Theory of pairing symmetry inside the Abrikosov vortex core

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by  Takehito Yokoyama

Theory of pairing symmetry inside the Abrikosov vortex core
Takehito Yokoyama1, Yukio Tanaka1 and Alexander A. Golubov2
1 Department of Applied Physics, Nagoya University, Nagoya, 464-8603, Japan and CREST
2 Faculty of Science and Technology, University of Twente, 7500 AE, Enschede, The Netherlands
(Dated: February 2, 2008)
We show that the Cooper pair wave function at the centerof an Abrikosov vortex with vorticity
mhasdifferentparitywithrespecttofrequencyfromthatinthebulkifmisanoddnumberandhas
thesameparityifmisanevennumber. Asaresult,inaconventionalvortexwith m=1,thelocal
8
density of states at the Fermi energy has a maximum (minimum) at the center of the vortex core
0
in even(odd)-frequency superconductor. We propose a scanning tunneling microscope experiment
0
2 using a superconductingtip to explore odd-frequency superconductivity.

n
a
The study of the mixed state in type-II superconduc- even-frequency pairing state may be generated around
J
tors has a long history and revealed a variety of physi- the vortex core in odd-frequency superconductor.
4
cal phenomena.[1] In the clean limit, low-energy bound InthisLetter,basedonthequasiclassicaltheoryofsu-
2
states (the Andreev bound states) are generated in the perconductivity, we develop a general theory of pairing

] vortex core due to the spatial structure of the supercon- symmetryinanAbrikosovvortexcoreincleansupercon-
n
ducting pair potential. [2, 3] One of the manifestations ductors, including odd-frequency superconductivity. We
o
of the bound states is the enhancement of zero-energy show that for a vortex with vorticity m in superconduc-
c
- quasiparticle density of states (DOS) locally in the core, tor, pairing function of the Cooper pair at the vortex
r
p observable as a zero-bias conductance peak by scanning centerhastheopposite(same)symmetrywithrespectto
u tunneling microscope (STM).[3, 4] However, despite ex- frequency to (as) that of the bulk if m is an odd (even)
s tensive studies of the vortex core, the issue of pairing integer. For a conventional vortex with m = 1, we show
.
t symmetry in the core remains unexplored. that zero energy local DOS is enhanced (suppressed) at
a
m Generally, superconducting pairing is classified into the center of the vortex core in even (odd)-frequency su-
even-frequency or odd-frequency state according to a perconductor. We further reveal that OSO p-wave pair-
-
d symmetry with respect to time. Due to the Fermi ing is generated at the center of the core of ESE s-wave
n statistics, even-frequency superconductors belong to the superconductor. On the other hand, in OSO p-wave su-
o
symmetry class of spin-singlet even-parity (ESE) or perconductor, ESE s-wave pairing state emerges at the
c
[ spin-triplet odd-parity (ETO) pairing state, while odd- center of the vortex core. Based on these results, we

frequency superconductors belong to the spin-singlet propose an experimental setup to explore odd-frequency
2
odd-parity(OSO)orspin-tripleteven-parity(OTE)pair- superconductivityby probinga localJosephsoncoupling
v
7 ing state. Although vortex core state of even-frequency by STM with superconducting tip.
6 superconductors has been well studied, that in odd- The electronic structure of the vortex core in a single
9 frequency superconductors has not been clarified yet. Abrikosov vortex in a clean superconductor is described
2 The possibility of the odd-frequency pairing state in bythequasiclassicalEilenbergerequations[10,11]based
.
0 various kinds of uniform systems was discussed in Refs. ontheRiccatiparametrizationofthequasiclassicalprop-
1 [5, 6], albeit its realization in bulk materials is still agator [12]. Along a trajectory r(x′) = r0+x′ vˆF with
07 controversial. On the other hand, the realization of unit vector vˆF parallel to vF, the Eilenberger equations
: the odd-frequency pairing state in inhomogeneous even- are generally represented in 4×4 matrix form.[13] For a
v frequency superconducting systems has recently been singlet (triplet) superconductor with ∆ˆ = ∆σy(σx) (σx
i
X proposed. It is established that odd-frequency pairing andσy arePauli’smatricesinspinspace)[14],theseequa-
r is induced due to symmetry breaking in such systems. tions are reduced to the set of two decoupled differential
a Inferromagnet/superconductorjunctions,odd-frequency equations of the Riccati type for the functions a(x′) and
pairing emerges due to the broken symmetry in a spin b(x′),
space.[7] It was recently realized that proximity-induced
odd-frequency pairing may be generated near normal ¯hvF∂x′a(x′)+ 2ǫn+∆†a(x′) a(x′)−∆ = 0,
metal/superconductorinterfacesduetothebreakdownof ¯hvF∂x′b(x′)−(cid:2)[2ǫn+∆b(x′)](cid:3)b(x′)+∆† = 0 (1)
translationalsymmetry[8]orinadiffusive normalmetal
where iǫ are the Matsubara frequencies, and ∆† =
n
attached to a spin-triplet superconductor.[9]
(−)∆∗ for even (odd)-frequency superconductor. For
Since an Abrikosov vortex breaks translational sym-
simple case of a cylindrical Fermi surface, the Fermi ve-
metry in a superconductor, one may expect the emer- locity can be written as vF =vF(e1cosθ+e2sinθ).
gence of an odd-frequency pairing state around the vor-
We choose the following form of the pair potential:
tex core even in a conventional spin-singlet s-wave su-
perconductor. Onthe other hand,one may imaginethat ∆(r,θ,E)=∆0χ(θ,E)F(r)exp(imϕ) (2)
2
with r = x2+y2 and exp(iϕ) = (x+iy)/ x2+y2.
TABLE I: Pairing symmetry in the vortex state.
Here,F(r)pdenotesthespatialprofileofthegapp,misthe
bulkstate vorticity symmetry at
vorticity,andχ(θ,E)isthesymmetryfunction. Also,we
m thecenter
introduce the coherence length ξ = h¯vF/∆0, the center (1) ESE odd OSO
ofa vortexis situatedat x=y =0, andexp(imϕ) is the
(2) ESE even ESE
phase factor which originates from the vortex. (3) ETO odd OTE
WeobtainpairingfunctionoftheCooperpair(anoma- (4) ETO even ETO
lous Green’s function) f as f = −2a/(1+ab). For the (5) OSO odd ESE
calculation of the local DOS normalized by its value in (6) OSO even OSO
the normalstate, the quasiclassicalpropagatorhas to be (7) OTE odd ETO
(8) OTE even OTE
integratedoverthe angleθ whichdefines the directionof
the Fermi velocity. The normalized local DOS in terms
of functions a and b is given by
(cid:3)(cid:5)(cid:4)
2π dθ 1−ab
N(r0,E)= Re , (3)
Z0 2π (cid:20)1+ab(cid:21)iǫn→E+iδ
where E denotes the quasiparticle energy with respect
to the Fermi levelandδ is aneffective scattering param-
eter that corresponds to an inverse mean free path. In
numericalcalculations throughoutthis paper, we will fix
this value as δ =0.1∆0.
First, we discuss the general property of the symme- (cid:31) (cid:31)
try at the vortex center. Vorticity and the symmetry
(cid:9)
ofsuperconductorwith respectto frequency cruciallyaf-
fect the symmetry of the Cooper pair at the core center.
(cid:10) (cid:11)(cid:12)(cid:13)(cid:14)
Consider a trajectory passing through the center of the
(cid:11)(cid:12)(cid:13)(cid:15) (cid:3)(cid:6)(cid:4)
vortex. By setting x′ = 0 at the vortex center, we get (cid:20)(cid:21)(cid:13)(cid:15)(cid:16)(cid:16)(cid:19)(cid:17)(cid:18)(cid:19)(cid:17)
b(x′,ǫ ) = −1/a(−x′,−ǫ ) from the Eilenberger equa- (cid:8)(cid:9)
n n
tionsforeven-frequencysuperconductorwithoddinteger
morodd-frequencysuperconductorwithevenintegerm. (cid:24) ! ! (cid:11)(cid:12)(cid:13)(cid:28)
fSriemqiulaenrlcyy, wsuepoerbctoanindubc(txo′r,ǫwn)ith=o1d/da(i−ntxe′g,e−rǫnm) oforreovdend-- (cid:27) (cid:11)(cid:20)(cid:21)(cid:12)(cid:13)(cid:13)(cid:28)(cid:29)(cid:30)(cid:16)(cid:19)
frequency superconductor with even integer m. Thus, (cid:20)(cid:21)(cid:13)(cid:29)(cid:17)
(cid:23)
at the vortex center x′ = 0, we get f(ǫn) = −f(−ǫn) (cid:3)(cid:7)(cid:4)
in the former case, while f(ǫ ) = f(−ǫ ) in the latter.
n n (cid:9)
Note that spin is conserved in the vortex state consid-
ered. Therefore,quitegenerally,foranoddintegermthe (cid:10) (cid:10)(cid:22)(cid:23) (cid:10)(cid:22)(cid:24) (cid:10)(cid:22)(cid:25) (cid:10)(cid:22)(cid:26) (cid:9)
(cid:0)
induced pairing at the vortex center has different sym- (cid:1) (cid:2)
metry with respect to frequency from that in the bulk
superconductor. On the other hand, for an even integer
m,theinducedpairingatthe vortexcenterhasthe same
FIG. 1: (color online) Results for ESE s-wave superconduc-
symmetry as that of the bulk. We summarize pairing
tor. (a) Normalized local DOS around the vortex at E = 0.
symmetry at the vortex center in Table I.
The center of the vortex is situated at x = y = 0. Spatial
For conventional s-wave case, there have been several dependenciesof(b) ESEand (c) OSOcomponentsat E =0.
studies of multi-vortex state with m≥1 [15, 16]. It was Only OSO components, Refpx and Imfpy, can survive inside
shownthatzeroenergypeakintheDOSonlyappearsfor thecore (near x=0).
odd number m at the vortex center [16]. This statement
is consistent with our result for the conventional s-wave
caseofχ(θ,E)=1becauseodd-frequencypairingstateis In general, the most realizable vorticity is m = 1.
generated only for odd integer m. The relation between Thus, in the following, we will study in detail two typ-
zeroenergypeakinDOSandodd-frequencypairingstate ical cases at m = 1 with ESE s-wave and OSO px-
will be discussed later. wave superconductors where we choose χ(θ,E) = 1 and
2
χ(θ,E)=(CcosθE/∆0)/[1+(E/∆0) ] with C =0.8,[9]
respectively. Also, spatial dependence of the gap is cho-
sen as F(r)=tanh(r/ξ).
3
(cid:3) (cid:4)
(cid:5) the core center, the magnitudes of f decrease rapidly,
except for s-wave one. Note that other angular momen-
tum components not shown in this figure are negligibly
small. We also find that the following representation of
anomalous Green’s function f holds at the center of the
core:
f =(Ref +iImf )(cosθ+isinθ)=f exp(iθ). (5)
px px px
(cid:11)(cid:12)(cid:15)
Thus,weseethatanomalousGreen’sfunctionatthecore
(cid:11)
center has chiral p-wave symmetry.
(cid:10)(cid:11)(cid:12)(cid:15) ! ! (cid:16)(cid:17)(cid:18)(cid:19)(cid:20) The enhancement of the local DOS in the presence
(cid:21)(cid:22)(cid:18)(cid:19)(cid:23) of odd-frequency pairing canbe understood, irrespective
(cid:10)(cid:11)(cid:12)(cid:14) (cid:3)(cid:6)(cid:4) (cid:16)(cid:17)(cid:18)(cid:24)(cid:25) of the detailed shape of ∆ by using the normalization
(cid:11)(cid:12)(cid:13) (cid:21)(cid:22)(cid:18)(cid:24)(cid:26) condition for the quasiclassical Green’s functions, g2 +
(cid:10)
ff¯ = 1. Indeed, since for odd-frequency pairing state,
(cid:11)(cid:12)(cid:27) theanomalousGreen’sfunctionf¯=−2b/(1+ab)atE =
0isgivenbyf¯(θ)=−f∗(θ)(seeRef.[13])andlocalDOS
(cid:11)(cid:12)(cid:14) is given by N(E) = −Reg, one can show that generally
(cid:31) (cid:31)
(cid:7)(cid:9)(cid:8) N(E = 0)> 1 since g2 = 1+|f|2 > 1. This means that
(cid:11) (cid:16)(cid:17)(cid:18)(cid:28) the emergence of the odd-frequency pairing is a physical
(cid:16)(cid:17)(cid:18)(cid:29)(cid:20)(cid:26)(cid:30)(cid:23)(cid:26) reason of zero energy peak of the local DOS inside the
(cid:21)(cid:22)(cid:18)(cid:29)(cid:20)(cid:23)
(cid:11)(cid:12)(cid:14) core. The manifestation of the odd-frequency chiral p-
(cid:10)
wavepairingstate atthe center ofthe vortexcoreis also
(cid:11)(cid:12)(cid:27) consistent with the experimental fact that the observed
(cid:10) (cid:11) (cid:15) (cid:14) (cid:13)
zero-bias conductance peak by STM at a vortex center
(cid:0)
(cid:1) (cid:2)
is very sensitive to disorder [17], since p-wave pairing is
fragile to impurity scattering.
Figure 2 depicts the results for OSO p -wave super-
x
FIG.2: (coloronline)ResultsforOSOpx-wavesuperconduc- conductor. ThelocalDOSaroundthe vortexatE =0is
tor. (a) Normalized local DOS around the vortex at E = 0.
showninFig. 2(a). Indramaticcontrasttotheresultfor
The center of the vortex is situated at x = y = 0. Spatial
the ESE s-wave, zero energy DOS is suppressed at the
dependenciesof (b)OSOand (c) ESEcomponentsat E =0.
core. The spatial dependencies of decomposed anoma-
OnlyESEcomponentscanexistinsidethecore(nearx=0).
lous Green’s function at E =0 are shown in Figs. 2 (b)
and (c). As is seen, only ESE pairing components ex-
Due to the broken translational symmetry of the sys- ist at the center of the core. For even-frequency pairing
tem,variouspairingstatesareexpectedtoemergearound state, f¯(θ) = f∗(θ) is satisfied at E = 0 and hence we
the vortex. In order to study possible pairing states, we get N(E = 0) < 1 which is consistent with Fig. 2 (a).
decomposeanomalousGreen’sfunctionf intovariousan- By comparing Figs.1 and 2, it is clear that zero energy
gular momentum components as follows localDOSN(0)hasamaximumatthecenterofthevor-
texcoreineven-frequencysuperconductor,whileithasa
f ∼=fs+fpxcosθ+fpysinθ+fdx2−y2cos2θ minimum at the core center in odd-frequency supercon-
ductor. This difference can be detected by STM.
+fdxysin2θ+ff1cos3θ+ff2sin3θ+..... (4)
As regardsthe candidate for the odd-frequency super-
Notethat all the above pairing components are spin sin- conductor,CeCu2Si2 andCeRhIn5 arepossiblematerials
glet. [6, 18]. In these systems, OSO state with p-wave sym-
Figure 1 shows the results for ESE s-wave supercon- metry is considered to be promising [6]. In the light of
ductor. The local DOS around the vortex at E = 0 the present theory, ESE s-wave pairing is expected to
is shown in Fig. 1 (a). As is well known, zero energy appear inside the vortex core. Based on this idea, we
peak appears in the core [3]. The spatial dependencies propose an experimental setup to verify the existence of
of decomposed anomalous Green’s function f at E = 0 odd-frequency pairing in bulk materials by using super-
areshowninFigs. 1(b)and(c). Interestingly,onlyOSO conducting STM where we use conventional s-wave su-
pairingcomponents,Ref andImf ,surviveatthecen- perconductorasaSTMtip[19]asshowninFig. 3. Local
px py
ter of the core. With the increase of the distance from JosephsoncurrentmeasuredinSTMexperimentwithsu-
4
(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7) (even)integer. Wehavealsoshownthatzeroenergylocal
(cid:8)(cid:9)(cid:8)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15) (cid:16)(cid:13)(cid:17)(cid:18) densityofstatesisenhanced(suppressed)atthecenterof
the vortex core for even(odd)-frequency superconductor.
(cid:22)(cid:19)(cid:21)(cid:20) Based on the obtained results, we proposed a scanning
tunneling microscope experiment using a superconduct-
(cid:23)(cid:9)(cid:23)(cid:16)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15) ing tip to detect local Josephson coupling in order to
(cid:10)(cid:24)(cid:16)(cid:15)(cid:18)(cid:25)(cid:26)(cid:27)(cid:28)(cid:24)(cid:25)(cid:29)(cid:26)(cid:18) explore and identify odd-frequency superconductor.
We acknowledgevaluable discussionwith C. Iniotakis,
(cid:0) (cid:1)
Y.AsanoandN.Hayashi. T.Y.acknowledgessupportby
the JSPS. This work was supported by Ministry of Ed-
ucation, Culture, Sports, Science and Technology,Grant
FIG.3: (coloronline)Suggestedexperimentalsetuptoprobe
No. 17071007and NanoNed Grant TSC7029.
thelocalJosephsoncouplingbetweensuperconductingtipand
superconductorwith Abrikosov vortex core.
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