Note on possibility of proximity induced spontaneous currents in superconductor/normal metal heterostructures
NNote on possibility of proximity induced spontaneous currents insuperconductor/normal metal heterostructures
Anatoly F. Volkov and Ilya M. Eremin
Institut f¨ur Theoretische Physik III, Ruhr-Universit¨at Bochum, D-44801 Bochum, Germany (Dated: March 2, 2021)We analyse the possibility of the appearance of spontaneous currents in proximated superconduct-ing/normal metal (S/N) heterostructure when Cooper pairs penetrate into the normal metal fromthe superconductor. In particular, we calculate the free energy of the S/N structure. We show thatwhereas the free energy of the N film F N in the presence of the proximity effect increases comparedto the normal state, the total free energy, which includes the boundary term F B , decreases. Thecondensate current decreases F N , but increases the total free energy making the current-carryingstate of the S/N system energetically unfavorable. PACS numbers:
Introduction . Penetration of Cooper pairs into the normal metal (N) in superconductor/normal metal (S/N) het-erostructures, provided the interface transparency is not too small, is a well-known effect [1–4]. This so-called proximityeffect (PE) is related to the Andreev reflections of electrons at the interface of the S/N bilayer [5]. In particular, thedepth of Cooper pairs penetration into the N in the diffusive case is equal to ξ N ∼ = (cid:112) D N / πT where D N = vl/ I c mayeven change sign [12–17], yielding the so-called π -junctions. Note, the change of sign in I c may also be achievedin conventional S/N/S multi-terminal Josephson junctions if the electric potential of the normal metal N is shiftedwith respect to the S counterparts [9, 18–21]. More recently, spectrum of Andreev bound states in S/N-multiterminalstructures with potentially non-trivial topological bands with Weyl points was also investigated [22, 23].Despite of these continuous research efforts in simple S/N systems and their derivatives, outlined above, the originof certain effects remains mysterious. For example, an interesting paramagnetic re-entrant effect (sometime calledMota effect) caused by spontaneous currents in S/N bilayer was observed in Refs.[24–27]. The authors of Ref.[28]proposed an explanation in terms of a repulsive interaction with a negative small coupling constant λ N i.e. assumingthe normal metal may acquires a gap, ∆ N , which sign is opposite to that in a superconductor, ∆. However, thepredicted paramagnetic response caused by spontaneous currents turned out to be too small because of the smallnessof the superconducting order parameter in N ∆ N ∼ λ N , and thus the origin of the Mota effect remains unclear [27].Note, the paramagnetism and spontaneous currents may occur in S/F system[29–32] or S/F/N structures [33, 34].However, there the origin of the paramagnetic effect should be quite different from that in S/N structures as in theformer it is related to internal exchange fields existing in the ferromagnet F and to the triplet Cooper pairs inducedin the film F by the PE [12–17]. In S/N/S Josephson junctions in a non-equilibrium [35] spontaneous currents arisewhen the Josephson current in S/N/S junctions changes sign [9, 18–21] but this situation then resembles the case ofS/F/S junctions with a negative Josephson current [12–17]. Therefore, the situation of the S/N bilayer in equilibriumrequires a separate study.In this paper, we consider a simple S/N bilayer heterostructure with a superconducting coupling constant in the Nlayer equal to zero, i. e., λ N = 0 and ∆ N = 0. We calculate the total free energy F S/N of the system that consistsof bulk terms F S and F N as well as the boundary term F B . Below the critical temperature T c , the energy F S ( F N )decreases (increases), respectively. On the contrary to F N , the boundary term F B decreases the total free energyin such a way that the contribution of the terms F N + F B is negative. The contribution F S + F B remains negativeas it is in the absence of the PE. The condensate current gives a positive contribution to both terms F N + F B and F S + F B making the current-carrying state unfavorable. Theory.
Frequently the analysis of the free energy ( F ) is performed using the Ginzburg-Landau free energy expan-sion, assuming the smallness of the order parameter ∆. This approach is not applicable to the considered heterostruc-ture because the superconducting order parameter ∆ N in the N film is assumed to be zero. On the other hand, a partof electrons in N condense due to the PE and therefore the free energy F N changes also in the superconducting state. a r X i v : . [ c ond - m a t . s up r- c on ] F e b SN x y FIG. 1: (Color online.) A schematic representation of the S/N bilayer structure.
Thus, in order to calculate the variation δF , we need to find first the quasiclassic matrix Green’s functions ˆ g in the Sand N regions using the boundary conditions and to express the free energy in terms of the functions ˆ g . We considera simple case of diffusive S/N structure when the function ˆ g obeys the Usadel equation [36]. In particular, the systemunder consideration is a bilayer which consists of S and N films with thicknesses d S,N , respectively as shown in Fig.1.The current is assumed to flow along the interface in the y -direction. We integrate out the phase χ ( y ) by making thetransformation ˆ g n = ˆ S † · ˆ g · ˆ S , where ˆ S = exp[( iQy/ τ ]. This means that the phase χ and the functions ˆ g n afterthe transformation depend only on the x coordinate and we drop the subscript ” n ” in what follows. We represent thematrix ˆ g in a standard form ˆ g = ˆ τ cos θ + ˆ τ sin θ , which is typically used in studying S/N structures [37–42] so thatthe normalization condition ˆ g · ˆ g = ˆ1 is automatically fulfilled. The function θ depends on x and obeys the Usadelequations in the S and N regions − D S ∂ xx θ S + 2 ω sin θ S −
2∆ cos θ S + ( D S P S /
2) sin(2 θ S ) = 0, S film (1) − D N ∂ xx θ N + 2 ω sin θ N + ( D N P N /
2) sin(2 θ N ) = 0, N film (2)where D S ; N are the diffusion coefficients in the S(N) films, ω is the Matsubara frequency, P = ∇ χ − π A / Φ is thegauge-invariant condensate momentum and Φ = hc/ e is the magnetic flux quantum. The Usadel equations arecomplemented by the standard Kurpiyanov-Lukichev boundary conditions for θ S,N [43] at the interface ∂ x θ S ( N ) = − κ B,S ( N ) sin( θ S − θ N ) | x =0 (3)where κ B,S ( N ) = 2 /R B σ S ( N ) , R B is the S/N interface resistance per unit area and σ S,N are the conductivities in theS and N films in the normal state. The order parameter ∆, which is non-zero in the S film, is determined by theself-consistency equation ∆ = λ (2 πT ) (cid:88) ω (cid:62) sin θ S ( ω ) (4)Note, Eq.(4) and Eqs.(1-2) are obtained by the variation of the total free energy F S and F N with respect to ∆ and θF S = 2 ν S (cid:90) − d S dx (cid:40) ∆ λ + 2 πT (cid:88) ω (cid:62) (cid:20) D S ∂ x θ S ) + ω (1 − cos θ S ) − ∆ sin θ + D S P − cos(2 θ S )) (cid:21) (cid:41) (5) F N = 2 ν N (cid:90) d N dx (cid:40) πT (cid:88) ω (cid:62) (cid:20) D N ∂ x θ N ) + ω (1 − cos θ N ) + D N P − cos(2 θ N )) (cid:21) (cid:41) (6)where ν , P , D are the density of states, momentum, and diffusion coefficient in either S or N film, respectively. We set∆ N equal to zero since we assume that λ N = 0. The energy F is counted from its value in the normal state, i.e. θ = 0.This expression for F N can be also derived from a more general expression for the free energy of a superconductorin the presence of an exchange field [44, 45]. We also note by passing that taking the variation of the sum of the F and the magnetic energy ( ∇ × A ) / π one obtains the London equation ∇ A = (4 π/c ) j , where j = − ( c/ π )Λ − L P .Here, Λ − L = [2 σ/ ( c (cid:126) )](2 πT ) (cid:80) ω (cid:62) sin θ ( ω )) is the inverse squared London penetration depth. In order to take intoaccount the boundary conditions (3), we need to add the boundary term F B [47, 48] to F S + F N so that the totalfunctional F is given by F = F S + F N + F B (7)In the following we solve Eqs.(1-2) for the functions θ S,N together with the self-consistency equation (4) and find aminimum of the free energy F as a function of the condensate velocity V = P /m . In a general case, this can be doneonly numerically. Here we restrict the analysis with the simplest case of a weak proximity effect when the Usadelequation for θ N θ S is weakly perturbed by the PE. The latter assumption isvalid if the condition δθ S (cid:46) ξ S / ( R B σ S ) (cid:28) ξ S ∼ = (cid:112) D S /
2∆ is a coherence length in S. Yet we dotake into account a suppression of the order parameter ∆ by the condensate flow. In the case of small suppressionof ∆, we find δ ∆ ∼ = − ( D S P S / (cid:80) ω (cid:62) ( ω /ζ ω ) / (cid:80) ω (cid:62) ζ − ω , where ζ ω = (cid:112) ω + ∆ . At low temperatures ( T (cid:28) ∆)the gap variation is δ ∆ ∼ = − D S P S /
2. Note that a strong suppression of ∆ by the condensate flow was studied inRefs.[44, 46]. In the absence of the PE and the condensate flow, one has sin θ S ≡ f S = ∆ /ζ ω and cos θ S = ω/ζ ω with ζ ω = (cid:112) ω + ∆ . The direct calculation of F S gives a well known result F S = − ν S ∆ d S / δF S caused by the condensate flow is δF S = ν S D S P S (2 πT )∆ (cid:80) ω (cid:62) ζ − ω . Thus, the energy F S of the S film with aspontaneous current is F S = − ν S d S ∆ − D S P S (2 πT ) (cid:88) ω (cid:62) ζ − ω (8)and as expected the condensate flow reduces the condensation energy.Next we evaluate the contribution to the free energy of the N film, F N . Linearized Eq.(2) has the form − ∂ xx θ N + κ q θ N = 0 (9)with a solution θ N ( x ) = κ B κ q f S cosh( κ q ( x − d N ))sinh( κ q d N ) , (10)where κ q = (cid:112) ω + q /ξ N , ξ N = (cid:112) D N / q = Qξ N and f S = ∆ /ζ ω . The solution describes correctly the condensateGreen’s function in N provided the condition R B > ρ N ξ N is fulfilled.In the limit of a weak PE the energy F N + F B can be written in the form F N + F B = ν N (2 πT ) D N (cid:88) ω (cid:62) (cid:40) (cid:90) d N dx
12 [( ∂ x θ N ) + κ q θ N ] + κ B [1 − cos( θ S − θ N )] (cid:41) (11)where the last term is the boundary free energy [47, 48]. Substituting the solution (10) into (11), we come to theformula for F N + F B and θ S one can easily calculate the F N + F B = ν N (2 πT )( D N κ B ) (cid:88) [ f S κ q tanh( κ q d N ) + 1 κ B (1 − cos θ S ) − f S κ q tanh( κ q d N ) ] (12)The first term in the figure brackets is the contribution of the bulk N region whereas the last term stems fromthe boundary contribution to the free energy. The second term is a reduction of the free energy due to the PE.One can see that the first term gives a positive contribution to the F and decreases with increasing the condensatevelocity V S ∼ q . However the boundary term (the last one) is twice larger than the first one and therefore the totalcontribution of the terms due to condensate current, Eq.(8,12), is positive. This means that the condensate currentreduces the free energy. Conclusions:
To conclude, we analyzed the free energy for S/N bilayer in the presence of the condensate current.We have shown that the bulk of the N film gives a positive contribution F N ( q ) to the free energy which decreaseswith increasing condensate velocity V ∼ q . However the contribution of boundary term F B to F is twice larger inmagnitude than F N ( q ) and is also negative as the contribution F S of the superconductor S. Therefore the total freeenergy F increases when condensate moves; this makes the current-carrying state unfavorable. Acknowledgements:
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Here we present the evaluation of the free energy F N in the N film. We integrate once Eq.(7) in the main text − D N ∂ x θ N ) + 2 ω (1 − cos θ N ) + D N P N − cos(2 θ N )] = 0 (S1)and assumed that d N (cid:29) ξ N, ∆ ≡ (cid:112) D N /
2∆ so that θ N = 0 at x = d N . Taking into account Eq.(S1), the energy F N can be written as follows F N = ν N (2 πT ) D N (cid:88) ω (cid:62) (cid:90) ∞ dx ( ∂ x θ N ) ≡ ν N (2 πT ) D N (cid:88) ω (cid:62) ˜ F N,ω , (S2)where ˜ ω = ω/ ∆ and the function ˜ F N,ω is defined as˜ F N,ω = (cid:90) ∞ dx ( ∂ x θ N ) = − (cid:90) θ N dθ N ( ∂ x θ N ) = (S3)= − (cid:90) θ N d ¯ θ N sin ¯ θ N (cid:113) κ ω + P cos ¯ θ N = (S4)= P (cid:34)(cid:112) a ω − t (cid:113) t + a ω + a ω ln 1 + (cid:112) a ω t + (cid:112) t + a ω (cid:35) (S5)where t ≡ cos ¯ θ N ≡ cos( θ N / | x =0 and a ω = κ ω /P . The parameter t is found from the boundary condition (cid:113) − t (cid:112) q/κ ω ξ N ) t = κ BN κ ω (cid:20) ∆(2 t − − ωt (cid:113) − t (cid:21) , (S6) Weak PE
Consider now a weak PE when the function θ N is small. In this case one can obtain a formula for F N for arbitrarythickness d N . At θ N (cid:28)
1, Eq.(2) in the main text can be linearised − ∂ xx θ N + κ N θ N = 0. (S7)where κ N = κ Nω + P , κ Nω = 2 ω/D N . The boundary conditions, Eq.(3), have the form ∂ x θ N = − κ B,N [sin θ S − θ N cos θ S ] | x =0 , (S8) ∂ x θ N = 0 | x = d N . (S9)where κ BN = 2 /R B σ N . The solution for Eq.(S7) obeying the condition (S8) is θ N ( x ) = κ BN κ N cosh( κ N ( x − d N ))cosh ϑ N D N sin θ S . (S10)where D N = tanh α N + ( κ BN /κ N )˜ ω/ √ ˜ ω + 1, α N = κ N d N , sin θ S = 1 / √ ˜ ω + 1. The energy of the N film, F N , is F N = 2 ν N (2 πT ) (cid:88) ω (cid:62) (cid:90) d N dx (cid:20) D N ∂ x θ N ) + 14 (2 ω + D N P ) θ N (cid:21) = (S11)= 2 ν N D N κ BN ξ N, ∆ (2 πT ) (cid:88) ω (cid:62) tanh α N D N (cid:112) ˜ ω + q ω + 1 (S12)In the limit of a thick N film ( d N (cid:29) ξ N, ∆ ) Eq.(S8) acquires the form F N = 2 ν N D N κ BN ξ N, ∆ (2 πT ) (cid:88) ω (cid:62) (cid:112) ˜ ω + q ω2