Andreev reflection is investigated in layered anisotropic normal metal /
superconductor (N/S) systems in the case of an energy gap \Delta in S not
negligible with respect to the Fermi energy E_F, as it probably occurs with
high critical temperature superconductors (HTS). We find that in these limits
retro-reflectivity, which is a fundamental feature of Andreev reflection, is
broken modifying sensitively transport across S/N interfaces. We discuss the
consequences for supercurrents in HTS Josephson junctions and for the midgap
states in S-N contactsComment: 4 pages, 4 figures, to be published in Phys. Rev. 

A. A. Golubov

A. F. Andreev

A. Furusaki

A. Poenicke

Alexander Golubov

C. C. Tsuei

C. W. J. Beenakker

D. B. Bailey

D. Dimos

E. L. Wolf

F. M. Tafuri

Francesco Tafuri

G. E. Blonder

I. I. Mazin

I. Zutic

J.-X. Zhu

L. Alff

M. Covington

M. J. M. de Jong

M. Sigrist

S. Kashiwaya

V. Z. Kresin

Y. Tanaka

Yu. S. Barash

English

arXiv

Andreev reflection is investigated in layered anisotropic normal-metal–superconductor (N/S) systems in the case of an energy gap (Δ) in S not negligible with respect to the Fermi energy (EF), as it probably occurs with high critical temperature superconductors (HTSC’s). We find that in these limits retroreflectivity, which is a fundamental feature of Andreev reflection, is broken modifying sensitively transport across S/N interfaces. We discuss the consequences for supercurrents in HTSC Josephson junctions and for the midgap states in S-N contacts

Golubov, Alexander

Tafuri, Francesco

NARCIS 

Andreev reflections in layered structures: Implications ifor high-Tc grain-boundary junctions

Andreev reflection is investigated in layered anisotropic normal-metal–superconductor ~N/S! systems in the
case of an energy gap (D) in S not negligible with respect to the Fermi energy (EF), as it probably occurs with
high critical temperature superconductors ~HTSC’s!. We find that in these limits retroreflectivity, which is a
fundamental feature of Andreev reflection, is broken modifying sensitively transport across S/N interfaces. We
discuss the consequences for supercurrents in HTSC Josephson junctions and for the midgap states in S-N contacts

GOLUBOV A

TAFURI, Francesco

Archivio Istituzionale della Ricerca - Università degli Studi della Campania "Luigi Vanvitelli"

Andreev reflection in layered structures: implications for high TC grain boundary Josephson junctions

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Andreev reflection in layered structures: Implications for high-
                    
                      
                        
                          
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University of Twente Research Information

PHYSICAL REVIEW B 1 DECEMBER 2000-IIVOLUME 62, NUMBER 22Andreev reflection in layered structures: Implications for high-Tc grain-boundaryJosephson junctionsAlexander GolubovDepartment of Applied Physics, University of Twente, 7500 AE Enschede, The NetherlandsFrancesco TafuriDipartimento di Ingegneria dell’Informazione, Seconda Universita` di Napoli, 81031 Aversa (CE), Italyand INFM-Dipartimento Scienze Fisiche dell’Universita` di Napoli ‘‘Federico II,’’ 80125 Napoli, Italy~Received 16 June 2000!Andreev reflection is investigated in layered anisotropic normal-metal–superconductor~N/S! systems in thecase of an energy gap (D) in S not negligible with respect to the Fermi energy (EF), as it probably occurs withhigh critical temperature superconductors~HTSC’s!. We find that in these limits retroreflectivity, which is afundamental feature of Andreev reflection, is broken modifying sensitively transport across S/N interfaces. Wediscuss the consequences for supercurrents in HTSC Josephson junctions and for the midgap states in S-Ncontacts.golinhecinssupe-niolomsbesineeretrddiorinter-enart inlogyeswen,cedac-hehetodofofatAndreev reflection~AR! is a scattering process occurrinat superconductor–normal-metal~S/N! interfaces which con-vert an electron incident on a superconductor into a hwhile a Cooper pair is added to the superconductcondensate.1 Because of conservation of momentum, thole is reflected back in the direction of the incoming eletron and all components of the velocity are substantiallyverted if the exchange momentum in the scattering procemuch less than the Fermi momentum. Retroreflection occwhenever the Fermi energy (EF) is much larger than the gavalue (D) ~Andreev approximation!. Such an approximationneglects that the retroreflected hole has in reality a differmomentumdk in the direction perpendicular to the S/N interface, which is proportional to the ratioD/EF .1,2 Thismeans that retroreflectivity is broken in some conditions athis will be one of the main issues of this paper.While predictions based on the Andreev approximatprovide accurate explanations in systems employingcritical temperature superconductors, in high-critical teperature superconductor~HTSC! structures the situation imore questionable. If we consider that the gap value couldin some directions of the order of 20 meV~one order ofmagnitude larger thanD of traditional superconductors! andthat the Fermi energy is roughly one order of magnitude lthanEF of traditional superconductors,3 it is interesting to gobeyond the Andreev approximation for systems employHTSC. Concepts based on AR have been widely usedinterpret properties of HTSC grain-boundary~GB! Joseph-son junctions~JJ’s!.4 Some interesting arguments have bedeveloped by taking into account unconventional ordparameter symmetry. Examples are given by the presenczero bound states in the density of states of YBa2Cu3O7 inthe ~110! crystallographic direction5,6 and more generally byphenomena associated with broken time-reversal symme7In this paper we demonstrate that the effects neglectethe Andreev approximation may determine an extremepression of Andreev reflection processes in some directat S/N interfaces and an enhancement of ordinary scattePRB 620163-1829/2000/62~ 2!/15200~4!/$15.00e,g--isrsntdnw-esgton-ofy.ine-nsg.The implications of these effects on bound states at infaces employing superconductors withd-wave-order-parameter symmetry are also considered. These phenomcan reveal several important features in charge transpoHTS JJ’s and enlighten some aspects of the phenomenoof the junctions within the framework of fundamental issuof HTSC’s, such as anisotropy. We stress that the effectconsider, being intrinsically related to Andreev reflectioprovides a microscopic explanation of an intrinsic enhanscattering at the S/N interface.Before taking into account an order parameter withd-wave symmetry,8 we consider a layered normal metal faing an isotropic superconductor with a high value of torder parameter typical for HTSC’s. This is illustrated in tjunction cross section scheme of Fig. 1~a! and in the three-dimensional view in Fig. 1~b!. An electron moving along theplanes tilted at an angleu1 with respect to the junction in-terface is reflected as a hole at an angleu2. Locally Andreevreflection tends to move quasiparticles out of plane andfavor in some way transport along thec axis. This counter-FIG. 1. ~a! Cross section of a S/N interface with N layerenormal metal and S isotropic superconductor with a high valuethe order parameter. An electron moving along the planes tiltedan angleu1 with respect to the junction interface is reflected ashole at an angleu2; ~b! three-dimensional view of~a!; ~c! crosssection of a S/N interface as in~a! being both electrodes layered.~ !Scheme of nonideal~001! GB JJ’s along with the supercurrentransport mechanism.15 200 ©2000 The American Physical Society-re-renee-Shendonthsscnonethinertoteperal-ngesshsonareec-n anhsonion,-oletheandpicesisia thee-re-kesofngfo-cetioneringPRB 62 15 201ANDREEV REFLECTION IN LAYERED STRUCTURES: . . .balances the fact that quasiparticle transport in HTSC’smuch favored along thea-b planes. We will present calculations and phenomenological predictions for the layestructures reported in Figs. 1~c! and 1~d!. These can be considered representative of~100! and nonideal~001! tilt GBJJ’s, respectively.9,10General formalism: Andreev reflection probability folarge values of theD/EF ratio. In order to describe chargtransport through a normal-metal–superconductor~N/S! in-terface, we have used the Blonder-Tinkham-Klapwijk~BTK!approach11 introducing some significant modifications. Isolving the Bogoliubov–de Gennes~BdG! equations for thewave functionscn,s in the N, S regions, we consider thterms of the order ofD/EF in the expressions of the wavvectors along the incoming~electronic! and reflected~hole!trajectories in N,ke,h5kFnA16E/EF, and along the transmitted trajectories without and with branch crossing ink65kFsA16AE22uD6u2/EF @see Fig. 1~a!#. The expres-sion fork6 is extended to the general anisotropic case, wthe magnitudeuD6u and the phasew6 of the gap function inS are specified on the trajectoriesk6 . We also generalize theBTK matching conditions forcn,s at the N/S interface:cn5cs,\22m~cs82cn8 !5S H1H2Dc~0!.HereH1,25*(U(x)2Ee,h)dx, U(x) is the interface barrierpotential,Ee,h are the kinetic energies of the electrons athe holes. In the WKB approximation barrier transmissicoefficients De,h} exp@22*AU(x)2Ee,h/\dx# and Dh,De , since the hole has less kinetic energy to overcomebarrier.12 In analogy with the BTK model, the dimensionlebarrier strengths for electrons and holes can be introduZ1,25A(12De,h)/De,h and generallyZ1,Z2 . This effectmay only hold for a realistic extended barrier12 ather thanfor a d-barrier.With these modifications, we solve the BdG equatioand find the probability of the Andreev and normal reflectiprocess for an electron incoming from the N side, resptively,A~E!52~a11a2!2b1b2uu22 v12 uug1g2u2u1eiw12d1d2v2v1eiw2u2, ~1!B~E!5Uu2u1g2h1eiw12v2v1d1h2eiw2g1g2u2u1eiw12d1d2v2v1eiw2U2. ~2!Here g1,25a1,21b1,272Z1,2, d1,25a2,12b1,262Z1,2,h1,25a1,27b172Z1 , a15 i (k1 /kFs), a25 i (k2 /kFs),b15 i (ke /kFs), b25 i (kh /kFs), u62 5 12 (11 iAuD62 u2E2/E), v62 5 12 (12 iAuD62 u2E2/E).This approach formally presents some analogies withformulation of the problem of spin-polarized tunnelingferromagnet-superconductor~F/S! junctions.13,14 A formalanalogy can be established, for example, between the Fmomenta in the spin subbands and the Fermi momenta inplanes (kuu5kn1) and across the planes (k'5kn2), respec-tively. By increasing the mismatch between them, in bcases the contribution to the current due to Andreev reflisd,needsc-emihehc-tion is reduced. The effects considered in the present paarise from the loss of retroreflectivity due to large gap vues, while in the F/S interfaces they are due to an exchafield in F.Josephson current: comparison with HTS JJ. The modifi-cation of the probability of the Andreev reflection procehas direct consequences on the calculation of the Josepcurrent carried by Andreev bound states@see Fig. 1~d!#. An-dreev bound states are localized in the barrier region andformed by an electron and a hole moving in opposite dirtions. As a consequence the momenta mismatch betweeelectron and a hole leads to a depression of the Josepcurrent. As a generic case, we consider tunnel SIS junctZ15Z25Z@1 and both electrodes ass-wave superconductors.Let an electron have an angleae relative to the planes. Asdiscussed in the Introduction, the Andreev reflected hmoves at an angleah different fromae . The anglesae,h arerelated by the conservation of the momenta parallel tointerface sin(a2ae)ke5 sin(a2ah)kh , where a5(p/22u1)is the angle between the planes and the interface normalthe electron ~hole! momenta, ke,h are given by (16EB /EF)ke,h225 cos2(ae,h2a)kuu221 sin2(ae,h2a)k'22 . HereEB is the Andreev bound-state energy, and the anisotroFermi surface is approximated by an ellipsoid with ax(k' , ki). The results do not depend qualitatively on thchoice.According to the formalism of Furusakiet al.15 the Jo-sephson current per conductance channel is expressed vamplitude of Andreev reflectiona(w,vn) in the barrier re-gion ~N!I s5eD2\ (n Fa~w,vn!ke 2 a~2w,vn!kh G ke1khAvn21D2 . ~3!Here vn5pT(2n11) and w is the phase difference. Thamplitudea(w,vn) is found from the solution of BdG equations and describes the multiple scattering in the barriergion. In a tunnel junctionEB5D. We neglect here a weaenergy dependence ofD due to nonconstant density of statin the relevant energy range.The angle dependence ofa(w,vn) in Eq. ~3! is mainlycontrolled by the scattering amplitudeteh in the electron-holechannel,a}teh(a,ae ,ah). The normal-state conductancethe junctionGN per channel is determined by the scatteriamplitude tee in the electron-electron channelGN}tee(a,ae). The explicit form of teh , tee depends on achoice of the shape of a potential barrier. For thed barrierthe BdG solution yieldsa}teh}kekhkFn22(11Z2)21, GN}eke2kFn22(11Z2)21. As it follows from the above set oequations forke,h , no real solution exists forkh for theanglesae.aeth , where the Andreev reflection process is prhibited. The threshold angleaeth sensitively depends onk' /ki andD/EF .I c and GN are obtained by considering all conductanchannels, i.e., integration over the angleae . The final resultdepends on the barrier shape, which controls the relabetweenae ,ah and trajectories in S regions. We considbelow an extended barrier for strong directional tunnelaround the normal direction~the tunneling cone effect!.12oinoa2li-mityhtucrecdithnbgtethbeeabflheofndmebesed.nc-9,ndl atos-ceionoutcesicthethinter-the-BArnthe15 202 PRB 62ALEXANDER GOLUBOV AND FRANCESCO TAFURIThe results of the numerical calculations ofI C(a) andGN(a) are shown in Fig. 2 for different values of theD/EFratio and Fermi mismatchk' /ki . We give evidence of aremarkable decrease ofI C with the increase of the anglea.The reason for this drop is the existence of both a threshangle aeth and a narrow tunneling cone. This result isqualitative agreement with experimental data obtainedGB JJ’s.9 An account of the distribution of tunnel angles inreal interface would broaden the sharp transitions in Fig.The general picture considered above can be also appto the geometry shown in Fig. 1~c! providing the same qualitative behavior. A rigorous treatment would require sofurther hypothesis on Cooper pair transport in thec directionthat is beyond the scope of this paper.The interplay between the effect of loss retroreflectivconsidered in this paper and well established effects sucinterface roughness or the coexistence of order paramewith different symmetry close to junction interfaces,4,16,18,19deserves further investigation. Interface roughness proda smooth dependence ofI C on the anglea, in contrast to thesharp dependences shown above.S (anisotropic superconductor) / N interface: conductanceand the problem of zero bound state.We consider anotheaspect typical of HTSC junctions related to Andreev refltion by introducing ad-wave order parameter for the S aninvestigating the origin of zero bound states~ZBS!. The ba-sic process is shown in Fig. 3, where ad-wave S faces aninsulator or a normal metal N’ along the~110! orientation~thea-b planes can be, in principle, rotated of an angle wrespect the S/N interface different from 45°). An electrocoming, for instance, along the direction of the positive loof the order parameter, first suffers an ordinary scatterinthe interface with a normal metal~N8! and then is Andreevreflected towards the negative lobe of the order parameThen the hole will experience an ordinary scattering atS/N8 interface and will be reflected towards the positive loof the order parameter, where it will be Andreev reflectagain. This process produces a constructive interferenceas a consequence a resonant state, formally describedpole in the Andreev amplitude Eq.~1!. This manifests itselfas a zero-bias peak in the density of states in S~zero-biasanomaly! ~ZBA!.5,17These arguments are essentially based on the retroreFIG. 2. Dependence ofI C andGN on the anglea between theplanes and the interface for different values of theD/EF ratio andthe Fermi mismatch (k' /ki)50.1 ~solid lines! and 0.3 ~dashedlines!. Experimental data for~100! tilt grain boundaries are showby dots.ldn.edeaserses-,eatr.edndy aec-tion property of Andreev reflection. On the other hand, tconstructive interference breaks down for high valuesD/EF : the electronic state created after two Andreev atwo normal reflections will not propagate along the satrajectory as the initial one. As a consequence ZBS willdamped and the low voltage conductance will be decreaWe have calculated the conductance of a tunnel NIS jution (Z15Z2@1) self-consistently by the method of Ref. 1taking into account the angle mismatch of electrons aholes and the self-consistent reduction of the pair potentiathe interface. The angle averaging was performed by choing ad-function potential barrier with an angular dependenof the transmission coefficientD(u)5D(0)cos2 u. The re-sults of calculations are presented in Fig. 3 for misorientatanglea5450 and different values of theD/EF ratio, givingsome evidence of broadening of the ZBA. We also pointthat in contrast with ZBS broadening due to surfaroughness,18–20 the mechanism we propose has an intrinnature mainly controlled by theD/EF ratio.The problem of the ZBA has also been investigated incase thatZ1 is different fromZ2. This difference is particu-larly meaningful in the formation of the ZBS, where boelectron and hole scattering processes across the sameface are involved. In Fig. 4 we report the conductance inregimeD/EF50 for fixed value ofZ150.5 and for differentvalues ofZ2 at temperatureT50 K. We notice the appearance of peaks at finite voltages and the removal of the Zfor some values ofZ2.Z1. This means that the crossoveFIG. 3. The conductance for different values of theD/EF ratioin a tunnel NIS junction. Inset: resonant state originating atinterface of ad-wave S facing an insulator or a normal metal N8along the~110! orientation.FIG. 4. The conductance for a fixed value ofZ150.5 and fordifferent values ofZ2 at temperature T5 0 K.ksicbfnrioallalapgyrgee ofin. AatesJ.I.hes’’PRB 62 15 203ANDREEV REFLECTION IN LAYERED STRUCTURES: . . .from ZBA to bound states at finite voltages can also taplace due to a different transparency of electrons and holethe S/N interface. Such a crossover has been also predfor ferromagnet-insulator–superconductor junctionsKashiwayaet al.14 Our result is obviously independent oany ferromagnetic electrode or barrier and only relies opossible barrier asymmetry for holes and electrons. Barasymmetry acts as a kind of filter which creates an electrhole imbalance, thus reducing the probability of the formtion of the ZBA.In conclusion, Andreev reflection has been theoreticainvestigated in layered anisotropic normal-metsuperconductor~N/S! systems in the case of an energy gJ..Beattedyaern--y-(D) in S not negligible with respect to the Fermi ener(EF). We have demonstrated that the combination of lagap and strong anisotropy leads to an intrinsic decreasthe critical current density as a function of the tilt angleHTSC Josephson junctions, as experimentally observeddamping of the resonances originating the zero bound sthas been also found.The authors would like to thank A. Barone, Z. Ivanov,Kirtley, M. Yu. Kupriyanov, A. Leggett, J. Mannhart, andI. Mazin for helpful discussions. F.T. was supported by tIstituto Nazionale di Fisica della Materia~INFM! under theproject PRA ‘‘High Temperature Superconductor Deviceand by a MURST COFIN98 program~Italy!.v.ys.. Bys.1A. F. Andreev, Zh. E´ksp. Teor. Fiz.48,1823~1964! @Sov. Phys.JETP19, 1228~1964!#.2C. W. J. Beenakker, Rev. Mod. Phys.69, 731 ~1997!.3V. Z. Kresin and S. A. Wolf, Phys. Rev. B41, 4278~1990!.4Y. Tanaka and S. Kashiwaya, Phys. Rev. B56, 892 ~1997!.5M. Covington, M. Aprili, E. Paraoanu, L. H. Greene, F. Xu,Zhu, and C. A. Mirkin, Phys. Rev. Lett.79, 277 ~1997!.6L. Alff et al., Phys. Rev. B55, R14 757~1997!.7M. Sigrist, Prog. Theor. Phys.99, 899 ~1998!; D. B. Bailey, M.Sigrist, and R. B. Laughlin, Phys. Rev. B55, 15 239~1997!.8C. C. Tsuei,et al., Phys. Rev. Lett.73, 593 ~1994!.9D. Dimos, P. Chaudhari, and J. Mannhart, Phys. Rev. B41, 4038~1990!.10F. M. Tafuri, F. Miletto Granozio, F. Carillo, A. Di Chiara, KVerbist, and G. Van Tendeloo, Phys. Rev. B59, 11 523~1999!.11G. E. Blonder, M. Tinkham, and T. M. Klapwijk , Phys. Rev.25, 4515~1982!.12E. L. Wolf, Principles of Electron Tunneling Spectroscopy~Clar-endon, New York, 1985!.13M. J. M. de Jong and C. W. J. Beenakker, Phys. Rev. Lett.74,1657~1995!; J.-X. Zhu, B. Friedman, and C. S. Ting, Phys. ReB 59, 9558 ~1999!; I. Zutic and O. T. Valls,ibid. 60, 6320~1999!; A. A. Golubov, Physica C326-327, 46 ~1999!; I. I.Mazin, Phys. Rev. Lett.83, 1427~1999!.14S. Kashiwaya, Y. Tanaka, N. Yoshida, and M. R. Beasley, PhRev. B60, 3572~1999!.15A. Furusaki, H. Takayanagi, and M. Tsukada, Phys. Rev. B45,10 563~1992!.16Yu. S. Barash, H. Burkhardt, and D. Rainer, Phys. Rev. Lett.77,4070 ~1996!.17Y. Tanaka and S. Kashiwaya, Phys. Rev. Lett.74, 3451~1995!.18Yu. S. Barash, A. A. Svidzinsky, and H. Burkhardt, Phys. Rev55, 15 282~1997!.19A. A. Golubov and M. Yu. Kupriyanov, Pis’ma Zh. E´ksp. Teor.Fiz. 69, 242 ~1999! @JETP Lett.69, 262 ~1999!#.20A. Poenicke, Yu. S. Barash, C. Bruder, and V. Istyukov, PhRev. B59, 7102~1999!.

Andreev reflections in layered structures:Implications ifor high-T<sub>c</sub> grain-boundary junctions

Andreev reflection is investigated in layered anisotropic normal-metal–superconductor ~N/S! systems in the
case of an energy gap (D) in S not negligible with respect to the Fermi energy (EF), as it probably occurs with
high critical temperature superconductors ~HTSC’s!. We find that in these limits retroreflectivity, which is a
fundamental feature of Andreev reflection, is broken modifying sensitively transport across S/N interfaces. We
discuss the consequences for supercurrents in HTSC Josephson junctions and for the midgap states in S-N
contacts

Golubov A.

Archivio della ricerca - Università degli studi di Napoli Federico II

Andreev reflection in layered structures: implications for high T_c
  grain boundary Josephson junctions

Andreev reflection in layered structures: implications for high T_c grain boundary Josephson junctions

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NARCIS

Archivio Istituzionale della Ricerca - Università degli Studi della Campania "Luigi Vanvitelli"

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University of Twente Research Information

Archivio della ricerca - Università degli studi di Napoli Federico II