Fragile charge order in the nonsuperconductingground state of the underdoped high-temperature superconductorsB. S. Tana, N. Harrisonb, Z. Zhub, F. Balakirevb, B. J. Ramshawb, A. Srivastavaa, S. A. Sabokc, B. Dabrowskic,G. G. Lonzaricha, and Suchitra E. Sebastiana,1

aCavendish Laboratory, Cambridge University, Cambridge CB3 OHE, United Kingdom; bNational High Magnetic Field Laboratory, Los Alamos NationalLaboratory, Los Alamos, NM 87545; and cPhysics Department, Northern Illinois University, DeKalb, IL 60115

Edited by Zachary Fisk, University of California, Irvine, Irvine, CA, and approved June 11, 2015 (received for review February 28, 2015)

The normal state in the hole underdoped copper oxide superconduc-tors has proven to be a source of mystery for decades. The measure-ment of a small Fermi surface by quantum oscillations on suppressionof superconductivity by high applied magnetic fields, together withcomplementary spectroscopic measurements in the hole underdopedcopper oxide superconductors, point to a nodal electron pocket fromcharge order in YBa2Cu3O6+δ. Here, we report quantumoscillationmea-surements in the closely related stoichiometric material YBa2Cu4O8,which reveals similar Fermi surface properties to YBa2Cu3O6+δ, de-spite the nonobservation of charge order signatures in the samespectroscopic techniques, such as X-ray diffraction, that revealed sig-natures of charge order in YBa2Cu3O6+δ. Fermi surface reconstructionin YBa2Cu4O8 is suggested to occur from magnetic field enhancementof charge order that is rendered fragile in zero magnetic fields be-cause of its potential unconventional nature and/or its occurrence as asubsidiary to more robust underlying electronic correlations.

superconductivity | strongly correlated electron systems |high-Tc cuprate superconductors | Fermi surface | charge order

The normal state of the underdoped copper oxide supercon-ductors has proven to be even more perplexing than the d-wave

superconducting state in these materials. At high temperatures inzero magnetic fields, the normal state of the underdoped cupratescomprises an unconventional Fermi surface of truncated “Fermiarcs” in momentum space, which is referred to as the pseudogapstate (1). At low temperatures in high magnetic fields, quantumoscillations reveal the nonsuperconducting ground state in variousfamilies of underdoped hole-doped copper oxide superconductorsto comprise small Fermi surface pockets (2–15). These small Fermipockets in YBa2Cu3O6+δ have been identified as nodal electronpockets (2, 3, 11, 16, 17) originating from Fermi surface re-construction associated with charge order measured by X-raydiffraction (18–20), ultrasound (21), nuclear magnetic resonance(22), and optical reflectometry (23). However, various aspects of theunderlying charge order and the associated Fermi surface re-construction remain obscure. A central question pertains to theorigin of this charge order, curious features of which include ashort correlation length in zero magnetic field that grows withincreasing magnetic field and decreasing temperature (20). It iscrucial to understand the nature of this ground-state order thatis related to the high-temperature pseudogap state and deli-cately balanced with the superconducting ground state. Here,we shed light on the nature of this state by performing extendedmagnetic field, temperature, and tilt angle-resolved quantumoscillation experiments in the stoichiometric copper oxide su-perconductor YBa2Cu4O8 (24). This material with double CuOchains has fixed oxygen stoichiometry, making it a model systemto study. YBa2Cu4O8 avoids disorder associated with the frac-tional oxygen stoichiometry in the YBa2Cu3O6+δ chains, whichhas been shown by microwave conductivity to be the dominantsource of weak-limit (Born) scattering (25).

Intriguingly, we find magnetic field- and angle-dependentsignatures of quantum oscillations in YBa2Cu4O8 (13, 14) thatare very similar to those in YBa2Cu3O6+δ, indicating a similarnodal Fermi surface that arises from Fermi surface re-construction by charge order with orthogonal wave vectors (16).However, the same X-ray diffraction measurements that show aBragg peak characteristic of charge order in YBa2Cu3O6+δ for arange of hole dopings from 0.084≤ p≤ 0.164 (19, 20, 26) have,thus far, not revealed a Bragg peak in the case of YBa2Cu4O8(19). We suggest that charge order enhanced by applied mag-netic fields reconstructs the Fermi surface in YBa2Cu4O8,whereas charge order is revealed even in zero magnetic fields inYBa2Cu3O6+δ because of pinning by increased disorder fromoxygen vacancies.

ResultsFig. 1 shows quantum oscillations in contactless conductivity (27)measured up to 90 T in YBa2Cu4O8 and at different tempera-tures from 1.3 to 8.0 K. The extended magnetic field range andincreased sensitivity compared with previous quantum oscillationexperiments (13, 14) enable precision measurements of thequantum oscillation frequency spectrum and effective quasipar-ticle mass of YBa2Cu4O8.Preliminary quantum oscillation measurements on YBa2Cu4O8

accessed two (13) to four (14) oscillation periods over a re-stricted magnetic field range for a magnetic field angle parallelto the crystalline c axis. A single quantum oscillation frequencyof 660 ± 30 T was reported, whereas the scatter of the quantumoscillation amplitude as a function of temperature precluded a

Significance

Results from quantum oscillation and spectroscopic measure-ments have suggested charge order as responsible for thecreation of a nodal electron pocket in the YBa2Cu3O6+δ andHgBa2CuO4+δ families of underdoped cuprates. However, thesituation in the pristine YBa2Cu4O8 family remains ambiguous.Our high-precision quantum oscillation measurements point toa very similar nodal electron pocket in the stoichiometric cup-rate family YBa2Cu4O8, despite the nonobservation of chargeorder signatures in diffraction experiments. Our findings inYBa2Cu4O8 indicate Fermi surface reconstruction in the holeunderdoped cuprates associated with the magnetic field en-hancement of charge order, the fragile low-magnetic fieldcharacter of which is reflected in the short correlation lengthreported in various materials families.

Author contributions: N.H. and S.E.S. designed research; B.S.T., N.H., Z.Z., F.B., B.J.R., A.S.,S.A.S., B.D., and S.E.S. performed research; B.S.T., N.H., and S.E.S. analyzed data; and N.H.,G.G.L., and S.E.S. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. Email: suchitra@phy.cam.ac.uk.

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determination as to whether a Lifshitz–Kosevich form is obeyedor an accurate extraction of a quasiparticle effective mass. Ourquantum oscillation measurements over an extended magneticfield range access more than seven oscillation periods, revealingfor the first time, to our knowledge, a pronounced quantumoscillation beat structure characteristic of multiple frequencies,very similar to YBa2Cu3O6.56 (6, 9, 28), with a dominant fre-quency of 640 T. Our precision measurements of quantum os-cillation amplitude as a function of temperature shown in Fig. 1B and C further reveal a distinctive Lifshitz–Kosevich form,characteristic of Fermi Dirac statistics. A fit to the Lifshitz–Kosevich form yields a quasiparticle effective mass of 1.8 (1)mein YBa2Cu4O8, which is, in fact, very similar to that measuredfor YBa2Cu3O6.56.Angular measurements as a function of tilt angle to the applied

magnetic field are required to identify the origin of the multiple-frequency spectrum that we observe in YBa2Cu4O8. Fig. 2 showsthe quantum oscillations in YBa2Cu4O8 measured up to a maxi-mum tilt angle of θ≈ 56°. A few key features are notable. First,the beat pattern at the 0° tilt angle persists up to high tilt angles.Second, the prominent Yamaji amplitude resonance (29) expectedat θ≈ 52° for a neck–belly warped Fermi surface in YBa2Cu4O8 isabsent up to the measured high tilt angles. The absence of aprominent Yamaji angle in YBa2Cu4O8 enables us to rule out aFermi surface where the observed frequency spread arises froma dominant neck and belly fundamental warping. Instead, both ofthe observed features are consistent with a quasi-2D Fermi sur-face, in which the multiple-frequency spectrum originates from asplitting of frequencies as opposed to a fundamental neck andbelly warping. Such a quasi-2D Fermi surface is similar to thatidentified in YBa2Cu3O6+δ (16, 28).Another clue as to the origin of the observed quantum oscil-

lations in YBa2Cu4O8 is obtained by inspecting the evolution ofthe amplitude and phase of the quantum oscillations as a functionof tilt angle. Fig. 3B shows the cross-correlation function betweenthe measured quantum oscillations in YBa2Cu4O8 and a phase-matched sinusoidal function with frequency 640 T, which is av-eraged over the indicated magnetic field range, referred to as the

correlator. We find that the correlator is very similar to thatpreviously measured for YBa2Cu3O6.56 (11, 16) and shown in Fig.3A. The correlator for YBa2Cu4O8 reveals a zero crossing andphase inversion of the quantum oscillation amplitude, signaling aspin zero at a tilt angle of θ≈ 48°. A spin zero feature arises ondestructive interference between quantum oscillations from twospin channels at certain special angles. At these spin zero angles,the ratio of the Zeeman energy to the cyclotron energy—cap-tured by the spin splitting factor Rs = cos½ðπ=2Þmk*g*=me cos θ�—crosses zero and inverts sign. Spin zero angles are located at

θsz = cos−1"

mk*g*ð2n+ 1Þme

#, [1]

where n is an integer, g* is the effective g factor, assumed here to beisotropic for simplicity (see Methods and Table 1 for the moregeneral case). me is the electron mass, and mk* is the effectivequasiparticle mass for B parallel to the crystalline c axis (30).A value of mk*g* can be extracted by the measurement of at

least two spin zeros, given the two unknown quantities n and g* ,as was done for YBa2Cu3O6.56 (16). To interpret the single spinzero observed at θszfirst ≈ 48° in YBa2Cu4O8, a comparison withYBa2Cu3O6.56 is instructive, in which case two spin zeros wereobserved at θszfirst ≈ 55° (10, 16) and θszsec ≈ 66° (16). Good agree-ment with experimental data is yielded by associating an indexvalue of n= 2 with the first spin zero, for parameters includinganisotropic g-factors (see Methods) given in Table 1. Parametersfor YBa2Cu4O8 are remarkably similar to those obtained in thecase of YBa2Cu3O6.56 (16), and are in good agreement withexperimental data (solid lines in Fig. 3).Given the similarities that we find between YBa2Cu4O8 and

YBa2Cu3O6+δ in terms of the measured multiple-quantum

Fig. 1. (A) Quantum oscillations measured in the contactless resistivity ofYBa2Cu4O8 for the magnetic field parallel to the c axis. (B) Fourier transformof the measured quantum oscillations at different temperatures. (C) Plot ofquantum oscillation amplitude as a function of temperature (symbols) ac-companied by a Lifshitz–Kosevich fit (red line).

Fig. 2. Quantum oscillations (colors) measured in the contactless resistivityof YBa2Cu4O8 for different angles of inclination (θ) of the magnetic field tothe c axis plotted as a function of 1

Bcos θ. Simulated quantum oscillations (gray)of a bilayer split quasi-2D Fermi surface model shown in Fig. 4 (16) are shownfor the parameters listed in Table 1. Inset shows a schematic of the crystal tiltangle to the magnetic field.

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oscillation frequency spectrum, split quasi-2D Fermi surface, andspin zero angles revealed by the correlator, we compare themeasured quantum oscillations in YBa2Cu4O8 with the Fermi

surface model fit to the measured quantum oscillations inYBa2Cu3O6+δ (11, 16, 17). We consider a model in which cyclo-tron orbits are associated with a nodal bilayer split Fermi surfacefrom charge ordering shown in Fig. 4. Here, the quantum oscil-lation frequency spread is associated with a splitting of the Fermisurface arising from tunneling between bilayers. Magnetic break-down tunneling in the nodal region where the splitting is smallestcan then give rise to a series of combination frequencies, whichis discussed in refs. 16 and 28. Figs. 2 and 3 show that thismodel can simulate the angular and magnetic field dependenceof the quantum oscillations measured in YBa2Cu4O8 reason-ably well. The parameters used for the model simulation are, infact, very similar to those used to simulate quantum oscillationsmeasured in YBa2Cu3O6.56 (Table 1) (16). The small stag-gered twofold warping included in the case of YBa2Cu3O6.56 isnot included for YBa2Cu4O8 given the restricted angularrange over which quantum oscillations can be accessed and thelikely weaker amplitude warping associated with a longer c axis.We note that there is some deviation from the model at a few ofthe highest angles. This deviation may be caused by additionallifting of degeneracy of frequency components from effects such asa distortion of the simple tetragonal crystal structure or otherdetails of Fermi surface geometry beyond those considered inthis model.

DiscussionOur findings, therefore, reveal closely similar quantum oscillationfeatures in YBa2Cu4O8 compared with YBa2Cu3O6.56, showing (i) asimilar quasiparticle effective mass from a Lifshitz–Kosevich fit tothe amplitude dependence as a function of temperature, (ii) aspread of multiple frequencies, yielding a prominent beat structure,(iii) an angular dependence of quantum oscillation frequenciesconsistent with Fermi surface splitting accompanied by magneticbreakdown rather than fundamental neck and belly warping, and(iv) a similar value of m*g* from the angular dependence and spinzero observed in the correlator. The common Fermi surface fea-tures point to the same origin of Fermi surface reconstruction inYBa2Cu4O8 as in YBa2Cu3O6+δ, which is particularly relevant giventhe nonobservation of charge order signatures thus far in YBa2Cu4O8(19). In the case of YBa2Cu3O6+δ, indications are that the electronicstructure comprises a nodal electron pocket from Fermi surface

Fig. 3. (A) Symbols represent the cross-correlation function between themeasured quantum oscillations in YBa2Cu3O6.56 and a phase-matched sinu-soidal function with frequency 534 T, which is averaged over the indicatedmagnetic field range. Reprinted from ref. 16. (B) Symbols represent thecross-correlation function between the measured quantum oscillations inYBa2Cu4O8 and a phase-matched sinusoidal function with frequency 640 T,which is averaged over the indicated experimental magnetic field range.Solid lines show the fits using the equation and parameters in Table 1.

Table 1. Parameters used to simulate the oscillatory waveform for a quasi-2D splitFermi surface model shown in Fig. 4 represented by the equationΨtwofold ≈∑6

j=1Nj[RMBRsRDRT ]j cos(2πFj=B cos θ− π+ϕ)

Parameter Description YBa2Cu4O8 YBa2Cu3O6.56

F0 Quantum oscillation frequency 639 T 534 TΔFtwofold Staggered twofold warping frequency — 15 TΔFsplit Bilayer splitting frequency 91 T 90 Tmk* Quasiparticle effective mass 1.8 me (fixed) 1.6 me (fixed)B0 Magnetic breakdown field 4.2 T 2.7 Tgk□* g-Factor 1 2.0 2.1gk◇* g-Factor 2 0.1 0.4ξ□ g-Factor anisotropy 1 1.6 1.4ξ◇ g-Factor anisotropy 2 0.8 0.2ϕ Phase −1.6c 0 (fixed)

Here, RMB is the magnetic breakdown amplitude reduction factor (defined in Methods), and Nj counts thenumber of instances that the same orbit is repeated within the magnetic breakdown network. RD is the Dingledamping factor, RT is the thermal damping factor, and Rs is the spin damping factor (defined in Methods). Thetabulated values used to simulate the quantum oscillation waveform yield good agreement with experiment as afunction of B and θ (Figs. 2 and 3). The effective mass is taken to be a fixed quantity, having been determinedindependently from temperature-dependent measurements (9). The parameters are the same for all of theorbits, except for those denoted by subscripts □ and ◇, which correspond to subsets of orbits as defined inthe text. The values of gkj* and ξj here represent parameters used for simulation rather than unique identifica-tions. The parameters used for YBa2Cu3O6.56 shown for comparison are taken from ref. 16.

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reconstruction by a superstructure with orthogonal wave vectorsQ1 = 2πð±δ1=a, 0Þ and Q2 = 2πð0, ±δ2=bÞ associated with chargeorder observed by X-ray diffraction and other experiments(where δ1 ≈ δ2 ≈ 0.3) (19, 20, 22).An unusual aspect of the charge order measured in

YBa2Cu3O6+δ is the short average correlation length, which hasbeen measured to be of the order of 100 Å in the basal plane andthe order of 10 Å along the c axis from X-ray diffraction mea-surements in a magnetic field of 17 T (20). The measured averagecorrelation length in the basal plane is comparable with the cy-clotron radius of 100− 200 Å in the magnetic field range of30− 100 T where quantum oscillations are measured (2–12).Moreover, given that the anisotropic Fermi velocity associatedwith the observed pocket has a magnitude along the c axis that isapproximately two orders of magnitude lower than in the basalplane (28, 31, 32), the component of the cyclotron trajectory alongthe c axis is also well within the c-axis correlation length. Themeasured charge ordering wave vectors with finite correlationlength measured up to 17 T are, therefore, consistent with theFermi surface reconstruction observed by quantum oscillations inan applied magnetic field. We note that the correlation lengthinferred from X-ray diffraction experiments represents an averageover all parts of the sample, thus representing a lower bound onthe correlation length associated with the most highly orderedregions of the sample, which are accessed, for instance, by quan-tum oscillation measurements. It has remained unclear thus farwhether the reported short correlation length is caused by thenucleation and pinning of charge order at impurity/defect sites orwhether it is because of the disruptive effects of impurities/defectsin an intrinsically long-range charge ordered state (33, 34).Our findings in YBa2Cu4O8, where impurity pinning centers are

reduced compared with YBa2Cu3O6+δ because of the fixed oxygenstoichiometry, suggest that charge order of inherently short cor-relation length arises in zero magnetic fields in YBa2Cu3O6+δ onaccount of impurity sites that act as pinning potential centers.Charge order in the well-known charge density wave materialNbSe2 has, for instance, been shown to initially develop aroundimpurities (35). In the case of YBa2Cu4O8, however, where such

impurity pinning centers are reduced, applied magnetic fields arerequired to tilt the balance of energy scales, such that chargesusceptibility is further enhanced, and charge order is revealed toreconstruct the Fermi surface.Our findings point to a fragile form of charge order in the non-

superconducting ground state of the underdoped cuprates, which isenhanced by applied magnetic fields to yield Fermi surface re-construction. An interesting question pertains to the origin of suchintrinsically fragile charge order. A contributing factor may be thepotential unconventional character of the observed charge corre-lations (36). Furthermore, charge order may arise as a corollary tomore robust correlations in the underdoped cuprates, such as un-derlying strong spin correlations (37). The interplay between thesecorrelations may be manifested as other order parameters, such asAmperean order (38), pair density wave order (39), d-wavecheckerboard order (40), and quadrupolar order (41, 42), whichhave also been proposed to appear in fluctuating form above thesuperconducting temperature.

MethodsQuantum oscillation simulations include conventional thermal, Dingle, andspin damping factors of the same form used for previous comparisons withquantum oscillations measured in the underdoped cuprates and other lay-ered families of materials (30).

The thermal damping factor is given by

RT =Xj

sinhXj,

where Xj =2π2kBmθj*T=ZeB, kB is the Boltzmann factor, and T is the tempera-

ture; mθj* =mkj* =cos θ (taken to be the same for all orbits, and the subscript j is

dropped) is determined by the projection of B perpendicular to theplanes (i.e., the projection parallel to the c axis in YBa2Cu3O6+x), and mkj*

refers to the value of mθj* when B is parallel to the crystalline c axis [taken to

be a fixed quantity determined independently from temperature-dependentmeasurements (9)].

The Dingle damping factor is given by

RD = exp�−

Λj

B cos θ

�,

where Λj is a damping factor, which is taken to be the same for all orbits,enabling us to drop the subscript j (30).

The spin damping factor is given by

Rs = cos

"π

2

mθj*

me

!gθj*

#,

where the anisotropic effective g factor has the form gθj* =gkj*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos2θ+ 1=ξjsin

2θq

.Here, gkj* refers to the value of gθj

* when B is parallel to the crystalline c axis,whereas ξj = ðgk* =g⊥

* Þ2 is the anisotropy in the spin susceptibility. Two sets ofanistropic g factors are considered for subsets of the orbits defined below. Be-cause of the multiple frequencies in the model and the restricted angular rangemeasured, it is not possible to uniquely identify the g factors. The values of gkj*

and ξj here represent parameters used for the simulation.A splitting of the Fermi surface arising from tunneling between bilayers

leads to two starting frequencies that are denoted as F1 = F0 − 2Fsplit andF6 = F0 + 2Fsplit. Magnetic breakdown tunneling (in the nodal region wherethe splitting is smallest) gives rise to a series of combination frequenciesF2, F3, F4, F5 as discussed in refs. 16 and 28. Only two sets of anisotropic gfactors are considered: orbits F1, F2, F4, F5, and F6, which undergo bothmagnetic breakdown tunneling and reflection, are approximated to havethe same g-factor gk◇* with anisotropy ξ◇, whereas orbit F3, which showsonly magnetic breakdown tunneling without reflection, is approximated tohave a common g-factor gk□* with anisotropy ξ□ (16). The magnetic break-down amplitude reduction factor is given by

RMB =�iffiffiffiP

p �lν� ffiffiffiffiffiffiffiffiffiffiffi1− P

p �lη,

in which lν and lη count the numbers of magnetic breakdown tunneling andBragg reflection events en route around the orbit having transmitted am-plitudes i

ffiffiffiP

pand

ffiffiffiffiffiffiffiffiffiffiffi1− P

p, respectively. The magnetic breakdown probability

is given by P = expð−B0=B cos θÞ, where B0 is the characteristic magneticbreakdown field (30).

Fig. 4. Brillouin zone cross-section showing a schematic of a nodal electronpocket created by charge ordering wave vectors Q1 = 2πð±δ1=a, 0Þ andQ2 =2πð0, ± δ2=bÞ (11, 16, 17). A shows the two Fermi surface cross-sections offrequency F1 = F0 − 2ΔFsplit and F6 = F0 + 2ΔFsplit. The gap separating bondingand antibonding surfaces is expected to be smallest at the nodes (31). A cutthrough the kz = 0 plane of the Brillouin zone shows the possible magneticbreakdown orbits (16, 28). B–D show the range of possible magnetic breakdownorbits (F2 = F0 −ΔFsplit, F3 = F0, F4 = F0, and F5 = F0 +ΔFsplit) as listed in Table 1.

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ACKNOWLEDGMENTS. We thank B. Keimer, S. A. Kivelson, M. le Tacon,P. A. Lee, C. Pépin, S. Sachdev, and T. Senthil for useful discussions. We alsothank the National High Magnetic Field Laboratory personnel, includingJ. B. Betts, Y. Coulter, M. J. Gordon, C. H. Mielke, M. D. Pacheco, A. Parish,R. D. McDonald, D. Rickel, and D. Roybal, for experimental assistance. B.S.T.,A.S., and S.E.S. acknowledge support from the Royal Society, the Winton Pro-gramme for the Physics of Sustainability, and the European Research Council(ERC) under European Union Seventh Framework Programme Grant FP/2007-2013/ERC Grant Agreement 337425. N.H., Z.Z., F.B., and B.J.R. acknowledge

support for high-magnetic field experiments from US Department of Energy(DOE), Office of Science, Basic Energy Sciences (BES)-Materials Sciences andEngineering (MSE) Science of 100 Tesla Programme. G.G.L. acknowledges sup-port from Engineering and Physical Sciences Research Council (EPSRC) GrantEP/K012894/1. Work at Northern Illinois University was supported by the In-stitute for Nanoscience, Engineering, and Technology. A portion of this workwas performed at the National High Magnetic Field Laboratory, which is sup-ported by National Science Foundation Cooperative Agreement DMR-0654118, the State of Florida, and the DOE.

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