Tunneling conductance and local density of states in time-reversal symmetry breaking superconductors under the influence of an external magnetic field

0.49 MB·English

by  Mihail A. Silaev

Tunneling conductance andlocal density ofstates intime-reversal symmetrybreaking
superconductors under the influence ofanexternal magnetic field
Mihail A. Silaev,1 Takehito Yokoyama,2 Jacob Linder,3 Yukio Tanaka,2 and Asle Sudbø3
1InstituteforPhysicsofMicrostructures,RussianAcademyofSciences,603950NizhnyNovgorodGSP-105,Russia
2Department of Applied Physics, Nagoya University, Nagoya,
464-8603, Japan and CREST, Nagoya, 464-8603, Japan
3Department ofPhysics, NorwegianUniversityofScienceandTechnology, N-7491Trondheim, Norway
(Dated:ReceivedJanuary12,2009)
9
0 Weconsiderdifferenteffectsthatarisewhentime-reversalsymmetrybreakingsuperconductorsaresubjectedto
0 anexternalmagneticfield,thusrenderingthesuperconductor tobeinthemixedstate. Wefocusinparticular
2 ontwotime-reversalsymmetrybreakingorderparameterswhicharebelievedtoberealizedinactualmaterials:
p+ıp′-waveandd+ıs-ord+ıd′-wave. ThefirstorderparameterisrelevantforSr2RuO4,whilethelatter
n
orderparametershavebeensuggestedtoexistnearsurfacesinsomeofthehigh-Tccuprates.Weinvestigatethe
a
interplaybetweensurfacestatesandvortexstatesinthepresenceofanexternalmagneticfieldandtheirinfluence
J
onboththetunnelingconductanceandthelocaldensityofstates.Ourfindingsmaybehelpfultoexperimentally
9
identifythesymmetryofunconventionaltime-reversalsymmetrybreakingsuperconductingstates.

]
n PACSnumbers:74.45.+c,74.20.Rp,74.25.-q
o
c
- I. INTRODUCTION tumspace p = p(cosθp,sinθp), g(θp) = cos(2θp +α) and
r
p g1(θp) = sin(2θp+α)whereα/2isananglemeasuringthe
u misorientation of crystalline symmetry axes and coordinate
Recently, considerable attention has been devoted to the
.s chiral superconducting phase which is believed to be real- axes. One obtainsdx2−y2-wavesymmetryof the mainorder
at izedinthep-wavetripletsuperconductor1Sr RuO . Thechi- parameter component for α = 0 and dxy-wave pairing for
2 4 α=π/2.Whiletheexperimentaldatasofarclearlyindicates
m
ral state of a p-wave superconductor corresponds to a non-
anorderparameterwhichbreakstime-reversalsymmetry,the
d- zero projection lz = ±1 of the Cooper pairs angular mo- questionofwhetherthesymmetryisd+ıs-ord+ıd′-wave
mentum l along the z axis, and thus breaks time-reversal
n remainsunsolved. Clearly, experimentalsignaturesthatmay
symmetry. The spatially homogeneous triplet order param-
o distinguishthesetwotypesofpairingswouldbehighlydesir-
c eter ∆ˆ = ∆0(d σˆ)iσˆy is described by the vector1 d(p) = able.
·
[ (0,0,p +iχp ),whichdependsonthedirectionofelectron
x y Oneoftheimportantfeaturesofunconventionalsupercon-
momentum p. Here ∆ is the bulk value of the order pa-
1 0 ductorsisthepossibilityfortheexistenceofsurfaceAndreev
v rameter, σˆ = (σˆx,σˆy,σˆz) is the vector of Pauli matrices of boundstates12,13,14. Theyoccurinthevicinityofthescatter-
2 conventionalspinoperators,andχ = 1 correspondsto the inginterfacebetweenasuperconductorandaninsulator,ifthe
±
2 two possible values of chirality. Also, chiral superconduct-
incidentand reflected quasiparticles(QP) with differentmo-
2 ing states can be associated with an admixture of two order
mentumdirectionsseedifferentphasesoftheorderparameter.
1 parameterscorrespondingto differentirreduciblerepresenta-
TheconsequenceoftheAndreevboundstatesformationisan
.
1 tionsofacrystalpointgroup.Naturallydifferentorderparam- increaseofthelocaldensityofstatesatthesurfaceresultingin
0 eter componentscan coexist in the vicinity of interfaces be- zero-biasconductancepeakanomalyobserved18intunneling
9 tweensuperconductorsandsurfacesduetobrokensymmetry
spectroscopy of high-T cuprates with d-wave symmetry of
:0 of the crystal group2,3,4,5. Among other possibilities of sub- superconductingpairingcaswellasin19 thep-wavetripletsu-
v dominant order parameter symmetry2 there are states which
perconductorSr RuO .Also,theAndreevboundstatesdeter-
i break time reversal symmetry3,4,5. The coexistence of order 2 4
X mine the anomalouslow–temperaturebehaviour of the Lon-
parametersshowsupinthelocaldensityofstates6,7,8,9aswell donpenetrationlength15 andtheJosephsoncriticalcurrentin
r
a asinthegenerationofspontaneouscurrentsflowingalongthe d-wave16andchiralsuperconductors17.
surfacesintimereversalsymmetrybreakingcases7,8.
Under the influence of an applied magnetic field, screen-
Time-reversal symmetry breaking order parameters have ing currents and vortices may be generated in a supercon-
been proposed to exist near surfaces10 and within vortex ductor. As a result, the spectrum of surface states acquires
cores11inhigh-T superconductors.Thisproposalstemsfrom aDopplershift,leadingtoasplittingofthezero-biasconduc-
c
the observation of a split zero-bias conductance peak in the tancepeak10. Abrikosovvorticeslocatednearasuperconduct-
absence of any applied magnetic field. In this case, it has ingsurfacegenerateanessentiallyinhomogeneoussuperfluid
been suggested that the relevant order parameter is either velocityfield,whichleadstoanon-trivialelectronicstructure
d + ıs- or d + ıd′- wave. The gap may then be written as ofthesurfaceboundstates20,21,22. Also,itwasrecently22pro-
∆ = ∆ g(θ )+ı∆ or∆ = ∆ g(θ )+ı∆ g (θ ),respec- posedthatthesameDopplershifteffectshouldleadtoachi-
0 p s 0 p d 1 p
tively,where∆ isanamplitudeofthemaincomponentand rality selectiveinfluenceof the magneticfield onthe surface
0
theadmixtureofanotherpairingsymmetryisdenotedbythe states in a p-wave chiral superconductor with broken time-
amplitudes∆ and∆ . Here, θ isapolarangleinmomen- reversalinvariance.Thequasiparticledensityofstates(DOS)
s d p
2
nearaflatsurfacewasshowntodependontheorientationof In the dx2−y2 + is wave case, one finds that εa0 =
magnetic field with respect to the chirality as well as on the ∆2cos2(2θ )+∆2, from which one infers that there are
0 p s
vorticity in case when the Abrikosov vortex is pinned near no subgapsurface states. This is qualitativelydifferentfrom
thesurfaceofsuperconductor. Additionally,insuperconduc- pthe d +is wave case. The interesting effects occur in the
xy
torsfeaturinggapnodes,such asin thecase in puredx2−y2- lattercase,sowefocusonthedxy+is(d)wavesymmetryin
symmetricsuperconductingcuprates,avanishingpairpoten- thefollowing,correspondingtoα=π/2.
tial in nodal directions results in important ramifications for ThetransformationofthesespectraduetotheDopplershift
thephysicsofthesystem11,23,24,25,26. effectisshowninFig.1. Tobedefiniteweassumethat∆ >
s
To understand the effect of an externally applied mag- 0, ∆ > 0andχ = 1. ConsideringtheDOSatFermilevel,
d
netic field on the surface DOS, let us consider a spectrum ν = ∂ε /∂k −1 ,inachiralp-wavesuperconductoronecan
ofAndreevboundstatesnearaflatsurfaceofatime-reversal seeth|ataitsdeyp|eεn=d0enceonthemagneticfieldismonotonic:it
symmetrybreakingsuperconductoroccupyingthehalf-space eitherincreasesordecreasesfordifferentfielddirections(see
x > 0. Below, we focuson the p+ıp-wave, dxy +ıs- and Fig.1a)asdiscussedinRef. 22.
dxy +ıdx2−y2 -wave cases for concreteness. We consider a AnotherbehaviouroftheDOSoccursinthecaseofad+is
model situation assuming spatially homogeneous gap func- wavesuperconductor. FromFig. 1bitfollowsthatfora cer-
tion,havingthefollowingforminmomentumspace: tainfielddirectiontherearenostatesattheFermilevelε=0
(red dashed lines in Fig.1b). For the opposite field direction
∆=∆ eiχθp (1)
0 (blue dash-dotted lines in Fig.1b), intersections of spectral
forp+ip-wave, brancheswiththeFermilevelappearwhenthesuperfluidve-
locity is large enough v > ∆ /p so that the value of
sy s F
∆=∆ sin(2θ )+ı∆ (2) | |
0 p s momentumprojectionattheintersectionpointissmallerthan
fordxy+is-waveand tthhee FDeOrmSiamt tohmeeFneturmmi|lkey∗v|el<shkoFu.ldTbheusz,eroonewchaenneHxpe<ctHth∗at,
∆=∆0sin(2θp)+ı∆dcos(2θp) (3) whereH∗ isthemagneticfieldvalueprovidingthecondition
v = ∆ /p tobefulfilled.
fordxy+idx2−y2-wavesuperconductors. | sOy|n th|e cso|ntrFary, in the d+ id wave case the DOS at the
AssumingthattheQParespecularlyreflectedatthesurface
Fermi level is non-zero even in the absence of a magnetic
ofthe superconductorwithina Dopplershiftapproach28, the
field. As can be seen from Fig. 1c (black solid lines) the
spectrumofthesurfacestatescanbeexpressedasfollows17,29:
spectralbranchesintersectthelevelε = 0atk∗ = k /√2.
εa =εa0+ǫd,whereεa0isapositionofenergylevelinzero Thetransformationofthespectrumduetotheymag±netiFcfield
magnetic field and ǫ = ~k v is the Doppler shift energy
d F s ofdifferentdirectionsisshowninFig. 1cbyreddashedlines
which is determined by a local field of superfluid velocity.
(H > 0)andbybluedash-dottedlines(H <0). Then,itcan
Thesuperfluidvelocityv nearasurfacehasonlyatangential
s beeasilyseenthatforH > 0thecoordinatesoftheintersec-
component, directed along the y axis v = (0,v ,0), and
s sy tionpointsk∗shifttowards k andforacertainvalueofthe
can be related to the density of supercurrent flowing along magnetic fieyld H > H∗ th±e DFOS at the Fermi level ε = 0
thesurfacejs = envs, whereeiselectronchargeandnisa
disappears.
concentrationofCooperpairs. Themagneticfieldisscreened
InthepresenceofanAbrikosovvortexnearthesurfaceof
in a superconductorat the Londonlength λ as follows B =
chiralsuperconductoranon-trivialstructureofthelocalden-
He−x/λ, where H is a value of magnetic field outside the
sityofstatesdistributionappearswhichdependsonthevortex
superconductor. Therefore, the superfluid velocity is v =
sy orientation22.AlongwiththeDopplershifteffect22,animpor-
(2e/mc)λH.
− tant modificationof the quasiparticle spectrum and the DOS
Ifthemagneticfieldisabsent,thespectrumofsurfacestates
can be obtained due to the overlapping of the surface states
isgivenby12,27
andthelow-energyQPstateslocalizedwithinthevortexcore,
ε =χ∆ k /k (4) foundin the pioneeringworkby Caroli-de GennesandMa-
0a 0 y F
tricon (CdGM)30. It was shown that QP states with energy
forachiralp-wave, lowerthanthebulksuperconductinggapvalue∆arelocalized
within the vortexcore at the characteristic scale of the order
ε =∆ sgn(k ) (5)
a0 s y
ofcoherencelengthξ andhavea discrete spectrumε (µ) as
v
forad +is-waveand afunctionofthequantized(half–integer)angularmomentum
xy
µ. Atsmallenergies ε ∆thespectrumforavortexwith
ε =∆ sgn(k )cos(2θ ) (6) | | ≪
a0 d y p vorticityM isgivenby
foradxy+idx2−y2-wave. Hereky isaprojectionofQPmo- ε (µ) Mµω, (8)
mentumalongthesurface.Theabovespectramaybeformally v ≈−
obtainedbysolving13 where k = p /~ and ω ∆ /(k ξ). For the most of
F F 0 F
∼
superconductingmaterials,includingSr RuO ,theinterlevel
∆(θ ) ∆(π θ ) 2 4
p p
= − . spacingω ismuchlessthanthesuperconductinggap∆since
ε ı ∆(θ )2 ε2 ε +ı ∆(π θ )2 ε2
a0− | p | − a0 a0 | − p | − a0 (kFξ) 1. Therefore,theCdGMspectrummaybeconsid-
(7) ≫
eredcontinuousasa functionof the impactparameterof the
p p
3
of surface and vortex states is determinedby the overlapin-
tegralJ ∆exp( a˜/ξ), where a˜ = a/cosθ and a is the
p
≈ −
distancefromthevortextothesurface.Takingacertainpoint
at the surface (see point A in Fig.2a) of the superconductor
one can obtain a relation between the angles and impact pa-
rameters of trajectories passing trough this point as follows
b=a˜sin(θ θ ).Thustheenergyofvortexcorestatescanbe
p
−
writtenasε = M(a˜/ξ)∆ sin(θ θ ). Then,fromEq.(10)
v 0 p
−
weobtainthespectrumtransformationshownqualitativelyin
Fig.2fortheparticularcaseofθ =0. Itiseasytoseethatfor
equal values of vorticity and chirality (Fig.2b) there appears
aminigapinquasiparticlespectrumneartheFermileveland
therefore the zero-energy DOS is suppressed. On the other
hand, in the case of opposite vorticity and chirality (Fig.2c)
thereisnominigapandtheDOSisnotsuppressed.
FIG.1:(Coloronline)Plotofthesurfacestatesspectrumfor(a)chi-
ral p+ip-wave superconductor, (b) d+is- waveand (c) d+id-
wavesuperconductors. Spectruminzeromagneticfieldisshownby
solidlines. Blueandredlinescorrespondtothespectrumtransfor-
mation due tomagnetic fielddirected along and opposite toz axis
correspondingly.
quasiclassicaltrajectoryb = µ/k andthedirectionofQP
F
−
momentumθ asinthefollowingform
p
ε (b,θ ) M∆(b/ξ). (9)
v p
≈
Inthecaseofachiralp-wavesuperconductor,thespectrumof
vortexcorestates differsfromthe CdGM result andis given
byEq. (8)with integerµ. Forthe d+is-andd+id-wave
superconductors the quasiclassical spectrum of vortex core
statesisgivenbyEq.(9)with∆(θ )= ∆2g2(θ )+∆2and
p 0 p s
∆(θ ) = ∆2g2(θ )+∆2g2(θ ). The discrete spectrum
p 0 p d 1 p p
is obtained by applying Bohr-Sommerfeld quantization rule
to the canopnical variables µ = k b and θ 23,31. It should FIG.2: SketchofQPtrajectoriesformingsurfaceandvortexstates
F p (a)andqualitativeplotofspectrumtransformationduetotheinter-
−
benotedthatwhenthesuperconductingorderparametercon- actionofsurfaceandvortexstatesinthecaseofachiralp-wavesu-
tains nodes, the quasiclassical expression (9) is invalid near perconductor;vorticityandchiralityhave(b)equaland(c)opposite
the nodal directions since energy states near the vortex core values.
are not truly localized, but rather ”leak” outthroughthe gap
nodes11,23,24,25,26. Thisisnotthecaseforussinceweconsider Inad+is-andd+id-wavesuperconductor,theinteraction
superconductingorderparameterswhicharegappedoverthe betweenvortexandsurfacestatescanalsoleadtonoticeable
entireFermisurface. effects,whichwillbediscussedlaterinthepresentpaper.
Tostudytheinteractionbetweenvortexandsurfacestates, Recently, it was pointed out that tunneling of quasiparti-
letusconsideranexampleofvortexpositionednearaflatsur- cles into vortex core states leads to a resonant enhancement
faceofchiralp+ip-wavesuperconductor.Comparingtheen- ofsubgapconductanceofnormalmetal/superconductor(N/S)
ergiesofsurfaceε (4)andvortexε (9)statesonecanseethat junction40. Inthecaseofchiralsuperconductors,suchatun-
a v
forcertainQPtrajectoriestheconditionofresonanceε =ε neling effectcan lead to either stimulation or suppressionof
a v
isrealized. Thusthespectrumtransformationinsuchalmost conductance,dependingonthedirectionofvorticity. Wewill
degeneratetwo-levelsystemisgivenbyasecularequation: showthatifvorticesarelocatedfarfromtheN/Sinterface,the
conductancefollowsthebehaviourexpectedfromtheDoppler
(ε ε )(ε ε )=J2. (10) shift approach. On the other hand, when the distance from
a v
− −
the vortex to the interface becomes comparable with coher-
Sinceweconsideralow-energyspectrum ε ∆ , thetra- encelengthξ thetunnelingintovortexcorestatescomesinto
0
| | ≪
jectories should pass close to the vortex center for the spec- play, leading to the peculiar nonmonotonic conductance de-
trum modification (10) to be effective. Then, the interaction pendenceonthe vortexcoordinatewith respectto the super-
4
conductingsurface. where ∆(θ ) describes the orbital symmetry of the super-
p
Thispaperis organizedas follows. In Sec. II, we givean conductingorderparameterinmomentumspace,whileΨ(r)
overviewof the theoreticalframeworkwhichis employedin describesits spatial dependenceboth magnitude-and phase-
this work, namely a Bogolioubov approach and a quasiclas- wise.
sicalEilenbergerapproach. InSec. III, we presentourmain ThelocalDOS(LDOS)canbeexpressedthroughtheeigen-
results for the influence of magnetic field on bound surface functionsoftheBdGequation(11)inthefollowingform32:
statesspectraandlocaldensityofstatesnearthesurface. We
discuss the transformation of surface states in the Miessner N(ε,r)= u (r)2δ(ε ε ), (14)
n n
stateofsuperconductoraswellastheeffectsofinterplaybe- | | −
n
tweensurfaceandvortexcorestates. Wegiveourconclusions X
inSec. IV. where u (r) is electron component of quasiparticle eigen
n
functioncorrespondingtoanenergylevelε . Theeigenfunc-
n
tionhastobenormalized: ∞ u (r)2+ v (r)2d2r =1.
−∞| n | | n |
II. THEORETICALAPPROACH We will also later employa quasiclassicalEilenbergerap-
RR
proachtostudythespatiallyresolvedDOS.Letusheresketch
OurfurtherconsiderationsarebasedontheBogoliubov–de theframeworkofthetreatmentwhichmakesuseoftheEilen-
Gennes (BdG) equations for particle– (u) and hole–like (v) bergerequation,followingthenotationofRefs.35,36. Itisnow
partsofthewavefunction,whichhavethefollowingform: convenient to solve the Eilenberger equation along trajecto-
ries along the Fermi momentum, and introducing a Ricatti-
1 e 2 parametrization for the Green’s function36. In this way, one
pˆ A u+∆ˆv = (ε+ε )u,
− 2m − c F obtains20
1 (cid:16) e (cid:17)2
2m pˆ+ cA v+∆ˆ†u = (ε−εF)v. (11) ~vF∂x′a(x′)+[2ω˜n+∆†a(x′)]a(x′) ∆=0,
−
Here ∆ˆ is a (cid:16)gap opera(cid:17)tor, A is a vector potential, pˆ = ~vF∂x′b(x′)−[2ω˜n+∆b(x′)]b(x′)+∆† =0, (15)
i(∂/∂ ,∂/∂ ),andr=(x,y)isaradiusvectorintheplane
− x y whereıω˜n =ıωn+mvF vs isaDoppler-shiftedMatsubara
perpendicularto the anisotropy plane. Hereafter we assume ·
frequencyand
theFermisurfacetobecylindricalalongthez–axisandcon-
sideramotionofQPsonlyinxyplane.
1 2e
In case of unconventional superconductors, the gap po- vs = ~ Φ A
2m ∇ − c
tential ∆ˆ is a non-local operator, so the BdG system effec- (cid:18) (cid:19)
tively becomes a very complicated integro-differentialequa-
is a gauge-invariantsuperfluid velocity where Φ(r) is a gap
tion. Another complexity arises from the broken spatial in- functionphase:Ψ(r)= ΨeiΦ.TheLDOSmaybeexpressed
variance of the superconducting gap in presence of vortices throughthescalarcohere|nc|efunctionsaandbasfollows20
near the N/S interface. A simplification can be obtained if
one considers a quasiclassical approximation, assuming that 2π dθ 1 ab
thewavelengthofquasiparticlesismuchsmallerthanthesu- N(ε)= Re − , (16)
perconducting coherence length (see e.g. Ref.34). Within Z0 2π n1+aboıωn→ε+ıδ
such an approximation, QP move along linear trajectories,
whereεisthequasiparticleenergymeasuredfromFermilevel
i.e. straight lines along the direction of QP momentum
and δ is a scattering parameter which accounts for inelastic
n = kFkF−1 = (cosθp,sinθp). Generally, the quasiclassi- scattering.
calformofthe wavefunctioncanbeconstructedasfollows:
To investigatethe transportpropertiesofN/Sjunction,we
(u,v) = eikF·r(U,V), where(U(r),V(r)) isa slowlyvary-
employ an approach similar to what was used in work by
ing envelope function. Then the system (11) reduces to a Bardeen,TinkhamandKlapwijk33. Theexpressionforthedi-
system of first-order differential equations along the linear
mensionlesszero-biasconductanceoftheN/Sjunctionmea-
trajectories defined by the direction of the QP momentum sured in terms of the conductance quantum e2/(π~) can be
n = kFkF−1 = (cosθp,sinθp). Introducing the coordinate writtenasfollows:
alongtrajectoryx′ =(n r)=rcos(θ θ)wearriveatthe
p
· −
followingformofthequasiclassicalequations: G π/2
G= sh [1 R (θ )+R (θ )]cosθ dθ , (17)
n 0 a 0 0 0
i~vF∂x′ +vF eA U +∆V =εU 2 Z−π/2 −
− · c
(cid:16)i~vF∂x′ +vF · ecA (cid:17)V +∆†U =εV, (12) wanhderAenRdrne(eθv0r)eflanecdtioRna(rθe0sp)eacrtievetlhye,θp0roibsatbhieliitniecsidoefntnaonrgmlea:l
(cid:16) (cid:17) kF = kF(cosθ0,sinθ0), characterizing the propagation di-
where the Fermi velocity is vF = n~kF/m. The pairing rection of quasiparticles, coming from the normal metal re-
potentialinEq.(12)maygenerallybewrittenas gion. The Sharvin conductance G = k L /π equals the
sh F y
total number of propagatingmodes determinedby the chan-
∆(r,θ )=∆(θ )Ψ(r), (13)
p p nelwidthL .
y
5
TheproblemofquasiparticlescatteringattheN/Sinterface III. RESULTS
isformulatedwithintheBdGtheory(11). Aninterfacialbar-
rier separating the N and S regions can be modeled by re- To illustrate the basic effect of how the interplaybetween
pulsivedeltafunctionpotentialW(x) = W0δ(x),parameter- theDoppler-shiftandthetime-reversalsymmetrybreakingof
izedbya dimensionlessbarrierstrengthZ = W0/~vF. The the superconductingorder parameter is manifested, we con-
boundaryconditionsattheN/Sinterfacethenread: sider a situation where an external magnetic field is applied
nearthesurface ofthe superconductoralongthe ˆz-axis, thus
[f(0)]=0, [∂xf(0)]=(2kFZ)f(0), (18) inducing a vector potential A in the superconductor which
drivestheshieldingsupercurrent.Inordertoproceedanalyti-
wheref =(u,v)and[f(x)]=f(x+0) f(x 0). cally,wemakethesimplifyingassumptionthatthesuperfluid
− −
Considering a zero-bias problem we will have to ana- velocityfieldisnearlyhomogeneousandthatthespatialvaria-
lyze only zero-energy excitations with ε = 0. For wave tionofthesuperconductingorderparameterneartheinterface
functions in S region corresponding to subgap quasiparti- issmall. Choosingarealgauge,wethenfindthattheRicatti-
cles, the following representation can be used: (U,V) = functionsaandbinEq.(15)maybewrittenas:20,22
eζ ei(η+Φ)/2,e−i(η+Φ)/2 , where ζ = ζ(s,b) and η =
η(s,b) are real-valued functions. Then, the quasiclassical a(θ)=s(θ)∆(θ), b(θ)=s(θ)∆∗(θ),
(cid:0) (cid:1)
equation(12)canbewrittenasfollows:
s(θ)=1/[ω˜ (θ)+ (ω˜ (θ))2+ ∆(θ)2], (23)
n n
| |
∂x′η+2|∆|cosη+2ǫd =0, (19) whereω˜ndependsonθthrougphtheDoppler-shift.Toevaluate
∂x′ζ+2∆ sinη =0. the LDOS in Eq. (16) at the surface, we need to take into
| | accountproperboundaryconditionsatx = 0. Assuming an
where ǫd(r) = ~kFvs is a Doppler shift energy. For impenetrable surface with perfect reflection, these boundary
wave functions decaying at the different ends of trajectory conditionsread:
(U,V)(x′ = )=0fromEqs.(19)weobtain:
±∞ asurface(θ)=a(π θ), bsurface(θ)=b(θ). (24)
−
η(x′ = )= π/2. (20) InsertingthisintotheexpressionfortheLDOS,weobtain
±∞ ±
Theboundaryconditions(18)modelthespecularlyreflect- 1
N(ε)=2Re 1. (25)
ikng(Nco/Ssθin,tesrifnaθce,),coaunpdlikn′gth=ewkav[ceossw(πithwθav)e,vsienc(toπrskθF)=]. nD1+a(π−θ)b(θ)Eoıωn→ε+ıδ −
F 0 0 F F 0 0
Thereforeiftheincidentelectronwaveis−ui =eikFr,th−enthe Theh...idenotesangularaveraging,whichwerestricttoan-
gles π/2 θ π/2 due to the surface. It maybe shown
reflectedelectronur andholevr waveswillhavetheform that −for a c≤hiral≤p-wave superconductor22, the zero-energy
u =U eik′Fr, v =V eikFr, DOSatthesurfacereads
r r r r
~k v
N(0)=1+ F sy +..., (26)
whereUr andVr aretheenvelopefunctions.Thus,eachpoint ∆0
(0,y)attheN/Sinterfaceliesontheintersectionoftwoqua-
siclassical trajectories, characterized by the angles θ = θ whileforpures-ord-wavesuperconductorsonefinds
p 0
and θ = π θ . Let us denote the distribution of phases
η(x′)palong t−hese0 trajectories as η (x′) and η (x′) corre- N(0)=C1+C2vs2y +..., (27)
+ −
spondingly.Usingtheboundaryconditionsweobtainthefol-
whereC andC arearbitraryconstants. Fromnumericalin-
lowingexpressionfortheconductance40: 1 2
vestigationsofEq.(25)atε=0,wefindthatthezero-energy
DOSmayquitegenerallybewrittenas
N Ly/2 π/2
G= 0 g(y,θ )cosθ dθ dy, (21)
0 0 0 N(0)=C +C v +..., (28)
2 1 2 sy
Z−Ly/2Z−π/2
wheneverthesuperconductingorderparametersi)breaktime-
whereg(y,θ )isgivenby
0 reversal symmetry and ii) support the presence of subgap
surface-bound states. This is the case both for the p +ıp -
2 x y
g(y,θ0)= (Z˜4+Z˜2)1 eiρ 2+1, (22) wwaevlleapsatihriengd+wısh-iwchavisebaenldiedve+dtıod-bwearveealpizaeirdinignsStrh2aRtuaOre4r,eals-
| − |
evant for the cuprates. In particular, tunneling spectroscopy
withZ˜ =Z/cosθ andρ(y,θ )=η η isdeterminedby measurements have indicated the presence of such a time-
0 0 − +
a difference of phases η (x′) and η (−x′) at the intersection reversalsymmetrybreakingorderparameternearsurfacesby
− +
point (0,y). To evaluate the conductance, one needs to find a split zero-bias conductance peak that was observed in the
the factor eiρ in Eq.(22) and then the reflection probabilities absenceofanexternalfieldinseveralexperiments.10.
bysolvingnumericallyEq.(19)withtheboundaryconditions InRef.35,itwaspointedoutthattheneglectofthegradient
inEq. (20). termintheEilenbergerequationisexpectedtobeareasonable
6
approximationaslongastheDoppler-shiftenergymv v
F s
·
issmallcomparedtothelocalgapenergy∆(θ). Thisapprox-
imationwouldthenfailclosetothevortexcoreorgapnodes
of∆(θ). Nevertheless,inthemodelcaseofspatiallyhomoge-
neousgap function and superfluid velocity field the gradient
terms in Eilenbergerequationcan be neglected in the whole
rangeofDopplershiftenergies.Howeverconsideringamodel
situationtheabovediscussionneverthelessservestoillustrate
ourmainqualitativeargument:namely,thatchirality-sensitive
effectsshouldbeexpectedinsuperconductorswithorderpa-
rametersthati)breaktime-reversalsymmetryandii)support
thepresenceofsubgapsurface-boundstates. Wenowproceed
todiscussthecasesofp +ıp -waveandd+ıs(d)-wavepair-
x y
inginmoredetail,sincethesearerelevanttoactualmaterials.
A. Surfacestatesinp+ipandd+is(d)superconductors
undertheinfluenceofmagneticfield.
In Fig.3, we show numerical plots of the surface LDOS
at the Fermi level given by Eq.(25) for the chiral p-wave
(Fig.3a),d+is-wave(Fig.3b)andd+id-wave(Fig.3c)cases
inawidedomainofsuperfluidvelocities.Thestructureofgap
functionsischosenintheformofEqs. (1)-(3)andtheparam-
etercharacterizinginelasticscatteringinEq.(16)ischosenas
δ = 0.1∆ . Weintroducethefollowingnotationsforthedif-
0
ferentcriticalvelocities: v = ∆ /(~k ),v = ∆ /(~k ) FIG.3: Plotofthenormalizedzero-energyLDOSN(0)for(a)p-
c 0 F cs s F
andvcd = ∆d /(~kF). | | wavesuperconductor with χ = 1, (b) d+ıs-wavecase with∆s =
As seen,| the| surface LDOS has sharp peaks at a certain 0.4∆0 and(c)d+ıd-wavecasewith∆d = 0.4∆0. Dashedlinesare
value of the superfluid velocity in all cases. We will show guides for eyes: thevertical ones denote positions of LDOSpeaks
and horizontal ones correspond to the level of normal metal DOS
below that peaked structure of LDOS is provided by bound
surfacestates. AnothercontributiontotheLDOScomesfrom N0.
thedelocalizedstatescorrespondingtothecontinuouspartof
QP spectrum. A delocalized state with zero energy ε = 0
bythesuperfluidvelocity,isgivenby
existsprovided(i) v > v incaseofchiralp-wavesuper-
sy c
| |
conductor, (ii) |vsy| > vcs and (iii) |vsy| > vcd in case of εa =χ∆0ky/kF +~vsyky (29)
d + is-wave and d + id-wave superconductors correspond-
ingly.Condition(i)isunlikelytoberealizedbecauseitmeans forthep-wavecase,
that the superfluid velocity is larger than the critical depair-
ing value. Conditions (ii) and (iii) can be realized, because εa =∆ssgn(ky)+~vsyky (30)
thevaluesv andv canbewellbelowthecriticaldepairing
cs cd ford+is-wavecaseand
velocity if the amplitude of additionalorder parameter com-
ponentsissmallenough. ε =∆ sgn(k )cos(2θ )+~v k (31)
a d y p sy y
ToanalyzethecontributiontoLDOSprovidedbythebound
surface states we will consider the domain of low energies for d + id-wave case. Consequently, the contribution from
ε ∆0. Byneglectingsmalldeviationsoftheelectronand Andreevboundstatestothezero-energyLDOSatthesurface
| |≪
hole momentum, the normalizedwave functionof QP local- ofachiralp-wavesuperconductorisgivenby
izedneartheboundarycanbewrittenas
1
N =N ,
a 0
u = 1 2 eikyysin(k x)e−x/(ξ˜cosθp), |vsy/vc+χ|
(cid:16)v(cid:17) (cid:18)i(cid:19)sξ˜|cosθp| x where N0 = m/(2π~2) is the normal metal LDOS per one
spin direction. For a chiral d + is superconductor, the be-
where(k ,k ) = k (cosθ ,sinθ ). Thiswavefunctionde-
x y F p p haviour of the LDOS is more complicated. Assuming that
cay in the superconductingside x > 0 at a characteristiclo-
∆ > 0, we obtain that the LDOS is zero for v >
calizationscaleξ˜isgivenbyξ˜= ~vF/∆0 forchiralp-wave ∆s /~k . Otherwise,itisgivenby sy
and ξ˜ = ~v /∆ sin(2θ ) for d + is- and d + id- wave − s F
F 0 p
superconductor|correspondin|glywithgapfunctionsgivenby vcsvc
N =4N .
Eqs.(2,3). ThespectrumoftheAndreevboundstates,shifted a 0 v2
sy
7
Onthecontrary,forthed+idcasetheLDOSiszeroifvsy > 1. p+ipwave
∆ /(~k )(for∆ >0)andotherwiseitisgivenby
d F d
In Fig.4 we show the LDOS profile near the surface of a
chiralp-wavesuperconductorinthepresenceofasinglevor-
∆ v
N =N 0 1+ cd . tex,positionedatsomedistanceafromthesurface.Whenthe
a 0
|∆d| vs2y +8vc2d vortex is positioned far from the surface a ≥ 2ξ the LDOS
 q  profilefollowsthebehaviour,expectedfromthepictureoflo-
calDopplershift22. Dependingontherelativevalueofvortic-
ItcanbeseenthatthesecontributionstoLDOShavepeaks
ityandchirality,thesurfaceLDOSiseitherincreased(Fig.4a)
at v = v for p-wave case. For d+is-wave and d+id-
sy c ordecreased(Fig.4b). Ananalyticalestimatewiththehelpof
wave superconductors the peaks are positioned at v =
sy spectrum Eq. (33) yields a following estimation of the am-
sgn(∆ )v andv =sgn(∆ )v correspondingly.Even
t−houghthse pcossitionsoyfthe peaksdarecddifferent,the dependen- plitudeofLDOSpeakin Fig.4a: ∆N/N0 = (1+Mχa)−1.
Atsmallerdistancesa 2ξ,thebehaviourofLDOSchanges
ciesofthesurfaceLDOSonthesuperfluidvelocity(andcon- ≤
drastically. Inthecaseofoppositevorticityandchirality,the
sequently on magnetic field) are very similar for d+is and
surface LDOS grows at a 2ξ, obviously due to the over-
d+id-wavesuperconductors.Therefore,itmightbedifficult ≤
lappingwith the peakof vortexcore states. In case of equal
todistinguishwhichcaseisrealizedexperimentally.
vorticityandchiralitythesameoverlappingoccurs,butonthe
Ontheotherhandtheconsideredmodelwithaspatiallyho-
contraryit leads to reductionof DOS, as it was discussed in
mogeneousgapfunctionΨ(r)=1isadequateonlywhenthe Sec. I. The peak of the LDOS at the surface discussed in
applied magnetic field is not too large. When the magnetic
Ref.22transformsintoadip-and-peakstructureasthevortex
field is large enough,it breaksthe Meissner state and gener-
comesclosetothesurface.
ates vortices near the surface of superconductor. Therefore,
we investigate the influence of vortices on the LDOS distri-
butionneara superconductingsurfaceas wellasonthe con-
ductanceofnormalmetal/superconductingjunctions.Wewill
showthatvorticeshavedifferenteffectontheconductancein
d+isandd+idcases.
B. Interplayofvortexandsurfacestatesinchiral
superconductors.
A chirality sensitive LDOS transformationdue to vortices
situated near the surface of a chiral p-wave superconductor
wasconsideredinRef.22. Itwasshownthatdependingonthe
chiralityandvorticityvalue, thesurfaceLDOSneariseither
enhancedorsuppressedupondecreasingthedistancefromthe
vortextothesurface.Incaseofd+is(d)superconductorsthe
transformationofLDOSprofileisalsosensitivetothevalueof
FIG. 4: (Color online) Plot of the normalized zero-energy LDOS
vorticity. Similarbehaviourisexpectedforaconductanceof
N(0)inthepresenceofavortexnearthesurfaceofachiralp-wave
normalmetal/chiralsuperconductorjunctioninthepresence
superconductor. (a)and(c)correspondtoequalvorticityandchiral-
ofvortices.
ity, (b) and (d) correspond to opposite vorticity and chirality. The
ToinvestigatetheinfluenceofasinglevortexontheLDOS distance from vortex to the surface is a = 2ξ for (a) and (b) and
profileandconductance,weassumethatatx > 0(supercon- a=ξfor(c)and(d).
ductingregion)thecoordinatedependenceoftheorderparam-
etermaybewrittenasfollows:
This is illustrated in Fig.5(a), where we plot the LDOS at
thesurfacepoint(0,0),whichisthenearestpointtothevortex
Ψ(r)=eiΦ. (32) inFig.4. Atlargedistancesa ξ theLDOSisamonotonic
≫
function of a, either increasing or decreasing depending on
Here, we considera modelsituationwherethemagnitudeof the relation between vorticity and chirality. At smaller dis-
the orderparameteris constant. Thephase distributionΦ(r) tancesa 2ξ,theextremumofLDOSappears.Inthecaseof
≤
consists of a singular part Φ (r) = arg(r r ) and a reg- oppositevorticityand chirality[lower curvein Fig.5(a)], the
v v
−
ularpartΦ (r), determinedbytheparticularmetastablevor- surfaceLDOSgrowsata 2ξ, dueto theoverlappingwith
r
≤
tex lattice configurationrealizing near the boundary. We as- the peakof vortexcore states. In case ofequalvorticityand
sumethattheregularpartofthephasedistributionisΦ (r)= chirality[uppercurveinFig.5(a)]thesameoverlappingleads
r
arg(r rav)correspondingtotheimagevortexsituatedat toreductionofLDOS.
− −
thepointrav =( 2a,0,0)behindtheN/Sinterface. To investigate the influence of vortices on the transport
−
8
properties of normal metal/ chiral p- wave superconductor anoppositeeffectoccurs:oneobtainsaconductancesuppres-
junction we solve the generic problem of the influence of a sioninsteadofenhancementandviceversa.Theoriginofthe
single vortex near the N/S surface on the zero-bias conduc- conductanceextremumisatunnelingofquasiparticlesintothe
tance of the junction. A numerical plot of the conductance vortexcorestates,orinotherwords,theoverlappingofvortex
G as a functionof a distance of vortex to the junction inter- andsurface boundstates. ComparingFigs.5(a) and 5(b)one
face is shownbythe solid linesin Fig.5(b)forequal(upper canseethattheconductanceingeneralfollowsthebehaviour
curve)andopposite(lowercurve)valuesofchiralityandvor- ofthesurfaceDOS.
ticity. TheconductanceisnormalizedtothevalueofSharvin
conductanceG =k L /π.
sh F y
2. d+isandd+idwave.
Inchirald+isandd+idsuperconductorstheLDOStrans-
formationappearstoalsobevorticitysensitive. IntheFig.6
FIG.5: (a) Plotofthenormalizedzero-energy LDOSatthepoint
onthesurfacewhichisclosesttothevortexcore. Differentcurves
correspondtodifferentvorticities.(b)Plotofthevortex-inducedcon-
ductanceinchiralp+ip-wavesuperconductorforequalandopposite
valuesofvorticityandchirality. Thestrengthofinterfacebarrieris
Z =5.LargedistancesasymptoticsforN andGareshownbydash
lines.
At large distances a ξ an analyticalestimation of con-
≫
ductance can be obtained by using a local Doppler shift ap-
proximationonthequasiparticlespectrum. Indeed,themodi-
ficationofthesurfacestatesenergyduetoasupercurrentflow-
ingalongtheboundaryofsuperconductorcanbewrittenas
FIG.6:(Coloronline)Plotofthenormalizedzero-energyLDOSpro-
ε (χ∆ +~v k )k /k , (33) fileN(0)inthepresenceofvortexnearthesurfaceofchirald+is-
a 0 sy F y F
≈ wavesuperconductor with∆s = 0.2∆0. (a),(c) and (b),(d) corre-
spond to different vortex orientations with respect to z axis. The
where k is a quasiparticle momentum along the surface,
χ= 1yisachiralityvalueandv =(M~/m)a/(y2+a2)is distance from vortex to the surface is a = 4ξ for (a) and (b) and
± sy a=2ξfor(c)and(d).
aprojectiononthesurfaceplaneofsuperfluidvelocitygener-
atedbythevortexandimageantivortex,M isvorticityvalue
we show the profile of zero-energy LDOS in the case when
and m is the electronmass. Itfollows fromEq.(33) that the
thevortexisplacedatadistanceofa=2ξfromaflatbound-
Dopplershifteffectleadstoachangeintheslopeofanoma-
ary of a d+is-wave superconductorcharacterized by a gap
lousbranch. Itiseasytoobtainthatinthiscasethefunction
functioninmomentumspacegivenbyEq.(2). Inthissection,
g(y,θ )inexpressionEq.(22)takesthefollowingform:
0 weusethenotationξ =~v /∆ .
F 0
2 OnecanseethatforonesignofvorticitythesurfaceLDOS
g(y,θ0)= 4(Z˜4+Z˜2)(ε /∆ )2+1. (34) showstwopeakswhicharesymmetricwithrespecttothevor-
a 0 tex position. As we have shown above, the large peaks in
surfaceLDOSappearwhentheenergycoincideswiththepo-
Thestraightforwardintegrationin Eq.(21)yieldsG = G +
0
δG,whereG =G (π/Z2)istheconductancewithoutvor- sitionofboundstatelevel.Foradifferentsignofthevorticity,
0 sh
there are no surface states at the Fermi level and the LDOS
texand
alongthesurfaceisaflatfunction.Anon-zerolevelofLDOS
2πξ inthiscase isprovidedbyinelasticscatteringwhichleadsto
δG/G = arctan(L /2a) (35)
sh ±Z2L y thesmearingtheQPenergylevels. ApplyingalocalDoppler
y
shiftapproach,whichholdsifthedistancefromvortextosur-
isavortex-inducedconductanceshift,wheretheupper(lower) faceisratherlarge(a ξ)onecaninterprettheresultsshown
≫
signcorrespondstoequal(opposite)vorticityandchirality. inFig.(6).
At distancessmaller than 2ξ, an extremumof the conduc- The coordinates y∗ of surface LDOS peaks can be es-
tanceappears. Uponplacingthevortexclosertothesurface, timated from the relation v = ∆ /p , where v =
sy s F sy
9
tivelydifferentforsandd-wavesymmetryoftheadditional
gapfunctioncomponent. Ford+is-wave, theconductance
has a sharp peak for one vortex orientation (upper curve in
Fig.8a)anditisa flatfunctionofa foranothervortexorien-
tation(lowercurveinFig.8a). Theoriginoftheconductance
enhancement is a formation of Andreev bound states at the
Fermilevelwhicharelocalizednearthesuperconductingsur-
face. AswasdiscussedintheIntroduction(seetheFig. 1b),
the zero-energyAndreev bound states can appear only for a
certain direction of superfluidvelocity flowing along the su-
perconductingsurfaceandifthevalueofthesuperfluidveloc-
ity is larger than a critical value v > ∆ /(~k ). For a
sy s F
| | | |
highinterfacebarrierZ 1applyinganapproximateanalyt-
≫
icalexpression(35) we find thata sharp increaseof conduc-
tanceinFig.8acanbedescribedbythefollowingexpression
16π ξ ∆ a 3/2
G/G = 0 1 +λZ−4,
sh 3Z2L ∆ − a∗
y s
| |(cid:16) (cid:17)
FIG.7:(Coloronline)Plotofthenormalizedzero-energyLDOSpro-
wherea∗ =ξ(∆ /∆ )andλ 1. Otherwise,ifa>a∗the
fileN(0)inthepresenceofvortexnearthesurfaceofchirald+id- 0 | s| ∼
conductanceismuchsmallersinceZ 1:
wave superconductor with ∆d = 0.2∆0 (a),(c) and (b),(d) corre- ≫
spond to different vortex orientations with respect to z axis. The 2
∆ 4
distance from vortex to the surface is a = 4ξ for (a) and (b) and G/G 0 .
sh ≈ ∆ 3Z4
a=2ξfor(c)and(d). (cid:18) s(cid:19)
Whenthe distancea isdecreasedfurther,theconductanceis
suppressed (see Fig.8a, upper curve). The decrease of con-
(M~/m)a/(y2 + a2) is a projection on the surface plane
ductancecanbeattributedtothegapattheFermilevelwhich
of superfluid velocity generated by the vortex with vortic-
appearsdue to the interactionof vortex and surface states in
ity M and image antivortex. It can be seen that for a >
a similar way as for the p+ip- wave case discussed in the
ξ(∆ /∆ ) the peak is situated at y∗ = 0 i.e. at the
0 s previoussection.
| |
surface point nearest to the vortex. Otherwise, we obtain
In a d + id-wave superconductor, zero-energy Andreev
y∗ = a 1 (a/ξ)(∆s /∆0). Comparing this estima- bound states may exist even in the absence of vortex. An
± − | |
tion with the numerical results in Fig.6, one observes a mi- asymptoticvalueoftheconductanceG ata ξ canbeob-
nor differepnce. For example, it follows from the estimation tainedusingtheexpression(35)asfollo0ws ≫
thattheLDOSpeaksshouldbepositionedaty∗ = 2.4ξfor
T∆hsis=dis0c.r2e∆pa0n,cbyuctainnbFeiga.t6trtihbueytedarteoltohceacteodmaptleyx∗s=h±ift±o2f.t0hξe. G0/Gsh =(cid:18)|∆∆d0|(cid:19)2√π2Z2.
energyε ε+iδ due to the effectivescattering parameter
→ Whenthevortexapproachesthesuperconductingsurface,the
δ = 0.1∆ whichwasusedinthenumericalcalculations. If
0 conductance is either suppressed (lower curve in Fig.8b) or
weincreasethedistancefromvortextosurfacea, theLDOS
slightlyenhanced(uppercurveinFig.8b).Thisbehaviourcan
peakswillmergewhena>ξ(∆ /∆ )(seeFig.6).
0 | s| be understoodby again using the Eqs.(35) with the Doppler
In Fig.7, we show the LDOS profile modulated by a vor-
shifted spectrum of Andreev boundstates (6). The decrease
tex placed at a distance of a = 2ξ from a flat boundary
(increase) of conductance corresponds to the transformation
of d + id superconductor. The structure of the gap func-
of spectrum shown qualitatively in Fig.1c by dash (dash-
tion was chosen in the form (3). Applying the approach
dotted)lines. Itispossibletoobtainananalyticalexpression
based on the local Doppler shift we obtain the similar ex-
forthevortex-inducedconductanceshiftata ξ inthefol-
pression for the coordinates of the peaks of surface LDOS: ≫
lowingform:
y∗ = a 1 (a/ξ)(∆ /∆ ). Fortheparticularvaluesof
d 0
± − | |
pyica∗arla=mple±otte2rpi.ns4ξ∆F,idwg.h7=icyh0∗.i2s∆m0uac4nhξd.leasTsh=tihsa2dnξisotchbritesapieansnetcdimyfracotaimnonanlyusimoelebdres- δG/Gsh =±2Zπ2 (cid:18)∆∆0d(cid:19)2 Lξy arctan(cid:18)L2ay(cid:19),
≈ ±
attributed to the effect of inelastic scattering, which appears where the upper and lover signs correspond to the different
tohavealargereffectind+id-wavecasethanindiscussed vortexorientations. Asthevortexapproachesthesurfacefur-
aboved+is-wavecase. ther,thereappearsanextremumoftheconductance.Suchbe-
AnumericalplotoftheN/Sjunctionconductanceasafunc- haviourcanbeexplainedbyaconductanceenhancementdue
tionofdistancefromvortextosurfaceisshowninFig.8forthe tothetunnelingofQPintothevortexcorestates,discussedin
d+isandd+idcases. Theconductanceisnormalizedtothe Ref.40. AsharpdecreaseoftheuppercurveinFig.8bcanbe
SharvinconductanceG =k L /π. ComparingFig.8aand attributedto the openingof an energygapat the Fermilevel
sh F y
Fig.8bonecanseethattheconductancebehaviourisqualita- duetotheinteractionofvortexandsurfacestates.
10
erablequalitativemodificationofboththetunnelingconduc-
tanceandtheLDOS.Whenthevortexislocatedatdistances
wellaboveacoherencelengthξfromthesurface,theDoppler-
shift produces an enhancement or suppression of the LDOS
dependingontherelativesignofthevorticityandthechirality
ofthesuperconductingOP.Thiseffectmaybedirectlyprobed
byfirstapplyinganexternalmagneticfieldinadirectionwhile
measuringtheLDOSandthenreversingthefielddirectionand
measuringagain.Whenthevortexislocatedveryclosetothe
surface(a distanceoforderξ orsmaller),thereisan overlap
betweenthevortexandsurfacestateswhicheffectivelycause
a dramatic change in the tunneling conductance and LDOS.
FIG.8: Plotsofthevortex-inducedconductanceincaseof(a)chiral
This effectis also sensitive to the relative sign of the vortic-
d+issuperconductorfor∆s=0.2∆0and(b)d+idsuperconductor ity and the chirality of the superconductingOP. The overlap
for∆d =0.2∆0. ThestrengthofinterfacebarrierisZ =5.Differ-
between these two sets of states results in either a strongly
entcurvesoneachplotscorrespond todifferentvortexorientations
enhancedorsuppressedtunnelingconductance/LDOSatzero
withrespecttothez-axis.
biasvoltage/zeroenergy.
We have demonstrated the aforementioned effects both
IV. SUMMARY
qualitativelyandquantitativelyforp+ip-,d+is-,andd+id-
wavesymmetries. Experimentally,thedistancefromthesur-
In summary, we have investigatedhow the tunneling con- facetotheclosestvortexcanbealteredbymodifyingthefield
ductanceandthelocaldensityofstates(LDOS)insupercon- strength. Allofourpredictionsshouldbepossibletotestex-
ductorsare affectedbytheinfluenceof anexternalmagnetic perimentallywithpresent-daytechniques.
field when the superconductingorderparameter(OP) breaks V. ACKNOWLEDGMENTS
time-reversal symmetry (TRS). This is directly relevant for
bothSr RuO ,wherechiralp+ip-wavepairingisbelievedto
2 4
berealized, andforthehigh-T cuprates,wherea d+is-or M.S. is grateful to Alexander S. Mel’nikov for numer-
c
d + id-wave OP has been suggested to exist near surfaces. ous stimulating discussions. This work was supported, in
In addition to breaking TRS, all of these OPs feature sur- part,byRussianFoundationforBasicResearch,byProgram
face bound zero-energy states at surfaces under appropriate ”Quantum Macrophysics” of RAS, and by Russian Science
circumstances(e.g.adominantd-waveOPinthed+is-wave Support and ”Dynasty” Foundations. J.L. and A.S. were
case). supported by the Norwegian Research Council Grant Nos.
We have shown how the Doppler-shift conspires with an 158518/431, 158547/431, (NANOMAT), and 167498/V30
interaction of vortex and surface states to produce a consid- (STORFORSK).T.Y.acknowledgessupportbyJSPS.
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