Motivated by recent discoveries of novel superconductors such as
Na$_x$CoO$_2\cdot y$H$_2$O and Sr$_2$RuO$_4$, we analysize features of
quasi-particle scattering due to impurities and interfaces for possible gapful
$d_{x^2-y^2}+id_{xy}$ and $p_x+ip_y$ Cooper pairing. A bound state appears near
a local impurity, and a band of bound states form near an interface. We
obtained analytically the bound state energy, and calculated the space and
energy dependent local density of states resolvable by high-resolution scanning
tunnelling microscopy. For comparison we also sketch results of impurity and
surface states if the pairing is nodal p- or d-wave.Comment: 4 pages, 4 figure

A. Tanaka

A.P. Mackenzie

A.V. Balatsky

B. Kumar

C. Honerkamp

C.C. Tsuei

C.R. Hu

D.E. Sheehy

D.J. Singh

D.K. Morr

E.W. Hudson

G. Baskaran

G.M. Zhang

H. Shiba

J.E. Hoffman

K. Ishida

K. McElroy

K. Sengupta

K. Takada

L. Yu

M. Matsumoto

M. Ogata

M.I. Salkola

P.W. Anderson

Qiang-Hua Wang

S.H. Pan

Y. Maeno

Z. D. Wang

English

arXiv

Crossref

Impurity and interface bound states in
                    
                      
                        
                          
                            d

Motivated by recent discoveries of novel superconductors such as NaxCoO2·yH2O and Sr2RuO4, we analyze features of quasiparticle scattering due to impurities and interfaces for possible gapful dx2–y2 + idxy and px + ipy Cooper pairing. A bound state appears near a local impurity, and a band of bound states form near an interface. We obtained analytically the bound-state energy, and calculated the space and energy dependent local density of states resolvable by high-resolution scanning tunnelling microscopy. For comparison we also sketch results of impurity and interface states if the pairing is nodal p or d wave.published_or_final_versio

Wang, ZD

Wang, QH

HKU Scholars Hub

Title Impurity and interface bound states in dx2–y2 + idxy and px +ipy superconductorsAuthor(s) Wang, QH; Wang, ZDCitation Physical Review B (Condensed Matter and Materials Physics),2004, v. 69 n. 9, p. 092502:1-4Issued Date 2004URL http://hdl.handle.net/10722/43411Rights Physical Review B (Condensed Matter and Materials Physics).Copyright © American Physical Society.PHYSICAL REVIEW B 69, 092502 ~2004!Impurity and interface bound states in dx2Ày2¿idxy and px¿ipy superconductorsQiang-Hua Wang1 and Z. D. Wang21National Laboratory of Solid State Microstructures, Institute for Solid State Physics, Nanjing University, Nanjing 210093, China2Department of Physics, University of Hong Kong, Pukfulam Road, Hong Kong, China~Received 18 November 2003; published 10 March 2004!Motivated by recent discoveries of novel superconductors such as NaxCoO2yH2O and Sr2RuO4, weanalyze features of quasiparticle scattering due to impurities and interfaces for possible gapful dx22y21idxyand px1ipy Cooper pairing. A bound state appears near a local impurity, and a band of bound states form nearan interface. We obtained analytically the bound-state energy, and calculated the space and energy dependentlocal density of states resolvable by high-resolution scanning tunnelling microscopy. For comparison we alsosketch results of impurity and interface states if the pairing is nodal p or d wave.DOI: 10.1103/PhysRevB.69.092502 PACS number~s!: 74.25.Jb, 71.27.1a, 74.20.2zRecently, Takada et al. discovered a novel superconductorNaxCoO2yH2O (x50.35) with a superconducting transitiontemperature Tc55K .1 A few features of this material bearstrong connection to cuprates: ~1! It has a layered structure.~2! As Cu21 in cuprates, Co41 atom is in a spin-1/2 stateaccording to first-principles calculation of Singh.2 Combinedwith the fact that the cobalt triangular lattice is frustrating toantiferromagnetic ordering, the new material offers a likelysituation for the physics of Anderson’s resonating valencebond ~RVB! theory.3 Soon after the discovery, theories basedon RVB physics4–7 and renormalization group analysis8 pre-dicted d1id8-wave pairing (d5dx22y2 and d85dxy), whileother theories suggest px1ipy-wave pairing derived fromthe weak ferromagnetic instability,9 in close analogy to thecase of Sr2RuO4.10 Identifying the pairing symmetry wouldbe a necessary step toward the understanding of the newsuperconductor. In this paper, we propose tunnelling mea-surements of impurity and interface states that are sensitiveto both the gap amplitude and the internal phase of the gapfunction. Such measurements have played invaluable roles inhigh-temperature superconductors in the context of nodald-wave pairing.11,12 Our main results are as follows. As aconsequence of the full gap as well as the internal phasedegrees of freedom of d1id8 and px1ipy Cooper pairs, abound state appears at any nonzero scattering intensity near alocal impurity and a band of bound states form near an in-terface. The bound-state energy is near the gap edge at weakscattering strength, and it approaches zero energy ~the Fermilevel! at increasing scattering strengths. We also calculatedthe energy and space dependent local density of states~LDOS!, whose rich features are directly resolvable by futurescanning tunnel microscope ~STM! and can help identify thepairing symmetry in the new superconductors. For compari-son, we also mention briefly the results of impurity and in-terface states for nodal p- and d-wave pairing.As usual the elastic scattering problem is best described interms of the retarded T-matrix formulation,G~ i , j !5G0~ i , j !1(a ,bG0~ i ,a !T~a ,b !G0~b , j !, ~1!where a ,b denotes the position of the impurities and all othernotations are standard. We suppressed the energy dependence0163-1829/2004/69~9!/092502~4!/$22.50 69 0925in Green’s functions, as it is conserved in elastic scattering.The T matrix is given by~T21!~a ,b !5~V21!~a ,b !2G0~a ,b !, ~2!where V(a ,b) is the general impurity potential that may beoff-diagonal. In our case, G, G0 and T are further 232 ma-trices in the particle-hole Nambu space. The scattering prob-lem is solved once G0 is known. The LDOS at site i is givenbyN~ i ,v!52Im@G11~ i ,i;v!1G22~ i ,i;2v!#/p ~3!with the energy argument v restored. A peak in N(i ,v) ap-pears if either G11(i ,i;v) or G22(i ,i;2v) diverges. Thiscorresponds to a bound/resonance state if Det@T21(6v)#50 occurs at real/complex energy v , behaving as aneigenstate/virtual state with finite lifetime. Due to the mixingof particle and hole in the presence of pairing, it is possiblethat there are two peaks in N(i ,v) but Det(T21)50 is sat-isfied at only one energy, or vice versa. In the followingdiscussion, we always count the bound/resonance states ac-cording to the peaks seen in the total density of statesN(i ,v).Let us write the gap function as , in the momentum space,Dk5Deiluk where D is the gap amplitude, uk is the azi-muthal angle of the vector k and l50,61,62 for gapful s-,p-, and d-wave pairing, respectively. We include the case ofs-wave pairing for comparison. The above pairing function isof simplified form, suitable near the normal-state Fermi sur-face, and suffices for qualitative discussion of low-energyquasiparticle states. Then G0(i , j)5G0(r) ~with r5ri2rj) isgiven byG0~r!5E d2k~2p!2v1s01eks31D(neinluksnv12 2ek22D2eikr5pN0S v1J0~kFr !s01DJl~kFr !(neinlursnDAD22v12,~4!©2004 The American Physical Society02-1BRIEF REPORTS PHYSICAL REVIEW B 69, 092502 ~2004!where v15v1i01, n56 , ek is the normal-state energydispersion, s0 is the 232 unit matrix, s1,2,3 are thePauli matrices, s65(s16is2)/2, and Jl(u)5*02pdu cos(lu)exp(iu cos u)/2p is the Bessel function. Inarriving at the above results, we have assumed a cylindricalFermi surface with Fermi vectors of magnitude kF , and con-stant density of states N0 near the Fermi level. We emphasizethat a particle-hole asymmetry is present in the normal stateDOS of NaxCoO2. We shall comment on such effects with-out going into details in the following qualitative discus-sions.I. SCATTERING FROM A LOCAL IMPURITYIn this case we set the impurity site at the origin, i.e., a5b50, and drop these indices in V5Vms01Vss3 andT215V212G0(0,0), where Vm ,s is the strength of ~classi-cal! magnetic/scalar potential. With G0 in Eq. ~4! at hand,the T matrix is now given byT215V211pN0AD22~v1!2~v1s01Dd0ls1!. ~5!One sees that Im(T21)→0 in the subgap regime v2,D2, sothat a true bound state could be generated since the conditionDet(T21)50 could be satisfied at real v .13 This should becontrasted to the resonant impurity state in the case of nodald-wave pairing for which the condition is met in general atcomplex v .14 The condition is governed by the dimension-less scattering strengths cm ,s5pN0Vm ,s . A few cases areclassified as follows.For p- and d-wave pairing, the off-diagonals1-component in T21 is zero. This is not an accidental resultfrom the adopted approximation, but rather a rigorous resultfrom the pairing symmetry, which forbids the on-site pairingamplitude @related to the anomalous part of G0(0)] to befinite. Consequently, both scalar and magnetic impurities cangenerate bound states. ~1! For a scalar impurity, Det(T21)50 is satisfied atFIG. 1. Density of states as a function of energy v and the radialdistance r off a scalar impurity. See the text for details.09250vb56uDu/A11cs2. ~6!In general this implies two peaks in the LDOS according toEq. ~3!. However, depending on the ratio vb2/D2 one of thepeaks may dominate over the other, with an associatedchange in the spatial dependence of the LDOS. We present afew examples of the energy and space dependent LDOS inFig. 1 for Cooper pairing with l51,2. For the weak impuritycase cs50.5 in Figs. 1~a! and 1~c!, vb is near the gap edge,the dominant peak is at the energy with opposite sign to cs ,and the corresponding DOS right at the impurity site is maxi-mal. In contrast, for the strong impurity case cs510 in Figs.1~b! and 1~d!, vb is approaching the Fermi level ~zero en-ergy!, the dominant DOS peak energy has the same sign asthat of cs , and the corresponding DOS is vanishing right atthe impurity site. Note that the cusplike feature at v56Daway from the impurity is just a feature of the bulk DOSN(v). ~2! For a magnetic impurity, Det(T21)50 is satisfiedatvb52sign~cm!uDu/A11cm2 . ~7!Since T21}s0 in this case, there are actually two peaks inDOS, according to Eq. ~3!, located symmetrically with re-spect to the Fermi level. Examples are shown in Fig. 2, incomparison to Fig. 1. By inspection, we see that except forthe symmetrical peaks, Fig. 2 are basically similar to Fig. 1.On the other hand, in both scalar and magnetic impuritycases the difference between l51 and l52 is mild. Thiswould pose difficulty for STM to resolve this quantum num-ber. Fortunately this can be resolved easily by other meanssuch as spin susceptibility measurements from the fact thatsinglet pairing (l52 here! forms a gap for spin excitationswhile triplet pairing (l51 here! does not.It is pertinent at this stage to comment on the effect ofparticle-hole asymmetry in the normal state Fermi surface.As can be seen from the derivation of G0, this would intro-duce a s3 component in G0(0,0), which effectively acts asan excess energy-dependent scalar potential in T21. There-FIG. 2. The same plot as Fig. 1 but for a magnetic impurity.2-2BRIEF REPORTS PHYSICAL REVIEW B 69, 092502 ~2004!fore, the effect is to modify the bound-state energy, and tobreak the symmetry of the bound-state energies in the case ofmagnetic scattering.II. SCATTERING FROM AN INTERFACEWe shall model an interface by an extended line of impu-rities. This could be fabricated by chemical erosion. It is toour advantage in that the T-matrix formalism can still beapplied. Since the unperturbed system at hand has rotationalsymmetry, the interface states should not depend on the sur-face normal direction nˆ , which we fix to be nˆ 5yˆ for defi-niteness. Due to the remaining translation symmetry alongthe x axis, we can do partial Fourier transforms of Eqs. ~1!,~2!, and ~4! with respect to x to find the reduced T-matrixequations at the x direction wave vector q asg~yi ,y j!5g0~yi2y j!1g0~yi!tg0~2y j!, ~8!g0~y !;2pN0Fv1cos pys01D(ncos~py1nluq!snG2pAD22v12,~9!t215v2112pN0pAD22v12~v1s01D cos luqs1!, ~10!where v5Vms01Vss3 is the same as the form of a singleimpurity, p5AkF2 2q25kFusin uqu and uq5arc cos(q/kF). Theconserved momentum q is suppressed in the arguments of g,g0 and t21 for brevity. The problem is reduced to an effec-tive single impurity scattering in one dimension. With theimplicit v and q arguments restored, the partial DOS is givenby N(v ,y ;q)52Im@g11(v ,y ;q)1g22(2v ,y ;q)#/p , andthe total density of states is N(v ,r)5*dqN(v ,y ;q)/2p ,which is independent of x due to the translation symmetry.~The integration over q should be cutoff at 6kF .) AgainDet(t21)50 would predict a bound state.FIG. 3. Dispersion of the positive bound-state energy as a func-tion of the wave vector along the interface. See the text for details.09250Although the above formulation is versatile to deal withany value of Vm ,s , we shall consider only the more likelyscalar interface with Vm50. It is easy from the above equa-tions that bound states occur at energies given byvb56DA4cs2cos2luq1kF2 sin2uq4cs21kF2 sin2uq , ~11!which clearly form two bands. ~Note that we have taken thelattice constant to be unity so that kF is dimensionless.! It isalso clear that no subgap bound states exist for s-wave pair-ing (l50).The dispersion of the positive bound-state energy is plotin Fig. 3 for weak (cs50.5, thick lines! and strong (cs510, thin lines! interface with p wave (l51, solid lines! andd-wave (l52, dashed lines! pairings. Here we have set kF5p/2 for calculation. One sees that the energy dispersesnear the gap edge for weak scattering interfaces, and it tendsto cover the whole subgap regime for strong interface scat-tering. Furthermore, the difference between p- and d-wavepairing is reflected in the number of minima, being identicalto l, in the dispersion.The spatial dependence of the LDOS near the interfacecan be calculated from the above theory. Examples areshown in Fig. 4. Consistent with the above bound-state en-ergy dispersion, the subgap states are near the gap edge ~ortend to cover the whole gap regime! for a weak ~or strong!scattering interface, distributed more or less symmetrically~or asymmetrically! with respect to the Fermi level. In thelimit of unitary scattering cs→‘ ~not shown here! the LDOSbecomes symmetrical in energy again. An interesting featurein Figs. 4~b! and 4~d! is that the peaks or bumps in energyoscillate with increasing distance from the interface, formingwavelike pattern in the energy-distance space. Moreover, thepeaks at lowest absolute energies in Fig. 4 can be related tothe dips in the dispersion relations in Fig. 3. This is becausethe dips corresponds to a large contribution to the density ofFIG. 4. Density of states as a function of energy v and thedistance d off an interface. See the text for details.2-3BRIEF REPORTS PHYSICAL REVIEW B 69, 092502 ~2004!states. It is also clear from Fig. 4 that the spatial profiledecays much more slowly than the single impurity cases inFigs. 1 and 2.We note that Matsumoto and Sigrist16 have addressed thequasiparticle states near a sample surface and a topologicaldomain wall ~with a p phase shift in the pairing gap! forpx1ipy-wave pairing in terms of quasiclassical theory. Theyfound that subgap states appear near the domain wall but notthe surface. Our interface is actually a nontopological do-main wall but with potential scattering.III. THE CASE OF NODAL p- AND d-WAVE PAIRINGAlong similar lines to that sketched above, we have alsoconsidered the impurity and interface states for nodal p- andd-wave pairing for comparison. The gap function may bewritten as Dk5D sin luk or Dk5D cosluk , depending onwhether one of the nodal or antinodal directions is along thex-axis. Note that there are only one nodal and one antinodaldirection for p-wave pairing. Due to limited space we sketchthe results without going into details.For the local impurity case, we found resonant energies atvr;6pD/@ lcm ,sln(4lcm,s /p)# for a scalar/magnetic impu-rity. This reduces to the known result in the case of nodald-wave pairing (l52).14 The new features for the case ofnodal p-wave pairing is that that LDOS pattern near the im-purity is twofold symmetric, forming stripelike features ex-tending along the antinodal direction, which should be com-pared to the fourfold symmetric pattern in the case of nodald-wave pairing.14,15,17,18On the other hand, the interface states depend on the in-terface orientation: ~1! If the ~scalar scattering! interface isalong one of the nodal directions, say xˆ , there are boundstates at energies vb56DukFsin uqsin luqu/A4cs21kF2 sin2uq.09250The definition of uq is the same as in Sec. II. All theseenergies approach zero in the unitary limit cs→‘ . The abun-dance of zero-energy states is due to the fact that in this limitquasiparticles reflect spectacularly from the interface, expe-riencing a sign change of the gap. The same physics is nicelydescribed in Refs. 19,20 in other contexts. ~2! Finally if theinterface is along one of the antinodal directions, redefinedalso as xˆ , there are resonant states exactly at vr56D cos luq irrespectively of the scattering strength. In factthis is equivalent to the case of s-wave pairing but with aq-dependent gap amplitude.IV. DISCUSSIONWe have only shown results for the cases l561,62 thatare relevant in the new superconductors, but the theory isclearly general for any integer value of l. There are somedetails missing in the theory, however. First, it does not takeinto account possible anisotropy in the normal-state Fermisurface. For example, in NaxCoO2, the Fermi surface has arounded hexagonal structure.2 Such anisotropy may causecorresponding anisotropic LDOS pattern around impurities.Second, Eq. ~4! is obtained by fixing the momentum on theFermi surface while integrating over energy. This possiblyleaves out an excess decay of G0 in space with the lengthscale j5vF /D . Apart from such details, our qualitative ana-lytical results are robust.ACKNOWLEDGMENTSThis was supported by NSFC ~Grants Nos. 10204011,10325416, and 10021001!, the Ministry of Science and Tech-nology of China ~Grant No. NKBRSF-G1999064602!, theURC fund of HKU, and the RGC grant ~Grant No.HKU7050/03P!.1 K. Takada, H. Sakurai, E. Takayama-Muromachi, F. Izumi, R.A.Dilanian, and T. Sasaki, Nature ~London! 422, 53 ~2003!.2 D.J. Singh, Phys. Rev. B 61, 13 397 ~2000!.3 P.W. Anderson, Science 235, 1196 ~1987!.4 G. Baskaran, Phys. Rev. Lett. 91, 097003 ~2003!.5 B. Kumar and B.S. Shastry, Phys. Rev. B 68, 104508 ~2003!.6 Qiang-Hua Wang, Dung-Hai Lee, and P.A. Lee,cond-mat/0304377 ~unpublished!.7 M. Ogata, J. Phys. Soc. Jpn. 72, 1839 ~2003!.8 C. Honerkamp, Phys. Rev. B 68, 104510 ~2003!.9 A. Tanaka and X. Hu, Phys. Rev. Lett. 91, 257006 ~2003!.10 Y. Maeno et al., Nature ~London! 372, 532 ~1994!; K. Ishidaet al., Phys. Rev. B 56, R505 ~1997!; A.P. Mackenzie et al.,Phys. Rev. Lett. 80, 161 ~1998!.11 C.C. Tsuei, J.R. Kirtley, C.C. Chi, L.S. Yu-Jahnes, A. Gupta, T.Shaw, J.Z. Sun, and M.B. Ketchen, Phys. Rev. Lett. 73, 593~1994!.12 S.H. Pan, E.W. Hudson, K.M. Lang, H. Eisaki, S. Uchida, andJ.C. Davis, Nature ~London! 403, 746 ~2000!; E.W. Hudson,S.H. Pan, A.K. Gupta, K.W. Ng, and J.C. Davis, Science 285, 88~1999!; J.E. Hoffman, K. McElroy, D.H. Lee, K.M. Lang, H.Eisaki, S. Uchida, and J.C. Davis, ibid. 297, 1148 ~2002!; K.McElroy, R.W. Simmonds, J.E. Hoffman, D.H. Lee, J. Orein-stein, H. Eisaki, S. Uchida, and J.C. Davis, Nature ~London!422, 592 ~2003!.13 For the s wave case l50, the theory predicts that a scalar impu-rity does not generate subgap bound states, but magnetic onedoes. See also L. Yu, Acta Phys. Sin. 21, 75 ~1965!; H. Shiba,Prog. Theor. Phys. 40, 435 ~1968!; M.I. Salkola, A.V. Balatsky,and J.R. Schrieffer, Phys. Rev. B 55, 12 648 ~1997!; D.K. Morrand N.A. Stavropoulos, Phys. Rev. B 67, 020502 ~2003!.14 A.V. Balatsky, M.I. Salkola, and A. Rosengren, Phys. Rev. B 51,15 547 ~1995!; M.I. Salkola, A.V. Balatsky, and D.J. Scalapino,Phys. Rev. Lett. 77, 1841 ~1996!.15 Qiang-Hua Wang, and Dung-Hai Lee, Phys. Rev. B 67, 020511~2003!.16 M. Matsumoto and M. Sigrist, J. Phys. Soc. Jpn. 68, 994 ~1999!.17 Qiang-Hua Wang, Phys. Rev. Lett. 88, 057002 ~2002!.18 G.M. Zhang, H. Hu, and L. Yu, Phys. Rev. B 66, 104511 ~2002!.19 C.R. Hu, Phys. Rev. Lett. 72, 1526 ~1994!; K. Sengupta, H.J.Kwon, and V.M. Yakovenko, Phys. Rev. B 65, 104504 ~2002!.20 D.E. Sheehy, I. Adagideli, P.M. Goldbart, and A. Yazdani, Phys.Rev. B 64, 224518 ~2001!.2-4

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Impurity and interface bound states in dx2–y2 + idxy and px + ipy superconductors

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Impurity and interface bound states in $d_{x^2-y^2}+id_{xy}$ and
  $p_x+ip_y$ superconductors

Impurity and interface bound states in $d_{x^2-y^2}+id_{xy}$ and $p_x+ip_y$ superconductors

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Impurity and interface bound states in dx2−y2+idxyd_{x^2-y^2}+id_{xy}dx2−y2​+idxy​ and px+ipyp_x+ip_ypx​+ipy​ superconductors

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Impurity and interface bound states in $d_{x^2-y^2}+id_{xy}$ and $p_x+ip_y$ superconductors