Magnetic field controlled charge density wave coupling in underdoped YBa2Cu3O6+x

The application of magnetic fields to layered cuprates suppresses their high-temperature superconducting behaviour and reveals competing ground states. In widely studied underdoped YBa2Cu3O6+x (YBCO), the microscopic nature of field-induced electronic and structural changes at low temperatures remains unclear. Here we report an X-ray study of the high-field charge density wave (CDW) in YBCO. For hole dopings ∼0.123, we find that a field (B∼10 T) induces additional CDW correlations along the CuO chain (b-direction) only, leading to a three-dimensional (3D) ordered state along this direction at B∼15 T. The CDW signal along the a-direction is also enhanced by field, but does not develop an additional pattern of correlations. Magnetic field modifies the coupling between the CuO2 bilayers in the YBCO structure, and causes the sudden appearance of the 3D CDW order. The mirror symmetry of individual bilayers is broken by the CDW at low and high fields, allowing Fermi surface reconstruction, as recently suggested.

C harge density wave (CDW) correlations 1 , that is, periodic modulations of the electronic charge density accompanied by a periodic distortion of the atomic lattice, have long been known to exist in underdoped La-based cuprate high-temperature superconductors 2,3 . More recently, it has been found that charge order is a universal property of underdoped high-temperature cuprate superconductors [4][5][6][7][8][9][10][11] . CDW correlations appear typically at temperatures well above the superconducting transition temperature T c . Cooling through T c suppresses the CDW and leads to a state, in which the superconducting and CDW order parameters are intertwined and competing [12][13][14] .
The application of magnetic fields suppresses superconductivity. In the case of underdoped YBa 2 Cu 3 O 6 þ x (YBCO), a number of changes in electronic properties have been reported in the field range BE10-20 T. For example, new splittings occur in NMR spectra 11,15 , ultrasound shows anomalies in the elastic constants 16 and the thermal Hall effect suggests that there is an electronic reconstruction 17 . At larger fields, B\25 T a normal state with quantum oscillations (QO) 18 and coherent transport along the c axis 19 is observed. The existence of QO, combined with a high-field negative Hall and Seebeck effect, is most easily understood in terms of electron pockets 9,[20][21][22][23] .
Fields BE10-20 T also cause changes in the CDW order that can be seen by X-ray measurements. Initial experiments 5 showed that a magnetic field causes an enhancement of the diffuse CDW scattering 5,8 . A recent X-ray free-electron laser experiment 24 has shown that a magnetic field of B\15 T induces a new CDW Bragg peak, with a propagation vector along the b axis, corresponding to an extended range of ordering along the c axis and an in-phase correlation of the CDW modulation between the neighbouring bilayers.
It is important to determine the nature of the CDW correlations induced by the magnetic field in YBCO and their relationship to the electronic properties. Of particular interest are the high-field CDW phase diagram and whether a field also induces new CDW order propagating along the a axis. We have therefore used hard X-ray scattering measurements to determine the evolution of the CDW correlations, with magnetic fields up to 16.9 T for several doping levels. Here we investigate the CDW for propagation vectors along the crystallographic a-and b-directions, allowing us to extend the pulsed-field measurements 24 and identify new field-induced anisotropies in the CDW. By measuring the profile of the diffuse CDW scattering as a function of field, we show that the CDW inter-bilayer coupling along the c axis is strongly field dependent. We also show that field-induced changes in the CDW can be associated with many of the anomalies 11,[15][16][17]25 observed in electronic properties. In particular, the B À T phase diagram has two boundary lines associated with the formation of high-field CDW order. Our data also provides insight into the likely high-field structure of the CDW (in the normal state) that is relevant to describe the Fermi surface reconstruction leading to QO.

Results
Charge density wave order in YBCO. The CDW correlations in the cuprates have propagation vectors with the in-plane components parallel to the Cu-O bonds and periodicities of 3 À 4a depending on the system 2,3,5,8 . YBCO shows a superposition of modulations localized near the CuO 2 bilayers, with basal plane components of their propagation vectors along both a and b: q a ¼ (d a ,0,0) and q b ¼ (0,d b ,0) with correlation lengths up to x a E70 ÅE20a. Both q a and q b CDWs have ionic displacements perpendicular to the CuO 2 bilayers combined with displacements parallel to these planes, which are p/2 out of phase 26 . These give rise to scattering along lines in reciprocal space given by Q CDW ¼ na* þ mb* þ cc*±q a,b , where n and m are integers. The distribution of the scattered intensity along c depends on the relative phase of the CDW modulations in the bilayers stacked along the c-direction. In zero magnetic field, there is weak correlation of phases in neighbouring bilayers and we observe scattered intensity spread out along the c* direction, peaked at cE0.5 À 0.6. This is illustrated by our X-ray measurements on YBCO 6.67 (P ¼ 0.123, T c ¼ 67 K and ortho-VIII CuO-chain ordering), shown in Fig. 1a,f. Note that the strong scattering around QB(13/8,0,0) in Fig. 1a,b is due the CuO-chain ordering, which does not change with field, and can be subtracted, as in Fig. 1c,d. By taking cuts through the data, we obtain the intensity of the CDW scattering versus c for the q a and q b positions (Fig. 1e,j).
Field-induced anisotropic CDW correlations. Figure 1 shows that the effect of applying a magnetic field is very different for two components (q a and q b ) of the CDW. For the q a component of the correlations (Fig. 1b), the rod of scattering becomes stronger with no discernible change in the c width or position of the maximum, that is, the correlations simply become stronger. In contrast, for the q b correlations, (Fig. 1i) we see two qualitative changes. First, at BE10 T, the rod of diffuse scattering becomes broader in c and its peak position begins to move to larger c. Second, at BE15 T, a new peak (shaded pink and first reported in ref. 24) appears centred on c ¼ 1, but only for the q b component. The new peak indicates that the sample has regions, where the CDW modulation is in phase in neighbouring bilayers and is coherent in three spatial directions. These regions would have a typical length along the c axis of x c E47 Å.
Structure of the three-dimensional CDW order. We measured the intensity of the new three-dimensional (3D) CDW order in 14 different Brillouin zones. These data (Supplementary Note 3; Supplementary Table 1 and 2) are consistent with the high-field CDW structure of an individual bilayer being unchanged from that determined at zero field 26 . Both low-and high-field structures break the mirror symmetry of a bilayer, but in the high-field structure (Fig. 2a,d), the atomic displacements in adjacent bilayers are in phase. Thus, the high-field order has q b ¼ (0,d b ,0); however, its structure yields zero CDW intensity for c ¼ 0 and nonzero for c ¼ 1 positions, as we observe ( Supplementary Fig. 4). The relationship between the CDW structures at low and high field is to be expected, since the coupling between the two CuO 2 planes in a bilayer will be stronger than coupling with another bilayer. For the other basal plane direction, no CDW signal was found at q ¼ (d a ,0,0) or (d a ,0,1) for Br16.9 T (Fig. 3c).
The phase diagram and 3D CDW precursor correlations. The c-dependent profiles in Fig. 1e,j contain information about the correlation between the phases of the CDW modulation in the bilayers stacked along the c axis. For B ¼ 0, the broad cE0.5 À 0.6 peaks in Fig. 1e,j for q a and q b indicate that the CDW phase is weakly anti-correlated between neighbouring bilayers. On increasing the field above BE10 T, the c-profile of the q b correlations evolves. The onset of this evolution can be seen as an increase in the intensity of the scattering at (0,4-d b ,1), see Fig. 4c, signalling the introduction of new c axis correlations. This change is accompanied by a growth of correlations along the b axis, as shown by the increase in the correlation length x b,c ¼ 1 measured by the peak width of scans parallel to b* through the (0,4-d b ,1) position ( Fig. 4e). We describe this state as 3D CDW precursor correlations. The onset temperature TE65 K of the precursor correlations at high field (B ¼ 16.5 T) may be determined from the increase in x b,c ¼ 1 and the scattering intensity at the (0,4-d b ,1) position (Fig. 4c,e). This allows us to designate a region of the B À T phase diagram (Fig. 5).
At higher fields, B\15 T, a peak (shaded pink in Fig. 1j) develops abruptly in the c-profile at c ¼ 1. The abrupt onset of the peak signals a rapid growth of the c axis correlation length x c (Fig. 4d,e). The growth of correlations in one spatial direction followed by growth in a second direction is typical of systems, with anisotropic coupling. Another CDW system that shows this behaviour 27 is NbSe 3 . Large correlated regions develop first in planes, where the order parameter is most strongly coupled. These act to amplify the coupling in the remaining direction. In case of YBCO 6.67 , the in-plane correlation length continues to grow down to low temperatures with x b,c ¼ 1 ¼ 80b ¼ 310 Å (at B10 K and 16.5 T). The c axis correlation length, however, saturates with x c,c ¼ 1 ¼ 47 Å at TE30 K. All these changes together signal the transition to a new phase (see Fig. 5 pink region), which we label 3D CDW order identified with a phase transition also seen in ultrasound 16 and thermal Hall effect 17 measurements. At the lowest temperatures, Tt25 K, we observe ( Fig. 4a) a suppression of the 3D CDW peak intensity signalling a competition between the superconducting and 3D CDW order parameters.
Previous X-ray 5,8 and NMR 25 measurements on YBCO 6.67 have shown that the weak anti-phase (c ¼ 1/2) CDW correlations appear at TE150 K. Further NMR anomalies in the form of line splittings 11,15 are observed at TE65 K for B ¼ 28.5 T and at BE10 T for T ¼ 2 K. These anomalies that are displayed on Fig. 5 appear to coincide with the onset of the 3D precursor correlations reported here. The fact that NMR sees similar transitions shows that the 3D CDW precursor correlations we observe are static on timescales t\0.1 ms. Correlations that are static 25 and short ranged are necessarily controlled by pinning with quenched disorder playing a role.
Doping dependence. We also studied other dopings of YBCO 6 þ x . For YBCO 6.60 with hole doping P ¼ 0.11 and ortho-II oxygen chain structure, a very similar onset field (Fig. 5) and c axis correlation length x c were found. In YBCO 6.51 and YBCO 6.75 , no 3D order was observed for Br16.9 T (Fig. 3a). However, we do observe the precursor movement of the CDW scattering to higher c implying that this structure is likely to appear at higher fields. Thus, the 3D order is most easily stabilized for doping around p ¼ 0.11-0.12 (Fig. 5b).   At B ¼ 0, the model shows that the broad cE0.5 À 0.6 peaks in Fig. 1e,j are due to weakly anti-correlated bilayers (Fig. 2b,e). The field evolution of the c-dependent profiles for q b (Fig. 1j), including the formation of the c ¼ 1 peak, may be modelled by a continuous variation of b and g from anti-phase coupling at low field to same-phase coupling at high field (Fig. 2e). The sign of b changes near the onset of the 3D order at BE15 T. Thus, we find that a c axis magnetic field can control the coupling between the CDWs in neighbouring bilayers. The field control of the coupling most likely arises through the suppression of superconductivity by field. Magnetic field strengthens the correlations along the a axis (Fig. 1e); however, it does not increase correlation lengths. Possible explanations for this difference in behaviour include the influence of the CuO chains promoting the b axis modulations or the chains pinning the a axis CDW modulations.
We conclude that the appearance of 3D CDW order corresponds to the onset of new c axis electronic coherence and hence electronic reconstruction. This is supported by thermal  (Fig. 4). Triangles are the Fermi surface reconstruction onset determined from thermal Hall coefficient 17 . Solid black squares indicate the onset of growing in-plane CDW correlation lengths (3D precursor correlations) determined from the variation of x b,c ¼ 1 (Fig. 4d,e). Hall conductivity measurements 17 that demonstrate Fermi surface reconstruction at the same field (Fig. 5a). At highest fields investigated, B ¼ 16.9 T, the structure of the CDW within individual bilayers involves the same breaking of mirror symmetry observed at zero field 26 , which has been posited to lead to Fermi surface reconstruction 28,29 .

Methods
Experimental details. Our experiments used 98.5 keV hard X-ray synchrotron radiation from the PETRA III storage ring at DESY, Hamburg, Germany. A 17 T horizontal cryomagnet 30 was installed at the P07 beamline. Access to the (h,0,c) and (0,k,c) scattering planes was obtained by aligning either the a-c axes or the b-c axes horizontally, with the c axis approximately along the magnetic field and beam direction. The samples were glued to a pure aluminium plate on which was mounted a Cernox thermometer for measurement and control of temperature.
With the high intensities of PETRA III, a small amount of beam heating of the sample was observed. By observing the effect of changes in beam heating (controlled by known attenuation) on the measured temperature of the 3D phase transition, we determined the effect of the beam on the sample temperature near 40 K. The sample heating at other temperatures was determined using the Cernox thermometer and a model of the heat flow from the sample to the aluminium plate. We estimate that there is an absolute uncertainty in our temperature determination of ±2 K. The relative temperature uncertainty is smaller than this. Four YBCO crystals with different in-planar doping and different oxygen chain structure were studied (Table 1). Except for the YBCO 6.60 sample, detailed descriptions of these crystals are found in refs 5,31,32. The YBCO 6.60 sample was studied with the scattering plane defined by (k,k,0) and (0,0,c). This configuration has the advantage that CDW modulations along both a and b axis directions could be accessed without reorienting the sample. The absence of c ¼ 0,1 CDW order along the a axis direction was checked using the (h,0,c) scattering plane.
Data analysis. h-and k-scans, as shown in Fig. 3c,d, are fitted with a single Gaussian function on a weakly sloping background. c-scans with a well-defined peak at c ¼ 1 (Figs 1j and 3a,b) are fitted using a two Gaussian functions. Correlation lengths x ¼ 1/s are defined by the inverse Gaussian s.d. s¼ s 2 meas À s 2 R À Á 0:5 .
The instrumental resolution s R -for a CDW reflection-was estimated at Bragg reflections near to the measured CDW reflections. Resolution-corrected correlation lengths are given in Table 1.
Simulation of scattering profiles. We use a simple Markov chain model 33 of order m ¼ 2 to interpret the diffuse and c ¼ 1 scattering profiles, for example, in Fig. 1j. A Markov chain is a stochastic series. Here we generate a series of two types of bilayer (A and B) corresponding to the phase of the displacement of the CDW in a given bilayer. We represent the bilayer type at position index i along the c-direction by a stochastic variable x i . This can take either the value x i ¼ 1, denoting bilayer type A, or x i ¼ 0 for type B. We create a series of bilayers starting, for instance, with an A bilayer, followed by a B. The probability P(x i ¼ 1) of adding an A bilayer at position i, preceded by x i À 1 and x i À 2 , in the series is given by: Clearly, P(x i ¼ 0) ¼ 1 À P(x i ¼ 1). Equation (1) is a recipe, using random numbers to represent the probabilities, to create a series of bilayers with a given amount of correlation built in. Let m A (m B ¼ 1 À m A ) be the fraction of A-type (B-type) bilayers in the series and P AA 1 the proportion of AA bilayer pairs separated by one lattice spacing. We choose a¼ 1 2 1 À b À g ð Þso that macroscopically m A ¼m B ¼ 1 2 . A number (NB500) of stochastic series x i (N site ¼ 100) subject to a given a, b and g are generated. The corresponding scattered intensity (assuming the single-unit cell structure from ref. 26) for each series is calculated and averaged. b and g are adjusted to give the best fit to the data and P AA 1 is calculated. The correlation length x b and x c of the (0,d b ,0) CDW order at the highest measured fields and lowest temperature are given in units of the lattice parameters b ¼ 3.87\AA(Å) and c ¼ 11.7 Å.