SUPERCONDUCTIVITY Magnetic field induced pair density r q ...eunahkim.ccmr.cornell.edu › sites ›...

SUPERCONDUCTIVITY

Magnetic field–induced pair densitywave state in the cuprate vortex haloS. D. Edkins1,2,3, A. Kostin1, K. Fujita1,4, A. P. Mackenzie3,5, H. Eisaki6,S. Uchida7, Subir Sachdev8, Michael J. Lawler1,9, E.-A. Kim1,J. C. Séamus Davis1,4,10,11*, M. H. Hamidian1,8*

High magnetic fields suppress cuprate superconductivity to reveal an unusual density wave(DW) state coexisting with unexplained quantum oscillations. Although routinely labeleda charge density wave (CDW), this DW state could actually be an electron-pair density wave(PDW). To search for evidence of a field-induced PDW, we visualized modulations in thedensity of electronic states N(r) within the halo surrounding Bi2Sr2CaCu2O8 vortex cores.We detected numerous phenomena predicted for a field-induced PDW, including twosets of particle-hole symmetric N(r) modulations with wave vectors QP and 2QP, with thelatter decaying twice as rapidly from the core as the former. These data imply that theprimary field-induced state in underdoped superconducting cuprates is a PDW, withapproximately eight CuO2 unit-cell periodicity and coexisting with its secondary CDWs.

Theory predicts that Cooper pairs with fi-nite center-of-mass momentum p = ℏQP(where ℏ is Planck’s constant h divided by2p) should form a state in which the densityof pairs modulates spatially at wave vector

QP (1, 2). In the phase diagram of underdopedcuprates, such a “pair density wave” (PDW) state(3–5), generated by strong local electron-electroninteractions (6–11), is anticipated to be anotherprincipal state, along with uniform supercon-ductivity. Numerous experimental observationsmay be understood in that context. For example,although intraplanar superconductivity appearsin La2-xBaxCuO4 at relatively high temperatures,interplanar superconductivity is strongly frus-trated (12), which is consistent with the exis-tence of orthogonal unidirectional PDW statesin each sequential CuO2 plane (3, 13, 14). More-over, themeasuredmomentum-space electronicstructure of the cuprate pseudogap phase is con-sistent with predictions that are based on abiaxial PDW (4). Reported breaking of time-reversal symmetry could be caused by a PDWwith inversion breaking (15–18). The field-inducedmomentum-space reconstruction and quantumoscillation phenomenology are potentially the

consequences of a PDW state (19–21), althoughthis view is not universal (22). At the highestfields, strong diamagnetism in torque magne-tometry (23) and supercurrents in dc transportmight also be understood as being due to afield-induced PDW state. Most recently, scannedJosephson tunneling microscopy allows directvisualization of cuprate pair densitymodulations(24). Taken together, these studies indicate thata fundamental PDW state may exist in under-doped cuprates, with the most common modelinvoked being an eight unit-cell (8a0) periodicmodulation of the electron-pair condensate.Such a PDW state clearly does not predomi-

nate at low temperature in zero magnetic field,where global d-wave superconductivity is robust.However, suppression of the superconductivityby highmagnetic fields generates a peculiar DWstate (25–32) along with exotic quantum oscil-lation phenomenology (33, 34). For type II super-conductors in general, application of a magneticfield generates quantized vortices. At the vorticesof a conventional d-wave superconductor, thefour zeros in the energy gap should generate aslowly decaying, star-shaped, zero-energy reso-nance oriented along the nodal (±1, ±1) direc-tions. For cuprates, however, strong N(r, E)modulations oriented along (1, 0); (0, 1) direc-tions have long been observed in the “halo” re-gion that surrounds the cuprate vortex core(35–38). Many theories hypothesize that thisphenomenon is a field-induced DW (5, 39–43),and some hypothesize that it is not a conven-tional CDW but a PDW (4, 5, 22, 43). This is a fun-damental distinction because the PDW and CDWare extremely different states in terms of theirbroken symmetries and many-body wave func-tions. Thus, to determine whether the primaryDW state induced by magnetic field in super-conducting cuprates is a PDW has recently be-come an urgent research challenge.To search for evidence of such a state, we

studied the field-induced modulations of the

density of electronic states N(r, E) within thehalo surrounding quantized vortex cores (35–38).Any periodic modulations of electronic structurecan be described by A(r) = AF(q)cos(Q · r + f0),where A(r) represents the modulating electronicdegree of freedom with amplitude A, Q is thewave vector, and F(q) is the modulation formfactor defined in terms of the angle q from the(1, 0) axis. An s-symmetry form factor FS(q) iseven under 90° rotations, whereas a d-waveform factor is Fd(q) is odd. Following (5), theorder parameters we considered are those ofhomogenous d-wave superconductivity D(r) =FSCDSC, with FSC = Fd, and that of a PDWDPDðrÞ ¼ FPDQP ðeiQP �r þ e�iQP �rÞ, with wave vec-torQP and either an Fs or Fd type of form factor[(44), section 1]. A field-induced PDW may beidentified from Ginzburg-Landau (GL) analysis(5, 22, 43) of the interactions between these twoorder parameters within vortex halos—regionsof suppressed but nonzero superconductivity thatsurround vortex cores (Fig. 1A). Given a genericGL free-energy density of the form

FA�SC ¼ FðDSCÞ þ FðDAÞ þ u1jDA j2jDSCj2 ð1Þ

where FðDSCÞ and FðDAÞ are the free-energydensities of a superconductor and of an alter-native repulsively coupled (u1 > 0) state DA, theobservation of coexistence of DA with DSC withinthe vortex halo [ (44), section 2] means that thetwo ordered states are almost energetically de-generate (39). Such a near degeneracy occursnaturally between a superconductor DSC and aPDWDQP that are made up of the same electronpairs. In this case, allowed N(r) modulationsgenerated by interactions between DSC and DAcan be found from products of these order pa-rameters that transform as density-like quan-tities. For example, the product of PDW anduniform SC order parameters

AQPºDQP D

�SC ⇒NðrÞºcosðQP � rÞ ð2Þ

results in N(r) modulations at the PDW wavevectorQP . The product of a robust PDWwith itself

A2QPºDQPD

�Q�P ⇒NðrÞºcosð2QP � rÞ ð3Þ

produces N(r) modulations occurring at 2QP.Thus, a key signature of a field-induced PDWwith wave vector QP in cuprate vortex halos(Fig. 1A) would be coexistence of two sets ofN(r)modulations at N(r) and at 2QP within eachhalo (Fig. 1B) (5, 22, 43).Within GL theory, substantial further infor-

mation can be extracted from measured ratesof decay of the inducedN(r) modulations awayfrom the vortex center and from the form fac-tors of these modulations within the vortex halo.For a field-induced PDW, the N(r, E) modula-tions at 2QP should decay with distance from thecore at twice the rate as those at QP. This is be-cause if DQP ¼ DQP ðjrj ¼ 0Þe�jrj=x , then DQPD�Q�Pdecays with jrj at twice the rate of DQPD�SC (Fig.1B). Current theory (22, 43) indicates that if theN(r, E) modulations at QP caused by D

QPD

�SC

RESEARCH

Edkins et al., Science 364, 976–980 (2019) 7 June 2019 1 of 5

1Laboratory of Atomic and Solid State Physics, Department ofPhysics, Cornell University, Ithaca, NY 14853, USA. 2Departmentof Applied Physics, Stanford University, Stanford, CA 94305,USA. 3School of Physics and Astronomy, University of St.Andrews, Fife KY16 9SS, Scotland. 4Condensed Matter PhysicsDepartment, Brookhaven National Laboratory, Upton, NY, USA.5Max-Planck Institute for Chemical Physics of Solids, D-01187Dresden, Germany. 6Institute of Advanced Industrial Scienceand Technology, Tsukuba, Ibaraki 305-8568, Japan.7Department of Physics, University of Tokyo, Bunkyo-ku, Tokyo113-0033, Japan. 8Department of Physics, Harvard University,Cambridge, MA 02138, USA. 9Department of Physics andAstronomy, Binghamton University, Binghamton, NY 13902,USA. 10Department of Physics, University College Cork, CorkT12R5C, Ireland. 11Clarendon Laboratory, Oxford University,Oxford, OX1 3PU, UK.*Corresponding author. Email: [email protected] (J.C.S.D.);[email protected] (M.H.H.)

on July 18, 2019

http://science.sciencemag.org/

Dow

nloaded from


exhibit s-symmetry form factor (Fs), this impliesthat the PDW order parameter DQP contains com-ponents with d-symmetry form factor (Fd), andvice versa [ (44), section 2]. These studies (22, 43)sustain the original GL approach (5) by showingthat an 8a0 PDWstabilized in the halo of a d-wavevortex core does indeed generate both an 8a0 anda 4a0 periodic chargemodulation therein.Overall,because a d-symmetry form factor PDW is typ-ically predicted for cuprates (6–11), its signaturewithin a vortex halo should be two sets of N(r)modulations occurring atQP and 2QP, both withs-symmetry form factor components and withthe amplitude of the 2QP modulation decayingtwice as rapidly as that at QP.To explore these predictions, we imaged scan-

ning tunneling microscope (STM) tip-sampledifferential tunneling conductance dI/dV (r, V) ≡g(r, E) versus bias voltage V = E/e and locationrwith sub–unit-cell spatial resolution; no scanned

Josephson tunneling microscopy (24) was involved.We measured slightly underdoped Bi2Sr2CaCu2O8samples [superconducting transition temperatureTc ~ 88K; hole doping p~ 17%] at temperature T=2K.We firstmeasured theN(r,E) at zero field andthen at magnetic field B = 8.25 T, in the identicalfield of view (FOV), using an identical STM tip(35). The former was subtracted from the latter toyield the field-induced changes dg(r, E, B) = g(r, E, B) – g(r, E, 0), which are related to thefield-induced perturbation to the density of statesas dN(r, E, B) º dg(r, E, B). This step ensuresthat the phenomena studied thereafter wereonly those induced by the magnetic field, withall signatures of the ubiquitous d-symmetry formfactor DW observed at B = 0 (45) having beensubtracted. Compared with our prior vortex halostudies (35), we enhanced the r-space resolutionusing smaller pixels and the q-space resolutionby using larger FOV (58 by 58 nm), increased the


Fig. 1. Schematic of field-induced uni-directional 8a0 PDW. (A) Diagram of the haloregion (gray) surrounding the vortex core (black)of a cuprate superconductor (SC). The CuO2plane orientation and Cu–Cu periodicity areindicated by using a dot for each Cu site. Withinthe halo, a unidirectional PDW modulation alongthe x axis with periodicity 8a0, characterized by

an order parameter DQP ðrÞ shown as red curve inthe top graph, is indicated schematically withred shading. (B) Solid curve indicates envelope

containing nonzero amplitude DQPD�SC of the N(r)

modulations caused by the interaction betweenthe SC and PDW order parameters, plottedalong the fine horizontal line in (A) through thevortex core. Dashed curve indicates the enve-

lope containing nonzero amplitude DQPD�Q�P of the

N(r) modulations caused by PDW itself, plottedalong the same fine line. For clarity, we ignorethe small region (less than 1 nm) at the core

where DQPD�SC must rise from zero as DSC does.

(C) Within a GL model, if the field-induced PDWhas a pure d-symmetry form factor, FP = Fd, thentwo sets of N(r) modulations should appeartogether. The first is N(r) º cos(QP · r) caused

by DQPD�S and indicated in Ñ(q) [the Fourier

transform of N(r)] with a solid red curve. The

second N(r) º cos(2QP · r) caused by DQPD

�Q�P is

indicated in Ñ(q) with a dashed red curve. Thedecay length for the 2QP modulation should behalf that of the QP modulation, meaning thatthe linewidth d(2Q)P of the 2QP modulation(dashed red) should be twice that of the QPmodulation, dQP (solid red). If the PDW has apure s-symmetry form factor, FP = FS, then adifferent pair of N(r) modulations should appeartogether. First is N(r) º cos[(QB – QP) · r],

caused by DQPD�S (solid blue line), and second

N(r) º cos(2QP · r), caused by DQPD

�Q�P (dashed

blue line). Here, QB is the Bragg wave vectorof the CuO2 unit cell.

Fig. 2. Four-unit-cell quasiparticle modula-tions at vortex halos in Bi2Sr2CaCu2O8.(A) Topographic image T(r) of BiO terminationlayer of our Bi2Sr2CaCu2O8 sample. Thedisplacement of every specific atomic site inthis field of view between zero field and B =8.25 Twas constrained by post processing of alllow- and high-field data sets to be less than10 pm [(45), section 3]. (B) Measured differentialtunnel conductance spectrum g(r, E = eV) ≡dI/dV(r, V) showing how to identify the symmetrypoint of a vortex core (dashed line).The full lineshows measured g(r, E = eV) at the identicallocation in zero field. Yellow-shaded region indi-cates low-energy Bogoliubov quasiparticle statesgenerated by the vortex. (C) Measureddg(r, 12 meV) = g(r, 12 meV, B = 8.25 T) –g(r, 12 meV, B = 0) showing typical examples ofthe low-energy Bogoliubov quasiparticle modula-tions (35–38) within halo regions surroundingfour vortex cores in Bi2Sr2CaCu2O8.

RESEARCH | REPORTon July 18, 2019


Dow

nloaded from


number of vortices per image, used distortion-corrected sublattice-phase-resolved imaging (45),and measured in a far wider energy range 0 <jEj < 80 meV [ (44), section 3].We next identified the location of every

vortex halo in dg(r, E, B) images using twowell-known phenomena: (i) suppression of thesuperconducting coherence peaks at the vortexsymmetry point (Fig. 2B) and (ii) appearanceof low-energy periodic conductance modula-tions (35–38) surrounding this point. A typicalsymmetry-point spectrum of the superconduct-ing vortex where maximum suppression of thesingle-particle coherence peaks occurs is shownin Fig. 2B; these peaks recover very rapidly as afunction of radius, so that robust d-wave super-conductivity signified by full coherence peakshas recovered within a radius of ~1 nm [(44),section 4]. At E = 12 meV, the typical halo ofconductance modulations we detected sur-rounding each vortex symmetry point (Fig. 2C)was in excellent agreement with previous studiesof modulations of low-energy quasiparticles,with q ≈ (±1/4, 0); (0, ±1/4)2p/a0 within theBi2Sr2CaCu2O8-x vortex halo (35–38). We fo-

cused on a different energy range 25 < jEj <50 meV because analysis of our dg(r, E) datarevealed major changes in this range. In Fig. 3A,we show measured dg(r, 30 meV) containing themodulations detected in the halo of each vortexcore. Fourier analysis of this dg(r, 30 meV) yieldsj~dgðq; 30 meVÞj and reveals a set of sharp peaks atq ¼ ðQxP;QyPÞ≈½ðT1=8;0Þð0; T1=8Þ�2p=a0, whichwe labelQP for reasons explained below (Fig. 3B).Similarly, there is a second set of weaker mod-ulations in ~dgðq; 30 meVÞ at q ≈ [(±1/4, 0); (0,±1/4)]2p/a0, which we label 2QP. The r-space am-plitude envelopes of theQP and 2QPmodulations(Fig. 3, C and D) reveal how these field-inducedphenomena are confined to the vortex haloregions only. Averaged over all vortices, themeasured amplitude j~dgðq; 30 meVÞj plottedalong (1, 0) in Fig. 3E discernibly discriminatesthe QP from the 2QP modulation peaks. Thus,we discovered strong, field-induced modula-tions of N(r, E), with period approximately 8a0coexisting with weaker modulations of periodapproximately 4a0, along both the (1, 0); (0,1) directions within every vortex halo. Theseparticle-hole symmetric phenomena exist with-

in the energy range 25 < jEj < 45meV [(44),section 5].To evaluate form factor symmetry for these

field-induced modulations [ (44), section 6], weseparated each such dg(r, E) image into threesublattice images (46): Cu(r, E), containing onlythe measured values of dg(r, E) at copper sites,and Ox(r, E) and Oy(r, E), containing only thoseat the x/y axis oxygen sites. All of the formfactors discussed here refer to modulationsin dg(r, E, B) and are not necessarily those ofthe order parameter of the field-induced statethat generates them. Complex-valued Fouriertransforms of theOx(r, E) andOy(r, E) sublatticeimages yield Õx(q, E); Õy(q, E). Then, modu-lations at anyQ having d-symmetry form factorFd generate a peak in ~D

dgðq;EÞ ≡ ~Oxðq;EÞ �~Oyðq;EÞ at Q, whereas those with s-symmetryform factor Fs generate a peak in ~Sdgðq;EÞ ≡½~Oxðq;EÞ þ ~Oyðq;EÞ� þ ~Cuðq;EÞ atQ. When weanalyzed the data in Fig. 3, A and B, in this wayusing measured ~S

dgðq; 30 meVÞ , the field-induced dg(r, E)-modulations occurring at q ≈(±QP, 0); (0, ±QP) and q ≈ (±2QP, 0); (0, ±2QP) allexhibited s-symmetry form factors (Fig. 3E).


Fig. 3. Field-induced s-symmetry form factor modulationswithin vortex halos. (A) Measured field-induced modulationsdg(r, 30 meV) = g(r, 30 meV, B = 8.25 T) – g(r, 30 meV, B = 0)in a 58 by 58 nm FOV. The simultaneously measuredtopographs T(r) at B = 8.25 T and 0 T are shown in fig. S2and (44), section 3, and demonstrate by the absenceof local maxima at q ≈ [(±1/8, 0); (0, ±1/8)]2p/a0 in their Fouriertransforms that the setup effect is not influencing observationsof dg(r, E) modulations at these wave vectors [(44), section 5].

(B) Amplitude Fourier transform j ~dgðq;30 meVÞj (square rootof power spectral density) of dg(r, 30 meV) data in (A).The q = (±1/4, 0); (0, ±1/4)2p/a0 points are indicated with blackcrosses. Four sharp maxima, indicated by QP, occur atq = (±1/8, 0); (0, ±1/8)2p/a0, whereas four broader maxima,indicated by 2QP, occur at q = (±1/4, 0); (0, ±1/4)2p/a0.(C) Measured amplitude envelope of the modulationsin dg(r, 30 meV) at QP showing that they only occurwithin the vortex halo regions. (D) Measured amplitudeenvelope of the modulations in dg(r, 30 meV) at 2QP, showingthat they also only occur within the vortex halo regions. (E) Measured

j ~dgðq;30 meVÞj along (0,0)-(1/2,0) [(B), dashed line], showing thetwo maxima in the field-induced N(r) modulations, occurring atby QP = 0.117 ± 0.01 and 2QP = 0.231 ± 0.01 (Fig. 4, A to D)(F) Amplitude Fourier transform of the d-symmetry form factormodulations in NðrÞ; j~Ddgðq;30 meVÞj; derived from measureddg(r, 30 meV) data in (A). Again, q = (±1/4, 0); (0, ±1/4)2p/a0 pointsare indicated with black crosses. Two maxima, labeled asQP, occur at q = (±1/8, 0); (0, ±1/8)2p/a0, whereas two broadermaxima, indicated by 2QP, occur at q = (±1/4, 0); (0, ±1/4)2p/a0, withboth sets oriented along the y axis. (G) Measured j~Ddgðq;30 meVÞjalong (0,0)-(1/2,0) [(F), dashed line], showing the maxima in the fieldinduced N(r) modulations occurring at QP and 2QP: A unidirectionald-symmetry form factor change density modulation, as observedextensively in zero field (46), would have such characteristics,as would contributions from an s-symmetry form factor PDW. Thesemodulations do not appear in Fig. 3 because, in that unprocesseddg(r, E) data, they occur at Q ≈ (0, ±7/8)2p/a0 and Q ≈ (0, ±3/4)2p/a0owing to their d-symmetry form factor (Fig. 4, A to D).



Dow

nloaded from


However, the measured ~Ddgðq; 30 meVÞ in Fig. 3,

F and G, also revealed that weaker d-symmetrydg(r, E)-modulations occur at q ≈ (0, ±QP) andq ≈ (0, ±2QP). They too are confined to the vortexhalo because the r-space amplitude-envelope ofthe 2QP-modulations in ~D

dgðq; 30 meVÞ is con-centrated there.

The overall measured amplitudes of j~dgðq;30 meVÞj derived from dg(r, 30 meV) in Fig. 3Aare shown in Fig. 4, A and B, plotted along the(1,0) and (0,1) directions of the CuO2 plane.Equivalent cuts of j~dgðq;�30 meVÞj derived fromdg(r, –30 meV) data are shown in Fig. 4, C andD. The four maxima at jqj ≈ 1=8, jqj ≈ 1=4, jqj ≈

3=4, and jqj ≈ 7=8 associated with field-inducedmodulations occur in Fig. 4, A to D. The measuredform factor of each set of modulations is identifiedby color code, red indicating s-symmetry and blueindicating d-symmetry. Although modulations atjqj ≈ 7=8 and jqj ≈ 3=4 (Fig. 4, A to D, blue) ap-pear subdominant, they do merit comment. First,


Fig. 4. Field-induced N(r) modulations indicative of a PDW state inthe vortex halo. (A and B) Amplitude Fourier transform j ~dgðq;30 meVÞjderived from dg(r, 30 meV) data are plotted along two orthogonal axesfrom (0,0)-(0,1) and (0,0)-(1,0), to reach both Bragg points. All fourlocal maxima, at QP and 2QP from the s-symmetry field-induced N(r)modulations, plus at 1 – QP and 1 – 2QP from the d-symmetry field-inducedN(r) modulations, may be seen. Measurement from these fits of theq-magnitude and width dq of the s-symmetry peaks at QP and

2QP yields QxP ¼ 0:117;QyP ¼ 0:129; 2QxP ¼ 0:231;2QyP ¼ 0:237;dQxP ¼ 0:020; dQyP ¼ 0:020; and d2QxP ¼ 0:034; d2QyP ¼ 0:035. (Inset)j ~dgðq;30 meVÞj. (C and D) As in (A) and (B) but at E = –30 meV.Measurement yields areQxP ¼ 0:115;QyP ¼ 0:128; 2QxP ¼ 0:239;2QyP ¼ 0:235;

dQxP ¼ 0:020; dQyP ¼ 0:020; and d2QxP ¼ 0:039; d2QyP ¼ 0:045. (Inset)j ~dgðq;�30 meVÞj. The s-symmetry field-induced N(r) modulations at QPand 2QP are almost perfectly particle-hole symmetric [(B) and (D), insets] in thesense that N(r, E > 25 meV) = N(r, E < –25 meV) for these two wave vectors.

(E) Fourier transform amplitude, ~jdgðqÞj; of measured dD(r) = D(r, 8.25 T) –D(r, 0) [(44), section 7]. The observed peaks revealing field-induced gapmodulation occur at points indistinguishable from QP. The peak along the(1, 1) direction occurs at the wave vector of the crystal supermodulation,where a modulation-induced PDW has long been identified. (F) Schematicrepresentation of a bidirectional PDW with a d-symmetry form factorinduced within a vortex halo that is consonant with the data in this workwhen considered in the context of vortex halo theory (5, 22, 43).



Dow

nloaded from


they are not inconsistent with an admixture ofs-symmetry and d-symmetry components in thePDW order parameter. However, these modu-lations may also represent a field-induced ver-sion of the unidirectional d-symmetry form factorN(r, E) modulation observed in zero field (45).But the predominant phenomena detected

are the two sets of s-symmetry form factormodulations at jQPj ≈ 1=8 and j2QPj ≈ 1=4 (Fig.4, A to D, red). Moreover, after subtraction of asmooth background, the widths dq of all jQPj ≈1=8 peaks are close to half of the j2QPj ≈ 1=4 peaks,as determined quantitatively by fitting as shownin Fig. 4, A to D. Averaged over the two direc-tions (1, 0) and (0, 1) and energies E = ±30 meV,we found that d(2QP) = (1.8 ± 0.2)d(QP) as ex-pected for a field-induced PDW (Fig. 1) (5, 22, 43).As additional evidence of a PDW, we searchedfor energy gap modulations in measured D(r) =DSC + DPcos(QP · r). Generally, in supercon-ductivity studies, the empirical D(r) is definedas half the energy separation of the coherencepeaks in N(r, E) (Fig. 2B, horizontal arrow), sothat field-induced changes to D(r) would herebe defined as dD(r) = D(r, 8.25 T) – D(r, 0) [(44),section 7]. When measured, this dD(r) yieldsa Fourier transform ~dDðqÞ, as shown in Fig.4E. This exhibits evidence for a field-inducedenergy-gap modulation at QP and not at 2QP,as would be expected specifically for a primaryfield-induced PDW at QP.Taken together, the results shown in Figs. 3

and 4 indicate that in Bi2Sr2CaCu2O8, a field-induced PDW state emerges within the haloregion surrounding each quantized vortex core.The principal experimental signatures are twosets of N(r) modulations occurring at QP and2QP, both being particle-hole symmetric, bothexhibiting principal amplitudewith s-symmetryform factor, with the amplitude of 2QP modu-lations decaying twice as rapidly as that of QPand with an apparently bidirectional structure,as shown schematically in Fig. 4F [(44), section8]. These phenomena occur in an energy range25 < jEj < 45 meV, as might be expected the-oretically for an 8a0 periodic PDW with energygap magnitude DQP occurring within that range.Several major implications stem from these ob-servations. First and foremost, the primary stateinduced by high magnetic fields in the super-conducting phase of cuprates is inferred to bea PDW with wave vector QP, accompanied bysecondary charge modulations at QP and 2QP.Second, the 8a0 periodicity points toward a

strong correlation-driven microscopic mech-anism for the PDW (6–11), in which case theform factor is generally predicted to have ad-symmetry (Fig. 4F). Third, because the PDWis generated by increasing magnetic field, ourdata imply that the high-field state of cupratesmight itself be a PDW state (4), and if so, it islikely phase fluctuating and intertwined withadditional CDW components. Last, putting allsuch conjectures aside, we emphasize that theexperimental observations reported in Figs. 3and 4 are in good, detailed, and quantitative agree-ment with theoretical models (5, 22, 43, 44) fora primary PDW with wave vector QP inducedwithin the cuprate vortex halo, which generatessecondary CDWs at QP and 2QP.

REFERENCES AND NOTES

1. P. Fulde, R. A. Ferrell, Phys. Rev. 135 (3A), A550–A563(1964).

2. A. I. Larkin, Y. N. Ovchinnikov, Sov. Phys. JETP 20, 762–769(1964).

3. E. Berg et al., Phys. Rev. Lett. 99, 127003 (2007).4. P. A. Lee, Phys. Rev. X 4, 031017 (2014).5. D. F. Agterberg, J. Garaud, Phys. Rev. B 91, 104512 (2015).6. A. Himeda, T. Kato, M. Ogata, Phys. Rev. Lett. 88, 117001 (2002).7. M. Raczkowski, M. Capello, D. Poilblanc, R. Fr’esard, A. M. Oleś,

Phys. Rev. B 76, 140505 (2007).8. K.-Y. Yang, W. Q. Chen, T. M. Rice, M. Sigrist, F.-C. Zhang,

New J. Phys. 11, 055053 (2009).9. F. Loder, A. P. Kampf, T. Kopp, Phys. Rev. B 81, 020511 (2010).10. P. Corboz, T. M. Rice, M. Troyer, Phys. Rev. Lett. 113, 046402

(2014).11. R.-G. Cai, L. Li, Y.-Q. Wang, J. Zaanen, Phys. Rev. Lett. 119,

181601 (2017).12. Q. Li, M. Hücker, G. D. Gu, A. M. Tsvelik, J. M. Tranquada,

Phys. Rev. Lett. 99, 067001 (2007).13. E. Berg, E. Fradkin, S. A. Kivelson, Nat. Phys. 5, 830–833 (2009).14. E. Berg, E. Fradkin, S. A. Kivelson, J. M. Tranquada,

New J. Phys. 11, 115004 (2009).15. C. Pépin, V. S. De Carvalho, T. Kloss, X. Montiel, Phys. Rev. B

90, 195207 (2014).16. H. Freire, V. S. de Carvalho, C. Pépin, Phys. Rev. B 92, 045132

(2015).17. Y. Wang, D. F. Agterberg, A. Chubukov, Phys. Rev. Lett. 114,

197001 (2015).18. D. F. Agterberg, D. S. Melchert, M. K. Kashyap, Phys. Rev. B 91,

054502 (2015).19. M. Zelli, C. Kallin, A. J. Berlinsky, Phys. Rev. B 84, 174525 (2011).20. M. Zelli, C. Kallin, A. J. Berlinsky, Phys. Rev. B 86, 104507

(2012).21. M. R. Norman, J. C. S. Davis, Proc. Natl. Acad. Sci. U.S.A. 115,

5389–5391 (2018).22. Z. Dai, Y.-H. Zhang, T. Senthil, P. A. Lee, Phys. Rev. B 97,

174511 (2018).23. F. Yu et al., Proc. Natl. Acad. Sci. U.S.A. 113, 12667–12672 (2016).24. M. H. Hamidian et al., Nature 532, 343–347 (2016).25. T. Wu et al., Nature 477, 191–194 (2011).26. J. Chang et al., Nat. Phys. 8, 871–876 (2012).27. D. LeBoeuf et al., Nat. Phys. 9, 79–83 (2012).28. S. Blanco-Canosa et al., Phys. Rev. Lett. 110, 187001 (2013).29. T. Wu et al., Nat. Commun. 6, 6438 (2015).

30. S. Gerber et al., Science 350, 949–952 (2015).31. J. Chang et al., Nat. Commun. 7, 11494 (2016).32. H. Jang et al., Proc. Natl. Acad. Sci. U.S.A. 113, 14645–14650

(2016).33. B. Vignolle, D. Vignolles, M.-H. Julien, C. Proust, C. R. Phys. 14,

39–52 (2013).34. S. E. Sebastian, C. Proust, Annu. Rev. Condens. Matter Phys. 6,

411–430 (2015).35. J. E. Hoffman et al., Science 295, 466–469 (2002).36. K. Matsuba et al., J. Phys. Soc. Jpn. 76, 063704 (2007).37. S. Yoshizawa et al., J. Phys. Soc. Jpn. 82, 083706 (2013).38. T. Machida et al., Nat. Commun. 7, 11747 (2016).39. S. A. Kivelson, D. Lee, E. Fradkin, V. Oganesyan, Phys. Rev. B

66, 144516 (2002).40. D. F. Agterberg, H. Tsunetsugu, Nat. Phys. 4, 639–642 (2009).41. J. D. Sau, S. Sachdev, Phys. Rev. B 89, 075129 (2014).42. M. Einenkel, H. Meier, C. P’epin, K. B. Efetov, Phys. Rev. B 90,

054511 (2014).43. Y. Wang et al., Phys. Rev. B 97, 174510 (2018).44. Materials and methods are available as supplementary materials.45. M. H. Hamidian et al., Nat. Phys. 12, 150–156 (2016).46. M. J. Lawler et al., Nature 466, 347–351 (2010).47. J. C. Davis, Supporting data for “Magnetic-field Induced

Pair Density Wave State in the Cuprate Vortex Halo” byS. D. Edkins et al., Harvard Dataverse (2019); doi: 10.7910/DVN/JDSN1W

ACKNOWLEDGMENTS

We acknowledge and thank D. Agterberg, P. Choubey, A. Chubukov,E. Fradkin, P. J. Hirschfeld, P. D. Johnson, D. H. Lee, P. A. Lee,C. Pepin, S. Sebastian, S. Todadri, J. Tranquada, and Y. Wangfor helpful discussions, advice, and communications. We aregrateful to S. A. Kivelson for crucial proposals on the completeset of PDW phenomena to search for within the vortex halo.Funding: S.U. and H.E. acknowledge support from a Grant-in-Aidfor Scientific Research from the Ministry of Science and Education(Japan); A.K. and K.F. acknowledge salary support from theU.S. Department of Energy, Office of Basic Energy Sciences, undercontract DEAC02-98CH10886. E.-A.K. acknowledges support fromthe U.S. Department of Energy, Office of Basic Energy Sciencesunder award DE-SC0018946. S.S. acknowledges support underNational Science Foundation under grant DMR- 1664842. J.C.S.D.acknowledges support from Science Foundation Ireland underaward SFI 17/RP/5445 and from the European Research Council(ERC) under award DLV-788932. J.C.S.D., S.D.E., and M.H.H.acknowledge support from the Moore Foundation’s EPiQS Initiativethrough grant GBMF4544. Author contributions: S.D.E., A.K.,and M.H.H. carried out the experiments; K.F., H.E., and S.U.synthesized and characterized the samples; M.H.H., S.D.E., andK.F. developed and carried out analysis; S.S., M.J.L., and E.-A.K.provided theoretical guidance; A.P.M. and J.C.S.D. supervisedthe project and wrote the paper, with key contributions from S.D.E.,K.F., and M.H.H. The manuscript reflects the contributions andideas of all authors. Competing interests: The authors declare nocompeting financial interests. Data and materials availability:The data files for the results presented here are available at (47).

SUPPLEMENTARY MATERIALS

science.sciencemag.org/content/364/6444/976/suppl/DC1Materials and MethodsSupplementary TextFigs. S1 to S8References (48–58)

5 February 2018; accepted 15 May 201910.1126/science.aat1773




Dow

nloaded from

https://dx.doi.org/10.7910/DVN/JDSN1Whttps://dx.doi.org/10.7910/DVN/JDSN1Whttps://science.sciencemag.org/content/364/6444/976/suppl/DC1http://science.sciencemag.org/

induced pair density wave state in the cuprate vortex halo−Magnetic field

Séamus Davis and M. H. HamidianS. D. Edkins, A. Kostin, K. Fujita, A. P. Mackenzie, H. Eisaki, S. Uchida, Subir Sachdev, Michael J. Lawler, E.-A. Kim, J. C.

DOI: 10.1126/science.aat1773 (6444), 976-980.364Science

, this issue p. 976Sciencespatially modulated.correspond to an exotic state called the pair density wave, in which the density of finite momentum Cooper pairs isused scanning tunneling spectroscopy to take a closer look into the halos. The results revealed that the patterns

et al.cores are surrounded by ''halos,'' where the density of electronic states exhibits a checkerboard pattern. Edkins Magnetic fields can cause the formation of vortices in a superconductor. In cuprate superconductors, the vortex

Decoding the halo pattern

ARTICLE TOOLS http://science.sciencemag.org/content/364/6444/976

MATERIALSSUPPLEMENTARY http://science.sciencemag.org/content/suppl/2019/06/05/364.6444.976.DC1

REFERENCES

http://science.sciencemag.org/content/364/6444/976#BIBLThis article cites 57 articles, 5 of which you can access for free

PERMISSIONS http://www.sciencemag.org/help/reprints-and-permissions

Terms of ServiceUse of this article is subject to the

is a registered trademark of AAAS.Sciencelicensee American Association for the Advancement of Science. No claim to original U.S. Government Works. The title Science, 1200 New York Avenue NW, Washington, DC 20005. 2017 © The Authors, some rights reserved; exclusive

(print ISSN 0036-8075; online ISSN 1095-9203) is published by the American Association for the Advancement ofScience

on July 18, 2019


Dow

nloaded from
http://science.sciencemag.org/content/364/6444/976http://science.sciencemag.org/content/suppl/2019/06/05/364.6444.976.DC1http://science.sciencemag.org/content/364/6444/976#BIBLhttp://www.sciencemag.org/help/reprints-and-permissionshttp://www.sciencemag.org/about/terms-servicehttp://science.sciencemag.org/

SUPERCONDUCTIVITY Magnetic field induced pair density r q ...eunahkim.ccmr.cornell.edu › sites ›...

Documents

Transcript of SUPERCONDUCTIVITY Magnetic field induced pair density r q ...eunahkim.ccmr.cornell.edu › sites ›...