We calculate the density of states (DOS) in a clean mesoscopic d-wave
superconducting quantum wire, i.e. a sample of infinite length but finite width
$N$. For open boundary conditions, the DOS at zero energy is found to be zero
if $N$ is even, and nonzero if $N$ is odd. At finite chemical potential, all
chains are gapped but the qualtitative differences between even and odd $N$
remain.Comment: 7 pages, 8 figures, new figures and extended discussio

A.A. Nersesyan

A.D. Zaikin

C. Pépin

D.C. Ralph

E. Fradkin

F. Braun

F.D.M. Haldane

J. Miller

K. Ziegler

M. Fogelstrom

P. J. Hirschfeld

P.W. Brouwer

T. Senthil

W. A. Atkinson

W.A. Atkinson

English

arXiv

Ziegler, Klaus

Atkinson, W. A.

Hirschfeld, P. J.

OPUS - Augsburg University Publication Server

Density of states width-parity effect ind-wave superconducting quantum wires

Ziegler, Klaus G.

OPUS Augsburg

PHYSICAL REVIEW B, VOLUME 64, 054512Density of states width-parity effect in d-wave superconducting quantum wiresK. Ziegler,1 W. A. Atkinson,2,3 and P. J. Hirschfeld21Institut fuÈr Physik, UniversitaÈt Augsburg, 86135 Augsburg, Germany2Department of Physics, University of Florida, P.O. Box 118440, Gainesville, Florida 326113Department of Physics, Southern Illinois University, Carbondale, Illinois 62901~Received 16 October 2000; revised manuscript received 19 April 2001; published 11 July 2001!We calculate the density of states ~DOS! in a clean mesoscopic d-wave superconducting quantum wire, i.e.,a sample of in®nite length but ®nite widthN. For open boundary conditions and half-®lling, the DOS at zeroenergy is found to be zero if N is even and nonzero if N is odd. At ®nite chemical potential, all chains aregapped but the qualtitative differences between even and odd N remain.DOI: 10.1103/PhysRevB.64.054512 PACS number~s!: 74.25.Bt, 74.25.Jb, 74.40.1kI. INTRODUCTIONIn recent years the fabrication of nanoscale devices hasheightened interest in solid-state quantum systems where thediscreteness of the level spacing plays an important role.Originally discussed only in semiconductor quantum dotsand quantum wires, following the work of Ralph et al.1many fascinating consequences of the level discretenesswere also observed in ultrasmall metal particles. These in-cluded the observation of a spectroscopic gap attributed topair correlations in the small grains, which depended on theso-called number parity of the sample, i.e., whether the num-ber of electrons in the grain was even or odd. The physics ofthe levels in such grains has been quite well understood.2Superconducting systems extended in one dimension but me-soscopic in another have also been studied.3In most of these studies, a gap with A1g or s-wave sym-metry has been assumed. However, for some practical pur-poses it may be of interest to use cuprate samples, which areknown to have dx22y2 symmetry. From a fundamental pointof view it is also expected that these systems will be differentbecause already in the bulk state there are states below themaximum gap scale; the pure d-wave bulk density of states~DOS! is linear in energy, r(E);uEu. We have studied theDOS of mesoscopic d-wave quantum wires and found a newkind of parity effect, not related to the number of particles inthe system, but to the parity of the number of chains acrossthe mesoscopic sample. In some ways, the effect is reminis-cent of a type of mesoscopic gap effect that has been recentlydiscussed in the context of single-wall carbon nanotubes,where the existence or nonexistence of a gap depends on theintrinsic twist or chirality induced while forming the tube.5While the d-wave systems we discuss may be considered tobe tubes if periodic boundary conditions are employed, theydo not break either time-reversal symmetry or spatial parity.Nevertheless, at half-®lling, we observe that if the width ofthe system N is even, there is a gap in the DOS whereas ifthe width is odd the density of states at zero energy r(0) isnonzero. As N is increased, both the even-N gap and the odd-N r(0) vanish as they must to give the bulk d-wave resultr(E);uEu. Although the DOS vanishes at E50 for all Nwhen mÞ0, there is still a pronounced even-odd effect in theDOS, and we argue that this effect should be observable inquantum wires fabricated from cuprate superconductors. We0163-1829/2001/64~5!/054512~7!/$20.00 64 0545speculate further that it may be of relevance to the study ofdisorder-induced pseudogaps in bulk d-wave superconduct-ors, which have been the subject of much controversy inrecent years.4II. MODELWe consider the DOS of a d-wave superconducting chain~DWSC! that is coupled to N21 other DWSC's. The chainsform a two-dimensional ~2D! system and the ®rst chain is atthe boundary. Similar problems were considered in previousstudies of random ūx systems 8 and quasi-one-dimensionaldisordered tight-binding models.9,10 In this paper we will ig-nore the effect of disorder and study the odd-even effect in apure system of coupled DWSC's. Such a behavior is alsoknown for a system of coupled spin chains, which has a gap~no gap! if the number of chains is even ~odd!.6,7There are several possible geometries one can assume forthis problem. The most natural, and the one we considerhere, is a wire parallel to the 100 ~or 010! crystal direction.In this geometry, the boundaries of the wire are aligned withboth the underlying CuO2 crystal and the gap maxima~which are tied to the crystal lattice!. Other geometries, suchas a 110 oriented wire, are expected to show similar even-odd effects to the ones described below, but these will bemixed with Andreev scattering resonances11 and thus will bemore dif®cult to interpret. In addition, we anticipate that the100 oriented wire is the most accessible to current technol-ogy.The Bogoliubov±de Gennes Hamiltonian of the 2D DWSH5~2¹21m !s31Ds1is de®ned on a square lattice with2¹2 f ~r !52t(j512@ f ~r1e j!1 f ~r2e j!#and the DWS order parameterD f ~r !5D0(j512~21 ! j@ f ~r1e j!1 f ~r2e j!# .©2001 The American Physical Society12-1K. ZIEGLER, W. A. ATKINSON, AND P. J. HIRSCHFELD PHYSICAL REVIEW B 64 054512With this de®nition, the gap in familiar k-space representa-tion is Dk52D0@cos(kx)2cos(ky)#. In the following calcula-tions we will measure energies in units of t for simplicity.Moreover, it is assumed that chains are in®nitely long andarranged parallel to the y axis of our 2D system.III. PERIODIC BOUNDARIESThe ®nite set of chains can be closed periodically in thexdirection by identifying the ®rst with the (N11)th chain.While this is not particularly physical, the calculation isfairly transparent and therefore worthwile for pedagogicalreasons. For this translational-invariant N-chain system theintegration in y direction ~i.e., along the chains! can be per-formed and gives the DOS at zero energyr~E50 !521pN (kx E2pp dky2p ImG0~k,1ie !5 lime→0er0N (n50N21 1Ae21@m14t cos~kx!#2, ~1!wherer052tpAt21D02A16t22m2.Here G0 is the Green's function for a clean 2D d-wave su-perconductor, kx52pn/N (n50,1, . . . ,N21), and we haveset the lattice constant to unity. First, we consider the DOS athalf-®lling (m50). We ®nd that r(E50) is ®nite for N amultiple of 4 since the allowed values of kx include kx5p/2 and 3p/2, which produce ®nite contributions to Eq.~1!. For other values of N, on the other hand, expression ~1!always vanishes. At m50, then, we haverN~E50 !5H 2r0 /N for N54k ,k51,2, . . .0 otherwise.This periodicity in N re ēcts the conditions necessary toform a standing wave in the x direction at the Fermi energy.For m50, it is the four nodal points at k5(6p/2,6p/2)that produce the standing wave. The wave function at E50is thus exactly the same as in the normal state; it is thereforenot surprising that r(E50) is only weakly dependent on D0.For given N, there will be a discrete set of m , whichproduces a ®nite density of states atE50. For generic m , thenodal eigenstates do not contribute at E50, but we expecton the basis of Eq. ~1! that a qualitative width-parity effectwill remain at nonzero E. As we shall see in the next section,numerical studies on realistic models ®nd a pronounced ef-fect of this type.IV. OPEN BOUNDARIESPeriodic boundary conditions are not realistic for typicalexperiments because the superconductor is a planar object.Therefore, a better choice is open boundaries if we have asituation with a ®nite number of chains. The DOS is calcu-05451lated numerically for N54,5,20,21, as shown in Figs. 1±5.Since we have just shown that D0 plays a minor role in theparity effect, we proceed ~for simplicity! under the assump-tion that D0 /t51 unless explicitly stated otherwise. In nu-merical work ~e.g., Fig. 6 below!, it is shown that this as-sumption does not change the qualitative results. From the®gures the alternating structure of the DOS for odd and evennumber of chains is immediately obvious, and may be char-acterized by the DOS of the end chain.A. Recursive methodSome insight into the dependence of these gaps and re-sidual DOS's on system size and other parameters may beobtained from analytical methods. First, the Hamiltonian Hcan be diagonalized with respect to y direction by a Fouriertransformation y→ky . Then the structure with respect to thex direction can be expressed in a matrix form for x51,2, . . . ,N asHÃN5S HNN HNN21 0 . . . 0HN21N HN21N21   A0    0A   H22 H210 . . . 0 H12 H11D .FIG. 1. Density of states for four and ®ve chains with D0 /t51 and m50.2-2DENSITY OF STATES WIDTH-PARITY EFFECT IN d- . . . PHYSICAL REVIEW B 64 054512For a compact notation we introduce a block-matrix repre-sentationHÃN :5S HNN HÃNN21HÃN21N HÃN21DwithHÃNN215~HNN21 ,0, . . . ,0 !, ~2!HÃN21N5S HN21N0A0D . ~3!The corresponding Green's function at zero energy, whichis required for the evaluation of the DOS, can then be written~after a shift H→H1ies0) asFIG. 2. Density of states for four chains: DOS of the end chainand average over all chains with D0 /t51 and m50.05451@HÃN21#NN5~HNN2HÃNN21HÃN2121 HÃN21N!215~HNN2HNN21@HÃN2121 #N21N21HN21N!21.The second equation is due to the special form of thematrices given in Eqs. ~2! and ~3!. Using GN5@HÃN21#NN therecurrence relation readsGN5~HNN2HNN21GN21HN21N!21. ~4!This is a continued-fraction representation of the 232 ma-trix GN , the Green's function of end chain labeled by N at agiven ky . It can be used even in the presence of randomterms along the x direction because no diagonalization of thematrix is necessary.Now we will apply the recursive method to evaluate theDOS of the end chain of N DWSC's. The calculation is sim-pli®ed greatly if we assumeD05t51 ~the more general caseis treated in numerical calculations!. In this special case wehaveFIG. 3. Density of states for ®ve chains: DOS of the end chainand average over all chains with D0 /t51 and m50.2-3K. ZIEGLER, W. A. ATKINSON, AND P. J. HIRSCHFELD PHYSICAL REVIEW B 64 054512HNN5S m12 cos ky1ie 22 cos ky22 cos ky 2m22 cos ky1ie D5~m12 cos ky!s322 cos kys11ies0[h1ies0~5!andHNN215HN21N5S 1 11 21 D 5s11s3 .The initial value ~single chain! isG15~H11!2152ies01~m12 cos ky!s322 cos kys1e21~m12 cos ky!214 cos2ky.B. Few chainsBy direct iteration of Eq. ~4! we can obtain the DOSanalytically for the end chain of a system of N chains, pro-vided we restrict consideration to m50. Using c52 cos ky ,for N51 we getFIG. 4. Density of states for 20 chains: DOS of the end chainand average over all chains with D0 /t51 and m50.05451r1~0 !5 lime→0epE2pp 1e212c2dky2p 51A8pin agreement with the result in Sec. III. For N52 we haver2~0 !5 lime→0epE2pp 1e212c212dky2p .From the pole structure of the integrand, we see that there isa gap 2Eg52A2'2.83. In the case of N53 there is again anonzero DOS at E50r3~0 !5 lime→0epE2pp e212c212~e212c2!214~e212c2!dky2p 512A8p.and for N54r4~0 !5 lime→0epE2pp e212c214~e212c2!216~e212c2!14dky2p ,which has a gap 2Eg52A32A5'1.75 ~see Fig. 1!. In gen-eral the energy gap observed depends on the size of the su-FIG. 5. Density of states for 21 chains: DOS of the end chainand average over all chains with D0 /t51 and m50.2-4DENSITY OF STATES WIDTH-PARITY EFFECT IN d- . . . PHYSICAL REVIEW B 64 054512perconducting order-parameter maximum D0 in a nonlinearway, although relatively simple expressions may be obtainedfor a small number of chains.The reader will notice that the even-odd effect describedhere differs from the periodic-boundary case, where gaplessstates arise for N a multiple of 4. This is expected, since theeffect we are reporting arises from self-interference of quan-tum wave functions in a nanoscale system. The path lengthfor a constructively interfering closed path in the x directionis 2L52np/kF for open boundary conditions, and L52np/kF for periodic boundary conditions.C. Many chains: Stationary behavior for N\`While in systems mesoscopic in both directions ~®nitelength L quantum wires!, even-odd parity effects in N areknown to survive the thermodynamic limit L→` ,10 theymust disappear as N→` when we recover the fully 2D sys-tem. This is evident from the numerical evaluation of theDOS for large N ~Figs. 4 and 5!, and we can see this analyti-cally by showing that as N→` , r(0)50 independent of N.A single iteration of Eq. ~4! yields a relation between theGreen's functions of an even ~or odd! number of chains,respectively,GN5~HNN2HNN21@HN21N212HN21N22GN22HN22N21#21HN21N!21. ~6!FIG. 6. Density of states for four and ®ve chains with D0 /t50.3 and m50.05451For the DWSC's this readsGN5HHNN2F12 ~s11s3!HN21N21~s11s3!2GN22G21J 21.For constant m50 we get from expression ~5!HN21N215h1ies0and12 ~s11s3!HN21N21~s11s3!52h1ies0such thatGN5~h1ies01@h2ies01GN22#21!21with the initial expressionsG151e212c2 ~h2ies0!andG25S 11 2e212c2D21G1 .The symmetric 232 matrix h can be diagonalized by anorthogonal transformation, leading to diagonal elementsl1/256A2c . This also diagonalizes the initial expressionsG1,2 as well as the recurrence relationgN12,j51l j1ie11l j2ie1gN , j~7!with diagonal elements gN ,1/2 of GN . Then the recursion ~7!has two ®xed pointsgÅj652z j*2 ~16A114/uz ju2!with z j5l j1ie52(21) jA2c1ie . For e.0 a positiveimaginary part of gN , j1 implies a positive imaginary part ofgN12,j1 . Since the imaginary parts of the initial values arepositive, only the ®xed pointgÅj1 can be reached in the caseunder consideration.The DOS of the end chain can be calculated from the®xed point and readsr~0 !521plime→0ImE2pp~gÅ111gÅ21!dky2p } lime→0e ln~1/e !50.This re ēcts the well-known result of the in®nite 2D DWSC,which has a linear pseudogap.2-5K. ZIEGLER, W. A. ATKINSON, AND P. J. HIRSCHFELD PHYSICAL REVIEW B 64 054512D. Realistic modelsThe preceding discussion has assumed a half-®lled tight-binding band and D0 /t51; both conditions correspond tospecial symmetries of the Hamiltonian that might be ex-pected to in ūence the form of the spectrum. The assumptionD0 /t51 was in fact made simply in order to obtain theanalytical results discussed in Secs. IVA±IVC, and it mayeasily be checked numerically ~Fig. 6! that for small valuesof D0 /t and m50, the qualitative even-odd effect in thechain width parity is still obtained. This is consistent with thequalitative results in the case of periodic boundaries.More signi®cant changes occur whenmÞ0. In Fig. 7 wesee that for an even number of chains, the full gap in allchains is preserved, while for an odd number, those chainsthat at half-®lling had a ®nite density of states have nowacquired a small gap. Analytic results for this case are com-plicated and not particularly enlightening, but it is clear thatfor m!D0, the small gap on the x5odd chains in the odd-Ncase is of the order of m itself. This is also evident by anal-ogy to Eq. ~1!, where the pole in the denominator of theintegral is shifted by the chemical potential.A more realistic parameter choice for the optimally dopedcuprates would correspond to a doping of 15%, or about m.0.3t for the simple tight-binding spectrum on a square lat-tice, and a much smaller gap magnitude, of order D050.1t .FIG. 7. Density of states for four and ®ve chains with D0 /t51 and m/t50.3.05451In this case the large gap is set by 2D0, but in the odd-chainsystems there is a much smaller gap on alternating chains,see Fig. 8. Thus the density of states still generally exhibits apronounced odd-even width-parity effect.V. CONCLUSIONSWe have exhibited, using both analytical and numericalmeans, a number-parity effect in the width N of a mesoscopicd-wave superconducting quantum wire, wherein a ®nite DOSis found at the Fermi level for odd N and zero DOS ~withenergy gap! is found for even N. The result should be ofsome practical interest in the not-so-distant future for nanos-cale d-wave wire structures. Superconductivity in single lay-ers of the cuprates has already been demonstrated and itseems plausible that a sample of controlled width might befabricated, and that a scanning-tunnel microscope experi-ment on such a sample would be able to observe the effectswe predict.It is interesting to note some unusual features of the sys-tem we treat here, to our knowledge for the ®rst time, in thepresence of impurities. First of all, although the supercon-ducting order parameter has d-wave symmetry, the systemmay ~for N even! have a full gap in the excitation spectrum.In such a situation, one may ask what is the effect of isolatedimpurities added to the system, and at least naively wouldFIG. 8. Density of states for four and ®ve chains with D0 /t50.1 and m/t50.3.2-6DENSITY OF STATES WIDTH-PARITY EFFECT IN d- . . . PHYSICAL REVIEW B 64 054512obtain single-impurity d-wave-like bound states in this gapwithout broadening arising from coupling to the quasiparticlecontinuum. Since the density of states in the odd-N systemshas an oscillatory behavior across the sample width, the ex-istence and lifetime of these states is expected to dependsensitively on their location in the wire as well.In addition, the observation we make here may be of in-terest to the study of the in ūence of disorder on the DOS offully 2D d-wave superconductors, a subject that has receivedintense attention recently.4 The result appears to be that theDOS is generically zero in a 2D d-wave superconductor atzero energy, but can be constant or divergent in cases mani-festing special symmetries. At the same time, Brouweret al.10 have shown in disordered quasi-1D systems with chi-ral symmetry that r(0) is zero or divergent according to thenumber parity of the wire width. It appears that the states in05451odd- or even-chain systems are modi®ed in different ways bythe combined effects of level repulsion and symmetry. Un-derstanding the differences in the odd or even approach tothe thermodynamic limit in the presence of disorder maygive insight into the physics of localization and DOS supres-sion in the 2D d-wave case, which is still poorly understood.We will address these questions in a subsequent work.ACKNOWLEDGMENTSThis work was supported by NSF Grants Nos. DMR-9974396 and INT-9815833 and an exchange grant fromDAAD. The authors are grateful to J. von Delft and C.Mudry for useful discussions. P.J.H. is grateful to the Insti-tute for Theoretical Physics for its hospitality during work onthe paper.1D.C. Ralph, C.T. Black, and M. Tinkham, Phys. Rev. Lett. 74,3241 ~1995!.2F. Braun and J. von Delft, Phys. Rev. Lett. 81, 4712 ~1998!.3A.D. Zaikin, D.S. Gobulev, A. van Otterlo, and G.T. Zimanyi,Phys. Rev. Lett. 78, 1552 ~1997!.4A.A. Nersesyan, A.M. Tsvelik, and F. Wenger, Nucl. Phys. B 438,561 ~1995!; Phys. Rev. Lett. 72, 2628 ~1994!; K. Ziegler, M.H.Hettler, and P.J. Hirschfeld, ibid. 77, 3013 ~1996!; Phys. Rev. B57, 10 825 ~1998!; T. Senthil, Matthew P.A. Fisher, LeonBalents, and Chetan Nayak, Phys. Rev. Lett. 81, 4704 ~1998!; T.Senthil and Matthew P.A. Fisher, Phys. Rev. B 60, 6893 ~1999!;C. PeÂpin and P.A. Lee, ibid. 63, 540502 ~2001!; W.A. Atkinson,P.J. Hirschfeld, A.H. MacDonald, and K. Ziegler, Phys. Rev.Lett. 85, 3926 ~2000!.5A. Kleiner and S. Eggert, cond-mat/0007244 ~unpublished!.6F.D.M. Haldane, Phys. Rev. Lett. 61, 1029 ~1988!.7E. Fradkin, Field Theory of Condensed Matter Systems ~Addison-Wesley, Redwood City, 1991!.8J. Miller and J. Wang, Phys. Rev. Lett. 76, 1461 ~1996!.9P.W. Brouwer, C. Mudry, B.D. Simons, and A. Altland, Phys.Rev. Lett. 81, 862 ~1998!.10P.W. Brouwer, C. Mudry, and A. Furusaki, Phys. Rev. Lett. 84,2913 ~2000!.11M. Fogelstrom, D. Rainer, and J.A. Sauls, Phys. Rev. Lett. 79,281 ~1997!.2-7

Density of states width-parity effect in d-wave superconducting quantum wires

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Density of states "width parity" effect in d-wave superconducting quantum wires

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