kaori tanaka, istvan robel and boldizsar janko- electronic structure of multiquantum giant vortex...

8/3/2019 Kaori Tanaka, Istvan Robel and Boldizsar Janko- Electronic structure of multiquantum giant vortex states in mesosc

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Electronic structure of multiquantum giant vortexstates in mesoscopic superconducting disksKaori Tanaka, Istvan Robel, and Boldizsar Janko

Materials Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439; Department of Physics and Engineering Physics,University of Saskatchewan, Saskatoon, SK, Canada S7N 5E2; and Department of Physics, University of Notre Dame, Notre Dame, IN 46556-5670

Communicated by Alexei A. Abrikosov, Argonne National Laboratory, Argonne, IL, February 19, 2002 (received for review January 11, 2002)

We report self-consistent calculations of the microscopic electronic

structure of the so-called giant vortex states. These multiquantum

vortex states, detected by recent magnetization measurements on

submicron disks, are qualitatively different from the Abrikosov

vortices in the bulk. We find that, in addition to multiple branches

of bound states in the core region, the local tunneling density of

states exhibits Tomasch oscillations caused by the single-particle

interference arising from quantum confinement. These features

should be directly observable by scanning tunneling spectroscopy.

Superconducting vortices are topological singularities in theorder parameter (1). In a bulk system, each vortex carries asingle flux quantum, whereas vortices with multiple flux quantaare not favorable energetically (2). In small superconductors,however, the situation may be different. Todays nanotechnologycan provide valuable insight into the nature of mesoscopicsuperconductors, whose linear dimensions can be comparable tothe coherence length or the inter-vortex distance of the Abri-kosov lattice. The following question then arises naturally: doessingle-quantum vortex matter survive the limit of decreasingsample size? More than 30 years ago, Fink and Presson (3) gavethe intriguing answer not always in their pioneering work ona related system: a thin cylinder in parallel field. They haveshown it theoretically, within the framework of the phenome-nological GinzburgLandau (GL) theory, and also providedexperimental evidence (4) for the existence of an enormoussuperfluid eddy current on the surface of a thin cylinder. Theycalled this state a giant vortex state.

Although the work of Fink and Presson was largely forgotten

for the next several decades, it nevertheless anticipated thepresent excitement in the field of nanoscale superconductivity.With recent advances in the controlled fabrication and study ofnanometer-scale superconductors, the concept of giant vortexwas brought back to focus by Moshchalkov and c oworkers a fewyears ago (5). Their experiments on mesoscopic squares andsquare rings have indeed revealed that small superconductors donot always favor many-vortex states reminiscent of the Abrikosov vortex lattice. The measured HT phase boundaries of thesesmall structures were explained in terms of giant vortex states inthe GL picture (5). Subsequent experiments on submicron disks(6) have further shown the existence of giant vortex states insidethe phase boundaries. Within the GL f ramework (6 10) some ofthe abrupt changes in the magnetization observed have beenattributed, e.g., to the collapse of a multivortex state into a giantvortex, or to transitions among different giant vortex states.

This phase of vortex matter has a single vortex occupying thesample, carrying multiple fluxoid (3, 11, 12) quanta. Such astate has no immediate analogue in bulk systems, and wouldonly be similar to vortex states predicted for artificially pat-terned structures (13, 14). Moshchalkov et al. (15) have alsosuggested that giant vortex states can cause the peculiarparamagnetic Meissner effect seen in granular and mesoscopicsuperconductors (16, 17), through the c ompression of the f luxtrapped in the sample. This effect has been seen experimen-tally in mesoscopic systems by Geim and coworkers (18). As itis apparent from these recent experiments, mesoscopic super-

conductors exhibit quantum phenomena that are not observ-able in bulk systems. Studying their unique properties is crucialnot only for potential applications but also for better under-standing of nanoscale superconductivity.

Despite the fact that the existence of giant vortex states hasbeen indicated more than three decades ago, and that theircounterpart in nanoscale superconductors has been under in-tense scrutiny in recent years, there has been no microscopic,self-consistent theoretical study of its electronic structure. In thispaper, we report the results of such a microscopic and self-consistent study of multiquantum giant vortex states in s-wavesuperconducting disks of submicron size, using the Bogo-liubovde Gennes (BdG) formalism (11). The spectroscopicproperties that we have obtained for such vortex states can beprobed directly by scanning tunneling microscopy (STM). Al-though GL studies give a good qualitative picture for a widerange of parameters, quantitatively reliable results are limited tothe range relatively close to the critical temperature and mag-netic field. Furthermore, analyzing the system from a micro-scopic point of view is essential to understanding superconduc-tivity on such a small scale. Latest experimental efforts aimed atSTM imaging of mesoscopic vortex matter have focused onNbSe2 samples (W. K. Kwok, unpublished data), because directSTM images of vortex states have been obtained only onhigh-quality single crystals of NbSe2 (20) and Bi2Sr2CaCuO8(2123). These compounds are highly two-dimensional and easyto cleave in situ, providing very clean surfacesa key ingredientfor successful STM imaging. Thus, anticipating STM measure-ments, we present in this paper self-consistent BdG results with

parameters corresponding to submicron NbSe2 disks.

Formulation

We consider an s-wave superconducting disk of radius R undera magnetic field perpendicular to the disk area, with a vortexcarrying m fluxoid quanta formed in the center. The system hascylindrical symmetry, and it is described by using cylindricalcoordinates (r, , z). In accordance with the experiments (ref. 6,and W. K. Kwok, unpublished data), we assume the diskthickness to be much smaller than the penetration depth.Consequently, the order parameter is assumed to be uniform inthe field direction (z), and the current density j as well as thevector potential A has no z component (7). In the gauge thatremoves thephase of the order parameter (11),i.e., (r) (r),we can write down the radial part of the BdG equations (13,

24, 25) as

Abbreviations: GL,GinzburgLandau;BdG, Bogoliubovde Gennes; STM,scanningtunnel-

ing microscopy; LDOS, local density of states.

To whom reprint requests should be addressed. E-mail: [email protected].

Afterthecompletionof thiswork,we becameawareof therecentresultsof A.S. Melnikov

and V. M. Vinokur (19), in which the possible application of mesoscopic disks as quantum

switches is studied. This work discusses the tunneling density of states in the core of giant

vortices in terms of quasiclassical calculations.

The publication costs of this article were defrayed in part by page charge payment. This

article must therefore be hereby marked advertisement in accordance with 18 U.S.C.

1734 solely to indicate this fact.

www.pnas.orgcgidoi10.1073pnas.082096799 PNAS April 16, 2002 vol. 99 no. 8 52335236


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z2

2med

2

dr2

1

r

d

dr 1

r2l z ec rAr

m

2

2

nr zUr nr xrnr nnr, [1]

where

nr unrvnr [2]is the radial quasiparticle amplitude. Here x andz are the Paulimatrices, me is the electron mass, and is the chemical potential.The angular momentum quantum number l is an integer when mis even, and a half-odd integer when m is odd (11). Thesingle-particle potential U(r) can incorporate the lattice poten-tial and inhomogeneity effects caused by impurities and sampleboundaries. To study quantum size and interference effects, weconsider a clean sample and take U(r) 0 inside the disk, whileincluding the periodic lattice potential in terms of the effectivemasses, mr and mz. Furthermore, we take mz mr, as justifiedfor highly anisotropic materials such as NbSe2, and neglect thedependence of Eq. 1 on the motion along the z direction.

In principle, because of finite thickness of the sample, thevector potential must depend on not only rbut also z, so that A A(r,z), and A(r) in Eq. 1 is an average ofA(r,z) over the disk

height L (7):

Ar 1

L L2

L2

dz Ar, z. [3]

However, for typical experimental parameters (ref. 6, and W. K.Kwok, unpublished data), the lateral sample size is much largerthan the thickness, and we therefore consider the case where the

vector potential is independent of the z coordinate; A(r,z) A(r). The order parameter and the current density j j(r) aregiven in terms of the eigenvalues and eigenfunctions of Eq. 1 as

r g n D

unrv*nr1 2fn [4]

jr emr

1r

nl m2 ec rArunr 2fn

l m2

e

crArvnr 21 fn , [5]

where g is the coupling strength for the electronelectronattraction, D is the Debye frequency, and fn f(n) is the Fermidistribution function. The vector potential is given in turn by thecurrent density through the Maxwell equation A (4c)j. We first solve Eq. 1 with initial guesses for (r) andA(r) and recalculate them from Eqs. 4 and 5, and repeat theprocess until self-consistency is acquired. The local tunnelingdensity of states and the differential conductance (13, 24) canthen be calculated by

Nr, E n

unr2E n vnr

2E n [6]

dIr, V

dV

n

unr2fn eV vnr

2fn eV .

[7]

Clearly, the differential tunneling conductance (Eq. 7) is a directprobe of the local density of states, N(r,E), provided that thetemperature is low enough. In the following paragraphs we will

present some results for the experimentally observable differ-ential conductance at low temperatures, but may refer to it as thelocal tunneling density of states (LDOS). We choose the pa-rameters corresponding to NbSe2: we take mr 2me, EF 37.3milli-electronvolts (meV), D 3.0 meV, and set the couplingstrength so that the bulk gap 01.12 meV.

The results are shown for disk radius R 500 nm. We haveinvestigated the size range of R 200600 nm and have foundqualitatively same features in the LDOS.

Results

In Fig. 1 we show the differential conductance for a vortex withm 4 (Fig. 1a) and m 5 (Fig. 1b) flux quanta, as a function

of voltage V and radial distance r from the disk center, fortemperature T 1 K.** In both cases, prominent sharp peakscan be seen near the vortex core and for low voltages fourpeaks in the former and five in the latter. Generally speaking, thenumber of low-bias conductance peaks corresponds to the

winding number m of the order parameter, which gives rise to mpeaks near the center (13, 25). This feature is in accordance with

**For data to beshownfor energieswell abovethe gapenergy,fast 1kF oscillationshavebeen removed by means of Fourier transform. In actual observations these fast oscilla-

tions will not be resolved.

Fig.1. Localtunnelingconductance asa functionof coordinaterandvoltage

V, for a giant vortex state with m 4 (a) and m 5 (b) flux quanta, sustained

in a superconducting disk with radius R 500 nm at temperature T 1 K.

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the index theorem established by Volovik (26) for the Caroli deGennesMatricon bound states in a vortex core. According tothis theorem, the quasiparticle spectrum of a vortex with windingnumber m has m branches of bound states, which cross zeroenergy as a function of angular momentum. These quasiparticlebranches also explain the evolution of them rows of conductancepeaks as one moves away from the core, as seen in Fig. 1, withdecreasing number of peaks one by one (13). In contrast to thesingly quantized case (27), one can see directly that as energyincreases, more states with higher angular momenta contributeto the density of states.

We illustrate finding this in Fig. 2, where a spatial map of theLDOS is taken for various fixed values of V. Clearly, withincreasing bias voltage, the density of states is redistributed fromthe core toward the sample boundaries.

Fig. 1 also reveals the presence of the so-called zero mode, i.e.,a peak around zero energy at the vortex core for odd m, and itsabsence for even m. The existence of a zero mode is, quitegenerally, linked to a sign change in the order parameter as afunction of some generalized coordinate (28). The zero-biaspeak is a signature of bound states for quasiparticles trapped bythe sign change. Here, in the given gauge, the order parameterchanges sign at the vortex core when m is odd, whereas it doesnot when m is even.

The low-bias peaks and zero modes discussed above aregeneral characteristics of the LDOS associated with the windingnumber of the order parameter. In addition to these, however,

we have found previously uncharacterized features in the LDOSthat are unique to giant vortex states in submicron disks. Thesefeatures are the oscillations seen above the gap energy in Figs.1 and 2c, and more clearly in the contour plot of Fig. 3. Theseoscillations are similar in origin to the so-called Tomaschoscillations discovered in a superconductor-normal metal junc-tion (2931) and reflect standing waves arising from theinterference of quasiparticle states. This interference ef fect is adirect consequence of strong confinement experienced by thesuperconducting quasiparticles because of the small system size.We dedicate the remainder of this paper to detailed discussionsof this effect.

When a vortex holds multiple f lux quanta m, the orderparameter vanishes around the center over a certain area - thelarger the m is, the larger the area (25). As the distance from the

center increases, the order parameter increases and recovers to

its bulk value eventually. In the case of NbSe2, because of theshort coherence length, the recovery happens relatively quickly.This recovery results in a well-defined superconducting region

with the constant order parameter 0 within the disk. At the diskboundaries, however, the order parameter is forced to vanish,and as a result, exhibits Friedel-like oscillations around its bulk

value near the surfaces. These oscillations have the largestamplitudes at the boundaries, and decay roughly over thecoherence length scale. Moreover, the smaller the system size,the larger these amplitudes are, relative to the bulk value. Thequasiparticles confined in the disk experience scattering by thislarge change in the order parameter at the surfaces. An electron-like quasiparticle is reflected back as a hole-like one and vice

versa, and the Tomasch effect results from the interferencebetween the electron-like and hole-like states in the supercon-

ducting region (31). The momenta of electron-like and hole-likequasiparticles for energy E are k (2mr)EF kF (vF), respectively, where E2 (r)2 with(r) 0 and kF is the Fermi momentum. At a given distanced from the surface, the LDOS oscillations in energy are deter-mined by (31) (En0) 1 n2(vF0d)2, where n is aninteger, and vF 0 151.15 nm for NbSe2. Furthermore, theinterference can be seen in the LDOS also as a function ofcoordinate (distance from the surface) for a given energy E. Theperiod of the oscillations in this case is given by d (vF0)(1(E0)2 1). The oscillation periods obtained in ournumerical results, as seen in Fig. 3, are in quantitative agreement

with these analytical expectations.In a superconductor with short coherence length, if the

winding number m and, consequently, the normal core area is

very large, the Tomasch effect may arise also from the vortexcore, as in a normalsuperconductor junction. For submicronNbSe2 disks with m up to 5, however, we have found that theLDOS is dominated by the Tomasch oscillations coming fromthe surfaces. Indeed, we have confirmed the LDOS oscillationscharacteristic of the Tomasch effect in terms of model calcula-tions in one and two dimensions, where the BdG equations aresolved without iteration with a step-function order parameter:(r) 0(r R2);0(r R2). In this case, the Tomasch effectcoming from the normalsuperconductor interface governs theLDOS structure, so that the oscillation period in energy (see En

Fig. 2. Spatial map of the local tunneling conductance for the entire disk in

thegiantvortex state of Fig. 1b, forvarious fixedvoltages. It canbe seen that

themaxima in thelocaldensityof statesgradually shifttoward theperimeter

of the disk as the voltage is increased.

Fig. 3. Contour plot of the LDOS in Fig. 1 b in the superconducting region,

where the Tomasch density of states oscillations occur.

Tanaka et al. PNAS April 16, 2002 vol. 99 no. 8 5235


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above) becomes larger as one approaches the interface (i.e., hered is the distance from the interface). Apart from this andenhanced amplitudes caused by a larger change in the order

parameter, the LDOS shows the same qualitative features asseen above (compare Figs. 3 and 4).

Conclusions

We have presented detailed, self-consistent calculations of themicroscopic electronic structure of giant vortex states. Webelieve that the most direct experimental evidence for theexistence of giant vortices can be provided by STM measure-ments of the local density of states in sub-micrometer super-

conductors capable of sustaining such vortex configurations. Wehave provided a spatial map of the tunneling density of states formultiquantum giant vortex states, and have identified severalsignatures that can be used to identify them with STM. We havefound that under extreme confinement the quantum interfer-ence arises among quasiparticle states and leads to experimen-tally observable Tomasch oscillations in the local density ofstates.

We thank Prof. A. A. Abrikosov, Dr. G. W. Crabtree, Dr. W. K. Kwok,Prof. F. Marsiglio, and Dr. O. Tchernyshyov for enlightening discussions.B.J. thanks Prof. W. Tomasch for comments and discussions on thepresented results. This research was supported by United States De-partment of Energy, Office of Science, under Contract no. W-31-109-ENG-38, and by the Natural Sciences and Engineering Research Councilof Canada.

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Fig. 4. Same as Fig. 3, but for the corresponding model calculation.

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kaori tanaka, istvan robel and boldizsar janko- electronic structure of multiquantum giant vortex...

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