. We also note the ther-moelectric response varies for two metallic terminals N and N . At high temperatures the corresponding voltagedifference δV = V − V is parametrically smaller anddecays faster than V in the case of symmetric structures(cf. Eq. (74)), whereas in asymmetric interferometers δV gets larger and may reach the same order of magnitudeas V .We also investigated the Josephson current I S flowingacross our hybrid structure between two superconductingterminals. Unlike in the equilibrium case, here both themagnitude of this current and its sign can be controllednot only by the phase difference χ and temperature, butalso by the temperature gradient T − T applied to nor-mal terminals of our device. Similarly to the thermo-electric signal, the Josephson current is determined bya sum of two different contributions – quasi-equilibriumand non-equilibrium ones. The first contribution takesa simple form aI J ( χ, T ) + bI J ( χ, T ), where a + b = 1and, in particular, a = b = 1 / I S in the limit T , (cid:29) E Th where it de-cays as a power law with increasing T , in contrast toexponentially decaying equilibrium Josephson current. Itis remarkable that in the presence of a temperature gra-dient I S can strongly exceed I J evaluated at any of thetwo temperatures T or T .This supercurrent stimulation effect is directly relatedto the presence of non-equilibrium low-energy quasiparti-cles inside our device which suffer little dephasing whilepropagating between superconducting terminals. Notethat a somewhat similar situation is encountered for theAharonov-Bohm effect in superconducting-normal metal-lic heterostructures [37, 38] and for the effect of super-current stimulation in long SNS junctions exposed to anexternal rf signal [39, 40]. In both these cases long-range phase coherence is also maintained due to non-equilibrium quasiparticles with energies below E Th prop-agating inside the structure almost with no phase relax-ation.Yet another interesting phenomenon is the possibilityof switching our device to the π -junction state by cre-ating non-equilibrium quasiparticles with the aid of atemperature gradient. Combining this effect with super-current stimulation one can realize a comparably large(cf., e.g., Eq. (1)) π -shifted Josephson current well ob-servable in the limit T , (cid:29) E Th . Previously a similarJosephson current inversion effect was discussed in thecontext of driving the electron distribution function outof equilibrium by applying an external voltage bias V x ˆ I a ˆ I b ˆ I ˆ I I ˆ I ˆ I ˆ I FIG. 7. Resistors connected in parallel (a) and in series (b). to normal terminals [10–13]. Unlike here, however, inthat case the magnitude of the supercurrent remains ex-ponentially small for eV x , T (cid:29) E Th [10]. Supercurrentstimulation with an external voltage bias can neverthe-less become possible in more complicated configurationswhere non-equilibrium quasiparticles are supplied, e.g.,by an extra normal terminal [41].Actually one can also combine both these physicalsituations by simultaneously exposing normal terminalsto temperature gradient and voltage bias. In this casethermoelectric voltages V , should obey an obvious con-straint V − V = V x and the current conservation con-dition I N + I N = 0 should hold. Elaborating the cir-cuit theory analysis developed here and observing theseadditional constraints it is straightforward to find bothinduced thermoelectric voltages V , V and the Joseph-son current I S as functions of the phase difference χ ,temperatures T , , bias voltage V x and other parame-ters. This task, however, goes beyond the frames of thepresent work.The results presented here demonstrate that low tem-perature transport properties of superconducting hybridnanostructures, such as Andreev interferometers, can becontrolled and manipulated by means of a temperaturegradient. This observation may create a variety of oppor-tunities for novel applications of such structures in suchfields as superconductivity-based quantum nanoelectron-ics, phase-coherent caloritronics and metrology. Appendix A: Kirchhoff rules for matrixconductances
Let us establish simple Kirchhoff rules for matrix con-ductances of metallic wires connected either in parallelor in series. Consider first a circuit displayed in Fig. 7a.In this case the total matrix current ˆ I equals to a sumof the matrix currents in each of the two branches con-nected in parallel, i.e. ˆ I = ˆ I a + ˆ I b . Employing Eq. (22)in both “a” and “b” branchesˆ G a, ˆ H (1) + ˆ G a, ˆ H (2) = − e ˆ I a , (A1)ˆ G b, ˆ H (1) + ˆ G b, ˆ H (2) = − e ˆ I b , (A2)3 ˆ I c ˆ I c ˆ I c ˆ I c ˆ I c ˆ I c
21 3 c ˆ I ˆ I ˆ I ˆ I c ˆ I c ˆ I c
21 3 ⇔ FIG. 8. Y-shaped and ∆-shaped circuits. we immediately obtainˆ G ˆ H (1) + ˆ G ˆ H (2) = − e ˆ I, (A3)where total conductances ˆ G and ˆ G readˆ G = ˆ G a, + ˆ G b, , ˆ G = ˆ G a, + ˆ G b, . (A4)We observe that these relations for matrix conductancesare identical to those for equivalent conductance of twonormal resistors connected in parallel.In the case of the wires connected in series (see Fig.7b) Eq. (22) can be written separately for each wiresegment. We haveˆ G ˆ H (1) + ˆ G ˆ H (2) = − e ˆ I, (A5)ˆ G ˆ H (2) + ˆ G ˆ H (3) = − e ˆ I, (A6)Combining these two equations, we obtainˆ G ˆ H (1) + ˆ G ˆ H (3) = − e ˆ I, (A7)where the matrix conductances ˆ G and ˆ G readˆ G − = ˆ G − − ˆ G − ˆ G ˆ G − , (A8)ˆ G − = ˆ G − − ˆ G − ˆ G ˆ G − . (A9)In the absence of superconductivity the matrix conduc-tances are symmetric ( ˆ G ij = ˆ G ji ) being proportional tothe unity matrix, cf. Eq. (25). In this limit Eqs. (A8)-(A9) simply reflect the standard rules for equivalent con-ductance of two normal resistors connected in series. Appendix B: Y- ∆ transformation It is well known that an Y-shaped circuit composed ofnormal resistors can be transformed to a ∆-shaped cir-cuit of such resistors and vice versa. This Y-∆ transfor-mation can be directly generalized to the case of matrixresistances considered here. Employing Eq. (22) for the Y-shaped circuit displayed in the left part of Fig. 8, wehave ˆ G c ˆ H + ˆ G c ˆ H c = − e ˆ I c , (B1)ˆ G c ˆ H + ˆ G c ˆ H c = − e ˆ I c , (B2)ˆ G c ˆ H + ˆ G c ˆ H c = − e ˆ I c . (B3)Making use of the matrix current conservation condition,we obtain an extra equation for the matrix currents ˆ I ic ˆ I c + ˆ I c + ˆ I c = 0 . (B4)In the case of the ∆-shaped circuit displayed in theright part of Fig. 8 the relations between the matrixcurrents and the matrix distribution functions take theform ˆ G ˆ H + ˆ G ˆ H = − e ˆ I , (B5)ˆ G ˆ H + ˆ G ˆ H = − e ˆ I , (B6)ˆ G ˆ H + ˆ G ˆ H = − e ˆ I , (B7)whereas the matrix currents ˆ I ic readˆ I c = ˆ I − ˆ I , (B8)ˆ I c = ˆ I − ˆ I , (B9)ˆ I c = ˆ I − ˆ I . (B10)It is easy to verify that the relations between the matrixcurrents ˆ I ic and the matrix distribution functions ˆ H i forthe ∆-shaped circuit governed by Eqs. (B5)-(B10) areequivalent to Eqs. (B1)-(B3) for the Y-shaped circuitprovided we setˆ G = − ˆ G c ( ˆ G c + ˆ G c + ˆ G c ) − ˆ G c , (B11)ˆ G = ˆ G c ( ˆ G c + ˆ G c + ˆ G c ) − ˆ G c , (B12)ˆ G = ˆ G c ( ˆ G c + ˆ G c + ˆ G c ) − ˆ G c , (B13)ˆ G = − ˆ G c ( ˆ G c + ˆ G c + ˆ G c ) − ˆ G c , (B14)ˆ G = − ˆ G c ( ˆ G c + ˆ G c + ˆ G c ) − ˆ G c , (B15)ˆ G = ˆ G c ( ˆ G c + ˆ G c + ˆ G c ) − ˆ G c , (B16)Note that the inverse transformation from the ∆-shaped circuit to the Y-shaped one is not always possible.Under the conditionˆ G − ˆ G ˆ G − ˆ G ˆ G − ˆ G = − , (B17)this inverse transformation can be formulated. In thiscase it reads ˆ G c = ˆ G − ˆ G + ˆ G ˆ G − ˆ G , (B18)ˆ G c = ˆ G − ˆ G + ˆ G ˆ G − ˆ G , (B19)ˆ G c = ˆ G − ˆ G + ˆ G ˆ G − ˆ G , (B20)ˆ G c = ˆ G − ˆ G + ˆ G ˆ G − ˆ G , (B21)ˆ G c = ˆ G − ˆ G + ˆ G ˆ G − ˆ G , (B22)ˆ G c = ˆ G − ˆ G + ˆ G ˆ G − ˆ G . (B23)4 Appendix C: Matrix conductance at subgap energies
At subgap energies | ε | < | ∆ | the kinetic equation (20)for the evolution operator ˆ U becomes singular in thevicinity of the interface between a superconducting ter-minal and a normal wire. This is because at such energiesone has ˆ G R = ˆ G A and, hence, the functions D L and Y vanish at the SN-interface together with the first deriva-tive of D L .In order to tackle this problem let us take a closer lookat the solution of Eq. (20) in the vicinity of this interface.Expanding the kinetic coefficients D L,T and Y to thelowest nonvanishing order in the distance to SN-interface x and making use of the identity ˆ G R (0) = ˆ G A (0), we getˆ D ≈ (cid:18) d T j ε x − j ε x d L x (cid:19) , (C1)where d T = D T (0), d L = D L (cid:48)(cid:48) (0) / j ε = Y (cid:48) (0).With this in mind the equation for the evolution operatorcan be solved explicitly, and we obtainˆ U ( x, ˜ x ) = (cid:18) d T d L + j ε x/ ˜ x − d L j ε ( x − ˜ x ) d T j ε (1 /x − / ˜ x ) d T d L + j ε ˜ x/x (cid:19) d T d L + j ε . (C2)The matrix conductance then readsˆ G x, ˜ x = σ A x − ˜ x (cid:18) d T j ε x − j ε ˜ x d L x ˜ x (cid:19) (C3)and, hence, we haveˆ G ,x = σ A x (cid:18) − d T j ε x (cid:19) , ˆ G x, = σ A x (cid:18) d T j ε x (cid:19) . (C4)Combining the above results with the relation for the twomatrix conductances connected in series one can easilyobserve that the matrix conductance of the whole wirehas the same structure as that in Eq. (44). Appendix D: Equilibrium Josephson current andspectral conductances
In equilibrium the supercurrent I J flowing across ourstructure can be derived from the equation I J ( T, χ ) = − σ A e (cid:90) j ε tanh ε T dε, (D1)where j ε is the spectral current defined in Eq. (12) andthe cross-section A equals to either A p or A S depend-ing on the wire in which j ε is evaluated. Accordingly,the task at hand is to evaluate anomalous Green func-tions F R and ˜ F R inside all normal wires interconnectingsuperconducting and normal terminals.In general this task can be accomplished only nu-merically since quasiclassical Usadel equations consti-tute a complicated system of coupled nonlinear differ-ential equations. However, in the high energy limit Us-adel equations can be simplified and resolved analytically x a x b x c S ∆ e iχ A S , L S A , L A A i A FIG. 9. An example of the network formed by quasi-one-dimensional normal wires attached to a superconducting ter-minal. for an arbitrary network of quasi-one-dimensional normalwires connected to each other and to normal or supercon-ducting terminals.Let us assume that electron energy ε exceeds the Thou-less energy for any wire segment. In this case the Usadelequation can be integrated straightforwardly from the su-perconducting terminal deep into normal wire network.At relatively short distances x (cid:28) L S away from a super-conducting terminal one can safely disregard the effectof all other terminals and the anomalous Green functioncan be expressed in the form F R = − i y (1 − y )(1 + y ) e iχ , y = a S e − √ − iε/Dx , (D2) a S ( ε ) = tan (cid:34)
14 arcsin | ∆ | (cid:112) | ∆ | − ε (cid:35) , (D3)where ∆ = | ∆ | e iχ is the order parameter in the corre-sponding superconducting terminal.At distances away from S -terminals exceeding (cid:112) D/ | ε | one can linearize the Usadel equation everywhere insidethe network of normal wires( F R ) (cid:48)(cid:48) − k F R = 0 , k = (cid:112) − iε/D (D4)and the anomalous Green functions take exponentiallysmall values F R = − ia S e − √ − iε/Dx e iχ . (D5)At larger distances from the superconducting terminalthe anomalous Green function F obeys Eq. (D4) whichcan be integrated step by step along the normal wirenetwork.As an illustration, let us consider an arbitrary normalwire segment x b x c with length L and cross section area A (see Fig. 9). A general solution of Eq. (D4) reads F R = C − e − k ( x − x b ) + C + e k ( x − x b ) , (D6)where one obviously has C + (cid:28) C − . Hence, with a goodaccuracy we can set C − = F R ( x b ) . (D7)The prefactor C + can be derived from the boundary con-ditions at the point x = x c which include the continuity5condition for the Green function F R combined with theequation A ( F R ( x c )) (cid:48) = (cid:88) i A i ( F Ri ( x c )) (cid:48) , (D8)where the index i enumerates all normal wires attachedto the wire segment x b x c at the node x c . As a result, weobtain C + = F R ( x b ) A − (cid:80) i A i A + (cid:80) i A i e − kL (D9)and, hence, F R ( x c ) = F R ( x b ) 2 AA + (cid:80) i A i e − kL . (D10)This equation together with Eqs. (D6), (D7), (D9) al-lows one to evaluate the anomalous Green functions ev-erywhere inside our structure.Obviously, the above analysis can easily be general-ized to the network of normal wires connected to sev-eral superconducting terminals. In this case the result-ing anomalous Green function everywhere inside this net-work is given by a simple superposition of the contribu-tions from each of these terminals.Turning back to the structure depicted in Fig. 1 andmaking use of the above results, in the high energy limit | ε | (cid:29) E Th we get F R = − κ A S e − kL S e iχ A S + A N + A p e − kx − κ A S e − kL S e iχ A S + A N + A p e − k ( L p − x ) , (D11)in the central wire and F R = − κ A S e − kL S e iχ A S + A N + A p e − kx − κ A S e − kL S e iχ A S + A N + A p e − kL p A p A S + A N + A p e − kx , (D12)in the wire connected to the normal terminal N . Here x is a coordinate along the wire from the crossing point p , and κ = 8 i/ (1 + √ F R is recovered from the expressions (D11) and (D12)by inverting their overall sign and performing the phaseinversion χ , ↔ − χ , . Substituting these results for F R and ˜ F R into Eq. (12) and evaluating the integralin Eq. (D1) we arrive at Eq. (75) for the equilibriumJosephson current in our structure.The spectral conductances G j and G Y N can be evalu-ated analogously. In the high energy limit we obtain G j = 1283 + 2 √ A S A p σ ( A S + A N + A p ) sin χ × k (cid:48) e − k (cid:48) L (cos k (cid:48) L − sin k (cid:48) L ) sgn ε (D13)and G Y N (cid:39) − G nN
643 + 2 √ A S A p ( A S + A N + A p ) × sin χe − k (cid:48) L sin k (cid:48) L p k (cid:48) L N sgn ε. (D14)where we define L = 2 L S + L p and k (cid:48) = (cid:112) | ε | /D .Finally, we specify the expression for ˜ G which entersEq. (82). It reads˜ G = E (cid:34)
14 ( G Y N ) (cid:48) G (cid:48) j G nS G Lp (0) + ( G (cid:48) j ) G nN G LN (0) G nS G nN [2 G Lp (0) + G LN (0)] + [ G (cid:48)(cid:48) S G np − G nS ( G Tp ) (cid:48)(cid:48) ] G nN G LN (0) − ( G nS + 2 G np ) G (cid:48) j ( G Y N ) (cid:48) G LN (0)( G nN + G nS + 2 G np )( G nS + 2 G np ) (cid:35) , (D15)where G (cid:48)(cid:48) S and ( G Tp ) (cid:48)(cid:48) represent the second derivatives ofthe corresponding spectral conductances with respect toenergy at ε = 0. Appendix E: Asymmetric interferometers
Let us apply our circuit theory approach to the analysisof subgap electron transport in Andreev interferometers6displayed in Fig. 1 making no assumption about theirsymmetry. Resolving the system of matrix equations (45)-(56) we can derive the expressions for the matrixcurrents as functions of the electron distribution func-tions inside the terminals. We obtainˆ I N = ˆ G N ˆ P (cid:34) ˆ G S + ˆ G S + ( ˆ G S + ˆ G N − ˆ τ + G j )( ˆ G p + ˆ τ G j / − ( ˆ G S − ˆ τ − G j ) (cid:35) ˆ H N + ˆ A ( ˆ H N − ˆ H N ) , (E1)ˆ I N = ˆ G N ˆ τ ˆ P T ˆ τ (cid:34) ˆ G S + ˆ G S + ( ˆ G S + ˆ G N + ˆ τ + G j )( ˆ G p − ˆ τ G j / − ( ˆ G S + ˆ τ − G j ) (cid:35) ˆ H N − ˆ τ ˆ A T ˆ τ ( ˆ H N − ˆ H N ) , (E2)where the matrices ˆ P and ˆ A readˆ P = (cid:34) ˆ G S + ˆ G S + ˆ G N + ˆ G N + ( ˆ G S + ˆ G N − ˆ τ + G j )( ˆ G p + ˆ τ G j / − ( ˆ G S + ˆ G N − ˆ τ − G j ) (cid:35) − , ˆ A = ˆ G N ˆ P ˆ G N . (E3)The currents I N , flowing in the normal wires connected to the terminals N and N are defined by the formula I N , = 12 (cid:90) I T , dε. (E4)Obviously, both these currents vanish provided the terminals are disconnected from any external circuit, i.e. theconditions (2) apply which determine the thermoelectric voltages V and V induced at these terminals. In the hightemperature limit T , (cid:29) E Th the calculation of V and V is simplified since the prefactors in front of the distributionfunctions h TN , entering the general expressions for I N , can be replaced by their normal state values. Then we get0 = I N = G nN G np ( G nS + G nS ) + ( G nS + G nN ) G nS ( G nS + G nS + G nN + G nN ) G np + ( G nS + G nN )( G nS + G nN ) V + G nN G nN G np [ G np ( G nS + G nS + G nN + G nN ) + ( G nS + G nN )( G nS + G nN )] ( V − V ) + 12 e (cid:90) A [ h LN − h LN ] dε, (E5)and0 = I N = G nN G np ( G nS + G nS ) + ( G nS + G nN ) G nS ( G nS + G nS + G nN + G nN ) G np + ( G nS + G nN )( G nS + G nN ) V − G nN G nN G np [ G np ( G nS + G nS + G nN + G nN ) + ( G nS + G nN )( G nS + G nN )] ( V − V ) + 12 e (cid:90) A [ h LN − h LN ] dε, (E6)where A and A denote the off-diagonal elements of the matrix ˆ A ˆ A = (cid:18) A A A A (cid:19) . (E7)Resolving Eqs. (E5) and (E6), we obtain V = − e (cid:90) [ G np ( G nS + G nS + G nN ) + ( G nS + G nN ) G nS ] A + G np G nN A G nN [( G nS + G nS ) G np + G nS G nS ] [ h LN − h LN ] dε, (E8) V = − e (cid:90) [ G np ( G nS + G nS + G nN ) + ( G nS + G nN ) G nS ] A + G np G nN A G nN [( G nS + G nS ) G np + G nS G nS ] [ h LN − h LN ] dε, (E9)Here with a good accuracy one can neglect the voltage dependence of the distribution functions h LN , replacing themby their equilibrium values h LN , = tanh[ ε/ (2 T , )]. It is convenient to identically rewrite the matrix elements A A in the form A = 1 Q n ( G nN ) ( G nS + G nN + 2 G np ) G nN G j / (cid:34) A − Q n ( G nN ) ( G nS + G nN + 2 G np ) G nN G j / (cid:35) , (E10) A = 1 Q n ( G nN ) ( G nS + G nN + 2 G np ) G nN G j / (cid:34) A − Q n ( G nN ) ( G nS + G nN + 2 G np ) G nN G j / (cid:35) , (E11)where we explicitly extracted G j -terms with the prefactors replaced by their normal state values. Here Q n = [ G np ( G nS + G nS + G nN + G nN ) + ( G nS + G nN )( G nS + G nN )][ G np ( G nN + G nN ) + G nN G nN ] (E12)is the normal state value of the function Q = det (cid:12)(cid:12)(cid:12)(cid:12) ˆ G S + ˆ G S + ˆ G N + ˆ G N ˆ G S + ˆ G N − ˆ τ + G j ˆ G S + ˆ G N − ˆ τ − G j − ( ˆ G p + ˆ τ G j / (cid:12)(cid:12)(cid:12)(cid:12) . (E13)With this in mind Eqs. (E8) and (E9) can be rewritten as V = G nN G nN G np + G nS [ G np ( G nN + G nN ) + G nN G nN ][ G np ( G nS + G nS ) + G nS G nS ] [ I J ( T , χ ) − I J ( T , χ )] − e (cid:18) T − T (cid:19) G nN [( G nS + G nS ) G np + G nS G nS ] (cid:90) (cid:40) [ G np ( G nS + G nS + G nN ) + ( G nS + G nN ) G nS ] A + G np G nN A − G nN G nN G nN (2 G np + G nS )[ G np ( G nN + G nN ) + G nN G nN ] G j (cid:41) εdε, (E14) V = G nN G nN G np + G nS [ G np ( G nN + G nN ) + G nN G nN ][ G np ( G nS + G nS ) + G nS G nS ] [ I J ( T , χ ) − I J ( T , χ )] − e (cid:18) T − T (cid:19) G nN [( G nS + G nS ) G np + G nS G nS ] (cid:90) (cid:40) [ G np ( G nS + G nS + G nN ) + ( G nS + G nN ) G nS ] A + G np G nN A − G nN G nN G nN (2 G np + G nS )[ G np ( G nN + G nN ) + G nN G nN ] G j (cid:41) εdε. (E15)Quasi-equilibrium contributions containing the differ-ence between the Josephson currents I J ( T , χ ) − I J ( T , χ )in Eqs. (E14) and (E15) coincide with the correspondingterms derived in Ref. [23].The supercurrent I S flowing in our circuit can be de-rived with the aid of the formula I S = 12 (cid:90) I TS dε. (E16)Here I TS is the corresponding component of the matrix current ˆ I S obtained from Eqs. (45)-(56) in the form e ˆ I S = − ( ˆ G S +ˆ τ + G j ) ˆ H N +( ˆ G S +ˆ τ + G j ) ˆ P (cid:104) ˆ G S + ˆ G S +( ˆ G S + ˆ G N − ˆ τ + G j )( ˆ G p + ˆ τ G j / − ( ˆ G S − ˆ τ − G j ) (cid:105) ˆ H N + ( ˆ G S + ˆ τ + G j ) ˆ P ˆ G N ( ˆ H N − ˆ H N ) . (E17)At high temperatures T , (cid:29) E Th the supercurrent canbe evaluated explicitly in exactly the same manner as thethermoelectric voltages V , . In this limit we obtain8 I S = ( G nS + G nS ) G nN ( G np ) + [ G nN G nN G nS + G nS G nS G nN ] G np + 12 G nS G nS G nN G nN [ G np ( G nN + G nN ) + G nN G nN ][( G nS + G nS ) G np + G nS G nS ] I J ( T , χ )+ ( G nS + G nS ) G nN ( G np ) + [ G nN G nN G nS + G nS G nS G nN ] G np + 12 G nS G nS G nN G nN [ G np ( G nN + G nN ) + G nN G nN ][( G nS + G nS ) G np + G nS G nS ] I J ( T , χ )+ 14 e (cid:18) T − T (cid:19) (cid:90) (cid:40) ( G nS + G nS ) G nN ( G np ) + [ G nN G nN G nS + G nS G nS G nN ] G np + 12 G nS G nS G nN G nN [ G np ( G nN + G nN ) + G nN G nN ][( G nS + G nS ) G np + G nS G nS ] × (cid:34) G nS ( G np + G nS ) A + G np A ( G nS + G nS ) G np + G nS G nS + P G S G Y N + P G S G LN + P G j G Y N + P G j G LN (cid:35) + ( G nS + G nS ) G nN ( G np ) + [ G nN G nN G nS + G nS G nS G nN ] G np + 12 G nS G nS G nN G nN [ G np ( G nN + G nN ) + G nN G nN ][( G nS + G nS ) G np + G nS G nS ] × (cid:34) P G S G Y N − P G S G LN + P G j G Y N − P G j G LN − G nS G np A + ( G np + G S ) A ( G nS + G nS ) G np + G nS G nS (cid:35)(cid:41) εdε. (E18)For completeness, let us also consider the low temper-ature limit T , (cid:28) E Th . In order to evaluate thermoelec-tric voltages V and V in this limit it suffices to replace G T and G L components of the matrix conductances bytheir zero energy values and set G j ≈ G (cid:48) j ε, G Y X ≈ ( G Y X ) (cid:48) ε, X = N , , p. (E19)With the same accuracy we can neglect products of the G j and G Y conductances since they have higher powerof energy. At zero energy G T components of the matrixconductances are known to exactly coincide with corre-sponding normal state conductances [3, 36, 37]. Withthis in mind one can demonstrate that Eqs. (E8) and(E9) remain applicable also in the low temperature limit.Evaluating A and A , we get V = π G LN (cid:8) G nS + G nS ) G np + G nS G nS ] G Lp ( G Y N ) (cid:48) + G nN (2 G np + G nS ) G LN G (cid:48) j + 2 G nN G nS G LN ( G Y p ) (cid:48) (cid:9) eG nN [( G nS + G nS ) G np + G nS G nS ][ G LN + G LN ) G Lp + G LN G LN ] ( T − T ) , (E20) V = π G LN (cid:8) − G nS + G nS ) G np + G nS G nS ] G Lp ( G Y N ) (cid:48) + G nN (2 G np + G nS ) G LN ( G j ) (cid:48) − G nN G nS G LN ( G Y p ) (cid:48) (cid:9) eG nN [( G nS + G nS ) G np + G nS G nS ][ G LN + G LN ) G Lp + G LN G LN ] ( T − T ) , (E21)where the spectral conductances G LN G LN and G Lp are evaluated at ε = 0.The Josephson current I S is evaluated in exactly the same way. 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