DOI: 10.1126/science.1201082, 426 (2011);333 Science

et al.A. MesarosNematicity in Cuprates

Unit-Cell−Topological Defects Coupling Smectic Modulations to Intra

This copy is for your personal, non-commercial use only.

clicking here.colleagues, clients, or customers by , you can order high-quality copies for yourIf you wish to distribute this article to others

  here.following the guidelines

can be obtained byPermission to republish or repurpose articles or portions of articles

  ): October 7, 2014 www.sciencemag.org (this information is current as of

The following resources related to this article are available online at

http://www.sciencemag.org/content/333/6041/426.full.htmlversion of this article at:

including high-resolution figures, can be found in the onlineUpdated information and services,

http://www.sciencemag.org/content/suppl/2011/07/20/333.6041.426.DC1.html can be found at: Supporting Online Material

http://www.sciencemag.org/content/333/6041/426.full.html#ref-list-1, 4 of which can be accessed free:cites 33 articlesThis article

http://www.sciencemag.org/content/333/6041/426.full.html#related-urls4 articles hosted by HighWire Press; see:cited by This article has been

http://www.sciencemag.org/cgi/collection/physicsPhysics

subject collections:This article appears in the following

registered trademark of AAAS. is aScience2011 by the American Association for the Advancement of Science; all rights reserved. The title

CopyrightAmerican Association for the Advancement of Science, 1200 New York Avenue NW, Washington, DC 20005. (print ISSN 0036-8075; online ISSN 1095-9203) is published weekly, except the last week in December, by theScience

on

Oct

ober

7, 2

014

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fr

om

on

Oct

ober

7, 2

014

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fr

om

on

Oct

ober

7, 2

014

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fr

om

on

Oct

ober

7, 2

014

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fr

om

on

Oct

ober

7, 2

014

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fr

om

on

Oct

ober

7, 2

014

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fr

om

Topological Defects Coupling SmecticModulations to Intra–Unit-CellNematicity in CupratesA. Mesaros,1,2* K. Fujita,2,3,4* H. Eisaki,5 S. Uchida,4 J. C. Davis,2,3,6 S. Sachdev,7

J. Zaanen,1 M. J. Lawler,2,8 Eun-Ah Kim2†

We study the coexisting smectic modulations and intra–unit-cell nematicity in the pseudogap states ofunderdoped Bi2Sr2CaCu2O8+d. By visualizing their spatial components separately, we identified 2ptopological defects throughout the phase-fluctuating smectic states. Imaging the locations of largenumbers of these topological defects simultaneously with the fluctuations in the intra–unit-cell nematicityrevealed strong empirical evidence for a coupling between them. From these observations, we proposea Ginzburg-Landau functional describing this coupling and demonstrate how it can explain thecoexistence of the smectic and intra–unit-cell broken symmetries and also correctly predict theirinterplay at the atomic scale. This theoretical perspective can lead to unraveling the complexities ofthe phase diagram of cuprate high-critical-temperature superconductors.

Electronic liquid crystals are proposed tooccur when the electronic structure of amaterial breaks the spatial symmetries of

its crystal lattice (1–8). In theory, nematic elec-tronic liquid crystals would preserve the latticetranslational symmetry but break the discreterotational symmetry, whereas smectic (striped)electronic liquid crystals would break both.

These concepts have played an important rolein theoretical considerations of the pseudogapphase of underdoped cuprates (1–8).

At hole densities ( p) below ~16%, cupratesexhibit d-wave superconductivity at lowest tem-peratures and the pseudogap phase above thesuperconductor’s critical temperature,Tc. Althoughit is unknown which broken symmetries (if any)

cause the pseudogap phase, both nematic andsmectic broken symmetry states have been re-ported in different underdoped cuprate com-pounds (9–18). Spin and charge smectic brokensymmetry (stripes) exists in La2–x–yNdySrxCuO4

and in La2-xBaxCuO4 when x ~ 0.125 (6, 9–12).On the other hand, broken 90°-rotational sym-metry is reported in underdoped YBa2Cu3O6+d

(13,15–17), underdopedBi2Sr2CaCu2O8+d (14, 18),and underdoped HgBa2CuO4+x (19). These statesare highly distinct: The former breaks both trans-lational symmetry with a finite wave vector →q ¼ →

S,where the magnitude of

→S is the wave number for

the modulation, and 90°-rotational symmetry(9–12), whereas the latter is associated with intra–

REPORTS

1Instituut-Lorentz for Theoretical Physics, Universiteit Leiden,2300 Leiden, Netherlands. 2Laboratory for Atomic and Sol-id State Physics, Department of Physics, Cornell University,Ithaca, NY 14853, USA. 3Condensed Matter Physics and Mate-rials Science Department, Brookhaven National Laboratory,Upton, NY 11973, USA. 4Department of Physics, University ofTokyo, Bunkyo-ku, Tokyo 113-0033, Japan. 5Institute of Ad-vanced Industrial Science and Technology, Tsukuba, Ibaraki305-8568, Japan. 6School of Physics and Astronomy, Universityof St. Andrews, North Haugh, St. Andrews, Fife KY16 9SS, UK.7Department of Physics, Harvard University, Boston, MA 02138,USA. 8Department of Physics, Applied Physics and Astronomy,Binghamton University, Binghamton, NY 13902–6000, USA.

*These authors contributed equally to this work.†To whom correspondence should be addressed. E-mail:eun-ah.kim@cornell.edu

Fig. 1. (A) Schematic image of an edge dislocationin a crystalline solid (solid circles indicate atomiclocations) and in the two-dimensional smecticphase of a liquid crystal (solid white lines indicatemodulation period). In both cases, it is the spatialphase of periodic modulations that winds around thedislocation core by precisely 2p. (Inset) Schematicimage of a superfluid or superconducting vortex over-lapped with its phase field, which winds by exactly2p. (B) Sub–unit-cell resolution image of the elec-tronic structure at the pseudogap energy Z(→r,e = 1).(Inset) Its Fourier transform of Z̃(→q,e = 1), whichdemonstrates that the →q-space electronic structurecontains two components, nematic [red circles at theBragg peaks, see (18)] and smectic (blue circles).The smectic peaks are centered at |

→Sx | = |

→Sy| =

0:72(2p /a0). White box is field of view (FOV) ofFig. 2, A and B. Tc of the sample is 50 K. (C) Spatialvariation of the electronic nematicity On(

→r,e = 1)in the same FOV as in (B). (Inset) The Bragg peakintensities are compared along x and y directions.(D) Spatial variation of the smectic electronic struc-ture modulations OS(

→r,e = 1) [see (18)].

A 0 2 π

π

π/2

0, 2 π

3π/2

xy

4 nm

C

On( r )

xy

4 nm

B

xy

4 nm

D

Os( r )

Qx

SxSy

Qy

0.01

50

Z~ (q

)

1.21.00.8q ( 2π / a0 )

Qx

Qy

1.610.47

+0.02-0.02-0.016 -0.002

22 JULY 2011 VOL 333 SCIENCE www.sciencemag.org426

unit-cell breaking of 90°-rotational symmetry(15, 18–20). A key challenge is therefore to un-derstand the interactions between these phenom-ena (9–27).

We consider the coexisting smecticmodulationsand intra–unit-cell nematicity in the pseudogap-energy electronic structure of the underdopedhigh-Tc superconductor Bi2Sr2CaCu2O8+d (18, 20)by using approaches derived from studies ofclassical liquid crystals. In those systems, fluctu-ating nematic and disordered smectic states coex-ist, and their dominant coupling can be capturedsuccessfully by using Ginzburg-Landau theory(22, 24, 25). The influence of 2p phase-windingtopological defects of the smectic was key to thosestudies. But the extension of such classical ideas toelectronic systems presents some new challenges.First, the intra–unit-cell C4-breaking observed atnanoscale in the cuprate pseudogap states (18, 20)is distinct from nematicity in a classical liquid crys-tal, because it has Ising symmetry resulting fromthe existence of the crystal lattice.Moreover,wheth-er 2p topological defects even exist within thecuprate pseudogap smectic states was unknown.

Topological defects are the fundamental emer-gent excitations when a new ordered phase isformed by breaking a continuous symmetry(21, 22). They are singular points or lines in theotherwise spatially continuous configuration ofthe order-parameter field. For example, whenthe order-parameter field is a complex functiony(→r) = y0e

iϕ(→r) of the position →r , the phase

ϕ(→r) winds by integer multiples of T2p aroundevery topological defect. Classic examples includethe quantized vortices in bosonic and fermionicsuperfluids (23) and the quantized fluxoids ofsuperconductors (22, 23) (Fig. 1A inset). Systemswith broken translational symmetry, such ascrystals or smectic liquid crystals, also exhibit 2pphase-winding topological defects. In a crystal,when a single line of atoms (Fig. 1A, black dots)terminates at an edge dislocation, nearby atomsare distorted away from their ideal lattice lo-cations, resulting in a spatially varying phase ofperiodic modulations that winds around thedislocation core by precisely 2p (22). In smecticliquid crystals, the equivalent topological de-fects are referred to as (smectic) dislocations.Again, each dislocation core is surrounded by aregion where the phase of the periodic (smec-tic) modulations (white lines in Fig. 1A) windsby exactly 2p. These topological defects areuniquely important in classical liquid crystals be-cause their properties reveal the dominant cou-pling between the nematic field and the smecticfield. In fact, quasi–long-range smectic-A orderin two dimensions is destroyed by this coupling,which lowers the energy cost of smectic dislo-cations, allowing their spontaneous appearanceat any temperature (22, 24, 25). We apply an anal-ogous theoretical approach to coexisting brokenelectronic symmetries in underdoped cuprates.

Spectroscopic imaging scanning tunneling mi-croscopy (SI-STM) allows visualization of electron-

ic broken symmetries in cuprates (18, 20, 26, 27) byusing atomically resolved spatial imagesof Z(→r,E) =[dI /dV (→r, E ≡ þeV )] / [dI /dV (→r, E ≡ −eV )],where dI /dV (→r,V ) is the spatially resolved dif-ferential tunneling conductance [supporting on-line material (SOM) a]. In underdoped cuprates,energy-independent symmetry breaking is vividin the nondispersive Z(→r,E) modulations at thepseudogap energy scale E ~ D1 (18, 20, 26–28).The coexistence of intra–unit-cell nematicity andsmectic modulations (18, 20) appears to be a ro-bust property of these electronic structure imagesof the cuprate pseudogap states, being virtually iden-tical in Bi2Sr2CaCu2O8+d and Ca2–xNaxCuO2Cl2(20) and unchanged from below to above Tc (27).

To separate the components of the E ~ D1electronic structure, each Z(→r,E ) image is firstdistortion-corrected to render the atomic sitesin a perfectly periodic array (18). Then, to dealwith the spatial disorder in D1(

→r ), E is rescaledlocally to e(→r ) ¼ E/D1(

→r ), yielding Z(→r,e); allthe broken symmetry phenomena of the pseudo-gap states then occur together in a single imageZ(→r,e = 1) (Fig. 1B and SOM a). Then, whenZ̃(→q,e = 1), the Fourier transform of Z(→r,e = 1),is calculated (Fig. 1B inset), it exhibits four sa-lient features: the Bragg peaks at→q =

→Qx and

→Qy

(red circles) and the smectic modulation peaks→q ¼ →

Sx and→Sy (blue circles). The phase-resolved

Bragg-peak Fourier components can then beused to detect intra–unit-cell symmetry breakingwithin each Z(→r,e = 1) image (18).

Fig. 2. (A) Smectic modulations along x directionare visualized by Fourier filtering out all the mod-ulations of Z(→r,e = 1) except those surrounding

→Sx,

in the FOV indicated by the broken boxes in Fig. 1Band in (C). (B) Smectic modulations along y direc-tion are visualized by Fourier filtering out all themodulations of Z(→r,e = 1) except those surround-ing

→Sy, in the FOV indicated by the broken boxes in

Fig. 1B and in (D). (C and D) Phase field ϕ1(→r ) and

ϕ2(→r ) for smectic modulations along x and y direc-

tion, respectively, exhibiting the topological defectsat the points around which the phase winds from 0to 2p (in the FOV same as in Fig. 1B). Depending onthe sign of phase winding, the topological defectsare marked by either white or black dots. The brokenred circle is the measure of the spatial resolutiondetermined by the cut-off length (3s) in extractingthe smectic field from Z̃(→q,e = 1). We did not markdefect-antidefect pairs when they are tightly boundby separation distances shorter than the cut-offlength scale.

low

high

B Ψ2( r→

)

2 nm

A Ψ1( r→

)

2 nm

4 nmD4 nmC

0

2π

www.sciencemag.org SCIENCE VOL 333 22 JULY 2011 427

REPORTS

We focus on intra–unit-cell “nematicity” de-fined by ⟨On(e)⟩ ≡ [Re Z̃(

→Qy, e) − Re Z̃(

→Qx, e)]/

Z(e), where Z(e) is the spatial average ofZ(→r, e),as a measure of the observed inequivalence be-tween x- and y-axis electronic structure withinthe CuO2 unit cell (18, 20, 27). A finite ⟨On(e)⟩implies that the C4v symmetry of an ideal CuO2

plane has been reduced at most to C2v sym-metry. There are eight symmetry reduction pos-sibilities for a system with full C4v symmetry;finite ⟨On(e)⟩ further restricts this to four. Infor-mation regarding further symmetry lowering (suchas inversion symmetry breaking) can determinethe actual symmetry of pseudogap states, butthose issues are beyond the scope of this paper.A coarse-grained image On(

→r, e = 1) represent-ing the local inequivalence of x- and y-axis elec-tronic structure (18) is presented in Fig. 1C. Thepanel shows how, althoughOn(

→r,e = 1) is strong-ly fluctuating at the nanoscale in very underdopedBi2Sr2CaCu2O8+d, it has a finite average valuewithin such a field of view.

The quite distinct properties of the smecticelectronic structure modulations at E ~ D1 can beexamined independently of the intra–unit-cellsymmetry breaking by focusing only on the in-commensurate modulation peaks

→Sx and

→Sy. A

coarse-grained image of the local degree of smec-tic symmetry breaking Os(

→r, e = 1) (Fig. 1D andSOM b) reveals the very short correlation lengthof the strongly disordered smectic (18, 20, 26–28).The amplitude and phase of two unidirectionalmodulation components (along x, y) within thebox in Fig. 1B can be further extracted, as shownin Fig. 2, A and B (29). To do so, we denote thelocal contribution to the

→Sx modulations at posi-

tion→r by a complex fieldy1(→r ). This contributes

to the Z(→r, e = 1) data as

y1(→r )ei

→Sx •→r þ y*1 (

→r)e�i→Sx •→r

≡ 2jy1(→r )jCos[→Sx • →r þϕ1(

→r )] ð1Þthus allowing the local phase ϕ1(

→r) of→Sx modu-

lations to be mapped similarly for the local phaseϕ2(

→r ) of the→Sy modulations. In Fig. 2, C and D,

we show images of ϕ1(→r ) and ϕ2(

→r ) derivedfrom Z(→r, e = 1). They reveal that the smecticphasesϕ1(

→r ) andϕ2(→r ) take on all values be-

tween 0 and T2p in a highly complex spatial pat-tern. Evenmore important is the detection of a largenumber of topological defects with T2p phasewinding. These are indicated by black (þ2p) andwhite (−2p) circles in Fig. 2, C and D, and occur inabout equal numbers (as one might anticipate fromthe likely macroscopic energy cost of an uncom-pensated dislocation). A typical example of an in-dividual topological defect (solid box in Fig. 2, AandC) is shown in Fig. 3, A andB. The dislocationcore (Fig. 3B) and its associated 2p phase windingare clearly seen (Fig. 3A). Moreover the amplitudeof y1(

→r ) or y2(→r ) always approaches zero near

each topological defect, as expected. These dataare all in close agreement with the theoretical expec-tations for quantum smectic dislocations (Fig. 1A).

Imaging the locations of these topologicaldefects (Fig. 2, C and D) simultaneously withthe intra–unit-cell nematicity (Fig. 1C) revealsanother key result. Figure 4A shows the lo-cations of all topological defects in Fig. 2, Cand D, plotted as black dots on the simultaneous-ly acquired image dOn(

→r) ≡ On(→r) − ⟨On⟩ rep-

resenting the fluctuations of the intra–unit-cellnematicity. By eye, nearly all the topologicaldefects appear located in (white) regions ofvanishing dOn(

→r ) ¼ 0. This can be quantified byplotting the distribution of distances of topolog-ical defects from the nearest zero of dOn(

→r ),thereby showing that they are far smaller than ex-pected if the topological defects were uncorre-lated with dOn(

→r) (Fig. 4A inset and SOM c).These data provide empirical evidence for a cou-pling between the smectic topological defects andthe fluctuations of the intra–unit-cell nematicityat E ~ D1.

To establish a Ginzburg-Landau (GL) modelrepresenting such a coupling, one needs to de-termine first whether the dOn(

→r ) fluctuations arecoupled to the phase or the amplitude of the

smectic modulations (30–33). Whether the mod-ulations are commensurate (periodic with wave-length rational multiple of a0) or incommensurateis key. For incommensuratemodulations, a smoothdeformation of the phase (Fig. 3A) costs a vanish-ingly small energy, whereas phase fluctuationsalways cost a finite energy for commensuratemodulations. On the other hand, fluctuations ofthe modulation amplitude (Fig. 3D) cost a finiteenergy in both cases (34). There are multiplereasons to conclude that we are dealing with in-commensurate modulations. First, the locationsof

→Sx and

→Sy are not necessarily at a commen-

surate point in→q space (Fig. 1B, inset), and they

change continuously with hole density (26)(Fig. 3C). More profoundly, a complex histo-gram of y1(

→r ) or y2(→r ) (Fig. 3D) shows little

predominant phase preference overall. At a fewhigh-amplitude locations (Fig. 3D), there is aparticular phase preference consistent with shortrange commensurate “nanostripes” (20). Howev-er the continuous winding around each defect(Fig. 2) is in clear contrast to discrete jumps whenonly specific values of phase are allowed (35).

Ψ1( r→

)B

1 nm

A

1 nm

low high0 2π0.90

0.85

0.80

0.75

0.70

0.65

0.60

|q→ |

(2π/

a 0)

0.200.160.120.080.04

p

Sx

Sy

-0.04

-0.02

0.00

0.02

0.04

ImΨ

1

-0.04 -0.02 0.00 0.02 0.04

Re Ψ1

DC

high

low

Fig. 3. (A) Phase field around the single topological defect in the FOV indicated by the solid box in Fig. 2,A and C. (B) Smectic modulation around the single topological defect in the same FOV, showing that thedislocation core is indeed at the center of the topological defect and that the modulation amplitude tendsto zero there. This is true for all the 2p topological defects identified in Fig. 2. (C) Doping dependence ofthe wavelength for the smectic modulations along wave vectors

→Sx and

→Sy (26). (D) Two-dimensional

histogram of real and imaginary components of the measured smectic field y1(→r ).

22 JULY 2011 VOL 333 SCIENCE www.sciencemag.org428

REPORTS

Hence, these observations support the incom-mensurate picture in which the smectic brokensymmetry exhibits free winding of the phase.Thus, our third advance is the demonstration thatthe simultaneously broken electronic symmetriesin the E ~ D1 states consist of intra–unit-cellnematicity coexisting with disordered and phasefluctuating smectic modulations.

Spatial patterns of coexisting smectic modu-lations and intra–unit-cell nematicity, as well astheir coupling, may be described most naturallyby a correspondingGL functional. For the locallyfluctuating

→Sx modulations represented byy1ð→rÞ,

the GL functional is

FGL[y1(→r )] = ∫d2r[axj∇xy1(

→r )j2 þayj∇yy1(

→r )j2 þ mjy1(→r )j2] ð2Þ

Here, ax ≠ ay and m are phenomenological GLparameters [assuming x and y directions are in-equivalent (18)]. FGL is a generalization of theGL free energy of a density modulation in onespatial dimension (22). It is similar to the GL freeenergy of a superfluid. As it is for superfluids,fluctuations in phaseϕ1(

→r ) enterFGL only throughthe spatial derivative terms because

j∇xy1(→r )j2 = [∇xjy1(

→r )j]2 þjy1(

→r )j2[∇xϕ1(→r )]2 ð3Þ

The absence of long-range smectic order (Figs.1D and 2) despite the finite modulation ampli-tudes (except within dislocation cores) impliesphase fluctuations play the predominant role insmectic disordering. Further, the finite densityof topological defects (Fig. 2) also indicates thatEq. 2 cannot provide a complete description ofthe phenomena. This is because an isolated topo-logical defect will cost an energy that grows as alogarithm of the system size and hence is unlikelyto occur. Yet we observe large numbers of iso-lated T2p topological defects (Fig. 2). Therefore,coupling to other degrees of freedommust reducethe energy of the smectic dislocations. For thecase of a classical nematic liquid crystal on theverge of freezing into a smectic-A, de Gennesdiscovered (24) a GL free energy describing howthe nematic fluctuations lower the energy costof smectic dislocations to a finite amount, thusallowing for the isolated topological defects toappear and resulting in destruction of quasi–long-range smectic order in two dimensions (24).With such a historical guide, we consider the in-terplay between the intra–unit-cell nematicity andincommensurate smectic modulations by in-cluding dOnð→rÞ fluctuations in the above GLfunctional.

When ⟨On⟩ ≠ 0 (Fig. 1C) (18), the local fluc-tuation dOn(

→r ) ≡ On(→r ) − ⟨On⟩ (Fig. 4A) is the

natural small quantity to enter the GL func-

tional [when ⟨On⟩ ¼ 0 possibly at higher dop-ings, the expansion should be in terms ofOn(

→r )with the appropriate symmetry]. Coupling tothe smectic fields can then occur either throughphase or amplitude fluctuations of the smectic.Here, we focus on the former, which means thatdOn(

→r ) couples to local shifts of the wave vec-tors

→Sx and

→Sy. Replacing the gradient in the x

direction by a covariant-derivative-like cou-pling gives

∇xy1(→r ) → [∇x þ icxd On(

→r )]y1(→r ) ð4Þ

and similarly for the gradient in the y direction, toyield a GL term coupling the nematic to smecticstates. The vector→c = (cx, cy) represents by howmuch the wave vector,

→Sx, is shifted for a given

fluctuationdOn(→r). Hence, we propose a GL func-

tional (for modulations along→Sx) based on sym-

metry principles and dOn(→r) and y1(

→r ) beingsmall:

FGL[dOn,y1] = Fn[dOn] þ

∫d2r[axj(∇x þ icxdOn)y1j2 þayj(∇y þ icydOn)y1j2 þ mjy1j2 þ…] ð5Þ

where … refers to terms we can neglect for thepresent purpose (SOM d). If we were to replace

Fig. 4. (A) Fluctuationsof electronic nematicitydOn(

→r,e = 1) obtainedbysubtracting the spatial av-erage⟨On(

→r,e = 1)⟩ fromOn(

→r,e = 1) (Fig. 1C). Thesimultaneously measuredlocations of all 2p topolog-ical defects are indicatedas black dots. They are pri-marily foundnear the lineswhere dOn(

→r,e = 1) = 0.(Inset) The distribution ofdistances between eachtopological defect and itsnearestdOn(

→r,e = 1) =0contour. This is comparedto the expected aver-age distance if there isno correlation betweendOn(

→r,e = 1) and the to-pological defect locations.There is a very strong tend-ency for the distance to thenearest dOn(

→r,e = 1) =0 contour to be small. Theboxes show regions thatare blown up in (E) and (F)and compared to simula-tions in (C) and (D). (B)Theoretical dOn(

→r,e = 1)predicted by the GL model in Eq. 5 (top) at the site of a single smectic topologicaldefect (bottom). The vector

→l lies along the zero-fluctuation line ofOn(

→r,e = 1). (Cand D) dOn(

→r,e = 1) obtained by numerical simulation using Eq. 5 and the ex-perimentally obtained topological defect configurations (black dots). Red broken

circle is themeasure of the spatial resolutiondeterminedby the cut-off length (3s) inextracting the smectic field. (See SOM f for the details of the numerical simulation).(E and F) MeasureddOn(

→r,e = 1) in the fields of view of (C) and (D). The achievedcross correlation is well above the lower bound for statistical significance (SOM f).

C

1 nm4 nm

A

-0.0043 +0.0043

B

δOn < 0

δOn > 0

+-

0 2π

l→

∇ →

ϕ( r→

)

low high

1 nm

E

low high

1 nm

F

1 nm

D

-0.0043 +0.0043

-0.0043 +0.0043

1086420

Cou

nts

806040200d ( pix )

dexdrand

www.sciencemag.org SCIENCE VOL 333 22 JULY 2011 429

REPORTS

→cdOn(→r ) by

2e

ℏ

→A(→r ) where

→A(→r ) is the electro-

magnetic vector potential, Eq. 5 becomes the GLfree energy of a superconductor; its minimizationin the long-distance limit yields

→A(→r ) =

ℏ

2e

→∇ϕ(→r )

and thus quantization of its associated magnet-ic flux (22, 23). Analogously, minimization ofEq. 5 impliesdOn(

→r ) =→l ⋅

→∇ϕ surrounding each

topological defect (SOM e). Here, the vector→l is

proportional to (ax; ay) and lies along the linewhere dOn(

→r ) = 0. The resulting key prediction isthat dOn(

→r ) will vanish along the line in the di-rection of

→l that passes through the core of the

topological defect, with On(→r ) becoming greater

on one side and less on the other (Fig. 4B). Addi-tional coupling to the smectic amplitude can shiftthe location of the topological defect away fromthe line of dOn(

→r ) = 0 (SOM e).To test whether this GL model correctly cap-

tures the observed (Fig. 4, A and B) dOn − yscoupling in Bi2Sr2CaCu2O8+d, we extend Eq. 5to include both

→Sx and

→Sy smectic modulations.

We then simulate the profile of dOn(→r ), treating

the phase and amplitude of smectic fields y1(→r )

and y2(→r ) (Fig. 2) as mean-field input that will

determine dOn(→r ) according to Eq. 5 (SOM e).

Figure 4, C and D, shows the overlay of topolog-ical defect locations within the small boxes in Fig.4AondOn(

→r ) as simulated byusingEq. 5 (SOMe).This demonstrates directly how the GL function-al associates fluctuations indOn(

→r ) with the smec-tic topological defect locations in the fashion ofFig. 4B. The close similarity between the mea-sured dOn(

→r) in Fig. 4, E and F, and the simu-lation in Fig. 4, C and D, with cross-correlationcoefficients of 56% and 62% demonstrates howthe minimal GL functional of Eq. 5 captures theinterplay between the measured dOn(

→r ) fluctua-tions (Fig. 4A) and disordered smectic modula-tions (Fig. 2). And, as expected with extrinsicdisorder (36), the GL parameters vary somewhatfrom location to location (SOM f). Indeed, asimultaneous “gapmap” (SOM g) shows vividlyhow much additional (probably dopant-atom-related) disorder coexists with the phenomenaanalyzed here.

Our results can lead to advances in understandingof coexisting and competing electronic phenomenain underdoped cuprates (9–20). By identifying 2ptopological defects within the phase-fluctuatingsmectic states and that they are associated withthe spatial fluctuations of the robust intra–unit-cellnematicity (18, 20), we demonstrated empiricallya coupling between these two locally brokenelectronic symmetries of the cuprate pseudogapstates. This allowed identification of a GL func-tional that explains how these phenomena coexistand predicts their interplay at the atomic scale.For example, the GL model explains why it ispossible for the intra–unit-cell nematicity to havefinite average ⟨On(

→r)⟩ ≠ 0 (Fig. 1C) even thoughthe smectic modulations are disordered (Figs. 2and 3) (18). This is because 2p topological defectsinduce fluctuations of dOn(

→r ) with respect to⟨On(

→r )⟩, but the dislocation cores sit close to lo-cations where On(

→r ) = ⟨On(→r )⟩ and thus do not

disrupt this state directly (SOM e). Perhaps mostimportantly, if the tendency for intra–unit-cellnematicity to coexist with a disordered electronicsmectic demonstrated here is ubiquitous to un-derdoped cuprates, which broken symmetrymanifests at the macroscopic scale (9–20) de-pends on the coefficients in the GL functionaland on other material-specific aspects, such ascrystal symmetry. Therefore, the GL model intro-duced here provides a good starting point toaddress these issues and, eventually, the interplaybetween the different broken electronic sym-metries and the superconductivity.

References and Notes1. J. Zaanen, O. Gunnarsson, Phys. Rev. B 40, 7391

(1989).2. S. A. Kivelson, E. Fradkin, V. J. Emery, Nature 393, 550

(1998).3. S. A. Kivelson et al., Rev. Mod. Phys. 75, 1201 (2003).4. S. Sachdev, Rev. Mod. Phys. 75, 913 (2003).5. M. Vojta, Adv. Phys. 58, 699 (2009).6. V. J. Emery, S. A. Kivelson, J. M. Tranquada, Proc. Natl.

Acad. Sci. U.S.A. 96, 8814 (1999).7. E.-A. Kim et al., Phys. Rev. B 77, 184514 (2008).8. E. Fradkin, S. A. Kivelson, M. J. Lawler, J. P. Eisenstein,

A. P. Mackenzie, Annu. Rev. Condens. Matter Phys. 1,153 (2010).

9. J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura,S. Uchida, Nature 375, 561 (1995).

10. J. M. Tranquada et al., Nature 429, 534 (2004).11. P. Abbamonte et al., Nat. Phys. 1, 155 (2005).12. Y.-J. Kim, G. D. Gu, T. Gog, D. Casa, Phys. Rev. B 77,

064520 (2008).13. X. F. Sun, K. Segawa, Y. Ando, Phys. Rev. Lett. 93,

107001 (2004).14. A. Kaminski et al., Nature 416, 610 (2002).15. B. Fauqué et al., Phys. Rev. Lett. 96, 197001 (2006).16. V. Hinkov et al., Science 319, 597 (2008);

10.1126/science.1152309.17. R. Daou et al., Nature 463, 519 (2010).18. M. J. Lawler et al., Nature 466, 347 (2010).19. Y. Li et al., Nature 455, 372 (2008).20. Y. Kohsaka et al., Science 315, 1380 (2007); 10.1126/

science.1138584.21. N. D. Mermin, Rev. Mod. Phys. 51, 591 (1979).

22. P. M. Chaikin, T. C. Lubensky, Principles of CondensedMatter Physics (Cambridge Univ. Press, Cambridge, 1995).

23. D. R. Tilley, J. Tilley, Superfluidity and Superconductivity(IOP Publishing, Bristol, UK, 1990).

24. P. G. de Gennes, Solid State Commun. 10, 753 (1972).25. J. Toner, D. R. Nelson, Phys. Rev. B 23, 316 (1981).26. Y. Kohsaka et al., Nature 454, 1072 (2008).27. J. Lee et al., Science 325, 1099 (2009).28. K. McElroy et al., Phys. Rev. Lett. 94, 197005 (2005).29. C. Howald, H. Eisaki, N. Kaneko, M. Greven,

A. Kapitulnik, Phys. Rev. B 67, 014533 (2003).30. A. Del Maestro, B. Rosenow, S. Sachdev, Phys. Rev. B 74,

024520 (2006).31. J. A. Robertson, S. A. Kivelson, E. Fradkin, A. C. Fang,

A. Kipitulnik, Phys. Rev. B 74, 134507 (2006).32. V. Cvetkovic, J. Zaanen, Phys. Rev. Lett. 97, 045701

(2006).33. M. Vojta, T. Vojta, R. K. Kaul, Phys. Rev. Lett. 97, 097001

(2006).34. G. Grüner, Density Waves in Solids (Addison-Wesley,

Boston, 1994).35. T. Choi et al., Nat. Mater. 9, 253 (2010).36. W. L. McMillan, Phys. Rev. B 12, 1187 (1975).Acknowledgments: We thank E. Fradkin, S. Kivelson,

M. Norman, J. P. Sethna, and J. Tranquada for helpfuldiscussions and communications. Theoretical studies atCornell were supported by NSF grant DMR-0520404 tothe Cornell Center for Materials Research and by NSFgrant DMR-0955822. Experimental studies are supportedby the Center for Emergent Superconductivity, an EnergyFrontier Research Center headquartered at BrookhavenNational Laboratory and funded by the U.S. Departmentof Energy under DE-2009-BNL-PM015, as well as by aGrant-in-Aid for Scientific Research from the Ministry ofScience and Education (Japan) and the Global Centersof Excellence Program for Japan Society for thePromotion of Science. The work at Harvard Universitywas supported by DMR-0757145. The work at LeidenUniversity was supported by the Nederlandse Organisatievoor Wetenschappelijk Onderzoek through a SpinozaAward. A.M. is grateful for the hospitality of E.-A.K.

Supporting Online Materialwww.sciencemag.org/cgi/content/full/333/6041/426/DC1SOM TextFigs. S1 to S3Table S1References

30 November 2010; accepted 2 June 201110.1126/science.1201082

Atmospheric Carbon Injection Linkedto End-Triassic Mass ExtinctionMicha Ruhl,1,2* Nina R. Bonis,1,3 Gert-Jan Reichart,4

Jaap S. Sinninghe Damsté,4,5 Wolfram M. Kürschner1,6

The end-Triassic mass extinction (~201.4 million years ago), marked by terrestrial ecosystem turnoverand up to ~50% loss in marine biodiversity, has been attributed to intensified volcanic activityduring the break-up of Pangaea. Here, we present compound-specific carbon-isotope data of long-chainn-alkanes derived from waxes of land plants, showing a ~8.5 per mil negative excursion, coincidentwith the extinction interval. These data indicate strong carbon-13 depletion of the end-Triassicatmosphere, within only 10,000 to 20,000 years. The magnitude and rate of this carbon-cycle disruptioncan be explained by the injection of at least ~12 × 103 gigatons of isotopically depleted carbonas methane into the atmosphere. Concurrent vegetation changes reflect strong warming and anenhanced hydrological cycle. Hence, end-Triassic events are robustly linked to methane-derived massivecarbon release and associated climate change.

The end-Triassic mass extinction (ETME)[~201.4 million years ago (1)], one of thefive major extinction events of the Phan-

erozoic (2), is marked by up to 50% marine bio-diversity loss and major terrestrial ecosystemchanges (2–5). This event closely matches a

22 JULY 2011 VOL 333 SCIENCE www.sciencemag.org430

REPORTS