Anisotropy of the Seebeck Coefficient in the Cuprate Superconductor YBa$_{2}$Cu$_{3}$O$_{y}$: Fermi-Surface Reconstruction by Bidirectional Charge Order

The Seebeck coefficient $S$ of the cuprate YBa$_{2}$Cu$_{3}$O$_{y}$ was measured in magnetic fields large enough to suppress superconductivity, at hole dopings $p = 0.11$ and $p = 0.12$, for heat currents along the $a$ and $b$ directions of the orthorhombic crystal structure. For both directions, $S/T$ decreases and becomes negative at low temperature, a signature that the Fermi surface undergoes a reconstruction due to broken translational symmetry. Above a clear threshold field, a strong new feature appears in $S_{\rm b}$, for conduction along the $b$ axis only. We attribute this feature to the onset of 3D-coherent unidirectional charge-density-wave modulations seen by x-ray diffraction, also along the $b$ axis only. Because these modulations have a sharp onset temperature well below the temperature where $S/T$ starts to drop towards negative values, we infer that they are not the cause of Fermi-surface reconstruction. Instead, the reconstruction must be caused by the quasi-2D bidirectional modulations that develop at significantly higher temperature.

The Seebeck coefficient S of the cuprate YBa2Cu3Oy was measured in magnetic fields large enough to suppress superconductivity, at hole dopings p = 0.11 and p = 0.12, for heat currents along the a and b directions of the orthorhombic crystal structure. For both directions, S / T decreases and becomes negative at low temperature, a signature that the Fermi surface undergoes a reconstruction due to broken translational symmetry. Above a clear threshold field, a strong new feature appears in S b , for conduction along the b axis only. We attribute this feature to the onset of 3D-coherent unidirectional charge-density-wave modulations seen by x-ray diffraction, also along the b axis only. Because these modulations have a sharp onset temperature well below the temperature where S / T starts to drop towards negative values, we infer that they are not the cause of Fermi-surface reconstruction. Instead, the reconstruction must be caused by the quasi-2D bidirectional modulations that develop at significantly higher temperature.
There is compelling evidence that this FSR is caused by charge-density-wave (CDW) order. Indeed, in all materials and at every doping where FSR has been detected, CDW modulations have also been observed by x-ray diffraction (XRD) [16][17][18][19][20][21] (except in Y124, where no XRD search has been reported). Having said this, the mechanism by which CDW order produces a small electron pocket in the Fermi surface of hole-doped cuprates remains a puzzle. This is because CDW order is thought to be unidirectional (or "stripe-like") in at least some cuprates and a unidirectional CDW modulation does not in general produce a closed electron pocket [22], at least not at "nodal" locations in the Brillouin zone, away from the anti-nodal pseudogap [23]. By contrast, bidirectional CDW order (with in-plane modulations along both high-symmetry directions of the tetragonal or orthorhombic lattice) readily produces a closed electron pocket at "nodal" locations [24,25].
This paradox has recently become vivid in the orthorhombic cuprate YBCO at p = 0.12, where XRD studies in high magnetic fields detect long-range three dimensional (3D) CDW order [26], with modulations that run only along the b axis [27,28], above a sharply defined threshold field that coincides with an anomaly in the sound velocity considered to be the thermodynamic signature of CDW order in YBCO [29]. Is this field-induced unidirectional CDW order causing the FSR in YBCO?
Here, we report measurements of the Seebeck coefficient S of YBCO along the a and b axes at p = 0.11 and p = 0.12, in magnetic fields high enough to reach the normal state. For both directions, we observe a negative S at low temperature, signature of the FSR giving an electron pocket. In addition, we detect a pronounced minimum in S b (H), not present in S a (H), whose onset field and temperature match the onset of the 3D unidirectional CDW order seen by XRD. However, since this onset temperature is well below the temperature where S / T starts to drop towards negative values, we infer that the primary cause of the FSR are the 2D bidirectional CDW modulations that develop in tandem with the gradual drop in S / T .

II. METHODS
Single crystals of YBa 2 Cu 3 O y (YBCO) were prepared by flux growth [30]. Their hole concentration (doping) p is determined from the superconducting transition temperature T c [31], defined as the temperature below which the zero-field resistance is zero. A high degree of oxygen order was achieved for samples with p = 0.11 (y = 6.54, ortho-II order, T c = 61.5 K) and p = 0.12 (y = 6.67, ortho-VIII order, T c = 65.4 K). The Seebeck coefficient S -the longitudinal voltage generated by a longitudinal thermal gradient -was measured, as described elsewhere [9], on two pairs of a-axis and b-axis YBCO samples, with dopings p = 0.11 and p = 0.12. S(H) was measured as a function of magnetic field up to H = 34 T, applied along the

III. RESULTS
In Fig. 1, the Seebeck coefficient of YBCO at p = 0.11 is plotted as S / T vs H, for several temperatures. Our data on S a agree well with previous measurements of the Seebeck coefficient in YBCO [8,9]. To our knowledge, there are no prior high-field measurements of S b in YBCO. We see that for both directions, S / T at high field goes from positive at high temperature to negative at low temperature, the signature that FSR is occurring upon cooling, resulting in a Fermi surface at low temperature that contains a small electron pocket [1]. Note that the magnitude of S / T at T → 0 ( −0.8 µV / K 2 ) is consistent with theoretical expectation [33] in the sense , if we use the Fermi temperature T F = 410 K measured by quantum oscillations in YBCO at p = 0.11 [2,34]. The isotherms of S b in Fig. 1(b) reveal a new and pronounced feature, essentially absent in S a . Indeed, on top of the same overall field and temperature dependence as observed in S a / T , S b / T exhibits an upturn at high field, producing a dip at H 18 T that deepens as temperature is reduced. In Fig. 2(a), we focus on this feature by comparing S a / T (blue) and S b / T (red) vs H at T = 20 K. At low field (up to about 16 T), both curves are identical: zero in the vortex-solid state, then slightly positive, followed by a dramatic drop to large negative values. At fields above 16 T, a striking anisotropy between the two directions appears, as a pronounced upturn develops in S b , but not in S a . We identify the field at which S b reaches a minimum as H Seebeck , equal to 19 ± 1 T at T = 20 K. Fig. 2(b) presents the same comparison at p = 0.12, in crystals with a different oxygen order (ortho-VIII instead of ortho-II). We observe a very similar Seebeck anisotropy, again characterized by an upturn in S b , appearing above H Seebeck = 16 ± 1 T.
To study the temperature dependence of H Seebeck in detail, we measured closely-spaced isotherms of S b up to 45 T, plotted in Fig. 3. We see that the minimum The green line is a guide to the eye ending at the zero-field Tc. Short-range 2D bi-directional CDW modulations (CDW-1) are detected by XRD throughout this phase diagram, both in the blue region above the red and green lines, as well as below (to the left of) those lines. In addition, long-range 3D unidirectional modulations (CDW-2) are detected in YBCO in a region very similar to the red region defined here, namely with an onset temperature T 47 K and field H 18 T at p = 0.11 [28], and T 47 K and H 15 T at p = 0.12 [27]. The red line is a guide to the eye.
in S b / T vs H is present at temperatures up to at least 30 K, remaining in roughly the same position. In Fig. 4, we plot H Seebeck on the H − T phase diagram of YBCO at p = 0.11 (yellow squares). It is essentially constant in temperature up to 30 K.
In Fig. 5(a), S a (T ) and S b (T ) measured at H = 34 T are compared directly, plotted as S / T vs T . We see that down to 45 K, the two curves are approximately parallel, with a roughly constant difference between them. Indeed, a smooth fit through the a-axis data (blue line) makes a good fit through the b-axis data if the line is simply shifted down rigidly (red line). Below 45 K, S a / T continues its monotonic decrease, but the anomalous feature in S b produces a striking departure of S b / T from its fit line (red), initially as a plateau which persists down to 30 K. To capture this extra anisotropy, we plot the difference between b-axis data and red fit line in the inset of Fig. 5(a). We see that it appears below T Seebeck = 47 ± 5 K. T Seebeck is plotted on the H −T phase diagram of Fig. 4, for three different fields.  (Fig. 2) shows up in the temperature dependence of S b / T initially as a plateau. Inset: Difference between the b-axis data points and the red line (fit) in the top panel (red dots). The green dots are obtained using the S b data of Fig. 1(b). The onset temperature for the extra anisotropy is T Seebeck = 47 ± 5 K (arrow). b) Sa / T vs T , in YBCO at p = 0.12, for three field values as indicated. At all fields, 2D bi-directional CDW modulations (CDW-1) are observed in the blue and red regions [36], while 3D uni-directional CDW order (CDW-2) is observed only in the red region and only when H > 15 T [27,28]. The dashed line is a smooth extension of the high-T data below its inflexion point at T = 130 K. The dotted line is a linear extension of the data below T = 40 K.

IV. DISCUSSION
The anomaly in S b we observe in YBCO at p = 0.11 is confined to a region of the H − T diagram (Fig. 4) that is essentially the same region where 3D unidirectional CDW order has been observed by XRD [26][27][28]. This order was detected in YBCO above an onset field H = 18 T at p = 0.11 [28] and above H = 15 T at p = 0.12 [26,27], in good agreement with H Seebeck = 19 ± 1 T and 16 ± 1 T at p = 0.11 and 0.12, respectively (Fig. 2). This value of H Seebeck at p = 0.11 is in agreement with the anomaly in the sound velocity [29] that marks the phase transition to CDW order (Fig. 4). Those field values are also in agreement with the threshold field detected in the thermal Hall conductivity κ xy [35], for both p = 0.11 (Fig. 4) and p = 0.12. Therefore, it is clear that H Seebeck coincides with the onset of 3D unidirectional CDW order.
The onset temperature for that order (T 47 K) [27,28] is not far below the onset of the NMR splitting associated with CDW order [37]. There is little doubt that T 47 ± 5 K coincides with the onset of 3D unidirectional CDW order.
The fact that we can clearly detect the onset of 3D unidirectional CDW order in the Seebeck coefficient allows us to examine whether it causes the FSR in YBCO. In Fig. 5(b), we plot S a / T vs T at p = 0.12 for H = 10, 16 and 34 T. We see that S a / T starts to deviate downward from its high-temperature behaviour below T 130 K, it peaks at 105 K and then it drops to become negative below ∼ 60 K. This is a gradual process, which starts in parallel with the gradual growth of short-range 2D CDW modulations seen in XRD below 140 K [19]. (Down to T c , both the Seebeck and the XRD intensity are independent of magnetic field.) The decrease in S / T upon cooling is the signature of the FSR that leads to the formation of a small electron pocket in the Fermi surface at low temperature, detected via quantum oscillations, whose Fermi energy is consistent with the value of S / T at T → 0 ( Fig. 5(b)) [9]. In other words, the entire evolution of S a / T vs T is quantitatively consistent (in temperature and in amplitude) with a scenario whereby FSR is caused by the 2D CDW modulations (CDW-1). The fact that this evolution is completely unaffected by the sharp onset of the 3D unidirectional CDW order at 47 K, measured in YBCO at the same doping (p = 0.12) and the same field (H = 16 T) [27], indicates that it doesn't play a fundamental role in causing the FSR. It appears to only confer an extra anisotropy.
The fact that the 2D CDW modulations are bidirectional, i.e. that they run along both the a and b directions in the CuO 2 planes of YBCO, provides a natural mechanism for the formation of a small electron pocket in the reconstructed Fermi surface [24,25], located in nodal positions where the states are believed to be in underdoped cuprates with an anti-nodal pseudogap. An analysis of the anomalies in the sound velocities concluded that the order responsible for the observed transition must be bi-directional [29].
Note that the CDW modulations observed in Hg1201 [20] are very similar to the 2D CDW modulations in YBCO, and they cause a very similar FSR [38], with negative Hall and Seebeck coefficients at low temperature [10]. Therefore, attributing the cause of the FSR to these 2D CDW modulations is consistent with the fact that so far no field-induced 3D CDW order has been observed in Hg1201.
Given that 2D CDW modulations exist in the superconducting state at H = 0 [18,19], one might ask: are there signatures of the FSR inside the superconducting phase, i.e. inside the green region of the H − T phase diagram (Fig. 4)? The answer is yes: in YBCO at p = 0.11, R H at T = 15 K is negative for all fields down to H = H vs 10 T, the field below which the vortex solid forms and R H = 0 [35]. So a negative R H is observed even when H < H Seebeck . In the vortex-liquid state between H vs and H c2 , the negative R H could come from states inside the vortex core.
On the other hand, the thermal Hall conductivity κ xy is dominated by d-wave quasiparticles outside the vortex cores. In YBCO at p = 0.12, κ xy is negative in the normal state just above T c [35], even in the limit H = 0, as is the electrical Hall conductivity [6]. Immediately below T c , κ xy becomes positive [35]. This sudden change of sign could be due to a sudden increase in the quasiparticle mean free path as the inelastic scattering is gapped out, as found in YBCO immediately below T c [39]. Because the correlation length of the 2D CDW modulations is rather short in YBCO (and even shorter in Hg1201), the longer electronic mean free path in the superconducting state may well average over the short-range CDW and wipe out the FSR. Increasing the field to suppress superconductivity makes κ xy negative again [35]. The threshold field at which this change of sign happens coincides with H Seebeck (Fig. 4), i.e. with the onset field for 3D CDW order. This can be understood as follows: 3D CDW order competes with superconductivity, its onset precipitates the demise of superconductivity, which causes a reduction in the mean free path, making the FSR by short-range 2D CDW modulations possible again. In other words, 3D CDW order triggers the transition out of the superconducting phase and this is where κ xy starts its transition from zero to its normal-state (negative) value [35].

V. SUMMARY
In summary, the Seebeck coefficient S of YBCO at p = 0.11 and 0.12 responds to two aspects of the complex CDW ordering in this material. First, as temperature is decreased from room temperature, S a / T deviates gradually downward from its dependence at high temperature in parallel with the gradual growth in the 2D bi-directional CDW modulations detected by XRD well above T c . S a / T decreases below T 100 K to eventually become negative, extrapolating to a large negative value at T → 0 that is quantitatively consistent with the small electron pocket in the normal-state Fermi surface detected by quantum oscillations at low temperature. We infer that the 2D bi-directional CDW modulations reconstruct the Fermi surface of YBCO, and produce the electron pocket. The same is true for Hg1201.
Secondly, a pronounced anomaly appears in S b below a temperature and above a field that are both consistent with the onset temperature and field of the 3D unidirectional CDW order detected in YBCO by high-field XRD at p = 0.11 and 0.12. We conclude that the extra anisotropy is due to that low-temperature order, which is not, however, the primary cause of the FSR. Nevertheless, given that the two types of CDW modulations (CDW-1 and CDW-2 in Fig. 4) have the same wavelength, they most likely have a common origin. It would be helpful to further elucidate the nature of their interplay.