We present a general theory of the proximity effect in junctions between
diffusive normal metals (DN) and unconventional superconductors in the
framework of the quasiclassical Green's function formalism. Various possible
symmetry classes in a superconductor are considered which are consistent with
the Pauli principle: even-frequency spin-singlet even-parity (ESE) state,
even-frequency spin-triplet odd-parity (ETO) state, odd-frequency spin-triplet
even-parity (OTE) state and odd-frequency spin-singlet odd-parity (OSO) state.
For each of the above four cases symmetry and spectral properties of the
induced pair amplitude in the DN are determined. It is shown that the pair
amplitude in a DN belongs respectively to an ESE, OTE, OTE and ESE pairing
state.Comment: 5 pages with one figur

A. A. Golubov

V. L. Berezinskii

Y. Tanaka

English

arXiv

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Theory of the Proximity Effect in Junctions with Unconventional Superconductors

We present a general theory of the proximity effect in junctions between diffusive normal metals (DN) and superconductors. Various possible symmetry classes in a superconductor are considered: even-frequency spin-singlet even-parity (ESE) state, even-frequency spin-triplet odd-parity state, odd-frequency spin-triplet even-parity (OTE) state and odd-frequency spin-singlet odd-parity state. It is shown that the pair amplitude in a DN belongs, respectively, to an ESE, OTE, OTE, and ESE pairing state since only the even-parity s-wave pairing is possible due to the impurity scattering

Tanaka, Y.

Golubov, Alexandre Avraamovitch

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University of Twente Research Information

Theory of the Proximity Effect in Junctions with Unconventional SuperconductorsY. Tanaka1,2 and A. A. Golubov31Department of Applied Physics, Nagoya University, Nagoya, 464-8603, Japan2CREST Japan Science and Technology Cooperation (JST), Nagoya, 464-8603, Japan3Faculty of Science and Technology, University of Twente, Enshede, The Netherlands(Received 9 June 2006; published 16 January 2007)We present a general theory of the proximity effect in junctions between diffusive normal metals (DN)and superconductors. Various possible symmetry classes in a superconductor are considered: even-frequency spin-singlet even-parity (ESE) state, even-frequency spin-triplet odd-parity state, odd-frequency spin-triplet even-parity (OTE) state and odd-frequency spin-singlet odd-parity state. It is shownthat the pair amplitude in a DN belongs, respectively, to an ESE, OTE, OTE, and ESE pairing state sinceonly the even-parity s-wave pairing is possible due to the impurity scattering.DOI: 10.1103/PhysRevLett.98.037003 PACS numbers: 74.45.+c, 74.20.Rp, 74.50.+rIt is well established that superconductivity is realizeddue to the formation of Cooper pairs consisting of twoelectrons. In accordance with the Pauli principle, it iscustomary to distinguish spin-singlet even-parity andspin-triplet odd-parity pairing states in superconductors,where odd (even) refer to the orbital part of the pair wavefunction. For example, s-wave and d-wave pairing statesbelong to the former case while p-wave state belongs to thelatter one [1]. In both cases, the pair amplitude is an evenfunction of energy. However, the so-called odd-frequencypairing states when the pair amplitude is an odd function ofenergy can also exist. Then, the spin-singlet odd-parity andthe spin-triplet even-parity pairing states are possible.The possibility of realizing the odd-frequency pairingstate was first proposed by Berezinskii in the context of3He, where the odd-frequency spin-triplet hypotheticalpairing was discussed [2]. The possibility of the odd-frequency superconductivity was then discussed in thecontext of various mechanisms of superconductivity in-volving strong correlations [3,4]. There are several experi-mental evidences [5] which are consistent with therealization of the odd-frequency bulk superconductingstate in Ce compounds [4,5]. In more accessible systems(ferromagnet/superconductor heterostructures with inho-mogeneous magnetization) the odd-frequency pairing statewas first proposed in Ref. [6] and then various aspects ofthis state were intensively studied [7]. At the same time, thevery important issue of the manifestation of the odd-frequency pairing in proximity systems without magneticordering received no attention yet. This question is ad-dressed in the present Letter.Coherent charge transport in structures involving diffu-sive normal metals (DN) and superconductors (S) wasextensively studied during the past decade both experimen-tally and theoretically. However, almost all previous workwas restricted to junctions based on conventional s-wavesuperconductors [8]. Recently, a new theoretical approachto study charge transport in junctions based on p-wave andd-wave superconductors was developed and applied to theeven-frequency pairing state [9,10]. It is known that in theanisotropic paring state, due to the sign change of the pairpotential on the Fermi surface, a so-called midgap Andreevresonant state (MARS) is formed at the interface [11,12].As was found in [9,10], MARS competes with the prox-imity effect in contacts with spin-singlet superconductors,while it coexists with the proximity effect in junctions withspin-triplet superconductors. In the latter case, it was pre-dicted that the induced pair amplitude in the DN has apeculiar energy dependence and the resulting local densityof states (LDOS) has a zero energy peak (ZEP) [10].However, the relation of this unusual proximity effect tothe formation of the odd-frequency pairing state was notyet clarified. Furthermore, there was no study of the prox-imity effect in junctions with odd-frequency superconduc-tors. The aim of the present Letter is to formulate a generaltheory of the proximity effect in the DN=S junctionsapplicable to any type of symmetry state in a supercon-ductor forming the junction in the absence of spin-dependent electronic scattering at the DN=S interface. Itwill be shown that for spin-triplet [spin-singlet] supercon-ductor junctions, odd-frequency spin-triplet even-parity(OTE) pairing state [even-frequency spin-singlet even-parity (ESE) pairing state] is generated in DN independentof the parity of the superconductor.Before proceeding with formal discussion, let us presentqualitative arguments illustrating the main conclusions ofthe Letter. Two constrains should be satisfied in the con-sidered system: (1) only the s-wave even-parity state ispossible in the DN due to isotropization by impurity scat-tering [1], (2) the spin structure of induced Cooper pairs inthe DN is the same as in an attached superconductor. Thenthe Pauli principle provides the unique relations betweenthe pairing symmetry in a superconductor and the resultingsymmetry of the induced pairing state in the DN. Namely,for even-parity superconductors, ESE and OTE states, thepairing symmetry in the DN should remain ESE and OTE.On the other hand, for odd-parity superconductors, even-frequency spin-triplet odd-parity (ETO) and odd-frequencyPRL 98, 037003 (2007) P H Y S I C A L R E V I E W L E T T E R Sweek ending19 JANUARY 20070031-9007=07=98(3)=037003(4) 037003-1 © 2007 The American Physical Societyspin-singlet odd-parity (OSO) states, the pairing symmetryin the DN should be OTE and ESE, respectively. The aboveresults are based on general properties and independent ofthe details of the geometry and the spin structure of thespin-triplet superconductors.The generation of the OTE state in the DN attached tothe ETO p-wave superconductor is of particular interest. Asimilar OTE state can be generated in superconductingjunctions with diffusive ferromagnets [6,7] but due todifferent physical mechanism. Although the symmetryproperties can be derived from the basic arguments givenabove, the quantitative model has to be considered to provethe existence of nontrivial solutions for the pair amplitudein the DN in each of the above cases.Let us start with the general symmetry properties of thequasiclassical Green’s functions in the considered system.The elements of retarded and advanced Nambu matricesĝR;A ĝ R;A  gR;A fR;AfR;A gR;A (1)are composed of the normal gR;r; ";p and anomalousfR;r; ";p components with spin indices  and . Herep  pF= j pF j , pF is the Fermi momentum, r and "denote coordinate and energy of a quasiparticle measuredfrom the Fermi level.The function fR and the conjugated function fR satisfythe following relation [13,14]: f R;r; ";p  fR;r;";p: (2)The Pauli principle is formulated in terms of the retardedand the advanced Green’s functions in the following way[13]: fA;r; ";p  fR;r;";p: (3)By combining the two above equations, we obtainfR;r; ";p  fA;r; ";p. Further, the definitions ofthe even-frequency and the odd-frequency pairing arefA;r; ";p  fR;r;";p and fA;r; ";p fR;r;";p, respectively. Finally we get f R;r; ";p  fR;r;";p (4)for the even-frequency pairing and f R;r; ";p  fR;r;";p (5)for the odd-frequency pairing. In the following, we willfocus on Cooper pairs with Sz  0 for the simplicity,remove the external phase of the pair potential in thesuperconductor and concentrate on the retarded part ofthe Green’s function. In the case of pairing with Sz  1our final results will not be changed. We consider a junc-tion consisting of a normal (N) reservoir and a supercon-ducting reservoir connected by a quasi-one-dimensionaldiffusive conductor (DN) with a length L much largerthan the mean free path. The interface between the DNand the superconductor (S) at x  L has a resistance Rband the N=DN interface at x  0 has a resistance Rb0 . ForRb0  1, the present model is reduced to the DN=S bilayerwith vacuum at the DN free surface. The Green’s functionin the superconductor can be parametrized as g"̂3 	f"̂2 using Pauli matrices, where the suffix 	 de-notes the right (left) going quasiparticles. g" and f"are given by g	"  gR;r; ";p, g"  gR;r; "; p,f	"  fR;r; ";p, and f"  fR;r; "; p, respec-tively, with p  pF= j pF j and pF  pFx; pFy. Usingthe relations (4) and (5), we obtain that f" f" for the even-frequency pairing and f" f" for the odd-frequency pairing, respectively,while g"  g" in both cases.In the DN region only the s-wave even-parity pairingstate is allowed due to isotropization by impurity scattering[1]. The resulting Green’s function in the DN can beparametrized by coŝ3 	 sin̂2 in a junction with aneven-parity superconductor and by coŝ3 	 sin̂1 in ajunction with an odd-parity superconductor. The function satisfies the Usadel equation [15] D@2@x2	 2i" sin  0 (6)with the boundary condition at the DN=S interface [9] LRd@@xjxL hF1iRb; (7) F1 2T1fS cosL  gS sinL2 T1 	 T1cosLgS 	 sinLfS; (8)and at the N=DN interface LRd@@xjx0 hF2iRb0; F2 2T2 sin02 T2 	 T2 cos0; (9)respectively, with L  jxL and 0  jx0. Here, Rdand D are the resistance and the diffusion constant in theDN, respectively. The brackets h. . .i denote averaging overthe injection angle  hF12i R=2=2 d cosF12R=2=2 dT12 cos; (10) T1 4cos2Z2 	 4cos2; T2 4cos2Z02 	 4cos2; (11)where T1;2 are the transmission probabilities, Z and Z0are the barrier parameters for two interfaces. Here gs isgiven by gS  g	 	 g=1	 g	g 	 f	f andfS  f	 	 f=1	 g	g 	 f	f for the even-paritypairing and fS  if	g  fg	=1	 g	g 	 f	ffor the odd-parity pairing, respectively, with g "="2  2qand f  =2  "2q.   for even-frequency paring and   odd" forodd-frequency pairing,  is the form factor withPRL 98, 037003 (2007) P H Y S I C A L R E V I E W L E T T E R Sweek ending19 JANUARY 2007037003-2	   and   .  is the maximum value ofthe pair potential for even-frequency pairing.In the following, we will consider four possible symme-try classes of superconductor forming the junction andconsistent with the Pauli principle: ESE, ETO, OTE, andOSO pairing states. We will use the fact that only the even-parity s-wave pairing is possible in the DN due to theimpurity scattering and that the spin structure of pairamplitude in the DN is the same as in an attached super-conductor.1. Junction with ESE superconductor.—In this case,f"  f" and g"  g" are satisfied.Then, fS"  fS"  fS and gS"  gS"  gSand we obtain for F1" F1" 2T1fS cosL"  gS sinL"2 T1 	 T1cosL"gS 	 sinL"fS:It follows from Eqs. (6)–(9) that sin"  sin" andcos"  cos". Thus the ESE state is formed in theDN, in accordance with the Pauli principle.2. Junction with ETO superconductor.—Now we havef"  f" and g"  g". Then, fS" fS"  fS and gS"  gS"  gS. As a result,F1" is given by F1"  2T1fS cosL" 	 gS sinL"2 T1 	 T1cosL"gS  sinL"fS:It follows from Eqs. (6) and (9) that sin"  sin" and cos"  cos". Thus the OTE stateis formed in the DN. Remarkably, the appearance of theOTE state is the only possibility to satisfy the Pauli prin-ciple, as we argued above. Interestingly, the OTE pairingstate can be also realized in superconductor/ferromagnetjunctions [6,7], but the physical mechanism differs fromthe one considered here.3. Junction with OTE superconductor.—In this casef"  f" and g"g". Then fS" fS" and gS"  gS" and one can show thatF1" has the same form as in the case of ETO super-conductor junctions. Then, we obtain sin"  sin" and cos"  cos". These relationsmean that the OTE pairing state is induced in the DN.4. Junction with OSO superconductor.—We havef"  f", g"  g" and fS" fS", gS"  gS". One can show that F1" takesthe same form as in the case of ESE superconductorjunctions. Then, we obtain that sin"  sin" andcos"  cos". Following the same lines as incase 1, we conclude that the ESE pairing state is inducedin the DN.We can now summarize the central conclusions as fol-lows:Symmetry ofthe pairing insuperconductorsSymmetry of thepairing in the DN1 Even-frequencyspin-singleteven-parity (ESE)ESE2 Even-frequencyspin-tripletodd-parity (ETO)OTE3 Odd-frequencyspin-tripleteven-parity (OTE)OTE4 Odd-frequencyspin-singletodd-parity (OSO)ESENote that for even-parity superconductors the resultingsymmetry of the induced pairing state in the DN is thesame as that of a superconductor (cases 1 and 3). On theother hand, for odd-parity superconductors, the inducedpairing state in the DN has symmetry different from that ofa superconductor (cases 2 and 4).To illustrate the main features of the proximity effect inall the above cases, we calculate the LDOS " realcos" and the pair amplitude f"  sin" inthe middle of the DN layer at x  L=2. We fix Z  1, Z0 1, Rd=Rb  1, Rd=Rb0  0:01 and ETh  D=L2  0:25.We start from junctions with ESE superconductors andchoose the s-wave pair potential with   1 (Fig. 1).The LDOS has a gap and the real (imaginary) part of f"is an even (odd) function of " consistent with the formationof the even-frequency pairing. In junctions with ETOsuperconductors, we choose px-wave pair potential with	    cos as a typical example. In this case, aunusual proximity effect is induced where the resultingLDOS has a zero energy peak (ZEP) [10]. The resultingLDOS has a ZEP [10] since g2" 	 f2"  1 and f" 0 becomes a purely imaginary number. This is consistentwith f"  f" and the formation of the OTE pair-ing in the DN. To discuss junctions with an odd-frequencysuperconductor we choose odd"  C"=1	 "=2 asthe simplest example of the " dependence of an odd-frequency superconductor pair potential. At "  , themagnitude of odd" becomes maximum. Here we chooseC< 1 when LDOS of bulk superconductor does not have agap around "  0. Let us first consider junctions with OTEsuperconductors and choose an s-wave pair potential as anexample. The resulting LDOS has a ZEP, in contrast tojunctions with ESE superconductors where the resultingLDOS has no ZEP. The formation of the ZEP is due to thesimilar reason in the ETO superconductor junctions, wheref"  0 is a pure imaginary number. Finally, let us dis-cuss junctions with OSO superconductors and choose anpx-wave pair as an example. In this case, the ESE pairing isinduced in the DN and f"  f" is satisfied. ThePRL 98, 037003 (2007) P H Y S I C A L R E V I E W L E T T E R Sweek ending19 JANUARY 2007037003-3resulting LDOS has a gap since f"  0 becomes a realnumber, in contrast to junctions with OTE super-conductors.In summary, we have formulated a general theory of theproximity effect in superconductor/diffusive normal metaljunctions. Four symmetry classes in a superconductor al-lowed by Pauli principle are considered: (1) even-frequency spin-singlet even-parity (ESE), (2) even-frequency spin-triplet odd-parity (ETO), (3) odd-frequencyspin-triplet even-parity (OTE) and (4) odd-frequency spin-singlet odd-parity (OSO). We have found that the resultingsymmetry of the induced pairing state in the DN is (1) ESE(2) OTE (3) OTE, and (4) ESE, respectively. The symmetryin DN is established due to the isotropization of the pairwave function by the impurity scattering and spin conser-vation across the interface. This universal feature is veryimportant to classify unconventional superconductors byusing proximity effect junctions.One of the authors Y. T. expresses his sincerest gratitudeto clarifying discussions with M. Eschrig and Ya. V.Fominov. Discussions with Y. Fuseya, K. Miyake, Yu. V.Nazarov, A. D. Zaikin, A. F. Volkov, and K. Efetov aregratefully acknowledged. This work is supported byGrant-in-Aid for Scientific Research (GrantNos. 17071007 and 17340106) from the Ministry ofEducation, Culture, Sports, Science and Technology ofJapan.[1] M. Sigrist and K. Ueda, Rev. Mod. Phys. 63, 239 (1991).[2] V. L. Berezinskii, JETP Lett. 20, 287 (1974).[3] A. Balatsky and E. Abrahams, Phys. Rev. B 45, 13 125(1992); M. Vojta and E. Dagotto, Phys. Rev. B 59, R713(1999); P. Coleman, E. Miranda, and A. Tsvelik, Phys.Rev. B 49, 8955 (1994).[4] Y. Fuseya, H. Kohno, and K. Miyake, J. Phys. Soc. Jpn.72, 2914 (2003).[5] G. Q. Zheng et al., Phys. Rev. B 70, 014511 (2004);S. Kawasaki et al., Phys. Rev. Lett. 91, 137001 (2003).[6] F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Phys. Rev.Lett. 86, 4096 (2001).[7] F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod.Phys. 77, 1321 (2005); I. Sosnin et al., Phys. Rev. Lett. 96,157002 (2006); A. Kadigrobov, R. I. Shekhter, andM. Jonson, Europhys. Lett. 54, 394 (2001); M. Eschriget al., Phys. Rev. Lett. 90, 137003 (2003); R. S. Keizeret al., Nature (London) 439, 825 (2006); Y. V. Fominov,A. A. Golubov, and M. Y. Kupriyanov, JETP Lett. 77, 510(2003).[8] W. Belzig et al., Superlattices Microstruct. 25, 1251(1999); A. A. Golubov, M. Yu. Kupriyanov, andE. Il’ichev, Rev. Mod. Phys. 76, 411 (2004).[9] Y. Tanaka, Y. V. Nazarov, and S. Kashiwaya, Phys. Rev.Lett. 90, 167003 (2003); Y. Tanaka et al., Phys. Rev. B 69,144519 (2004).[10] Y. Tanaka and S. Kashiwaya, Phys. Rev. B 70, 012507(2004); Y. Tanaka, S. Kashiwaya, and T. Yokoyama, Phys.Rev. B 71, 094513 (2005); Y. Tanaka et al., Phys. Rev. B72, 140503(R) (2005); Y. Asano, Y. Tanaka, andS. Kashiwaya, Phys. Rev. Lett. 96, 097007 (2006).[11] Y. Tanaka and S. Kashiwaya, Phys. Rev. Lett. 74, 3451(1995); S. Kashiwaya and Y. Tanaka, Rep. Prog. Phys. 63,1641 (2000); T. Löfwander, V. S. Shumeiko, andG. Wendin, Supercond. Sci. Technol. 14, R53 (2001).[12] L. J. Buchholtz and G. Zwicknagl, Phys. Rev. B 23, 5788(1981); J. Hara and K. Nagai, Prog. Theor. Phys. 76, 1237(1986); C. Bruder, Phys. Rev. B 41, 4017 (1990); C. R. Hu,Phys. Rev. Lett. 72, 1526 (1994).[13] J. W. Serene and D. Rainer, Phys. Rep. 101, 221 (1983).[14] M. Eschrig, Phys. Rev. B 61, 9061 (2000).[15] K. D. Usadel, Phys. Rev. Lett. 25, 507 (1970). −0.3 0 0.301ESE  (s −wave) superconductorε/∆ρ(ε)(a)−0.3 0 0.3−101f (ε)−0.3 0 0.301ETO (px−wave) superconductor(b)−0.1 0 0.1−1005−0.3 0 0.301OTE (s−wave) superconductorρ(ε)(c)−0.3 0 0.3−202−0.3 0 0.301OSO(px−wave) superconductorρ(ε)(d)−0.3 0 0.3−101ρ(ε)f(ε)f (ε)ε/∆ε/∆ε/∆ε/∆ε/∆ε/∆ε/∆f(ε)RefRefRefRefImfImfImfImfFIG. 1. Local density of states " and pair amplitude f" atthe center of the DN, x  L=2 is plotted. Ref and Imf denotethe real and imaginary part of f". The pairing symmetry of thesuperconductor is (a) ESE, (b) ETO (c) OTE, and (d) OSO,respectively. For (c) and (d), we choose C  0:8. The resultingsymmetry of f" is (a) ESE, (b) OTE (c) OTE and (d) ESE,respectively.PRL 98, 037003 (2007) P H Y S I C A L R E V I E W L E T T E R Sweek ending19 JANUARY 2007037003-4

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Theory of the proximity effect in junctions with unconventional superconductors

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