Table Of ContentConductance enhancement due to the resonant tunneling into the subgap vortex core
states in normal metal/superconductor ballistic junctions.
M. A. Silaev1
1Institute for Physics of Microstructures, Russian Academy of Sciences, 603950 Nizhny Novgorod, GSP-105, Russia
(Dated: February 1, 2008)
We investigate the low-energy quantum transport in ballistic normal metal-insulator -
superconductor junction exposed to a magnetic field creating Abrikosov vortices in the supercon-
8 ductingregion. Within theBogolubov- de Gennes theory we show that thepresence of thesubgap
0 quasiparticle states localized within thevortex cores nearthejunction interface leads tothestrong
0 resonantenhancementofAndreevreflectionprobability,andthenormal-tosupercurrentconversion.
2
The corresponding increase of charge conductance is determined by the distance from the vortex
n chain to thejunction interface, which can becontrolled bythe applied magnetic field.
a
J PACSnumbers: 74.25.QP,74.25.Fy,73.40.Gk
1
1
I. INTRODUCTION in which several small tunnel junctions were attached to
] the mesoscopic superconductor to simultaneously mea-
n
sure the charge transport at the different points of the
o Recently, the investigation of transport properties of
sample. The measuredtransport characteristicswere re-
c normal metal- superconducting (N/S) hybrid structures
- lated to the local density of states (DOS), depending on
r has attracted a considerable interest. In the classical the local supercurrentdensity and hence on the configu-
p work by Blonder,Tinkham and Klapwijk1 it was shown,
u ration of the vortex system. Generally, when there is a
that at the energies below the superconducting gap ∆
s 0 uniform supercurrent flowing at the superconductor, the
t. thechargetransportcanonlyberealizedviatheAndreev excitationspectrumacquiresaDopplershiftbythevalue
ma rperfloeccetsisoninawthtihche tNh/eSelienctterrofnascew.itThhloiswisenaertgwieos-pεa<rti∆cle vspF,wherevsisasuperfluidvelocity,pF isaFermimo-
0 mentum. In this case, the minimal excitation energy is
incident from the normal metal (N) are reflected at the
d- N/S interface as holes traversing the backward trajec- givenbyEmin =∆0−vspF. Atlargedistancesfromthe
vortex core (r ξ, where ξ is a coherence length) the
n tories (and vice versa). In accordance with the charge ≫
o Doppler shift model givesquite a goodapproximationof
c conservation law, at the superconducting region (S) the the quasiparticle (QP) spectrum with v = ~/2mr (see
Cooperpairsareformedandthenormalcurrentconverts s
[ Ref.[4]). As long as the superfluid velocity is small com-
intothe supercurrent. Forthe perfectly transparentN/S
pared to the critical value ∆ /p , the Doppler shift of
3 interface the charge doubling due to Andreev reflection 0 F
v the gap edge results in the reduction of the height and
results in enhancement of the subgap conductance by a
8 broadeningof the superconducting DOS peak5. Close to
factor of two compared with the corresponding normal
4 the vortex core (r ξ) where v ∆ /p , the gap edge
5 state conductance. Being the two-particle process An- E goes to zero∼. However, isn∼this0 caFse the Doppler
1 dreev reflection is strongly suppressed due to QP scat- min
shift model does not hold: it completely misses one of
. tering at the layer of insulator separating the N and S
9 the remarkable features of the vortex state: the pres-
regions. Indeed, in case of the small interfactial bar-
0 ence of low-energy QP states localized within the vortex
7 rier transparency T 1, the Andreev reflection proba-
≪ core. Thesevortexcorestateswerefoundinthepioneer-
0 bility, and therefore the conductance is proportional to
ing work by Caroli- de Gennes and Matricon (CdGM)6
: T2 which is smaller by the factor T as compared to the
v within the more rigorous approach based on the quasi-
i single- electron case. classicalsolution of Bogoliubov-de Gennes (BdG) equa-
X
The phenomenon of subgap conductance suppression tions. It was shown that QP states with energy lower
r
a results in the remarkable dependence of transport prop- than the bulk superconducting gap value ∆0 are local-
erties of N/S structures on the spatial distribution and ized within the vortex core and have the discrete spec-
symmetry of the superconducting gap. For example, re- trumε as afunction of the quantized(half–integer)an-
µ
cently the chargecurrentmeasurementswereutilizedfor gular momentum µ. This spectrum of the CdGM states
thedirectobservationofmultivortexstructuresinmeso- varies from ∆ to ∆ as µ changes from to + ,
0 0
scopic superconductors2. Mixed state of mesoscopic su- crossing zero when−µ changes its sign. At sm−a∞ll energ∞ies
perconductors is formed by a small number of vortices ε ∆ the spectrum is given by ε µǫ , where
0 µ 0
andrevealsa richvarietyofexoticvortexconfigurations, |k| ≪= p /~ and ǫ = ∆ /(k ξ). For conv≈en−tional super-
F F 0 0 F
such as vortex molecules and multiquantum giant vor- conductors,theinterlevelspacingǫ ismuchlessthanthe
0
tices, realizing in such samples due to the quantum con- superconducting gap ∆ since (k ξ) 1, therefore the
0 F
finementoftheCooperpairsmotion3. InRef.2thephase CdGMspectrumcanbeconsideredco≫ntinuousasafunc-
transitions between different vortex configurations were tion of the impact parameter of quasiclassicaltrajectory
observed by the multiple-small-tunnel-junction method,
2
b = µ/k . The presence of the QP states bounded in port properties of wide N/S junction so that its trans-
F
−
the vortex core was confirmed in scanning tunnel spec- verse dimension L is much larger than the Fermi wave
y
troscopy (STS) experiments by the observation of the length L λ = 2π/k , therefore the trajectory di-
y F F
zero-bias conductance peak at the vortex core7. The vergence ca≫n be neglected. Also we assume that L is
y
analogous effect of the zero-bias conductance enhance- much larger than the distance from the vortices to the
ment due to the resonanttunneling into the midgapsur- N/S interface a and the characteristic intervortex dis-
face states was thoroughly studied in high temperature tance L . The important point is that the total QP re-
v
D-wave superconductors8. flectionprobabilitiescanbe foundasasumofindividual
vortex contributions. Indeed, the intervortex distance is
Thegoalofourworkistodevelopatheorytocalculate
muchlargerthantheFermiwavelengthsinceL ξ and
the zero-bias conductance of N/S junction in case when v ≥
k ξ 1. Therefore, the quasiclassical trajectories (ex-
the external magnetic field generates vortices in the S F ≫
cept those with θ = π/2) can pass through not more
region near the N/S interface. We consider the charge p ±
than one vortex core. For some directions of QP mo-
transport across the direction of applied magnetic field.
mentum two resonant trajectories (i.e. passing through
The N and S regions are assumed to be separated by
the vortex cores)are coupled by the normalreflectionat
the interfacial barrier, suppressing the Andreev reflec-
the interfacial barrier. Assuming the specularly reflect-
tion and the electron transport. We predict the strong
ingbarrierthiscouplingoccursforthemomentumangles
enhancement of the Andreev reflection due to the reso-
at the narrow angle domains near θ =arctan(nL /2a),
nanttunnelingofQPthroughthebarrierintotheCdGM p v
where n is integer. The width of the resonant angle do-
states localized within the vortex cores. The essential mains δθ Te−2a/ξ π is determined by the width of
physics of this effect is analogous to the one which takes ∼ ≪
the resonant vortex core levels. Therefore, the contribu-
place in double-barrier resonant tunneling diode9. The
tionofsuchtrajectoriestothe amplitude ofthe reflected
resonanttunnelingindouble-barrierquantumwellstruc-
QP waves is negligible. Then, evaluating the amplitude
tures occurs if the energy of incident QP wave coincides
of the QP wave we can separate the resonant trajecto-
with the resonant energy, then the reflection probability
riescoupledwithboundedQPstateslocalizedwithinthe
iseffectivelysuppressedduetotheinterferenceoftheQP
different vortex cores. Since different trajectories do not
waveswithinthequantumwell. Inourcasethequantum
interfere with each other, the resulting reflection prob-
well is represented by the vortex core and the bounded
abilities and the conductance can be found as a sum of
lowenergyQPstateconsistsofcoupledelectronandhole
contributions from individual vortices.
waves of almost the same amplitude. Therefore, if the
The dimensionless conductance (further we will mea-
incident electron has resonant energy, then the reflected sure it in terms of the conductance quantum e2/π~) in-
electron wave is suppressed and the hole wave leaking
duced by a single vortex at zero temperature can be es-
fromthevortexcorecontributestotheAndreevreflection timated as follows: G N e−2a/ξT. Here N k ξ
probability. The importantdifference betweenoursitua- v ∼ r r ∼ F
is the number of transverse modes of the N/S junction
tion and the double- barrier diode case is that the spec-
which effectively interact with the QP states bounded
trum of bound states is very dense with the characteris-
within the vortex core (ξ is the characteristic transverse
tic interlevel spacing ǫ0 ∼∆0/(kFξ) being much smaller size of the vortex core). The factor e−2a/ξT is the one-
than the bulk energy gap ∆ . At the same time, the
0 particle tunneling probability through the barrier con-
broadening of this levels due to the finite barrier trans-
sisting of the insulating layer at the N/S interface and
parency can be estimated as δE ∆ T e−2a/ξ, where T
∼ 0 the superconducting slab of the thickness a. The total
is the transparencyof the interfacialbarrierand the fac-
vortex-induced conductance is the sum of the individual
tor e−2a/ξ is due to the exponential decay of subgap QP
vortex contributions G = n G , where n = L /L is
atthesuperconductingslabofthethicknessa,whichisin vt v v v y v
the total number of vortices near the N/S interface, L
factthedistancefromthevortextotheN/Sinterface,ξis v
is the intervortex spacing and L is the transverse size
the superconducting coherence length. Hereafter in this y
of the junction. The resonantmechanism of Andreev re-
paperwewillneglectthediscretenessoftheboundenergy
flection exists along with the usual non-resonant scheme
levels assuming that T e−2a/ξ (k ξ)−1. In fact this
≫ F involving the two-particle tunneling through the interfa-
conditionisnotveryrestrictivesincek ξislargeinmany
F cial barrier with the probability T2. The corresponding
superconducting materials, for example k ξ 3 102 in
Nb and k ξ 104 in Al. Neglecting thFe d∼iscr·eteness zero-bias conductance can be estimated as G0 ∼ N0T2,
of the speFctru∼m of bounded QP states we can use the where N0 = kFLy/π is the total number of transverse
modesinN/Sjunction. Therefore,thetotalconductance
quasiclassical approximation of QP quantum mechanics
oftheN/Sjunctioninmagneticfieldcanbeevaluatedas
(see e.g. Ref.10). Within such approximation QP move
follows:
along linear trajectories, i.e. the straight lines along the
directionofQPmomentumn=kFkF−1 =(cosθp,sinθp). G=αN0T2+nvβNre−2a/ξT, (1)
Note, that for the N/S point contacts of atomic size in
magnetic field it is necessary to take into account the wherethecoefficientsα,β 1. Then,forthe strongbar-
∼
non-quasiclassical divergence of the electron and hole rierT 1,the conductanceinducedby the vortexchain
trajectories11. Inthepresentworkweconsiderthetrans- with sp≪acing L becomes dominant at a<a , where the
v c
3
threshold distance a (ξ/2)ln(ξ/TL ) can be much
c v
∼
largerthanthevortexcoresizeandthecoherencelength.
The parameters of the vortex lattice can be estimated
asfollows: a,L φ /B,whereB is the averagemag-
v 0
neticfieldofthes∼uperconductingsampleandφ =π~c/e
0
p
is the flux quantum. Then, the magnetic filed depen-
dence of the vortex-induced second term in Eq.(1) con-
ductance is given by:
G N B e−2√Hc2/BT, (2)
vt 0
∼ H
r c2
where H φ /ξ2 is the upper critical magnetic field
c2 0
∼
of the superconductor. Note, that at zero magnetic field
B = 0 the conductance is G N T2 and at B = H
0 c2
∼
theconductanceismuchlarger: G N T. Therefore,we
0
∼
suggestthatifthesurfacebarrierishighT 1,itshould
be possible to observe in experiment the≫field-induced FIG.1: GeometryofN/SjunctionandsketchofQPtrajecto-
increase of the conductance according to the Eq.(2). ries. ThewidthofthejunctionisLy.Externalmagneticfield
directed along z axis introduces vortices in superconductor.
In the present work we do not take into account the
Thedistance from thefirst vortexchain totheN/S interface
subgap QP states which exists near the surface of su-
is a, and the intervortex spacing is Lv. The electrons are in-
perconductor due to the Mejssner screening of external
jectedwiththeincidentangleθ0 andexperienceAndreevand
magnetic field12. Such QP levels lie higher at the energy
normal reflection at theN/S interface.
scalethantheCdGMstatesifthedensityofsupercurrent
nearthe surfaceislessthenthe criticalvalue. Therefore,
as long as the zero-bias conductance is considered the
similar to the model profile of the isolatedvortex core13:
influence of the surface QP states can be neglected.
D (r) = r/ r2+ξ2, where ξ is the coherence length.
The paper is organized as follows. In Sec. II a de- v
The phase distribution Φ(r) consists of a singular part
scriptionofthe model andthe basic equations aregiven. p
Φ (r) = arg(r) and a regular part Φ (r), determined
In Sec. III we solve the scattering problem to find the v r
by the particular metastable vortex lattice configuration
Andreev and normal reflection probabilities. Sec. IV is
realizing near the boundary.
devotedto the conductance calculationandSec.V to the
discussion of obtained results. Finally, conclusions are In principle, the method developed in the present pa-
given in Sec. VI. per is applicable to the arbitrary order parameter phase
distributioncorrespondingto the metastable vortexcon-
figuration. At first, we solve a generic problem of the
II. MODEL AND BASIC EQUATIONS influence of a single vortex near the N/S surface on the
zero-biasconductanceofthejunction. Thevortexstabil-
ity condition given by the London model, requires van-
ShownontheFig.(1)istheschemeoftheN/Sjunction
ishingregularpartofthesuperfluidvelocityatthevortex
withvortexlinesintheSregionparalleltotheN/Sinter-
position: ( Φ (2π/φ )A)(r=0)=0, where A is the
face. For the sake of simplicity we assume that there is ∇ r − 0
vectorpotential. Onthemicroscopiclevel,thiscondition
onlyonequantizedQPmodeinz directionandtakeinto
is necessary for the existence of the CdGM QP states
account only the QP motion in xy plane, perpendicular
forming the vortex phase singularity14,15. The next step
to the vortex lines.
is the summation of the individual vortex contributions
Considering one vortex from the array,the coordinate
to the conductance which add independently. The par-
system is chosen so that the z axis coincides with the
ticular vortex configuration near the boundary depends
vortex line and the origin at the xy plane coincides with
on many factors, such as the random pinning potential,
vortex phase singularity point. Neglecting the suppres-
geometry of the superconducting sample, magnetization
sion of the superconductivity near the N/S interface due
history, etc. To estimate the dependence of the conduc-
to the proximity effect we assume that at x > a (su-
− tance on the magnetic field we consider the model sit-
perconducting region) the order parameter can be taken
uation assuming that the vortices near the boundary of
as follows:
superconductorsitperiodicallyonchainwithanintervor-
∆(r)=∆0Dv(r)eiΦ(r), (3) tex spacing Lv at a distance a from the N/S interface.
We take a and L as external parameters of the order
v
Here ∆ is the gapvalue far fromthe vortexcore,D (r) a,L φ /B,whereB istheaveragemagneticfieldof
0 v v 0
∼
and Φ(r) are the dimensionless profile and the phase of the superconducting sample. The influence of the next
p
theorderparameter. TheparticularformofD (r)isnot vortex chains on the conductance can be neglected due
v
essentialforourconsideration,thereforeitcanbechosen to the rapid decay of the QP tunneling probability with
4
the growing distance from vortices to the N/S interface. where ǫ = E/∆ , ξ = ~V /∆ is the coherence length,
0 F 0
Theexpressionforthedimensionlesszero-biasconduc- D (r) and Φ(r) are the dimensionless magnitude and
v
tance of the N/S junction can be written as follows(see phase of the order parameter and F(r)=(πξ/φ )n A.
0
·
Ref.1): Consideringthezero-biasproblemwewillhavetoana-
lyze only the zero-energy excitations with ǫ=0. Within
N π/2 the normal metal region at x < a, neglecting the in-
G= 0 (1 R (θ )+R (θ ))cosθ dθ , (4) −
n 0 a 0 0 0 fluence of magnetic field on the QP motion, the quasi-
2 Z−π/2 − classical equation (8) becomes trivial: ∂ fˆ(s,b) = 0. It
s
means that the envelope function is constant along the
where R (θ ) and R (θ ) are the probabilities of nor-
n 0 a 0
mal and Andreev reflection respectively, θ is the inci- trajectories.
0
dent angle: kF = kF(cosθ0,sinθ0). The total number Obviously, this is not the case at the superconducting
of propagating modes N is determined by the channel region x > a, where the electron and hole waves are
−
coupled. Note that for the wave functions at the S re-
width: N = k L /π. The problem of QP scattering at
0 F y
the N/S interface is formulated within the BdG theory. gioncorrespondingtothezeroenergythefollowingrepre-
The equationfor the electronand hole wavescoupled by sentation can be used10: fˆ= eζ ei(η+Φ)/2,e−i(η+Φ)/2 ,
the superconducting gap ∆(r) reads as follows: where ζ = ζ(s,b) and η = η(s,b) are real-valued func-
(cid:0) (cid:1)
tions. Then, the quasiclassical equation (8) can be writ-
ten as follows:
Hˆ (r) ∆(r)
0 Ψˆ(r)=EΨˆ(r), (5)
∆∗(r) Hˆ0(r)! ξ∂ η+2D cosη+ǫ =0, (9)
− s v d
Here
1 e 2 ξ∂ ζ+2D sinη =0. (10)
Hˆ (r)= pˆ A E +V(x) s v
0 F
2m − c −
(cid:16) (cid:17) where ǫd(r) = ~kFvs/∆0 is the dimensionless Doppler
with pˆ = i~ , Ψˆ(r)=(u(r),v(r)). The interfacial bar- shift energy. For the wave functions fˆ± decaying at the
rier separ−atin∇g the N and S regions is modeled by the different ends of the trajectory fˆ±(s = ) = 0 from
±∞
repulsive delta potential V(x) = Hδ(x), parameterized Eq.(10) we obtain:
by the dimensionless barrierstrengthZ =H/~V 1. The
F
boundary conditions at the N/S interface are: η±(s=±∞)=±π/2. (11)
As we will see below, the main contribution to the
Ψˆ( a) =0, (6) enhanced Andreev reflection probability comes from the
−
trajectories with small impact parameters b ξ, pass-
h i
| |≪
ing through the vortex core. For such trajectories ne-
∂ Ψˆ( a) =(2k Z)Ψˆ( a), (7) glecting the nonsingular part of the superfluid velocity
x − F − nearthe vortexcorethe analyticalsolutionofEq.(9)can
h i be obtained following the results of Ref.16:
where [f(x)]=f(x+0) f(x 0).
− −
To overcome the complexity of the scattering prob-
lem coming from the broken spatial invariance of the tanη± =0.5 A−±1e−2K(s) A±e2K(s) , (12)
−
superconducting gap, we treat the Eq.(5) within qua-
(cid:16) (cid:17)
siclassical approximation. Generally, the quasiclassical A± =γ(b)(sgn(s) 1), where γ(b)= ωb,
∓ −
form of the wave function can be constructed as follows:
Ψˆ(r) = eikF·rfˆ(r), where fˆ(r) = (U(r),V(r)) is a slow
s
varying envelope function. Then the system (5) reduces K(s)= 1 D (s′)ds′ = (s/ξ)2+1 1,
to the system of the first-order quasiclassical equations ξ v −
Z
along the linear trajectories defined by the direction of 0 p
the QP momentum n = kFkF−1 = (cosθp,sinθp). Each
∞
trajectory is specified by the angle θ and the impact 1 D (s)
p ω = v e−2K(s)ds.
parameter b = rsin(θ θp), where θ is the polar angle: ξ s
r= r(cosθ,sinθ).Int−roducingthecoordinatealongtra- Z0
−
jectorys=(n r)= rcos(θ θ)wearriveatthefollow-
p
· − −
ing form of the quasiclassical equation: Hˆfˆ= ǫfˆ, with III. SCATTERING PROBLEM: NORMAL AND
the hamiltonian: ANDREEV REFLECTION PROBABILITIES
The boundary conditions (6,7) determine the specu-
Hˆ = iξσˆ ∂ +F +D (σˆ cosΦ σˆ sinΦ), (8) larly reflecting N/S interface, coupling the waves with
z s v x y
− −
5
wave vectors kF = kF(cosθ0,sinθ0), and k′F = where χ(θ,θ0)=η− η+ and Z˜ =Z/cosθ0.
−
kF(cos(π θ0),sin(π θ0)). Therefore if the incident Note, that for the small impact parameters b± ξ
electron w−ave is u =−eikFr, then reflected electron u the factors eiη± can be obtained analytically |wit|h≪the
i r
and hole v waves will have the form help of Eq.(12) as follows:
r
ur =Ureik′Fr, vr =VreikFr, eiη+ =iJ −i(θ−θ0), (17)
J +i(θ θ )
0
where and U (b,s) and V (s,b) are the envelope func- −
r r
tions. Eachpoint( a,y)atthe N/Sinterfaceliesonthe
−
intersection of two quasiclassical trajectories, character- J i(θ+θ )
izedbytheanglesθ =θ andθ =π θ .Fromthesim- eiη− = i − 0 , (18)
p 0 p 0 − J +i(θ+θ )
− 0
pletrigonometryitiseasytoseethatthe impactparam-
eters of these trajectories are b+ = asin(θ0 θ)/cosθ where J = e−2K(a/cosθ)cosθ/(2ωa). For the small an-
− −
θan=d b−ar=cta−n(ays/ian)(θi0s +theθ)p/ocloasrθancgolrer.esTphoendcionogrldyi,nwatheeroef gsilmesu|lθta|,n|θe0o|u≪slyξy/iaeldthinege:quations (17) and (18) are valid
−
the intersection point is s = acos(θ θ )/cosθ and
+ 0
− −
s− =acos(θ+θ0)/cosθforthetrajectoriescharacterized eiχ = J2+θ2−θ02−2iθ0J. (19)
bTyhetnh,etahnegrleeflseθcpti=onθp0roabnadbθilpit=iesπa−reθg0ivceonrrbeys:pondingly. −J2+θ2−θ02+2iθ0J
a α
R (θ )= U (θ,θ )2(cosθ)−2dθ (13) IV. VORTEX-INDUCED ZERO-BIAS
n 0 r 0
Ly Z−α| | CONDUCTANCE
a α Now,usingthe expressions(15, 16)forthe amplitudes
R (θ )= V (θ,θ )2(cosθ)−2dθ, (14) of reflected waves and reflection probabilities (13, 14) it
a 0 r 0
Ly Z−α| | ispossibletofindthezero-biasconductance. Introducing
the function g(θ,θ ) = 1 U (θ ,θ)2 + V (θ ,θ)2 the
wheretheintegrationisdoneovertheN/Sinterface,α= 0 −| r 0 | | r 0 |
expression for the dimensionless conductance (4) reads
arctan(Ly/2a), Ur(θ,θ0) = Ur(b−,s−) and Vr(θ,θ0) = as follows:
V (b ,s ).
r + +
Following the usual procedure, to find the reflected
mwaavtechatmheplNituadnedsSUrre(gθio,nθ0s)soaluntdioVnrs(aθt,θth0)eNon/eSinneteedrfsacteo. G= kFa α(cosθ)−2dθ π/2 g(θ,θ0)cosθ0dθ0, (20)
Fortheenvelopefunctionstheboundaryconditions(6,7) 2π Z−α Z−π/2
yield:
where α = arctan(L /2a). It is convenient also to intro-
y
duce here a local conductivity, i.e. the conductance per
1+U =eiη+/2C++eiη−/2C−, unit length of the N/S surface:
r
k π/2
F
σ(θ)= g(θ,θ )cosθ dθ .
Vr =e−iη+/2C++e−iη−/2C−, 2π Z−π/2 0 0 0
Employing Eqs.(15,16) we obtain:
(1 U )+2iZ(1+U )=eiη+/2C+ eiη−/2C−,
r r
− −
2
g(θ,θ )= . (21)
0 (Z˜4+Z˜2)1 eiχ 2+1
V (1+2iZ)=e−iη+/2C+ e−iη−/2C−, | − |
r
−
If the applied magnetic field is zero and the super-
C+,C− are arbitrary constants and η± = η±(s±,b±), conductor is homogeneous χ = π, we obtain g(θ,θ ) =
0
where η±(s,b) arethe solutions ofEq.(9) with boundary g (θ ), where g (θ ) = (1/2)(Z˜2 + 1/2)−2. Then, the
conditions (11) along the trajectories with θ = θ and 0 0 0 0
p 0 vortex-induced part of the conductivity is given by
θ =π θ fortheupperandlowersignscorrespondingly.
p 0
−
Solving this system we obtain: k π/2
F
σ (θ)= g (θ,θ )cosθ dθ ,
v v 0 0 0
(1 eiχ)(Z˜2 iZ˜) 2π Z−π/2
U (θ,θ )= − − , (15)
r 0 − 1+Z˜2(1 eiχ) where g =g g :
− v − 0
(Z˜4+Z˜2) 4 1 eiχ 2
e−iη+ g = −| − | .
Vr(θ,θ0)= 1+Z˜2(1 eiχ), (16) v 2(Z˜2+1/2)2 (Z˜(cid:0)4+Z˜2)1 eiχ(cid:1)2+1
| − |
−
(cid:16) (cid:17)
6
To start the analysis of Eq.(21) let’s note that for the Within such model the condition of vanishing current
low surface barrier Z˜ → 0 we get gv(θ,θ0) = 0. In this throughtheN/Sinterface(∂xΦ−(2π/φ0)Ax)=0issat-
case all the incident QP undergo Andreev reflection and isfied automatically, and the vortex stability is achieved
the zero-bias conductance is the same as in case of ho- by setting a = φ0/B. The numerical plot of the func-
mogeneous superconductor: G = 2N0. As the barrier tion σv(θ)/σv0 pat different distances a from the vortex
becomeshigher,theAndreevreflectionissuppressedand to the interface is presented on the inset at Fig.(2). The
the conductance is reduced. maximum value σv(θ =0) with good accuracy coincides
The function g (θ,θ ) reaches its maximum with the analytical estimation given by (23). The coef-
v 0
ficient β in Eq.(24) is found to be nearly constant as a
Z˜4+Z˜2 functionofa: itdecreasesslightlyfromβ 0.6ata=2ξ
g =2 ≃
vm (Z˜2+1/2)2 toβ 0.4ata=5ξ.OntheFig.(2)forthevariousmag-
≃
nitude of barrier strength we plot the ratio σ¯ /σ of the
v 0
if 1 eiχ =0. In fact this condition determines the res- average vortex-induced conductivity σ¯v = Gv/ξ to the
| − |
onanttrajectories,correspondingto the zero-energyvor- conductivity of the N/S junction in the absence of vor-
texcorestatesmodifiedbythenormalreflectionfromthe tices given by:
interfacialbarrier. The resonanttrajectoriesshouldpass
through the vortex core therefore the function gv(θ,θ0) σ = kF π/2 g (θ,θ )cosθ dθ .
has a sharp peak at θ0 ≈ ±θ. The width of this peak 0 2π Z−π/2 0 0 0 0
is determined by the barrier strength and the distance
from the vortex to the surface. For the small angles
θ , θ ξ/a with the help of Eq.(19) we obtain: V. DISCUSSION
0
| | | |≪
g J2θ2 Considering the strong barriers Z 1 Eq.(24) can be
gv(θ0,θ)= (Z˜2+1/2)2(θ02vm−θ02−0 J02)2+J02θ02, (22) twhreitbteanrraiesrGtrvan=spβar(kenFcξy)eT−2=K((a1)T+,Zw2≫h)e−r1e weZ−in2t.roAdusicmed-
≈
where J = e−2K(a)/(2ωa) (ξ/a)e−2a/ξ, which is a ple understanding of this result can be obtained within
0 ∼ the framework of the tunneling hamiltonian approach17.
smallparametersinceJ 1fora ξ.Themaximumof
g (θ,θ ) determined by E≪q.(22) lie≥s at θ2 =θ2+J2. Theconventionalexpressionforthetunnelingconductiv-
v 0 0 0 ity of the N/S junction at zero temperature reads:
Employing Eq.(22) it is easy to compute the vortex-
inducedpartoftheconductivityσ (θ)atthe smallangle
v σ =σ ν/ν , (25)
n 0
domain θ ξ/a. The main contribution to integral
| | ≪
overθ comesfromthesmallvicinityofthepointθ =θ.
0 0 where ν is a local superconducting DOS at the Fermi
Then with good accuracy we obtain: σ =σ , where
v v0 level and σ is a normal state tunneling conductivity.
n
FortheS-wavesuperconductorthetransformationofvor-
Z4+Z2
σ =k J . (23) tex core states near the surface can be neglected at first
v0 F 0(Z2+1/2)3 approximation18. Then the local DOS near the surface
at the Fermi level is determined by the density of vortex
At larger angles the function σ (θ) can be evaluated
v core states ν (r) given by19,20:
only numerically. Numerical calculation described be- v
low shows that σ (θ) is maximal at θ = 0 and steadily
v 1 2π
decreases to zero as θ π/2. (see inset on Fig.2). ν (r)= fˆ(r,θ )2δ(ǫ k rsin(θ θ ))dθ , (26)
| | → v 2π | p | 0 F − p p
Then,the resultingconductanceinducedbyasinglevor- Z0
tex G =a α (cosθ)−2σ (θ)dθ is given by:
v −α v where fˆ(r,θ ) is the envelope of the QP wave function.
p
GRv =β(kFξ)e−2K(a)(ZZ24++1Z/22)3, (24) Fthoernthνev(Crd)GMνw0(aξv/erf)uen−c2tKio(rn)s,|wfˆ(hre,rθep)ν|02 ∼=em−2/K~(2r)iksFt/hξe,
∼
two-dimensional DOS at the normal metal. Substitut-
where β =(2ωξ)−1 α (cosθ)−2σ (θ)/σ dθ 1. ing ν = ν (r = a/cosθ) and σ Tk we obtain the
−α v v0 ∼ v n ∼ F
To evaluate the conductance rigorously, we find the conductivity: σ k TJ(θ), which coincides to the or-
F
factor eiχ in Eq.(21R) and then the reflection probabili- der of magnitud∼e with expression (23) if Z 1 and
≫
tiessolvingnumericallyEq.(9)withboundaryconditions θ 1. Integrating σ(θ) over the N/S interface we ar-
| | ≪
(11). We assume that the regular part of the phase dis- riveat expression(24) for the conductance with factor β
tribution is Φr(r)=−arg(r−rav) corresponding to the givenby: β ∼ −ααexp(2K(a)−2K(a/cosθ))(cosθ)−1dθ.
image vortex situated at the point rav = ( 2a,0,0) be- Note, that although yielding the qualitatively right an-
− R
hind the N/S interface. The vector potential is chosen swer for the vortrex-induced conductance, the tunneling
as A = B[z0 (r r0)]/2, where r0 = ( a,0,0) is the hamiltonian approach drops out the contribution from
× − −
point at the boundary between the vortex and the im- the non-resonantAndreevreflectionwiththe probability
age vortex, and z0 is the unit vector along the z axis. of the order T2. As we will see below, the contributions
7
oftheresonantandnon-resonantAndreevreflectionscan motionofQPiscertainlyaffectedbyimpurityscattering.
be comparable even when the surface barrier is rather The influence of impurities can be neglected completely
strong. Certainly, the tunneling hamiltonian approach assuming that the life time of the vortex core states due
fails to provide the answer if the barrier strength is not to the finite barrier transparency is much shorter than
very high, when the influence of vortices on the conduc- the relaxationtime ofthe QPmomentum: ~/δE τ,or
≪
tanceisreducedandthenon-resonantAndreevreflection
prevails. On the Fig.3a we show in logarithmic scale the le (Te−2a/ξ)−1ξ, (28)
≫
vortex induced conductivity σ¯ as a function of the bar-
v wherel =V τ isanelasticmeanfreepathofQPatthe
rier strength Z for the several values of the distance a. e F
S region. This condition certainly can be fulfilled if the
At small values of Z the function σ¯ (Z) grows σ¯ Z2
v v ∼ barrier transparencyis not very high, i.e. T 1 and the
in accordance with the estimation (23). At larger values ∼
vortex chain is situated not far from the N/S interface,
ochfatnhgeesbatorrσi¯er strZe−ng2.thBuZt≫at t1hethseambeehtaimvieouthreocfoσ¯nvd(uZc)- so that the factor Te−2a/ξ is not very small. Otherwise,
tivity withouvt∼vortices at Z 1 behaves as σ Z−4. ifthe condition(28)is notfulfilled, the impurityscatter-
≫ 0 ∼ ing will modify the conductance. The simplest approach
Therefore, the ratio σ¯ /σ is monotonically growing as
v 0 to estimate the conductance in this situation is based
a function of the barrier strength Z proportional to Z2
on the tunneling hamiltonian, yielding the Eq.(25) for
(see Fig.3b).
the conductivity. Due to the impurity scattering the lo-
Let’s have a look at the expression for the total con-
cal superconducting DOS differs from that given by the
ductance of the N/S junction, which has quite a simple
Eq.(26), which is valid for the clean case l ξ. In par-
form if Z 1. Neglecting edge effects end summing up e ≫
≫ ticular, the sharp peak at r = 0 is smeared, therefore
the individual vortex contributions we obtain:
at the center of vortex the DOS is smaller as compared
G=(8/15)N T2+n β(k ξ)e−2K(a)T, (27) to the clean case. But at the distances r > ξ from the
0 v F
vortex core (e.g. at the N/S interface) the DOS is not
where nv = Ly/Lv is the total number of vortices near suppressed by the impurities even if le ∼ ξ. On the
the N/S interface. The obtained expression for the total contrary, it is even larger than in the clean case due to
conductance (27) consists of two terms. The first term the smearing of the DOS peak at the center of vortex21.
G N T2 coincides with the conductance of the N/S Therefore,in caseofthe ratherhighimpurity concentra-
0 0
∼
junction at zero magnetic field. The factor T2 is deter- tiononthe S side the dependence ofthe vortex-induced
mined by the probability of the sequential tunneling of conductance on the magnetic field is still described by
the incident and reflected QP through the high interfa- the Eq.(2). Another important point is the influence of
cialbarrier. The secondterm is the total vortex-induced impurities on the non-resonant part of the conductance,
conductance G = n G n (k ξ)e−2K(a)T; it comes i.e. the first term in Eq.(1). In particular, the interfer-
vt v v v F
∼
from the tunneling of the incident QP into the zero en- enceofQPwavesreflectedfromtheinterfacebarrierand
ergyCdGMstatesinsidethevortexcore. Indeeditiseasy impurities on the N side of the N/S junction can also
to see that e−2K(a)T, where K(a)= (a/ξ)2+1 1, is result in the low-bias conductance enhancement, known
−
the one- particle tunneling probability through the in- as reflectionless tunneling (see Ref. 22 and references
p
terfacial barrier and the superconducting layer of the therein). In experiments where the reflectionless tun-
thickness a with slightly suppressedgap due to the pres- neling effect was observed24 the condition le > ξ was
ence of the vortex. The factor k ξ is the number of fulfilled. In this case the critical value of magnetic field
F
resonant transverse modes for a single vortex. The suppressingthereflectionlesstunneling23 Hc ∼φ0/(12le2)
vortex-induced conductance G prevails over G when was much less than the upper critical field of the super-
vt 0
a < a , where the critical distance a is determined by conductor. In our case the applied magnetic field must
c c
a (ξ/2)ln(L /Tξ).The parametersofthe vortexcon- be strong enough to create the dense vortex lattice in
c v
≈
figuration, such as the intervortex spacing Lv and the superconductor: B ∼ Hc2 ≫ Hc. Therefore, under the
distance a from the vortex array to the boundary of sameexperimentalconditionsas inRef.24 the reflection-
superconductor are determined by the magnetic field, less tunneling effect is absent in the range of magnetic
therefore the conductance of the N/S junction can be fields that we are interested in.
controlled by the magnetic field. Using Eq.(2) we ob-
tain that the critical magnetic field H when G G
c vt 0
is determined by the following transcendental equ∼ation VI. CONCLUSION
ln(x/T) = 2/x, where x = B/H . Taking for ex-
c2
ample the barrier strength Z = 5, we obtain that the To summarize, we have investigated the low-energy
p
critical field is B 0.5H , and the critical distance charge transport in N/S junction across the direction of
c c2
∼
a 1.5ξ. Therefore, the influence of the resonant vor- applied magnetic field. We have found the strong en-
c
∼
texcorestatesontheconductancecanbecomesignificant hancement of the zero-bias conductance due to the res-
whenthemagneticfieldislessthentheuppercriticaland onant tunneling of the incident QP into the subgap vor-
vortices are quite far from the N/S interface. tex core states. The effect is most sound if the conven-
Finally, we should note that in real N/S junctions the tionalchannel of Andreev reflectionis suppressedby the
8
high interfactial barrier. Note that usually, the vortex
core states are investigated in STS experiments, where
the charge transport is measured along the direction of
magnetic field. For the alternative to the STS methods
now we can suggest the transport measurements in pla-
nar structure with wide-area N/S contacts. The vortex
induced conductance that we have studied depends ex-
ponentially on the distance from the vortex chain to the
N/S interface and therefore can be effectively controlled
by the external magnetic field. Also for the possible ex-
perimentalsetup one canconsiderthe mesoscopicsuper-
conductingsample withlateraltunneling junctions, such
as in Ref.2. Since the vortex -induced conductance is
proportional to the number of vortices, the conductance
vs magnetic field dependence should reveal pronounced
stepsmarkingtheswitchofthetotalvorticityofthesam-
FIG.2: Plotoftheratioσ¯v/σ0 oftheaveragevortexinduced ple.
conductivity to the conductivity in the absence of vortices.
VII. ACKNOWLEDGEMENTS
CurvesfromtoptobottomcorrespondtoZ =5,4,3,2. Inset:
function σv(θ)/σv0 for a/ξ =2 (open circles), a/ξ =3 (filled
circles) a/ξ=5 (asteriskes); Z =2.
It’s my pleasure to thank Dr. Alexander S. Mel’nikov
for numerous stimulating discussions and help in prepa-
ration of this paper. Also I am grateful to Ekaterina
FIG. 3: (a): Plot of the average vortex-induced conductiv- Ezhova for help with numeric calculations. This work
ity σ¯v as a function of the barrier strength Z in logarithmic was supported, in part, by Russian Foundation for Ba-
scale. (b): Plotoftheratioσ¯v/σ0 asafunctionofthebarrier sic Research, by Program “Quantum Macrophysics” of
strength Z in logarithmic scale. Curves from top to bottom RAS, and by Russian Science Support and “Dynasty”
correspond to a/ξ=2, 2.5, 3, 3.5.
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