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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
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Edkins et al., Science 364, 976–980 (2019) 7 June 2019 1 of
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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.)
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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
Edkins et al., Science 364, 976–980 (2019) 7 June 2019 2 of
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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.
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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).
Edkins et al., Science 364, 976–980 (2019) 7 June 2019 3 of
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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).
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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,
Edkins et al., Science 364, 976–980 (2019) 7 June 2019 4 of
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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).
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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.
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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
Edkins et al., Science 364, 976–980 (2019) 7 June 2019 5 of
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-
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
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