Using a model Hamiltonian with d-wave superconductivity and competing
antiferromagnetic (AF) interactions, the temperature (T) dependence of the
vortex charge in high T_c superconductors is investigated by numerically
solving the Bogoliubov-de Gennes equations. The strength of the induced AF
order inside the vortex core is T dependent. The vortex charge could be
negative when the AF order with sufficient strength is present at low
temperatures. At higher temperatures, the AF order may be completely suppressed
and the vortex charge becomes positive. A first order like transition in the T
dependent vortex charge is seen near the critical temperature T_{AF}. For
underdoped sample, the spatial profiles of the induced spin-density wave and
charge-density wave orders could have stripe like structures at T < T_s, and
change to two-dimensional isotropic ones at T > T_s. As a result, a vortex
charge discontinuity occurs at T_s.Comment: 5 pages, 5 figure

A. Ghosal

B. Lake

C. S. Ting

Ch. Renner

D.I. Khomskii

D.P. Arovas

E. Demler

G. Blatter

H.D. Chen

J. Kolacek

Jian-Xin Zhu

K. Kumagai

K.M. Lang

M. Franz

M. Ichioka

M. Takigawa

N. Hayashi

N. Ogata

P. Lipavsky

R.I. Miller

S.H. Pan

T. Nagaoka

V.F. Mitrovic

Y. Wang

Yan Chen

Z. D. Wang

Z.D. Wang

English

arXiv

Crossref

Temperature dependence of vortex charges in high-temperature superconductors

Using a model Hamiltonian with d-wave superconductivity and competing antiferromagnetic (AF) interactions, the temperature (T) dependence of the vortex charge in high-T c superconductors is investigated by numerically solving the Bogoliubov-de Gennes equations. The strength of the induced AF order inside the vortex core is T dependent. The vortex charge could be negative when the AF order with sufficient strength is present at low temperatures. At higher temperatures, the AF order may be completely suppressed and the vortex charge becomes positive. A first-order-like transition in the T-dependent vortex charge is seen near the critical temperature T AF. For an underdoped sample, the spatial profiles of the induced spin-density wave and the charge-density wave orders could have stripelike structures at TT s. As a result, a vortex charge discontinuity occurs at T s.published_or_final_versio

Chen, Y

Wang, ZD

Ting, CS

HKU Scholars Hub

Title Temperature dependence of vortex charges in high-temperaturesuperconductorsAuthor(s) Chen, Y; Wang, ZD; Ting, CSCitation Physical Review B - Condensed Matter And Materials Physics,2003, v. 67 n. 22, p. 2205011-2205014Issued Date 2003URL http://hdl.handle.net/10722/43386Rights Creative Commons: Attribution 3.0 Hong Kong LicenseRAPID COMMUNICATIONSPHYSICAL REVIEW B 67, 220501~R! ~2003!Temperature dependence of vortex charges in high-temperature superconductorsYan Chen,1 Z. D. Wang,2,3 and C. S. Ting11Texas Center for Superconductivity and Department of Physics, University of Houston, Houston, Texas 77204, USA2Department of Physics, University of Hong Kong, Pokfulam Road, Hong Kong, China3Department of Material Science and Engineering, University of Science and Technology of China, Hefei 230026, China~Received 25 February 2003; published 3 June 2003!Using a model Hamiltonian with d-wave superconductivity and competing antiferromagnetic ~AF! interac-tions, the temperature ~T! dependence of the vortex charge in high-Tc superconductors is investigated bynumerically solving the Bogoliubov–de Gennes equations. The strength of the induced AF order inside thevortex core is T dependent. The vortex charge could be negative when the AF order with sufficient strength ispresent at low temperatures. At higher temperatures, the AF order may be completely suppressed and thevortex charge becomes positive. A first-order-like transition in the T-dependent vortex charge is seen near thecritical temperature TAF . For an underdoped sample, the spatial profiles of the induced spin-density wave andthe charge-density wave orders could have stripelike structures at T,Ts , and change to two-dimensionalisotropic ones at T.Ts . As a result, a vortex charge discontinuity occurs at Ts .DOI: 10.1103/PhysRevB.67.220501 PACS number~s!: 74.20.2z, 74.25.JbFor the past several years, there have been intensive stud-ies on the vortex physics in high-temperature superconduct-ors ~HTS!. It is well established now that the d-wave super-conductivity ~DSC! with a competing antiferromagnetism~AF! order plays an important role in determining the vortexstructure of HTS. Many theoretical studies1–10 have shownthat the AF order may appear and coexist with the underlyingvortices. Experimental facts including neutron-scattering,11moun spin rotation measurement,12 and nuclear-magnetic-resonance ~NMR! experiments13,14 have provided a strongsupport to the existence of AF order inside the vortex core inappropriately doped HTS. On the other hand, the electricalcharge associated with the vortex in superconductors has alsobeen paid considerable attention both theoretically15–20 andexperimentally.21,22 Based on the BCS theory, Blatter et al.16pointed out that for an s-wave superconductor, the vortexcharge is proportional to the slope of the density of states atthe Fermi level. However, the NMR and nuclear quadrupoleresonance ~NQR! measurements on YBa2Cu3O7 andYBa2Cu4O8 ~Ref. 21! seemed to obtain results for the vortexcharge, contradictory to that predicted from the existing BCStheory with respect to both the sign and the order of magni-tude. In addition, if the impact of vortex charge on the mixedstate Hall signal is considered,15 the sign estimated from Halleffect experiments22 for various HTS materials disagreeswith the prediction of BCS-type theory. In view of thesepoints, together with the fact that the strong electron corre-lation with the d-wave superconducting pairing was not con-sidered for the vortex charge, we carried out the previousstudies20 at zero temperature and found that the vortexcharge in HTS is strongly influenced by the presence of theinduced AF order in the vortex core. The charge carried by avortex is always positive for a pure d-wave superconductor,and it becomes negative when there is sufficient strength ofAF order in the vortex core.20 Since most relevant experi-mental phenomena have been observed at finite T not closeto zero, including the sign change of the mixed state Halleffect,22 it is necessary and valuable to study theoretically thevortex charges at finite T, which so far have not been ad-dressed in the literature.0163-1829/2003/67~22!/220501~4!/$20.00 67 2205In this paper, we shall investigate the vortex charge as afunction of T for both optimally doped and underdoped HTSin detail. Our calculation will be based on the model Hamil-tonian on a square lattice with a nearest-neighboring attrac-tive interaction VDSC describing the DSC and an on-siteCoulomb repulsion U representing the competing AF order.We shall adopt a well-developed numerical scheme23 tostudy the vortex structure. Our numerical results at finite Tconfirm that the sign of the vortex charge is still determinedby the AF order in the vortex core, being consistent with ourprevious work.20 The vortex charge could be transformedfrom negative to positive values as T is increased to a criticalvalue TAF such that the AF order is completely suppressed atT.TAF , and the transition seems to be a first-order-like. Foran underdoped sample, we show that the symmetry of theinduced spin-density wave ~SDW! and the charge-densitywave ~CDW! may change from stripelike to two-dimensional-like at a critical temperature Ts . A discontinuityof the T-dependent vortex charge also appears at Ts .The effective mean-field Hamiltonian describing bothDSC and SDW orders on a two-dimensional lattice can berepresented asH52 (i , j ,st i jc is† c js1(i ,s~Unis¯ 2m!cis† cis1(i , j~D i jc i↑† c j↓† 1H.c.!, ~1!where cis† is the electron creation operator and m is thechemical potential. In the presence of magnetic field B per-pendicular to the plane, the hopping integral can be ex-pressed as t i j5t0exp@i(p/F0)*rjriA(r)dr# for the nearest-neighboring sites i and j. A Landau gauge A5(2By ,0,0) ischosen. The two possible SDW and DSC orders in cupratesare defined as D iSDW5U^ci↑† ci↑2ci↓† ci↓& and D i j5VDSC^ci↑c j↓2ci↓c j↑&/2, where U and VDSC represent, re-spectively, the interaction strengths for the two orders. The©2003 The American Physical Society01-1RAPID COMMUNICATIONSYAN CHEN, Z. D. WANG, AND C. S. TING PHYSICAL REVIEW B 67, 220501~R! ~2003!mean-field Hamiltonian ~1! can be diagonalized by solvingthe resulting Bogoliubov–de Gennes equations self-consistently,(j S Hi j ,s D i jD i j* 2Hi j ,s¯* D S u j ,snv j ,s¯n D 5EnS ui ,snv i ,s¯n D , ~2!where the single-particle Hamiltonian Hi j ,s52t i j1(Unis¯ 2m)d i j , and ni↑5(nuui↑n u2 f (En), ni↓5(nuv i↓n u2@12 f (En)# , and D i j5(VDSC/4)(n(ui↑n v j↓n*1v i↓* u j↑n )tanh(En/2kBT), with f (E) as the Fermi distribu-tion function. The DSC order parameter at the ith siteis D iD5(D i1ex ,iD 1D i2ex ,iD 2D i ,i1eyD 2D i ,i2eyD )/4, where D i jD5D i jexp@i(p/F0)*ri(ri1rj)/2A(r)dr# and ex ,y denotes the unitvector along x and y directions. In our calculation, the relatedparameters are chosen as t5a51, VDSC51.2, the linear di-mension of the unit cell of the vortex lattice is Nx3Ny540320. This choice corresponds the magnetic field B.37 T.In the absence of the magnetic field and for optimallydoped sample (d50.15), the parameters are chosen in such away that only DSC ~with Tc;0.164) prevails in the system.Our numerical results with U52.2 indeed show that the AForder is induced at low temperature and is suppressed whenT is larger. The spatial profiles of the superconductivity orderparameter, staggered magnetization, and the charge densitynear the vortex core are plotted at two different temperaturesin Fig. 1. Here, the staggered magnetization of the inducedAF or SDW order is defined as M si 5(21) iD iSDW/U . The AForder is generated in the region where the DSC order param-eter is partially suppressed. The low-temperature results atT50.02 are shown in Figs. 1~a! to 1~c!, where the AF orderis nucleated and spreads out from the core center, and thevortex charge is negative. The induced AF order behavessimilar to a two-dimensional SDW with the same wavelengthin the x and y directions. The higher-temperature results atT50.05 are shown in Figs. 1~d! to 1~f!, where the AF orderis completely suppressed and the vortex charge becomespositive. The appearance of the SDW order pinned by thevortex lattice at low T strongly enhances the net electrondensity ~or depletion of the hole density! at the vortex core asshown in Fig. 1~c!. The present result is consistent with thatof Ref. 20 where the doping and U-value dependences ofvortex charges are examined. Both works reach the sameconclusion: the vortex charge is negative when a sufficientstrength of SDW order is induced, while it becomes positivewith the suppression of the SDW order due to increasing theT or doping level, or decreasing the U value.Figure 2~a! plots the phase diagram of T versus U forpositively ~hole-rich! and negatively ~electron-rich! chargedvortices. It is obvious that the AF vortex core can easily beinduced in the low-temperature regime or with a stronger AFinteraction U, while the pure DSC core tends to exist in thehigh T or smaller U. The electron density inside the core ishigher than the average density in the low-temperature re-gion, while the electron density becomes lower than the av-erage in the high-temperature region. There exists a clearboundary between these two phases. To estimate the vortex22050charge, we first determine the vortex size by examining thespatial profile of DSC order parameter. Next, we sum overthe net electron density inside the vortex core. In Fig. 2~b!,we depict the chemical potential versus T at a fixed dopinglevel (d50.15 with U52.4 and U52.2). The result ~solidline! for U52.4 exhibits a first-order-like transition at TAF;0.077, indicating the existence of induced AF order belowthis critical temperature. For U52.2, the result ~dashed line!FIG. 1. Spatial variations of the DSC order parameter D iD @~a!and ~d!#, staggered magnetization M is @~b! and ~e!#, and net electrondensity ni @~c! and ~f!# in a 20320 lattice. The left panels @~a!–~c!#and the right panels @~d!–~f!# are for T50.02 and T50.05,respectively. The averaged electron density is fixed at n¯50.85and U52.2.FIG. 2. ~a! Phase diagram of interaction strength U versus tem-perature T for positively and negatively charged vortices at optimaldoping d50.15. ~b! Temperature dependence of chemical potentialfor U52.4 and U52.2.1-2RAPID COMMUNICATIONSTEMPERATURE DEPENDENCE OF VORTEX CHARGES IN . . . PHYSICAL REVIEW B 67, 220501~R! ~2003!seems to show a very weak discontinuity which can hardlybe seen in Fig. 2~b!. The above finding may be closely re-lated to the experimental results for underdoped TlBa2Cu4O8~Ref. 14! where the vortex core with induced AF order belowTAF was reported. Also from Fig. 2~b!, a discontinuity in theslope of the T-dependent chemical potential at TAF for bothcases can be clearly observed. We expect that the local den-sity of states ~LDOS! of the vortex core should have doublepeaks5,7,10,25,26 near zero bias for T,TAF ; while for T.TAF only a single broad zero-bias peak can appear in theLDOS as one of the essential characteristics of DSC.23 Itwould be interesting to measure this temperature evolution inthe LDOS by future scanning tunnel microscopy ~STM! ex-periments. Also, interestingly, even though the origin of theHall sign anomaly in HTS is still debatable,24 the vortexcharge could make an additional contribution to the signchange in the mixed state Hall conductivity.15 As a result, wepredict that a sign reversal in the Hall signal may occur at Tnot too far below TAF .In Fig. 3, we plot the vortex charge number Qv /e as afunction of T for three typical U values, where e,0 is thecharge of an electron. The T dependence of the induced stag-gered magnetization M sc at the vortex core center has alsobeen examined in previous studies27,28 where the induced AForder is shown to disappear for T.TAF . From Fig. 3 and forU52.4, one can clearly see an abrupt jump in the number ofvortex charge Qv /e as T varies around TAF;0.077, which isconsistent with the critical temperature as shown in Fig. 2~b!.This positive to negative vortex charge transition seems to befirst-order-like. The magnitude of the discontinuity reducesto one-third when U52.2 at TAF;0.04. Here, the disconti-nuity occurs at the positive to positive vortex charge transi-tion, not the positive to negative transition as studied in thecase of U52.4. This is because slightly below TAF , theinduced AF order is too weak to make the vortex chargenegative. For a pure DSC case with U50, the vortex chargeQv is always positive and its magnitude first increasesFIG. 3. The temperature dependence of the number of vortexcharge Qv /e for U52.4 ~solid line!, U52.2 ~dotted line!, andU50 ~dashed line!. The averaged electron density is fixedat n¯50.85.22050slightly with T and then decreases to zero as T approaches toTc . The critical temperature TAF is U-value dependent orsample dependent. The larger U corresponds to larger TAF .These results confirm that the vortex charge is strongly in-fluenced by two competing effects: the suppression of theDSC order at the core center leading to the depletion ofelectrons, and the induction of the AF order which favors theaccumulation of electrons. The condition for the negativevortex charge to appear depends solely on whether there issufficient AF order inside the vortex core. We emphasize thatour results are robust and indifferent to band parameters, andshould give a qualitative description on the vortex physicsin HTS.In the following, we are going to investigate the tempera-ture dependence of the vortex charge for the underdopedHTS (d50.12). For U52.4, the spatial profiles of the stag-gered magnetization and electron density near the vortexcore at three different temperatures are plotted in Fig. 4. Asdiscussed in our previous work,7 the parameters used to ob-tain Fig. 4 yield stripelike structures in the DSC, SDW, andCDW order parameters at very low-temperature. In thepresent work, the DSC and SDW orders coexist in under-doped sample below Tc , which implies TAF.Tc . So thequestion arises: will these stripelike structures persist withincreasing temperature? From our calculations, the spatialprofiles of these order parameters may go through three dif-FIG. 4. Spatial variations of the local electron density ni forvarious temperatures ~a! T50.0, ~b! T50.11, and ~c! T50.17. Thestrength of the on-site repulsion U52.4. The averaged electrondensity is fixed at n¯50.88.1-3RAPID COMMUNICATIONSYAN CHEN, Z. D. WANG, AND C. S. TING PHYSICAL REVIEW B 67, 220501~R! ~2003!ferent symmetries as T increases. At T50, the spatial distri-bution of SDW (M s) and CDW (ni) orders exhibit quasi-one-dimensional stripelike behavior @see Figs. 4~a! and 4~b!#.When T is raised to T50.11 above a characteristic tempera-ture Ts (;0.1), the symmetry of the patterns change to iso-tropic and two dimensional @see Figs. 4~c! and 4~d!#. AboveTc (;0.16) and at T50.17, the residual AF and electrondensity become uniform, as shown in Figs. 4~e! and 4~f!. TheAF order vanishes when T.TAF .Finally, we display the temperature evolution of the vor-tex charge number Qv /e for the underdoped case d50.12with U52.4 and U52.2 in Fig. 5. Comparing with Fig. 3,the magnitudes of vortex charges for both cases in Fig. 5 atlow-temperatures become larger as a result of stronger AForder near the vortex core in underdoped samples. For U52.4, the Qv /e versus temperature curve ~see the solid line1 D.P. Arovas et al., Phys. Rev. Lett. 79, 2871 ~1997!.2 N. Ogata, Int. J. Mod. Phys. B 13, 3560 ~1999!.3 E. Demler et al., Phys. Rev. Lett. 87, 067202 ~2001!.4 M. Ichioka et al., J. Phys. Soc. Jpn. 70, 33 ~2001!.5 Jian-Xin Zhu, and C.S. Ting, Phys. Rev. Lett. 87, 147002 ~2001!.6 H.D. Chen et al., Phys. Rev. Lett. 89, 137004 ~2002!.7 Yan Chen and C.S. Ting, Phys. Rev. B 65, 180513~R! ~2002!.8 Yan Chen et al., Phys. Rev. B 66, 104501 ~2002!.9 M. Franz et al., Phys. Rev. Lett. 88, 257005 ~2002!.10 A. Ghosal et al., Phys. Rev. B 66, 214502 ~2002!.11 B. Lake et al., Science 291, 1759 ~2001!; B. Lake et al.,cond-mat/0104026 ~unpublished!.12 R.I. Miller et al., Phys. Rev. Lett. 88, 137002 ~2002!.13 V.F. Mitrovic et al., Nature ~London! 413, 501 ~2001!;cond-mat/0202368, Phys. Rev. B ~to be published 1 June 2003!.14 K. Kakuyanagi, K. Kumagai, Y. Matsuda, and M. Hasegawa,cond-mat/0206362 ~unpublished!.FIG. 5. The vortex charge number Qv /e for U52.4 ~solid line!and U52.2 ~dotted line! as a function of temperature T. The aver-aged electron density is fixed at n¯50.88.22050in Fig. 5! exhibits a discontinuity at Ts;0.1, which reflectsthe characteristic temperature between two different symme-tries. For U52.2, the spatial distribution patterns for SDWand CDW are always isotropic and two-dimensional withTAF;0.09, where an appreciable discontinuity in Qv /e ~seethe dotted line in Fig. 5! is present.Recently, STM experiments29,30 showed the existence ofinhomogeneities in the electron density even for the opti-mally doped Bi2Sr2CaCu2O81x , there the hole density var-ies from 0.1 to 0.2. From our previous paper20 and thepresent study, we found that there may exist two types ofvortex cores, either negatively charged or positively charged.By increasing T, the number of negatively charged vorticesdecreases, while that of positively charged vortices increases.This may have a profound effect on the mixed state transportproperties in HTS. We have also examined the effect of thelong-range Coulomb interaction on the charge distributionsusing the Poisson equation. Our results show that its influ-ence on the vortex charge is weak and there is no qualitativechange in our conclusion. Finally, we wish to point out thatour numerical study has been performed in a strong magneticfield. So far we are not able to extend this calculation to amuch weaker field. However, according to the extrapolationscenario,20 the magnitude of the induced AF order followsroughly a linear relationship with B. The strength of the AForder and the magnitude of vortex charge are estimated to bemuch smaller as B is reduced to the experimental value(;9.4 T). Despite this difficulty, our numerical approachshould give a consistent description to the behavior of vortexcharges in HTS.We are grateful to Professor S. H. Pan, B. Friedman, andDr. J. X. Zhu for useful discussions. This work was sup-ported by the Robert A. Welch Foundation and by the TexasCenter for Superconductivity at the University of Houstonthrough the State of Texas. Z.D.W. also appreciates the sup-port from the RGC grant of Hong Kong ~Grant No.HKU7092/01P!, the Ministry of Science and Technology ofChina under Grant No. 1999064602, and the URC fundof HKU.15 D.I. Khomskii and A. Freimuth, Phys. Rev. Lett. 75, 1384 ~1995!.16 G. Blatter et al., Phys. Rev. Lett. 77, 566 ~1996!.17 N. Hayashi et al., J. Phys. Soc. Jpn. 67, 3368 ~1998!.18 J. Kolacek et al., Phys. Rev. Lett. 86, 312 ~2001!.19 P. Lipavsky et al., Phys. Rev. B 66, 134525 ~2002!.20 Yan Chen et al., Phys. Rev. Lett. 89, 217001 ~2002!.21 K. Kumagai et al., Phys. Rev. B 63, 144502 ~2001!.22 T. Nagaoka et al., Phys. Rev. Lett. 80, 3594 ~1998!.23 Y. Wang and A.H. MacDonald, Phys. Rev. B 52, R3876 ~1995!.24 Z.D. Wang et al., Phys. Rev. Lett. 72, 3875 ~1994!.25 Ch. Renner et al., Phys. Rev. Lett. 80, 3606 ~1998!.26 S.H. Pan et al., Phys. Rev. Lett. 85, 1536 ~2000!.27 M. Takigawa et al., Phys. Rev. Lett. 90, 047001 ~2003!.28 Yan Chen, Jian-Xin Zhu, and C.S. Ting, cond-mat/0302114 ~un-published!.29 S.H. Pan et al., Nature ~London! 413, 282 ~2001!.30 K.M. Lang et al., Nature ~London! 415, 412 ~2002!.1-4

Temperature dependence of Vortex Charges in High Temperature
  Superconductors

Temperature dependence of Vortex Charges in High Temperature Superconductors

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