Critical doping for the onset of Fermi-surface reconstruction by charge-density-wave order in the cuprate superconductor La{}_{2-x}Sr{}_{x}CuO{}_{4}

Critical Doping for the Onset of Fermi-Surface Reconstruction by Charge-Density-Wave Order in the Cuprate Superconductor La$ _{2-x} $Sr$_{x} $CuO$ _{4}$


The Seebeck coefficient of the cuprate superconductor LaSrCuO (LSCO) was measured in magnetic fields large enough to access the normal state at low temperatures, for a range of Sr concentrations from to . For , 0.12, 0.125 and 0.13, decreases upon cooling to become negative at low temperatures. The same behavior is observed in the Hall coefficient . In analogy with other hole-doped cuprates at similar hole concentrations , the negative and  show that the Fermi surface of LSCO undergoes a reconstruction caused by the onset of charge-density-wave modulations. Such modulations have indeed been detected in LSCO by X-ray diffraction in precisely the same doping range. Our data show that in LSCO this Fermi-surface reconstruction is confined to . We argue that in the field-induced normal state of LSCO, charge-density-wave order ends at a critical doping , well below the pseudogap critical doping .


Since the discovery of quantum oscillations Doiron-Leyraud et al. (2007) and a negative Hall coefficient  LeBoeuf et al. (2007) in the cuprate superconductor YBaCuO (YBCO), it has become clear that the Fermi surface of underdoped YBCO undergoes a reconstruction at low temperature that produces a small electron pocket Taillefer (2009), in a doping range from to LeBoeuf et al. (2011). This Fermi-surface reconstruction (FSR) was also detected as a sign change in the Seebeck coefficient , going from positive at high temperature to negative at low temperature Chang et al. (2010). A strikingly similar change of sign in observed in the cuprate LaEuSrCuO (Eu-LSCO) Laliberté et al. (2011) suggested that the stripe order known to exist in Eu-LSCO Fink et al. (2011) – a combination of charge-density-wave (CDW) and spin-density-wave (SDW) modulations – is responsible for the FSR in both materials. The observation of CDW modulations in YBCO by NMR Wu et al. (2011) and X-ray diffraction (XRD) Ghiringhelli et al. (2012); Chang et al. (2012) confirmed this conjecture, and demonstrated that it is the CDW (and not the SDW) modulations that cause the FSR.

In YBCO, the drop in and begins at a temperature  that peaks at (Fig. 1a). The drop is attributed to the CDW modulations detected by XRD Hücker et al. (2014); Blanco-Canosa et al. (2014) and NMR Wu et al. (2015) below a temperature  in the same doping range as the FSR LeBoeuf et al. (2011), with  also peaking at (Fig. 1a).

In HgBaCuO (Hg1201), high-field measurements of Hall and Seebeck coefficients revealed a similar FSR Doiron-Leyraud et al. (2013), confirmed by the observation of quantum oscillations Barišić et al. (2013) and again attributed to XRD-detected CDW modulations Tabis et al. (2014). All this suggests that CDW modulations and the associated FSR are generic properties of hole-doped cuprates in the vicinity of . A major outstanding question is : Up to what critical doping  do CDW modulations extend in the phase diagram (Fig. 1), in particular in the field-induced normal state at ? In this context, the material LSCO offers a powerful platform, since good crystals can be grown with up to 0.3 and beyond. CDW modulations have been observed in LSCO with XRD, at Croft et al. (2014); Thampy et al. (2014); Christensen et al. (2014), but there is little information about the associated FSR.

In this Article, we report high-field measurements of the Seebeck coefficient in LSCO single crystals at several dopings, which show that becomes negative in the normal state at low temperature in precisely the doping range where CDW modulations are detected by XRD.  is also found to be negative in that range. The FSR in LSCO is therefore very similar to the FSR in YBCO and Hg1201. Our data show that the FSR does not extend above , strong evidence that CDW order in LSCO ends at a critical doping  . This implies that in the normal state of LSCO the phase of CDW order ends well before the pseudogap phase, which ends at the critical doping (ref. Cooper et al., 2009).

Figure 1: Temperature-doping phase diagram of the cuprate superconductors YBCO (a) and LSCO (b). The superconducting transition temperature  is drawn as a black line. Charge density-wave (CDW) modulations are detected by X-ray diffraction below  (green triangles) in YBCO (up triangles (Hücker et al., 2014), down triangles (Blanco-Canosa et al., 2014)) and LSCO (up triangles (Croft et al., 2014), down triangle (Christensen et al., 2014)). Spin-density-wave (SDW) modulations are detected by neutron diffraction below  (blue squares) in YBCO (Haug et al., 2010) and LSCO Chang et al. (2008); Kofu et al. (2009); Wakimoto et al. (1999); Lake et al. (2002); Kimura et al. (1999). When plotted as vs , the normal-state Seebeck coefficient peaks at a temperature  (full red circles) before it drops at low temperature due to Fermi-surface reconstruction (YBCO, ref. Laliberté et al., 2011; LSCO, this work, Figs. 3 and 4). A similar  can also be defined for the Hall coefficient (open red circles), below which drops at low temperature (YBCO, ref. LeBoeuf et al., 2011).
Figure 2: Isotherms of the Seebeck coefficient in LSCO, plotted as vs magnetic field , at various temperatures, as indicated, for six samples, with (a), (b), (c), (d), (e), and (f). For and 0.13, at high decreases at low temperature, to reach negative values. For , also decreases at low temperature, below 15 K. This decrease is the signature of FSR. In contrast, for and 0.15, at the highest measured field keeps increasing with decreasing temperature down to the lowest temperature. This shows that there is no FSR at those dopings, at least down to 4 K and 9 K, respectively. The same is true at , at least down to 15 K.
Figure 3: Seebeck coefficient of LSCO, plotted as vs temperature , measured in a magnetic field (open circles), 16 T (full circles) and 34 T (squares), for four samples, with and 0.125 (a), and and 0.15 (b). The data in panel b are normalized to their value at  K. All data points represent the normal state, for which the solid lines are a guide to the eye, except the lowest point for each of and (panel b). For these two points, the isotherms are still going up the superconducting transition (Fig. 2). The dashed lines are an extension of the normal-state behavior based on extrapolating those isotherms beyond 34 T.  marks the temperature below which decreases at low temperature (arrow), in some cases to reach negative values, as seen here for . This decrease is the signature of Fermi-surface reconstruction (FSR). Note how the data for and split below  K, with the former dropping at low due to FSR and the latter showing no decrease, and hence no FSR (at least down to 9 K).

Methods.– Single crystals of LSCO were grown by the flux-zone technique with Sr concentrations , 0.11, 0.12 and 0.13 at the University of Bristol, and 0.125 at the University of Tokyo, and 0.15 at Tohoku University. Samples were cut in the shape of rectangular platelets, with typical dimensions 0.5 mm 1.0 mm  mm. The hole concentration (doping) is taken to be . The (zero-resistance) superconducting transition temperature of the 8 samples is  , 20.2, 26.2, 27.5, 28.0, 32.3, 37.2, and 36.5 K for , 0.085, 0.11, 0.12, 0.125, 0.13, 0.144, and 0.15, respectively. The Seebeck coefficient was measured, as described elsewhere Laliberté et al. (2011), at Sherbrooke (all samples) up to  T, at the National High Magnetic Field Laboratory (NHMFL) in Tallahassee up to  T ( and 0.15) and up to  T (), and at the Laboratoire National des Champs Magnétiques Intenses (LNCMI) in Grenoble up to  T ( and 0.144). The Hall coefficient of samples with , 0.12, 0.125 and 0.13 was measured, as described elsewhere LeBoeuf et al. (2011), at Sherbrooke in  T. All crystals have an orthorhombic crystal structure and they are twinned. The thermal gradient or electrical current was applied in the basal plane, while the magnetic field was applied along the axis.

Figure 4: Same as in Fig. 3, for samples with (yellow), (blue), (red) and (green), measured at  T (full circles), 17.5 T (open squares) and 44 T (full squares). The data in panel b are normalized to their value at T = 100 K. FSR is clearly observed in all four samples, as a drop in at low temperature. Inset of panel b: Isotherm at  K for , showing that becomes increasingly negative with increasing field, demonstrating that the negative is a property of the normal state.
Figure 5: Comparison of LSCO (red) and YBCO (green) at . a) Temperature dependence of the X-ray intensity associated with the CDW modulations, normalized at , detected in LSCO Croft et al. (2014) and YBCO Chang et al. (2012). Lines are a guide to the eye. The cusp is at . b) Normal-state Seebeck coefficient of LSCO (this work) and YBCO Laliberté et al. (2011), measured in a magnetic field as indicated, plotted as vs .  is the temperature below which drops to reach negative values at low temperature (arrow), the signature of Fermi-surface reconstruction (FSR). This  is plotted as full circles in Fig. 1. Lines are a guide to the eye. c) Hall coefficient of LSCO at  T and YBCO at  T LeBoeuf et al. (2007), plotted as , where is the electron charge and the volume per planar Cu atom.  is the temperature below which drops to reach negative values at low temperature (arrow), another signature of FSR.  is plotted as open circles in Fig. 1a LeBoeuf et al. (2011).

Seebeck coefficient.– In Fig. 2, the Seebeck data for 6 samples are plotted as vs for several temperatures. We see that for (Fig. 2c) and (Fig. 2d), becomes negative at high field and low temperature. This shows that a negative is a property of the normal state of LSCO at these dopings, as in YBCO, Eu-LSCO and Hg1201. At , we see that at high field decreases when the temperature drops below  K (Fig. 2e). In contrast, no such decrease is observed at , down to the lowest temperature (Fig. 2f). At , increases steadily with decreasing at high field, down to the lowest temperature (Fig. 2a). This is also true at (Fig. 2b). Although here our data only goes to 20 T, the crossing of the lowest isotherms shows that keeps increasing down to  K, at least.

In Figs. 3 and 4, we plot vs , at high field. In Fig. 3a, we see that the drop in at to negative values starts below a temperature   K. This is also the case at and 0.13 (Fig. 4a). In Fig. 4b, we compare data on 3 samples taken in identical conditions, at  T. (Although the LSCO sample with was only measured up to 18 T, at  K is increasingly negative with increasing (inset of Fig. 4b), confirming that a negative is a property of the normal state also at that doping.) The location of the peak in vs is seen to decrease from   K at , to   K at , to   K at . Those  values are plotted on the phase diagram of LSCO in Fig. 1b. Raising the doping further, we observe that  continues its steady descent. Indeed, at , now peaks at   K (Fig. 3b). Extrapolating this trend yields   at (Fig. 1b). Our data at confirm this, with showing no decrease down to at least 9 K (Figs. 2f and 3b). This shows that FSR in LSCO ends at a critical doping  .

At , the normal-state increases monotonically with decreasing , down to our lowest temperature (Fig. 3a). There is clearly no FSR at that doping. At , although we only measured up to 18 or 20 T, we observe that at  T increases as , at least down to 15 K (Fig. 2b). So here   K. In Fig. 1b, we plot  vs for our 8 samples, with their uncertainty, and thereby delineate the region where FSR occurs in the phase diagram of LSCO. We see that the FSR region peaks at and is confined between and  .

Hall coefficient.– In Fig. 5c, the Hall coefficient of our LSCO crystal with , measured at  T, is plotted as  vs . We see that drops below  K and becomes negative below  K. Data for our crystals with , 0.125 and 0.13 are very similar, also negative at low , all in excellent agreement with prior low-field data on single crystals of LSCO with Suzuki et al. (2002). (The absence of a negative  in previous high-field data on thin films of LSCO Balakirev et al. (2009) may be due to the higher disorder of such samples.) A similar drop in  has been seen in Eu-LSCO Cyr-Choinière et al. (2009) and in LaNdSrCuO (Nd-LSCO) Noda et al. (1999), when ; in both materials, it is closely linked to the onset of CDW order.

Discussion.– Taken together, the negative Hall and Seebeck coefficients in the normal state of LSCO are conclusive evidence of FSR in this material, in the vicinity of . This adds to the previous three cases, namely YBCO, Eu-LSCO and Hg1201. In all 4 cases, the FSR occurs in a region of the phase diagram where CDW modulations have been detected by XRD (Fig. 1). The link between CDW and FSR is robust.

It is instructive to compare LSCO and YBCO. The two phase diagrams are similar (Fig. 1). In both cases,  and  peak at , and the region of FSR is confined to similar ranges – from to in LSCO and from to in YBCO LeBoeuf et al. (2011). In Fig. 5, we compare data for LSCO and YBCO directly, at . The CDW modulations detected by XRD emerge below a temperature twice as high in YBCO compared to LSCO (Fig. 5a) :   K in YBCO vs   K in LSCO. Correspondingly, the FSR is detected at a temperature twice as high in YBCO compared to LSCO, with   K in YBCO vs   K in LSCO (Fig. 5b). All this suggests that CDW ordering is a stronger tendency in YBCO than in LSCO. Intriguingly, the superconducting transition temperature  is roughly twice as high in YBCO as compared to LSCO (see cusp in Fig. 5a). This raises the interesting possibility that the same underlying mechanism, perhaps magnetic, fuels both superconductivity and CDW order Efetov et al. (2013).

Given that FSR in LSCO ends at  , we infer that this is also where CDW order ends. This is consistent with recent XRD measurements that detect no CDW modulations in LSCO at Croft and Hayden (). (The same consistency is observed at , where again no CDW modulations are detected by XRD Croft and Hayden ().) We thus arrive at a key information : the CDW phase in LSCO ends at the critical doping  .

This is distinctly below the critical point where the pseudogap phase is believed to end in LSCO, at , as determined from the normal-state resistivity measured in high magnetic fields Cooper et al. (2009). This clear separation reveals that the pseudogap phase is not caused by the CDW ordering. Instead, it suggests that CDW order is a secondary instability of the pseudogap phase. A very similar separation was recently observed in YBCO from high-field Hall effect measurements, with   and Badoux et al. (2016). This strongly suggests that a separation of  and is a generic property of cuprates.

Summary.– Our high-field measurements of the Seebeck coefficient in the cuprate superconductor LSCO reveal that its normal-state Fermi surface undergoes a reconstruction at low temperature, in the doping range . In analogy with the cuprates YBCO, Eu-LSCO and Hg1201, we attribute this FSR to the CDW modulations detected by XRD in the very same doping range. Combined with XRD data on LSCO, our Seebeck data make a compelling case that CDW modulations disappear at , so that the field-induced non-superconducting ground state of LSCO above has no CDW order. Because the pseudogap phase in the normal state of LSCO extends up to , we infer that the pseudogap is not tied to CDW ordering. Instead, the CDW modulations appear to be a secondary instability of the pseudogap phase.

Acknowledgements.– We thank J. Chang, N. E. Hussey, M.-H. Julien, P. A. Lee, B. J. Ramshaw, S. Sachdev, J. L. Tallon, and G. A. Sawatzky for stimulating discussions. A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by the National Science Foundation Cooperative Agreement No. DMR- 1157490, the State of Florida, and the U.S. Department of Energy. Another portion of this work was performed at the Laboratoire National des Champs Magnétiques Intenses of the CNRS, member of the European Magnetic Field Laboratory. L.T. acknowledges support from the Canadian Institute for Advanced Research (CIFAR) and funding from the National Science and Engineering Research Council of Canada (NSERC), the Fonds de recherche du Québec - Nature et Technologies (FRQNT), the Canada Foundation for Innovation (CFI) and a Canada Research Chair.


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