Introduction

The enhancement of spin-stripe order by a magnetic field in La2−xSrxCuO4 (ref. 1) and the enhanced modulation of the local density of states (LDOS) around the vortex cores in Bi2Sr2CaCu2O8+δ (Bi-2212) (ref. 2) are milestone results that have promoted the idea of electronic ordering that competes with superconductivity3,4,5,6,7. Recently, the discovery of charge order in YBa2Cu3Oy appearing only for fields perpendicular to the copper-oxide planes sufficiently strong to be detrimental to superconductivity8, as well as the subsequent observation of related charge density wave (CDW) correlations9,10,11, has provided further evidence of competition. The unambiguous observation of charge order, without spin order, in this compound with a low level of disorder is significant as it reveals the ubiquity of charge-ordering tendencies in the normal state of cuprate superconductors. Furthermore, it gives unprecedented opportunities to follow, specifically by NMR, how charge order arises and develops starting deep in the superconducting state where transport techniques are inoperative, through the vortex-melting field, Hmelt where ultrasound measurements are blurred, up to field values comparable to Hc2, which have so far been out of reach for scattering and tunnelling techniques.

Here, we show with NMR that charge order, that is, a static, long-range, spatial modulation of the charge density, emerges above a threshold magnetic field in the vortex-solid state and we show how this result can be related to earlier evidence of competing orders, thereby highlighting universal aspects of the competition between superconducting and charge orders in cuprates. These results constrain theories relating cuprate superconductivity to the charge instability.

Results

Field dependence of charge order

Charge order in YBa2Cu3Oy modifies the NMR lineshapes of some of the copper and oxygen (63Cu and 17O) sites in CuO2 planes8 (Fig. 1). Here, we adopt the simplest description of these modifications, namely a line splitting8. Should the actual lineshape in the charge-ordered state be more complex than a simple splitting, this would affect the discussion of the exact pattern of charge order, but not the conclusions of this article, which are independent of such details. The spectral modifications induced by the charge order are relatively small, so that detecting a possible departure from a splitting or hypothetical field-dependent modifications (for instance, due to a field-dependent ordering wave-vector) is currently beyond our experimental resolution.

Figure 1: 17O NMR evidence of charge order in YBa2Cu3O6.56.
figure 1

Magnetic field-induced modifications of the highest-frequency quadrupole satellite of O(2) sites (lying in bonds oriented along the a axis) at T=2.9 K and doping level p=0.109. f0 (~40–160 MHz) is the frequency of the centre of the shown spectrum. Continuous lines are fits with one Gaussian function at 6.2 and 9.6 T and with two Gaussian functions (each shown as a dotted line) at higher fields. See Methods section for more information about the 17O spectra.

The net line splitting involves a splitting Δνmagn of magnetic hyperfine origin and a splitting Δνquad of electric quadrupole origin. As νquad at Cu and O sites in the cuprates is a linear function of p (ref. 12), we take Δνquad to be, in the first approximation, a measure of a charge density difference, that is, the amplitude of the charge order. The expression ‘charge order’ will be used here generically, with no regard given to its specific morphology, such as, uni- or bi-directional, or its microscopic origin such as Fermi surface instability, electron–phonon coupling or strong correlation effect.

The sharp temperature dependence of the NMR splitting below the onset temperature Tcharge (ref. 6) (Fig. 2a) and recent ultrasound measurements13 already indicate that charge order occurs through a phase transition. However, up to now, the field dependence for T<<Tcharge has not been accessed. The new central result is our observation at T≈3 K of a sharp square-root-type increase of Δνquad, Δνquad(H−Hcharge)1/2, starting above a threshold field, Hcharge≈10.4 T in a sample having p=0.109 and ortho-II oxygen order (Fig. 3a). Qualitatively similar results, albeit covering a smaller field range, were obtained for three other samples from either 63Cu or 17O NMR (Fig. 3b-d). This constitutes the first example of a (apparently second order) quantum phase transition, controlled by the magnetic field, from a homogeneous d-wave superconductor to a superconductor with charge order. Remarkably, the low values of the magnetic hyperfine shift K at low T (Fig. 2c) reveal a vanishing of the spin susceptibility, which persists in high fields. Charge ordering thus leaves the pseudogap intact.

Figure 2: Temperature-induced transition towards charge order in the pseudogap state.
figure 2

(a) Quadrupole part of the splitting of the highest-frequency O(2) quadrupole satellite for p=0.109 and p=0.125 (see Methods for details) as a function of temperature in fields of 27.4 and 28.8 T, respectively. The lines are guides to the eye. (b) Transition temperature Tcharge showing a maximum around hole-doping p=0.115–0.12. The thick trace is a guide to the eye. (c) Magnetic hyperfine shift 17K of O(2,3) sites for p=0.109 and H||c. No anomalous change of 17K is observed across Tcharge. At low temperature in the charge-ordered state, the maximum shift variation Δ17K≈0.01% between 12 and 28.5 T (yellow region) represents a minor change compared to the decrease Δ17K≈0.12% between T=300 K and 60 K associated with the pseudogap42. The pseudogap is thus essentially unaffected by the occurrence of charge order. This result agrees with the relatively modest size of the field-induced changes in the 63Cu relaxation rate 1/T1 (ref. 8). The field dependence of 17K below Tc arises from the density of nodal states in a d-wave superconductor43. Error bars represent s.d of the fit parameters.

Figure 3: Quantum phase transition to the charge-ordered state.
figure 3

(a) Quadrupole part of the splitting of the O(2) line shown in Fig. 1 for p=0.109 and T≈3 K (see Methods for details). The thick red trace is a guide to the eye. The thin black line is a fit to (HHc)0.5. (b) Quadrupole splitting of 63Cu(2F) (planar Cu sites below oxygen-filled chains) for p=0.104 (ortho-II) at T=2 K. (c) Full linewidth at half maximum of the 63Cu central line for p=0.12 (ortho-VIII) at T=1.4 K. Because of the broad and complex Cu spectra in this sample, the splitting of the 63Cu satellite line could not be observed8. However, the width of the central line reflects the modifications of the lineshape due to charge order modulating the hyperfine fields8. (d) Quadrupole part of the splitting of the 17O(2) line for p=0.125 (ortho-VIII) at T=2.1 K. (e) Hole-doping dependence of the onset field Hcharge as determined from a fit of the data to (H−Hcharge)0.5 (thin black lines in ad). The minimum of Hcharge near p=0.115–0.12 parallels that of the vortex-melting field Hmelt (T→0), which has been argued to reflect the upper critical field Hc2 (T→0) (ref. 15). Error bars represent s.d. in the fit parameters.

Relationship with other probes of charge order in YBa2Cu3Oy

The finite value of Hcharge suggests that there is no static long-range charge order in zero field, in agreement with the interpretation of X-ray results in terms of CDW fluctuations9. We note that our Hcharge≈9.3 T for p=0.12 corresponds approximately to the field above which the intensity and the width of the superlattice peaks in X-ray measurements10 become larger in the low-temperature limit (T=2 K) than at 66 K, that is at the zero-field Tc. This suggests that Hcharge corresponds to a threshold in the screening of the CDW correlations by the superconducting regions of the sample (see next section for a more precise interpretation of this threshold). On the other hand, sound velocity data13 for p=0.108 suggest Hcharge≈18 T (c11 mode) and Hcharge≈16±2 T (other modes), both larger than Hcharge≈10.4±1.0 T in NMR for p=0.109. It is possible that NMR somewhat underestimates Hcharge if pre-transitional effects modify the lineshape below the real Hcharge. This should lead to some caution regarding the precise value of Hcharge but it does not affect the conclusions of this paper.

Relationship with vortex physics

For all samples, Hcharge≈9–15 T is found to be lower than the melting transition of the vortex lattice that takes place at Hmelt >20 T for T ≤5 K (ref. 14). The charge-ordering transition thus occurs inside the vortex-solid phase. As the vortex cores represent normal regions of radius ξSC within the superconductor, it is expected15,16,17 that the charge fluctuations detected above Tc (refs 9, 10, 11) continue to develop at low temperatures within the cores where they escape the competition with superconductivity. As suggested by LDOS modulations in Bi-2212 (ref. 2), halos of incipient charge order are centred on the cores and they extend over a typical distance ξcharge>ξSC (Fig. 4). On increasing the field, the long-range, static, charge order may be expected to appear when these halos start to overlap. This should occur at Hcharge=Φ0/(2πξcharge2), as the halo density equals the density of vortices whose cores start to overlap at the upper critical field, Hc2=Φ0/(2πξSC2). Owing to our observation of a field-induced transition to the charge-ordered state, this prediction is now confirmed by experiments for the first time: Hcharge=9.3±1.3 T for p=0.12 (ortho-VIII) translates into ξcharge=16a, where a is the planar Cu–Cu distance. This is to be compared to ξcharge≈19a measured at H=9 T and T=2 K by X-ray diffraction for the same doping level10. Despite the obvious simplistic nature of the description (for instance, neither a coupling between CuO2 planes nor an in-plane anisotropy of ξcharge is considered), this agreement suggests that this picture is indeed the correct starting point for explaining the field-induced transition. This is the second central result of this work.

Figure 4: Halos of incipient charge order centred on vortex cores.
figure 4

The yellow tubes represent the vortex cores of radius given by the superconducting coherence length ξSC. Black and white halos represents sites with high and low charge density. The actual pattern of the charge density modulation has no role in the discussion of this paper, so its schematic representation is arbitrary here. Such halos of incipient charge order start to overlap at Hcharge and thus induce long-range order. This description, inspired from scanning tunnelling microscopy results in Bi-2212 (ref. 2), is quantitatively consistent with our results in YBa2Cu3Oy (see text). ξcharge is the typical length over which the charge density is correlated, thus defining halos of diameter 2ξcharge around the vortex cores (image used with permission; CNRS Alpes—service communication—LRF).

Doping dependence of charge order around p=0.11–0.12

On increasing the field further in the p=0.109 sample, Δνquad saturates at fields of 30–35 T (Fig. 3a). Remarkably, this field scale is similar to Hc2, as defined from transport measurements in these samples14,18,19,20, indicating that the growth of the amplitude of the charge order is controlled by the decrease of the superconducting order parameter. Although the precise value of Hc2 is a matter of debate21, a dip of Hc2 near p=0.115–0.12 and values in the range 24–60 T appear to be robust conclusions. Furthermore, this dip of Hc2 is paralleled by both a minimum of Hcharge (Fig. 3e) and by a maximum of Tcharge near p=0.115–0.12 (Fig. 2b). These correlations are again a manifestation of the competition between superconductivity and charge order and they are consistent with charge order being strongest near p=0.115–0.12 (ref. 22), which is within the superconducting dome.

This last observation is, however, puzzling. On one hand, there is an obvious parallel with the stabilisation of charge stripes of period λ≈4a at a similar (but not strictly identical) doping of p=0.125=1/8 in compounds of the La-214 family such as La2−xBaxCuO4 and possibly in Bi-2212 as well23. This similarity would be natural if the CDW in YBa2Cu3Oy (p≈0.12) also had λ≈4a. On the other hand, if the charge order is incommensurate with λ≈3.3a (refs 9, 10, 11), there is no obvious reason why it should be more stable near p≈0.12, especially because the chain-oxygen order does not seem to have a prominent role in determining Hcharge given our finding of similar Hcharge values in ortho-II (p=0.109) and ortho-VIII (p=0.12) samples (Fig. 3e).

Discussion

Even if our understanding of the charge order in YBa2Cu3Oy (and perhaps of the 1/8 problem in general) is incomplete, the results reported here reveal an outstanding universality of magnetic field effects in underdoped cuprates. Indeed, a competition between superconductivity and charge order has also been argued6,15,16,17 to provide a natural explanation of the field-enhanced spin and charge orders in La-214 (refs 1, 3, 4, 5, 6) and LDOS modulations in Bi-2212 (refs 2, 7). Despite possible differences in the morphology of the charge order that may depend on the crystallographic structures and on the level of disorder in different families of cuprates, the above experiments and the results in YBa2Cu3Oy are all consistent with the idea that a magnetic field applied perpendicular to the CuO2 planes generates vortices around which fluctuating or weakly pinned, short-range charge order is revitalised. The long-range order that should follow from the charge instability of the normal state is initially hindered by superconductivity, but it eventually sets in when a sufficiently high density of vortices is reached. On the other hand, when charge order and superconductivity already coexist in zero field, the charge order is simply enhanced by the field6.

This field-tuned competition differs from the simple coexistence of CDW order and superconductivity in, for example, NbSe2 for which the onset of superconductivity occurs well below the CDW transition and does not affect CDW order. Therefore, the magnetic field has no effect on the CDW24. The competition in cuprates is apparently also different from the coexistence of two adjacent phases in the phase diagram of other unconventional superconductors: in this case, the transition to the competing phase can take place at a much higher temperature than superconductivity, which occurs near the verge of this competing phase25. Here, in contrast, the maximum of Tcharge (and Hcharge) occurs at p=0.11–0.12 within the superconducting dome and the transition temperatures are similar (Tc~Tcharge), indicating that the two orders have very close energy scales near p=0.11–0.12. Such near-degeneracy suggests that, although competing, charge order and superconductivity are joint instabilities of the same normal (pseudogap) state in this doping range.

Despite the mounting evidence for a charge-ordering instability in virtually all cuprate families2,6,8,9,10,11,23,26,27,28,29,30, there is still a long way to go before elucidating the importance of this instability in determining the properties of the cuprates. Neither the possibility of an intertwining of the charge and superconducting order parameters31 nor the possibility of a direct relationship between the two orders32,33 can be addressed by the present results. Furthermore, more work is needed to understand whether the interplay between charge ordering and superconductivity is fundamentally, or only superficially, different from that in other systems showing CDW and superconducting orders in their phase diagrams34,35. The most pressing question for clarifying these issues is now to determine how far the charge correlations extend in the temperature versus doping phase diagram.

Methods

Samples

High-quality, oxygen-ordered (Supplementary figure S1), detwinned single crystals of YBa2Cu3Oy were grown in non-reactive BaZrO3 crucibles from high-purity starting materials36. Two samples were enriched with the oxygen-17 (17O) isotope that, unlike 16O, possesses a nuclear spin. Table 1 summarises the properties of the four samples studied in this work. Note that the p-values are obtained on the basis of the values of the superconducting transition temperature Tc, as measured by SQUID, using the standard calibration37. As this calibration was established for samples with 16O, the isotope effect on the transition temperature Tc of the 17O-enriched samples must be taken into account. As 16O→18O exchange produces a change ΔTc≈−2 K in the Y-123 system with Tc values around 60 K (ref. 38), the hole content of the two 17O-enriched crystals was estimated using a Tc value corrected by ΔTc=+1 K with respect to their Tc value measured by SQUID. This is based on the standard expression for the isotope effect ΔTc/Tc=−αΔM/M, where M is the isotope mass and α is the isotope effect exponent (α≈0.27 here in YBa2Cu3Oy). The correction for 16O→17O exchange is thus half of that for 16O→18O.

Table 1 Composition and characteristics of the samples reported in this study.

NMR spectra of quadrupolar nuclei

The resonance frequency of a given (mm−1) transition for a nucleus of spin I >1/2 is given by the sum of magnetic hyperfine and electric quadrupole contributions39:

with−I+1≤mI, vmagn=(1+Kzz)vref, where Kzz is the component of the hyperfine magnetic shift tensor along the magnetic field direction and vref=γH is a reference frequency (such as the resonance frequency of the bare nucleus in vacuum or the resonance frequency of the nucleus in a substance without unpaired electrons).

To first order in perturbation (although not negligible, the second order is not necessary for the simple qualitative background that we intend to provide here),

where is the quadrupole frequency, and , where V is the electrostatic potential at the nucleus position and is the corresponding electric field gradient tensor. The principal axes (X, Y, Z) of the tensor are defined such that |XX|≤|YY|≤|ZZ| and the asymmetry parameter η=(XXYY)/ZZ. θ is the polar angle between the magnetic field direction z and ZZ and φ is the azimuthal angle in the (x, y) plane perpendicular to the field. For planar 63Cu, Z is the crystalline c axis, while for planar 17O, Z is the Cu–O–Cu direction (that is, a for O(2) and b for O(3)). Supplementary figure S2 shows a sketch of a typical quadrupole-split NMR spectrum.

Understanding precisely how the charge density modulation affects the electric field gradient and thus modifies Δνquad=f(νQ,η,θ,φ) requires complex ab initio calculations that are beyond the scope of this article. As νquad at Cu and O sites in cuprates is a linear function of p (ref. 13), we take Δνquad to be, in the first approximation, a measure of a charge density difference, which is the amplitude of charge order.

NMR methods

Experiments were performed in the LNCMI-Grenoble resistive magnets M1, M9 and M10 as well as in the NHFML hybrid magnet. Standard spin-echo techniques were used with heterodyne spectrometers. Spectra were obtained at fixed magnetic fields by combining Fourier transforms of the spin-echo signal recorded for regularly spaced frequency values40.

The 27Al NMR reference signal from metallic aluminium was used to calibrate the external magnetic field values. The 17K measurements were performed with H||c, on the central line, that is the (−1/2↔1/2) transition, where O(2) and O(3) sites overlap. The 17K values thus represent average values for these two sites. Neither the very weak and broad 17O signal from the chains nor the sharp signal from apical 17O sites significantly affected the determination of the position of the O(2,3) central line. 17K values are given with respect to the resonance frequency of the bare nucleus and they are in excellent agreement with earlier works41.

In order to determine the 17O line-splitting, the magnetic field was tilted by θ=16° off the c axis in order to separate O(2) from O(3) satellite transitions. In that case, the quoted magnetic field values correspond to the c axis component (that is they are corrected by a factor cos(16°)=0.961 with respect to the total external field values), which is justified by the disappearance of charge order when Hc (ref. 8).

We report here the field-induced modifications of the O(2) NMR lines, which are those sites from Cu–O–Cu bonds aligned along the a axis, which is perpendicular to the chain direction b. Clear field-induced spectral modifications are also observed for O(3E) and/or O(3F) sites (planar sites in bonds along b, below empty and filled chains, respectively), but these could not be easily analysed as the O(3E) and O(3F) lines overlap. A complete account and interpretation of 17O NMR spectra in the charge-ordered state is beyond the scope of the present work and will be published separately (Wu et al., in preparation).

Analysis of NMR spectra

The separation of the magnetic hyperfine, Δνmagn, and quadrupole, Δνquad, contributions to the total line splitting was performed by reproducing the experimental positions of 63Cu(2F) or 17O(2) lines with a simulation based on an exact diagonalisation of the nuclear-spin Hamiltonian. For 17O(2), it is also possible to extract Δνquad by subtracting the total splitting Δνtotal(1)=Δνmagnνquad of the (1/2↔3/2) satellite from the total splitting Δνtotal(2)=Δνmagn+2Δνquad of the (3/2↔5/2) satellite. However, Δνtotal(1) is not always experimentally accessible. For p=0.09, the saturation of Δνquad and Δνmagn above ~30 T is confirmed by the linear field dependence of the total splitting ΔννquadνmagnνquadKzz γH, and by the perfect overlap of the lineshapes of the 63Cu central transition at all fields >30 T when plotted in a frequency scale normalised by field.

No stable fit of the temperature dependence of Δvquad could be performed close to Tcharge. Nevertheless, we found that the temperature dependence of the guide to the eye shown in Fig 2a for the p=0.125 data also matches very well the data for p=0.104 (ref. 8) and p=0.109. We thus used this guide to determine the transition temperature of the charge-ordered state, Tcharge.

Hcharge is determined by fitting Δvquad data to (H-Hcharge)0.5, so that the result does not depend on the points for which Δvquad is assumed to be zero. The value of Hcharge is largely determined by the curvature of Δvquad versus H: that is, the data points for which the large splitting results in relatively small error bars contribute as much as the points close to Hcharge, which have larger error bars. The data for the p=0.109 sample were fit to (HHcharge)α and the fit resulted in α=0.49±0.09.

Additional information

How to cite this article: Wu, T. et al. Emergence of charge order from the vortex state of a high temperature superconductor. Nat. Commun. 4:2113 doi: 10.1038/ncomms3113 (2013).