Abstract
The interplay between superconductivity and any other competing order is an essential part of the long-standing debate on the origin of high-temperature superconductivity in cuprate materials1,2. Akin to the situation in the heavy fermions, organic superconductors and pnictides, it has been proposed that the pairing mechanism in the cuprates comes from fluctuations of a nearby quantum phase transition3. Recent evidence for charge modulation4 and its associated fluctuations5,6,7 in the pseudogap phase of YBa2Cu3Oy makes charge order a likely candidate for a competing order. However, a thermodynamic signature of the charge-ordering phase transition is still lacking. Moreover, whether the charge modulation is uniaxial or biaxial remains controversial. Here we address both issues by measuring sound velocities in YBa2Cu3O6.55 in high magnetic fields. We provide the first thermodynamic signature of the competing charge-order phase transition in YBa2Cu3Oy and construct a field–temperature phase diagram. The comparison of different acoustic modes indicates that the charge modulation is biaxial, which differs from a uniaxial stripe charge order.
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In most La-based cuprate superconductors, static order of both spin and charge (so-called stripe order) has been unambiguously identified by spectroscopic and thermodynamic probes1,2. At low temperature, magnetic fields weaken superconductivity and at the same time reinforce the magnitude of such orders8,9,10. As the superconducting transition temperature (Tc) in the La-based materials is substantially lower than in other cuprate materials, it has been argued that stripe order is detrimental to high-temperature superconductivity. In underdoped YBa2Cu3Oy (YBCO), there is now compelling evidence of competing order even though Tc = 94 K at optimal doping. The discovery of quantum oscillations11 combined with the negative Hall12 and Seebeck13 coefficients at low temperature has demonstrated that the Fermi surface of underdoped YBCO undergoes a reconstruction at low temperature and consists of at least one electron pocket. A comparative study of thermoelectric transport in underdoped YBCO and in La1.8−xEu0.2SrxCuO4 (a cuprate in which stripe order is well established) has been interpreted as charge stripe order causing reconstruction of the Fermi surface at low temperature14. High-field nuclear magnetic resonance (NMR) measurements have revealed that the translational symmetry of the CuO2 planes in YBCO is broken by the emergence of a modulation of the charge density at low temperature4. In addition, NMR measurements show that the modulation is observed above a threshold magnetic field and suggest that charge order is most likely uniaxial4. In zero field, long-range charge fluctuations in YBCO were recently observed with resonant soft X-ray scattering (RSXS) up to 150 K and 160 K for p = 0.11 (ref. 5) and p = 0.133 (ref. 6), respectively, whereas hard X-ray scattering experiments suggest that a charge order develops below 135 K for p = 0.12 (ref. 7). All measurements identify charge fluctuations at two wave vectors corresponding to an incommensurate periodicity of approximately 3.2 lattice units. The identification of a thermodynamic phase transition is thus important to determine where long-range charge order exists in the phase diagram and particularly whether static order occurs only in high magnetic fields.
Here we report sound velocity measurements, a thermodynamic probe, in magnetic fields large enough to suppress superconductivity. The sound velocity is defined as , where ρ is the density of the material, ci j = ∂2F/∂ ɛi∂ ɛj (ref. 15), F is the free energy and ɛi is the strain along direction i (in the contracted Voigt notation). Changes in the elastic constants ci j are expected whenever a strain-dependent phase transition occurs. Owing to their high sensitivity, sound velocity measurements are a powerful probe for detecting such phase transitions, in particular charge ordering in strongly correlated electron systems16.
We have measured several elastic constants (see Supplementary Table S1 for the description of the elastic modes) in high magnetic fields in an underdoped YBCO6.55 sample with Tc = 60.7 K corresponding to a hole doping p = 0.108 (ref. 17). Figure 1a,b shows the field dependence of the relative variation of the sound velocity Δvs/vs corresponding to the c11 mode, at different temperatures. At T = 4.2 K, the softening of the elastic constant at the vortex lattice melting field Bm≈20 T corresponds to the first-order melting transition from a vortex lattice to a vortex liquid18,19 (see Supplementary Information for more details). At T = 29.5 K, this anomaly shifts to lower field (Bm≈5 T) and because the pinning potential becomes less effective, the magnitude of the change of c11 at the melting transition becomes smaller20. At T = 29.5 K and above Bm, a sudden increase of the elastic constant can clearly be resolved at Bco = 18 T, which corresponds to a thermodynamic signature of a phase transition. Whereas Bco is almost temperature independent at low temperature, it increases rapidly between 35 and 50 K (see red arrows in Fig. 1b). For T≥50 K, no change of c11 can be resolved up to the highest field. Owing to the difference in the temperature dependence of Bm and Bco, the phase transition at Bco cannot originate from vortices. Figure 2 shows the phase diagram in which both Bm and Bco deduced from sound velocity measurements are plotted as a function of temperature. The identification of this phase stabilized by the magnetic field above Bco is straightforward. High-field NMR measurements in YBCO at similar doping have shown that charge order develops above a threshold field Bco>15 T and below TcoRMN = 50±10 K (ref. 4). Given the similar field and temperature scales, it is natural to attribute the anomaly seen in the elastic constant at Bco to the thermodynamic transition towards the static charge order.
The phase diagram in Fig. 2 shares common features with the theoretical phase diagram of superconductivity in competition with a density-wave order21 (see discussion in the Supplementary Information). For T below 40 K or so, static charge order sets in only above a threshold field of 18 T, akin to the situation in La2−xSrxCuO4 (x = 0.145) in which a magnetic field is necessary to destabilize superconductivity and to drive the system to a magnetically ordered state9. Close to the onset temperature of static charge order, Tco, the threshold field Bco sharply increases and the phase boundary tends to become vertical. This is in agreement with the theoretical phase of competing order with superconductivity that predicts that superconducting fluctuations have no significant effect on charge order in this part of the phase diagram.
We now turn to the analysis of the symmetry of the charge modulation. In the framework of the Landau theory of phase transitions, an anomaly in the elastic constant occurs at a phase transition only if a coupling in the free energy Fc = gm nQm ɛn (where m and n are integers and gm n is a coupling constant) between the order parameter Q and the strain ɛ is symmetry allowed, that is, only if Qm and ɛn transform according to the same irreducible representation22. In Fig. 3 we compare the field dependence at T = 20 K of four different modes c11, c44, c55 and c66 that display an anomaly at Bco. To explain the presence of such coupling for all these modes, we rely on group theory arguments. YBCO is an orthorhombic system (point group D2h), and given the even character of the strains we have only to consider the character table of point group D2 shown in Table 1.
To represent the different symmetric charge modulations that transform according to each irreducible representation of the point group D2 and to determine to which acoustic mode they couple, we will follow the procedure of ref. 16 and use the projection operator in the basis defined by the four Cu atoms at the corner of the CuO2 plane (see Supplementary Information). This simplification is justified because the charge order is an intrinsic property of the CuO2 plane4,6. The four kinds of symmetric charge distribution allowed in this basis are shown in Supplementary Fig. S5. In Fig. 4, we illustrate them with concrete charge modulation patterns in the CuO2 plane with an arbitrary periodicity. A uniaxial charge modulation along the a axis should induce an anomaly in c44 but not in c55. The very similar field dependence of c44 and c55 around Bco (Fig. 3) indicates a biaxial charge modulation both along the a axis (Qa) and along the b axis (Qb). As the irreducible representation and ɛ6 transforms according to B1, a further coupling βa bQaQbɛ6 explains the anomaly seen in c66. A similar coupling exists for c11 because (i = 1,2 or 3).
As a thermodynamic bulk probe with a typical energy scale below 1 μeV, sound velocity measurements establish that static charge order sets in below Tco≈45 K in magnetic fields above Bco = 18 T in YBCO at a doping level p = 0.108. For a nearby doping p = 0.11 (non-ortho-II ordered sample), zero-field RSXS measurements show that charge fluctuations appear at T = 150 K (ref. 5). The fluctuating character of the charge order at high temperature is consistent with recent resonant ultrasonic measurements in YBCO at similar doping level. Whereas anomalies in the sound velocity are detected at the pseudogap temperature, T*≈240 K, and at the superconducting transition temperature Tc, no further phase transition is observed in zero field23. An important aspect of these fluctuations is their biaxial character inferred from both RSXS (refs 5, 6) and X-ray7 measurements. However, those techniques cannot determine whether there is a single charge density wave (CDW) with biaxial modulation or whether there are domains of two uniaxial CDWs. From sound velocity measurements, the latter scenario seems unlikely because for the anomaly in c66 to exist, charge modulations along the two directions should in principle coexist within the same sample volume. The biaxial character of the CDW observed in our study and in X-ray measurements supports the scenario in which the static charge ordering occurring at low temperature and finite magnetic field is a consequence of the freezing of the charge fluctuations observed in zero field at high temperature. A biaxial CDW seems to differ from the conclusion drawn from NMR measurements, which have been interpreted as a uniaxial modulation of the charge density along the a axis4. However, an extra charge modulation along the b axis, as reported here, is not excluded from the NMR spectra. The biaxial nature of the CDW differs from the phenomenology of stripe order, which corresponds to a uniaxial charge order, as observed in La-based cuprates1,2. This is further supported by the absence of spin order down to the lowest temperature at this doping level as inferred from NMR measurements4. It raises the questions of the origin of the Tc anomaly close to p = 0.12 in underdoped YBCO and of the peculiar behaviour of the transport properties at high fields around this doping level24. These questions and the interplay between biaxial CDW and high-temperature superconductivity call for further investigations. The biaxial character of the charge order has another important consequence, namely on the topology of the reconstructed Fermi surface. In the absence of spin order, as inferred from NMR (ref. 4), uniaxial order by itself cannot produce electron pockets in the reconstructed Fermi surface25, unless one assumes that the Fermi surface at high temperature already breaks C4 symmetry26. On the other hand, no particular assumption is needed to produce electron pockets when the Fermi surface is folded along two perpendicular directions27.
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Acknowledgements
We thank M-H. Julien, S. Kivelson, B. Lüthi, R. Ramazashvili, G. Rikken, L. Taillefer, B. Vignolle, M. Vojta and S. Zherlitsyn for useful discussions. We acknowledge experimental support from A. Mari, D. Rickel and the LNCMI staff. Research support was provided by the French ANR SUPERFIELD, Euromagnet II, the Canadian Institute for Advanced Research and the Natural Science and Engineering Research Council.
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D.L. and C.P. performed the high-field measurements. S.K. provided instrumental support for the d.c. field measurements. W.N.H., R.L. and D.A.B. prepared the samples (crystal growth, annealing, de-twinning). D.L. analysed the data. D.L. and C.P. wrote the manuscript and C.P. supervised the project.
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LeBoeuf, D., Krämer, S., Hardy, W. et al. Thermodynamic phase diagram of static charge order in underdoped YBa2Cu3Oy. Nature Phys 9, 79–83 (2013). https://doi.org/10.1038/nphys2502
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DOI: https://doi.org/10.1038/nphys2502
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