Table Of ContentLow-lying Quasiparticle Excitations around a Vortex Core in Quantum Limit
N. Hayashi, T. Isoshima, M. Ichioka, and K. Machida
Department of Physics, Okayama University, Okayama 700, Japan
(Submitted 26 September1997)
8
9
9 quantum-limit aspects of the single vortex in s-wave su-
Focusing on a quantum-limit behavior, we study a sin-
1 perconductors and to discuss a possibility for the obser-
gle vortex in a clean s-wave type-II superconductor by self- vation of them.
n
consistentlysolvingtheBogoliubov-deGennesequation. The
a The present study is motivated by the following re-
discreteenergylevelsofthevortexboundstatesin thequan-
J centexperimentalandtheoreticalsituations: (1)Theso-
tum limit is discussed. Thevortexcore radiusshrinksmono-
6 called Kramer and Pesch (KP) effect [1,3,7,8]; a shrink-
tonically up to an atomic-scale length on lowering the tem-
1
peratureT,andtheshrinkagestopstosaturateatalower T. age of the core radius upon lowering T (to be exact, an
The pair potential, supercurrent, and local density of states anomalous increase in the slope of the pair potential at
]
n around the vortex exhibit Friedel-like oscillations. The local thevortexcenteratlowT)isnowsupportedbysomeex-
o density of states has particle-hole asymmetry inducedby the periments[9]. The T dependence ofthe coresizeisstud-
c vortex. These are potentially observed directly bySTM. iedbyµSRonNbSe andYBCO[10],whichis discussed
- 2
r later. The KP effect, if confirmed, forces us radically al-
p PACS number(s): 74.60.Ec, 61.16.Ch, 74.72.-h
ter the traditional picture [11] for the vortex line such
u
s as a rigid normal cylindrical rod with the radius ξ0. (2)
t. Growing interest has been focused on vortices both in The scanning tunneling microscopy (STM) experiment
a on YBCO by Maggio-Aprileet al. [12], which enables us
m conventional and unconventional superconductors from
todirectlyseethespatialstructureofthelow-lyingquasi-
fundamental and applied physics points of view. This is
- particle excitations around the vortex, arouses much in-
d particularly true for high-T cuprates, since it is essen-
c
n tial that one understands fundamental physical proper- terest. They claim that surprisingly enough, there exist
o only a few discretized bound-state levels in the vortex
ties of the vortices in the compounds to better control
c core, i.e., the vortex is almost “empty.” It resembles
various superconducting characteristics of some techno-
[
logical importance. Owing to the experimental devel- our naive image for conventional s-wave superconduc-
1 tors where in the quantum limit a few quantized levels
opments, it is not difficult to reach low temperatures
v of the bound states remain inside the bulk energy gap
of interest where distinctive quantum effects associated
6
∆ . (3) The theoretical situation on this subject [12] is
6 with the discretized energy levels of the vortex bound 0
stillveryconfusing;Some[13]claimthatthebound-state
1 states are expected to emerge. The quantum limit is
1 realized at the temperature where the thermal smear- energylevels arenot discretizedfor d-wavepair,but dis-
0 cretizedfor s-wavepair. Some[14]claimthe discretized-
ing is narrower than the discrete bound state levels [1]:
8 likestructureevenfortheformer. Fors-wavecase,where
T/T ≤ 1/(k ξ ) with ξ =v /∆ the coherence length
9 c F 0 0 F 0
the formulationoftheproblemis welldefined,weshould
/ (∆0 the gapatT =0)andkF (vF)the Fermiwavenum-
t establishourunderstandingofthevortexstructureinthe
a ber (velocity). For example, in a typical layered type-II
m superconductor NbSe with T = 7.2 K and k ξ ∼ 70, quantumlimit. (4) Lastly,we are motivatedby a curios-
2 c F 0
ity; Previously we have calculated the local density of
- the quantum limit is reached below T < 50 mK. As for
d states (LDOS) for s-wave pair on the basis of the qua-
the high-T cuprates, the corresponding temperature is
c
n siclassical (Eilenberger) theory [15], successfully applied
rather high: T <10 K for YBa Cu O (YBCO).
o 2 3 7−δ to the STM observations on NbSe done by Hess et al.
c Important microscopic works to theoretically investi- 2
[16]. We are particularly interested in what happens in
: gate the quasiparticlespectralstructure arounda vortex
v
in a cleanlimit areput forthby Caroliet al. [2], Kramer LDOS at further lower T, say, below 50 mK deep into
i
X and Pesch [1], and Gygi and Schlu¨ter [3]. The low-lying the quantum limit, at which it may be now feasible to
r excitationsareessentialtocorrectlydescribelow-T ther- perform STM experiments.
a Prompted by these motivations, we self-consistently
modynamic and transport properties in the vortex state
solve the Bogoliubov-de Gennes (BdG) equation, which
(or the mixed state). These include anomalous electric
is one of the most fundamental microscopic equations
or thermal Hall conductivity [4] and mysterious obser-
of superconductivity and contains fully quantum effects.
vations of the quantum magnetic dHvA oscillations [5];
We start with the BdG equations for the quasiparticle
various topics are debated intensively [6]. Yet there has
wave functions u (r) and v (r) labeled by the quantum
been no serious attempt or quantitative calculation to j j
number j:
explore deep into the quantum regime.
The purposes of the present paper are to reveal the
1
−1 ∇2−E u (r)+∆(r)v (r)=E u (r), alsoexhibits a weak oscillationaroundr =0.2–0.5ξ0. It
F j j j j
(cid:20)2kFξ0 (cid:21) is difficult to see the oscillation in H(r), because it is
obtained by integrating j (r) via the Maxwell equation
θ
∇× H = 4πj(r), resulting in a smeared profile. It is
−1 c
− ∇2−EF vj(r)+∆∗(r)uj(r)=Ejvj(r), (1) also seen that the position of the maximum of jθ(r) be-
(cid:20)2k ξ (cid:21)
F 0 comesshorteras T decreases. These features quite differ
from those obtained within the Ginzburg-Landau (GL)
ina dimensionless form,where∆(r) is the pairpotential
framework [11,18].
andE (=k ξ /2)theFermienergy. Thelength(energy)
F F 0
The T dependence of ξ (T) for various k ξ values is
scaleismeasuredbyξ (∆ ). Foranisolatedsinglevortex 1 F 0
0 0
shown in Fig. 3. Coinciding with Kramer and Pesch
in an extreme type-II superconductor, we may neglect
[1] for s-wave pair and Ichioka et al. [8] for d-wave
the vector potential in Eq. (1). The pair potential is
pair, ξ (T) decreases almost linearly with T, that is,
determined self-consistently by 1
ξ (T)/ξ ∼ T/T except at extremely low T. An im-
1 0 c
portant difference from these quasiclassicaltheories [1,8]
∆(r)=g u (r)v (r){1−2f(E )} (2)
j j∗ j
appears at lower T. At a lower T < T ≃ T /(k ξ ),
X 0 c F 0
|Ej|≤ωD wherethe quantumlimit is realized,the shrinkageofthe
with the Fermi function f(E). Here, g is the coupling core size stops to saturate, and the saturated value is
constant and ωD the energy cutoff, which are related by estimated as ξ1/ξ0 ∼(kFξ0)−1.
a BCS relation via the transition temperature T and Accordingto the µSRexperimentaldata[10],the core
c
the gap ∆0. We set ωD = 10∆0. The current den- radiusinNbSe2 showsastrongT dependence,whilethat
sity is given by j(r) ∝ Im j f(Ej)u∗j(r)∇uj(r)+{1− in YBCO with Tc=60 K is almost T-independent below
∼0.6T . This seemingly contradicting result can be un-
f(E )}v (r)∇v (r) . We cPonsi(cid:2)der an isolated vortex un- c
j j j∗
derstood as follows. The strong T dependence in NbSe
derthefollowingco(cid:3)nditions. (a)Thesystemisacylinder 2
is the usual KP effect corresponding to the curves for
with a radius R. (b) The Fermi surface is cylindrical,
larger k ξ in Fig. 3. At lower T than T estimated
appropriate for the materials such as NbSe and high-T F 0 0
2 c
as ∼100 mK (k ξ ∼ 70), the shrinkage must saturate
cuprates. (c)Thepairinghasisotropics-wavesymmetry. F 0
(the above experiment is done above ∼2 K). As for the
Thus the system has a cylindrical symmetry. We write
the eigenfunctions as u (r) = u (r)exp i(µ− 1)θ and YBCO data, since the estimated kFξ0 is small (∼4 [12]
j nµ 2 for YBCO with T =90 K), the saturation is already at-
v (r)=v (r)exp i(µ+1)θ with∆(r)=(cid:2)∆(r)exp(cid:3)−iθ c
j nµ 2 tained at a relatively high T such as shown in Fig. 3.
in polar coordinat(cid:2)es, where(cid:3)n is a radial quantum(cid:2)num(cid:3)-
Thus the absence orweaknessofthe KP effect in YBCO
ber and the angular momentum |µ| = 1,3,5,···. We
2 2 2 is simply attributable to the factthat the quantum-limit
expand the eigenfunctions in terms of the Bessel func-
temperature T is quite high.
tions J (r) as u (r) = c φ (r) and v (r) = 0
m nµ i ni i|µ−12| nµ Reflectingthe shrinkageofthe coreradius,the bound-
idniφi|µ+21|(r)withφimP(r)= RJm+√12(αim)Jm(αimr/R), state energies Eµ increases as T decreases. This T-
Pi=1,2,···,N, and α is the i-th zero of J (r) . The dependent E shift, due to the KP effect, and its satu-
im m µ
(cid:0)BdGisreducedtoa2N×2N matrixeigenvaluepr(cid:1)oblem rationatlower T may leadto a nontrivialT dependence
[3]. Our system is characterized by k ξ , which is a key in thermodynamic and transport properties.
F 0
parameter of the present problem. In Fig. 4, we plot the energy levels E of the low-
µ
In Fig. 1, the calculated spatial variation of ∆(r) is lying bound states µ = 1,3,···,13 as a function of
2 2 2
displayed for various T. It is seen that as T decreases, kFξ0, at sufficiently(cid:0)low T (T/Tc = 0.(cid:1)01) where increas-
the core size ξ1 defined by ξ1−1 = limr 0∆(r)/r shrinks ing of the energy levels saturates. It is seen that in
→
and the oscillatory spatial variation with a wave length large-k ξ region, the bound states densely pack inside
F 0
∼1/k becomes evident in ∆(r) [1,3]. The physical rea- the gap ∆ , allowing us to regard them as continuous
F 0
son for this Friedel-like oscillation lies in the following ones. This is the case where the quasiclassical approx-
facts. All eigenfunctions u (r) and v (r) contain a imation [1,8] validates. In small-k ξ region, where the
nµ nµ F 0
rapid oscillation component with 1/k . At lower T the quantum effect is important even at high T, only a few
F
lowestbound states, whose oscillationamplitude is large bound states remain within the low-energy region. We
nearthecore,dominatephysicalquantities. Wenotethat find that even in small-|µ| region, the spacing between
theoscillatorybehaviorcanalwaysappearsatsufficiently the energy levels E is not constant,but rather becomes
µ
lowT irrespectiveofvaluesofk ξ . Wealsomentionthat narrower as |µ| increases. The often adopted formula
F 0
asimilaroscillatoryspatialvariationaroundavortexcore E /∆ =2µ/(k ξ )or2µ/(k ξ )duetoCaroliet al.[2],
µ 0 F 0 F 1
in the Bosecondensate of 4He is found theoretically,due or E /∆ = (2µ/k ξ )ln[ξ /2ξ ] by Kramer and Pesch
µ 0 F 0 0 1
to the roton excitations [17]. in the limit ξ ≪ ξ [1] do not satisfactorily explain our
1 0
The associated supercurrent j (r) and the field H(r) self-consistent results. Instead, our result is empirically
θ
areshowninFig.2. Reflectingtheaboveoscillation,j (r) fittedtoaformulaE /∆ =(0.5/k ξ )ln[k ξ /0.3]for
θ 1/2 0 F 0 F 0
2
large k ξ as shown in the dotted curve in Fig. 4. 12), and a borocarbide LuNi B C [19] with T =16 K,
F 0 2 2 c
In Fig. 5, the spectral evolution, i.e., the spatial ξ ≃80 ˚A are the best candidates to check our results.
0
variation of LDOS, which is calculated by N(r,E) ∝ In summary, we have analyzed the vortex core struc-
|u (r)|2f (E −E )+|v (r)|2f (E +E ) , is shown ture and the related quasiparticle energy spectrum by
j j ′ j j ′ j
Pfor(cid:2)kFξ0=8 at low temperature T=0.05Tc.(cid:3) It is well self-consistentlysolvingtheBdGequationforanisolated
contrasted with that of the higher T case by Gygi and vortex in a clean s-wave type-II superconductor, focus-
Schlu¨ter [3] (see, for comparison, Fig. 15 in Ref. [3] ingonthelow-T quantumeffects. Wehavefoundthefar
where k ξ ∼ 70 and T ≃ 0.13T , calculated under richer structure in the pair potential, supercurrent, and
F 0 c
the two-dimensional Fermi surface). As lowering T, be- LDOS than what one naively imagines from the corre-
causeofthequantumeffects,thethermallysmearedspec- sponding calculations done athigh T ork ξ ≫1 [3,15],
F 0
tral structure drastically changes and becomes far finer and pointed out experimental feasibility to observe it.
one around the vortex. The spectra are discretized in- Thewidelyusedworkinghypothesisforthevortexcore
side the gap and consist of several isolated peaks, each of a rigid normal rod with the radius ξ [11] must be
0
of which precisely corresponds to the bound states E cautiously used for the clean superconductors of inter-
µ
|µ| = 1,3,··· . Reflecting the oscillatory nature of the est: the magnetic field distribution probed by neutron
2 2
e(cid:0)igenfunctions(cid:1)uµ(r)andvµ(r)withtheperiod1/kF,the diffraction [20] or µSR [10] through the magnetic form
spectralevolutionalsoexhibitstheFriedel-likeoscillation factor analysis based on the GL theory must be taken
as seen from Fig. 5. with caution. Detailed investigations of various myster-
To show clearly the particle-hole asymmetry of the ies associated with the vortices, e.g., the thermal Hall
LDOS of Fig. 5, which is another salient feature, we conductance [4] belong to future work.
present in Fig. 6 the spectra at the vortex center r = 0 We would like to thank J. E. Sonier and A. Yaouanc
and0.2ξ WecanbarelyseetheasymmetryinWangand for useful discussions.
0
MacDonal(cid:2)d[13] seeFig. 3(a)inRef.[13] . Atthe cen-
FIG. 1. The spatial variation of the pair potential ∆(r)
ter r = 0, the bo(cid:0)und-state peak with E (cid:1)(cid:3)appears only
1/2 normalizedby∆0 aroundthevortexforseveraltemperatures
atE >0side,becausetheeigenfunctionu (r =0)6=0,
1/2 and kFξ0 =16. The length r is measured by ξ0.
whichconsistsoftheBesselfunctionJ (r =0)(6=0),and
0
all others for |E | < ∆ vanish at r = 0. At r = 0.2ξ ,
µ 0 0
the other bound-state peaks are seen. The particle-hole FIG. 2. The current distribution normalized by
asymmetry in the vortex bound states appears even if cφ0/(8π2ξ03κ2) for several temperatures, where φ0 is the flux
the normal-state density of states is symmetric. These quantum and κ (≫1) is the GL parameter. The inset shows
features are subtle [3] or absent [15] in the previous cal- the field distribution normalized by φ0/(2πξ02κ2). The tem-
culations. This asymmetry around the vortex is quite peratures are thesame as in Fig. 1, and kFξ0=16.
distinctive, should be checkedby STM experiments, and
may be crucial for the Hall conductivity in the mixed
state. FIG. 3. The T dependence of the vortex radius ξ1 nor-
malized by ξ0 for several kFξ0 (= 1.2, 2, 4, and 16 from top
Let us argue some of the available experimental data
to bottom).
in the lightof the presentstudy. The lowestbound state
levelE /∆ isestimatedbyMaggio-Aprileetal.[12]for
1/2 0
YBCOwithT =90K(E =5.5meVand∆ =20meV),
c 1/2 0 FIG. 4. The lowest seven bound-state energies Eµ, nor-
yielding kFξ0 ∼ 4. Since it implies that ξ0 is only of malized by ∆0, as a function of kFξ0, at enough low temper-
the order of the crystal-lattice constant, we should cau- atureT/Tc =0.01. Thedotted lineis a fittingcurve(seethe
tion that Maggio-Aprile et al. [12] take their data for text).
the spectral evolution every 10 ˚A apart near the core,
thus the important spatial information on LDOS might
be lost. So far the existing STM data [12,16,19] taken FIG. 5. The spectral evolution N(E,r) at T/Tc = 0.05
at the vortex center are almost symmetric about E=0, and kFξ0 = 8. It is normalized by the normal-state density
e.g., on NbSe at T=50 mK [16]. The reason why the of states at the Fermi surface. E and r are measured by ∆0
2
and ξ0, respectively.
so-calledzero-biaspeakiscenteredjustsymmetricallyat
E=0 is that k ξ is large and T is too high to observe
F 0
the quantum effects.
FIG.6. ThelocaldensityofstatesN(E,r)atr=0(solid
We emphasize that in any clean s-wave type-II super-
line) and 0.2ξ0 (dotted line). T/Tc =0.05 and kFξ0=8.
conductors at appropriately low T < T ≃ T /(k ξ ) ,
0 c F 0
one can observe these eminent chara(cid:0)cteristics associate(cid:1)d
with the quantum effects. For example, a typical A-15
compound V Si [5] with T =17 K, ξ ≃ 60 ˚A (k ξ ≃
3 c 0 F 0
3
[1] L. Kramer and W. Pesch, Z. Phys. 269, 59 (1974).
[2] C.Carolietal.,Phys.Lett.9,307(1964);Phys.Kondens.
Mater. 3, 380 (1965).
[3] F. Gygi and M. Schlu¨ter, Phys.Rev.B 43, 7609 (1991).
[4] K.Krishanaetal.,Phys.Rev.Lett.75,3529(1995).See
also, K. Krishana et al.,Science 277, 83 (1997).
[5] See, e.g., R. Corcoran et al., Phys. Rev. Lett. 72, 701
(1994).
[6] See, e.g., A. van Otterlo et al., Phys. Rev. Lett. 75,
3736 (1995); N. B. Kopnin and G. E. Volovik, cond-
mat/9706082;A.I.LarkinandYu.N.Ovchinnikov,cond-
mat/9708202. These authors try to understand vortex
dynamics in terms of the quasiparticle excitations in a
vortex core.
[7] G. E. Volovik, JETP Lett. 58, 455 (1993).
[8] The KP effect for d-wave pair: M. Ichioka et al., Phys.
Rev.B 53, 15316 (1996).
[9] S.G. D¨ottinger et al.,Phys.Rev. B 55, 6044 (1997).
[10] J.E.Sonier et al.,Phys.Rev.Lett.79, 2875 (1997); 79,
1742 (1997).
[11] A. L. Fetter and P. C. Hohenberg, in Superconductivity,
editedbyR.D.Parks(MarcelDekker,NewYork,1969),
Chap. 14.
[12] I.Maggio-Aprileetal.,Phys.Rev.Lett.75,2754(1995).
[13] Y. Wang and A. H. MacDonald, Phys. Rev. B 52, 3876
(1995).
[14] Y.Moritaet al.,Phys.Rev.Lett.78,4841(1997);Phys.
Rev. Lett. 79, 4514 (1997); Europhys. Lett. 40, 207
(1997). See also, M. Franz and M. Ichioka, Phys. Rev.
Lett.79, 4513 (1997).
[15] N. Hayashi et al., Phys. Rev. Lett. 77, 4074 (1996);
Phys. Rev. B 56, 9052 (1997). For vortex lattice case,
M. Ichiokaet al.,Phys.Rev. B55, 6565 (1997).
[16] H.F.Hesset al.,Phys.Rev.Lett.62, 214 (1989); Phys.
Rev. Lett. 64, 2711 (1990); Physica B 169, 422 (1991);
H.F. Hess, Physica C 185-189, 259 (1991).
[17] S. Giorgini et al., Phys. Rev. Lett. 77, 2754 (1996), and
earlier references therein.
[18] H.E. Brandt, Phys. Rev.Lett. 78, 2208 (1997).
[19] Y.De Wilde et al.,Phys. Rev.Lett. 78, 4273 (1997).
[20] M. Yethiraj et al., Phys. Rev. Lett. 78, 4849 (1997); M.
R.Eskildsen et al., Phys.Rev.Lett. 79, 487 (1997).
4
1
0.8
0.6
)
r
(
∆
⁄
0.4 T T = 0.01
c
0.2
0.4
0.2 0.6
0
0.2 0.4 0.6 0.8 1
r
Fig.1 Hayashi et al.
30
25
20
)
r
(
H20
15
)
r 0 0.2 0.4 0.6 0.8
(
θ r
j
10
0
0 0.2 0.4 0.6 0.8 1
r
Fig.2 Hayashi et al.
1.2
0.9
1
ξ
0.6
0.3
0
0 0.2 0.4 0.6
⁄
T T
c
Fig.3 Hayashi et al.
1
0.8
0.6
µ
E
5/2
0.4
3/2
0.2
µ
= 1/2
0
0 4 8 12 16
ξ
k
F 0
Fig.4 Hayashi et al.
10
r = 0
1/2
0.2
8
µ
= -1/2
6 3/2
)
r
,
E
(
N
4
2
0
-1.5 -1 -0.5 0 0.5 1 1.5
E
Fig.6 Hayashi et al.
