Charge density waves and Fermi surface reconstruction in the clean overdoped cuprate superconductor Tl2Ba2CuO6+δ

Hall effect and quantum oscillation measurements on high temperature cuprate superconductors show that underdoped compositions have small Fermi surface pockets whereas when heavily overdoped, a single much larger pocket is found. The origin of this change in electronic structure has been unclear, but may be related to the high temperature superconductivity. Here we show that the clean overdoped single-layer cuprate Tl2Ba2CuO6+δ (Tl2201) displays CDW order with a remarkably long correlation length ξ ≈ 200 Å which disappears above a hole doping of pCDW ≈ 0.265. We show that the evolution of the electronic properties of Tl2201 as the doping is lowered may be explained by a Fermi surface reconstruction which accompanies the emergence of the CDW below pCDW. Our results demonstrate importance of CDW correlations in understanding the electronic properties of overdoped cuprates. The origin of the Fermi surface reconstruction that occurs in cuprate superconductors as hole doping increases remains unclear. Here, the authors observe long range charge density wave (CDW) order in the overdoped single-layer cuprate Tl2Ba2CuO6+δ, which then disappears above a hole concentration 0.265, suggesting a correlation between Fermi surface reconstruction and the emergence of the CDW.

T he normal state electronic properties of the hole doped cuprate superconductors evolve dramatically with doping. This evolution, in part, evidences the emergence and disappearance of competing ordering tendencies and their fluctuations, namely those associated with the pseudogap, charge and spin 1 . Fluctuations associated with these orders are enhanced close to their putative quantum critical points and have been conjectured to play an important role in producing the high superconducting transition temperature in these materials.
The pseudogap corresponds to a loss of electronic states near the Fermi energy 2,3 and in most, if not all, cuprates occurs for doping p < p ⋆ ≈ 0.19 3,4 . In this regime a charge density wave (CDW) order is also found [5][6][7][8][9][10][11] . In YBa 2 Cu 3 O 6+x (YBCO), charge order 5,6 is the obvious candidate responsible for the reconstruction of the Fermi surface 12,13 resulting in closed pockets observed by quantum oscillations 14 . The onset of the CDW 5,6 also naturally explains 12 a sign reversal observed 15 in the Hall number n H at low-temperature for doping p < 0.16. At slightly higher doping, n H in the high-field, low-temperature limit n 1 H undergoes a rapid increase from p → 1 + p over a narrow doping range 0.16 < p < 0.20, which has been suggested to be caused by the closing of the pseudogap 16,17 . In the overdoped cuprate Tl 2 Ba 2 CuO 6+δ (Tl2201), a similar p → 1 + p transition is observed in n 1 H which onsets at a higher doping p ≃ 0.25, where thermodynamic probes suggest that there is no pseudogap [18][19][20][21] . Furthermore as, prior to the present work, there were no reports of a CDW in overdoped Tl2201, a universal microscopic origin of the transition in n 1 H remains to be found. Here we use Cu-L 3 edge resonant inelastic x-ray scattering (RIXS) to observe a CDW in heavily overdoped Tl2201. The doping onset of the CDW coincides with the decrease in n 1 H in Tl2201 suggesting these phenomena are linked.
The single-layer cuprate, Tl2201, is exceptionally electronically clean and is the only hole doped cuprate in which the large (1 + p) Fermi surface has been observed by quantum oscillations (QO) [22][23][24] . The shape of the Fermi surface for the most overdoped compositions (p ≥ 0.27) has been determined by angle-dependent magnetoresistance 25,26 , QO and angle resolved photoemission spectroscopy (ARPES) 27 and is in excellent agreement with conventional density functional theory 24 . Previously, there have been no reports of charge or other ordering in this overdoped material. However CDWs have been seen in other overdoped cuprates. In La 2−x Sr x CuO 4 (LSCO), the CDW has been seen to persist into the overdoped regime (up to p = 0.21) 28 , and a second CDW phase, disconnected from the CDW at lower doping, was observed in overdoped (Bi,Pb) 2.12 Sr 1.88 CuO 6+δ (Bi2201) 29 . In overdoped Bi 2 Sr 2 CaCu 2 O 8+δ (Bi2212) no CDW is seen directly 30 although there is phonon softening which may be a precursor of CDW formation.

Results
Charge density wave order in Tl 2 Ba 2 CuO 6+δ . Charge density wave correlations in underdoped superconducting cuprates [5][6][7][8][9][10][11] have ordering wavevectors with in-plane components along the CuO bonds i.e. (δ, 0) and (0, δ), where 0.23 ≲ δ ≲ 0.33 r.l.u. In the absence of a large magnetic field or uniaxial stress, a 2D CDW develops below T CDW . This state corresponds to a weak anticorrelation of the phase of the CDW in neighbouring CuO 2 planes and produces a peak in the scattered intensity at halfinteger L positions such as (δ, 0, 2.5). In this paper, we label reciprocal space in reciprocal lattice units (r.l.u.) where Q = Ha ⋆ + Kb ⋆ + Lc ⋆ . We use RIXS (see Methods) to study the charge correlations in three overdoped superconducting samples. Our samples have critical temperatures T c = 56, 45 and 22 K, corresponding to hole doping p = 0.23, p = 0.25 and p = 0.28 21 . Throughout the paper we probe CDW correlations with L = 2.5, which allows us to scan a wide range of H (see Supplementary Note 1, Supplementary Fig. 1 and Supplementary Table 1). Figure 1a shows a typical RIXS energy (E) dependent spectrum obtained in this experiment showing dd excitations, corresponding to a transition between 2p → 3d states, near E = 2 eV and features at lower energy which may be fitted (see Methods) with an elastic peak, a non-elastic peak near 60 meV and a broad peak centred at about 150 meV. We believe the broad 150 meV peak is due to paramagnon spin fluctuations previously observed in Tl2201 31 and the 60 meV peak is due to a phonon. Figure 2 shows data such as that in Fig. 1 for the p = 0.25 (T c = 45 K) sample compiled into H − E colour maps. The spectra can be fitted to the three components mentioned above. The elastic intensity, obtained by integrating the RIXS intensity between −100 < E < 100 meV (Fig. 2c) is peaked at H ≈ ±0.31. These elastic features are also seen in Fig. 2a, and we interpret them as being due to charge density wave order as explained below. The elastic scattering is also peaked at H = 0 due to specular scattering from disorder in the sample surface. Note that charge scattering is generally enhanced for grazing incident-geometry (denoted by negative H) and the vertical polarisation used here. The energy of the fitted phonon peak is plotted as white circles (Fig. 2a).
Comparing the dispersion and the sin 2 ðπHÞ intensity variation ( Fig. 2b) to theory 32 we conclude that we observe the CuO bondstretching mode previously seen in Cu-L RIXS on cuprates 33 . The sin 2 ðπHÞ intensity variation can understood in terms of a momentum dependent electron-phonon coupling 28,34 . Phonon softening has been observed in other cuprates such as La 2−x Sr x CuO 4 28 for H > δ. Our data is consistent with a softening of up to 10%, but our experiment is not able to say definitively whether there is phonon softening or not and further work is required (see Supplementary Note 2 and Supplementary Figs. 2, 3). In order to further characterise this CDW feature we investigated other reciprocal space positions and performed non-resonant x-ray scattering. Figure 3a shows elastic intensity scans obtained by integrating the RIXS intensity over the energy range −100 < E < 100 meV for the sample with doping p = 0.23 at a temperature T = T c = 56 K. The open circles show a scan along (H, 0, 2.5) yielding a peak at H = −0.31 r.l.u. This peak is not present when scans are made along ðζ cos ϕ; ζ sin ϕ; 2:5Þ, with ϕ = 10 ∘ or 45 ∘ , nor for non-resonant scattering when the incident photon energy was detuned from resonance by 5 eV. From this, we conclude that the data is consistent with a CDW with an in-plane propagation vector of (δ, 0) as observed in other cuprates. A high-energy (500 < E < 900 meV) inelastic ring-like feature in (H, K) was observed in Bi 2 Sr 2 CaCu 2 O 8+δ , which was interpreted as evidence for dynamical charge fluctuations 35 . We find no evidence for a feature like this in our ζ scans (see Supplementary Note 3 and Supplementary Fig. 4).
Temperature and doping dependence, correlation length.  Fig. 1b, c believed to be a phonon and the intensity of the feature is plotted in b, along with a line proportional to sin 2 ðπHÞ in red. c The elastic intensity obtained by integrating the RIXS intensity between −100 < E < 100 meV, and dividing by the integral between 1000 < E < 3000 meV. Peaks at H ≈ ±0.31 are due to the charge density wave. d-f are RIXS intensity maps of the peak centred at H ≈ −0.31 at different temperatures. CDW for three dopings at various temperatures. At T ≈ T c , the samples with dopings p = 0.23 and p = 0.25 exhibit peaks centred at δ = 0.309 ± 0.003 and 0.310 ± 0.003 r.l.u. respectively, while no peak is observed for the p = 0.28 sample. We measured a detailed T-dependence of the CDW peak in the p = 0.25 (T c = 45 K) sample (Fig. 3c) to determine the variation of the CDW peak intensity I CDW and correlation length ξ (see Fig. 4), which we define as the reciprocal of the σ parameter of the Gaussian fitted to the CDW peak in H. The CDW peak intensity and ξ increase below T = 160 K, however, there is evidence that they remain finite with ξ = 38 ± 7 Å to the highest temperature investigated T = 200 K. Thus Tl2201 seems to have precursor CDW order as seen in other cuprates such as LSCO 10,36 and YBCO 37 . As the temperature is lowered below T = 160 K the correlation length increases until T c is reached, at which point it decreases, suggesting that the superconductivity and the CDW interact. Similar behaviour is observed in other cuprates 6,36 . For the p = 0.23 sample, only two temperatures were measured, T = T c = 56 and 160 K. In Fig. 3b, we see that the H = −0.31 peak has disappeared at 160 K peak placing an upper bound T CDW of 160 K for this composition. While the small peak away from H = −0.31 is most likely spurious scatting, it could also be caused by a temperature dependent CDW incommensurability, as seen in LBCO 38 . More investigation would be required to confirm this. For p = 0.28 (Fig. 3c), no peak is seen even at T c = 22 K where the signal is expected to be maximal. From our measurements, we deduce the phase diagram shown in Fig. 5c, where the black circles represent T CDW (p), (for p = 0.23 it is an upper bound) and the critical doping of the CDW is p CDW = 0.265 ± 0.015, corresponding to T c = 35 ± 13 K.

Discussion
Charge density wave order has been observed in many underdoped (p < 0.16) cuprates [5][6][7][8][9][10][11] , where the ordering wavevectors have been measured as 0.23 ≲ δ ≲ 0.33. CDW order has recently been observed by x-ray diffraction in two overdoped materials. In (Bi,Pb) 2.12 Sr 1.88 CuO 6+δ (Bi2201), re-entrant charge order, which is disconnected from the CDW below critical doping is observed 29 and a CDW is observed in overdoped La 1.79 Sr 0.21 CuO 4 36 . The behaviour of the peak observed in overdoped Bi2201 29 differs from Tl2201 in that it remains strong up to 250 K and it is unaffected by the superconductivity. In contrast, the effect of varying temperature on the peak observed in overdoped LSCO (p = 0.21) 36 is qualitatively similar to Tl2201 in that its height drops considerably above T c , leaving a small broad component at higher temperatures, and the peak height is reduced on entering the superconducting state. The wavevectors of the CDWs in overdoped Tl2201 (δ ≈ 0.31) and LSCO (δ ≈ 0.24) are similar to those found in underdoped cuprates (0.23 < δ < 0.33). In the case of LSCO, δ does not change much with doping 36 and δ for Tl2201 (0.31) is similar to the values found for underdoped YBCO 5,6 where 0.31 < δ < 0.33. In this experiment we probe the CDW at L = 2.5. Unfortunately, scans at different L values were not possible due to experimental constraints (see Supplementary Note 1), thus the detailed L-dependence is unknown. We note, however, that in YBCO 6 and LSCO 10 the CDW is known to be peaked at L equal to halfinteger positions such as (δ, 0, n + 1/2). Likely mechanisms causing the corresponding c-axis correlations are the interaction of the CDW with pinning sites 39 caused by dopant atoms/ quenched disorder or the interactions between the CDW in neighbouring planes 40 .
The CDW order in Tl2201 shows a long correlation length for cuprates (in the absence of 3D order induced by magnetic field or strain) of ξ = 196 ± 17 Å for p = 0.25 at T = T c = 45 K. In LSCO (x = 0.21) 36   is likely limited by quenched disorder induced pinning which may be low in Tl2201, as evidenced by its long electronic meanfree-path 24 . We can compare the H-integrated area of the CDW Bragg peaks normalised by the dd excitations (see Fig. 1) to other cuprates. We find (see Supplementary Note 4, Supplementary Table 2, and Supplementary Fig. 5) that the H-integrated area for p = 0.23 Tl2201 is comparable to p = 0.12 LSCO at T ≈ T c . Figure 5a, b shows the Fermi surface of Tl2201 (p = 0.25) determined from the tight-binding fit of Platé et al. 27 based on ARPES, with the chemical potential shifted to give a FS area corresponding to 1 + p = 1.25. We place Q CDW on the same figure at positions where it connects FS states. Johannes and Mazin 41 have emphasised that Fermi surface nesting rarely determines CDW order other than in quasi-1D systems. Hence it is not surprising that our Q CDW does not connect the nested states across the X position. We note, however, that it does approximately connect Fermi surface states along Γ − M, which have no k z dispersion 25 and therefore are nested along k z . However, this condition does not appear to predict Q CDW in other cuprates. Our result is also inconsistent with models (e.g. ref. 8,42 ) that connect Q CDW with hotspot positions, where the FS intersects the magnetic zone boundary (dashed line in Fig. 5a). Thus it appears that determining Q CDW requires a strong coupling theory of the electron interactions 43 .
The appearance of the CDW below p = 0.28 and its increase in amplitude as the doping is lowered (see Fig. 5d) correlates with marked changes in the electronic properties as a function of p, in particular n 1 H , as shown in Fig. 5d. Within Boltzmann transport theory, n 1 H is simply related to the effective number of carriers and so a reduction in n 1 H could be explained by a reconstruction of the Fermi surface, such as that produced by a CDW.
If the CDW were sufficiently coherent we would expect a reconstruction of the Fermi surface. We have modelled this by adding potentials due to the CDW to a tight-binding model of Tl2201 (see Methods). The result is shown in Fig. 6a. Here, for simplicity, we have assumed a commensurate bi-axial order with |Q CDW | = 1/3 (compared to |Q CDW | ≈ 0.31 in our experiment) and a very small gap (2 meV). To first approximation, the calculation translates the FS by multiples of Q CDW along the a and b axes, with gaps appearing where sections of the original FS are connected by Q CDW . In this commensurate case, no gaps appear at the other places where the Fermi surfaces cross and so electrons will follow the trajectory of the original FS. The reconstructed FS therefore consists of electron and hole pockets along with open sections of FS as outlined by the coloured lines in Fig. 6a.
Although these pockets have a much lower volume than the original unreconstructed Fermi surface, in the limit that the CDW gap is small, calculations (see Supplementary Note 5 and Supplementary Fig. 6) show that n H will not be strongly affected. However, as the gap grows the electron pocket will shrink and eventually disappear and n H will be reduced to values~p. The behaviour is similar to the model of Storey 17 for an antiferromagnetic (pseudogap) reconstruction. Hence, the evolution of n 1 H with p in Tl2201 (Fig. 5d) might be explained by a growth in the CDW gap as p is reduced. This is supported by the observed growth of the x-ray intensity (Fig. 5d) but further x-ray data, at lower p, is required to test this proposition.
In YBCO, the occurrence of the CDW with p is accompanied by a change in sign of n H 15 , whereas for Tl2201 n H remains positive for all p 21 . In Fig. 6b we show a similar calculation where the Fermi-level has been adjusted to give p = 0.12. It can be seen that the p = 0.25 case has larger hole pockets and smaller electron pockets compared to the p = 0.12 case. In underdoped (p = 0.12) YBCO, the hole pocket volume is further reduced by the emergence of the pseudogap 13 . The change in sign of n H in YBCO is explained by the electron pocket contribution being larger than the hole pocket (see Methods). For Tl2201, calculations (see Supplementary Note 5 and Supplementary Fig. 6) show that the larger size of the hole pockets and the absence of a pseudogap mean that n H remains positive even when the CDW gap is large.
The large unreconstructed Fermi sheet centred on the M point in Fig. 5b results in quantum oscillations with frequency F ≈ 18 kT which have been observed in Tl2201 for p ≥ 0.27 22,24 . The observation of QO is dependent on electrons completing coherent orbits around the Fermi surface. Thus the disappearance of the 18 kT frequency for p < 0.27 (see Fig. 5c) is consistent the CDW causing additional scattering of the orbiting electrons at specific k vectors. Note that no obvious concomitant increase in the resistivity at p CDW is observed 21 , so the additional scattering must be quite localised in k. The non-observation of lower-frequency QO for p < 0.27 23,24 in Tl2201 suggests that the CDW coherence is not sufficient for electrons to complete cyclotron orbits around the smaller reconstructed pockets with the available magnetic field. This is a much stronger criterion than that required to see reconstruction in the Hall effect 44 .
Spin and charge order or low-frequency fluctuations persist into the overdoped region of the cuprate phase diagram. Lowenergy incommensurate spin fluctuations 45 and charge density wave order 36 are seen in overdoped La 2−x Sr x CuO 4 (p ≈ 0.21). It is likely that the spin fluctuations are also present in overdoped Tl2201 although they are difficult to measure. Here we show that charge density wave order with a long correlation length is present in Tl2201 at higher dopings disappearing at doping p CDW = 0.265 ± 0.015, corresponding to T c = 35 ± 13 K. The CDW appears in a part of the phase diagram where there is no pseudogap, hence showing the two phenomena are not linked, at least in Tl2201. As in other cuprates 6 , superconductivity in Tl2201 causes the CDW correlation length to shorten and hence the CDW is affected by the superconductivity.
Overdoped Tl2201 has a number of interesting quasiparticle properties including an increase in strength of a linear-in-T component to the resistivity as doping is reduced from the edge of the superconducting dome 21,46 , quasiparticle scattering rates which are highly anisotropic 26 , a strong variation of Hall number with doping 21 and a magnetoresistance which is much larger than predicted by Boltzmann theory and is insensitive to impurities and magnetic field direction 47 . We have argued that the CDW order is a potential explanation for the variation of the Hall number with doping. Future work is needed to establish whether the other properties mentioned above can be understood in terms of the spin and charge correlations, and the associated scattering of quasiparticles near the Fermi surface.

Methods
Sample growth and preparation. Single crystals of Tl2201 were grown by a selfflux method, similar to that described in ref. 48 . The three samples measured roughly 300 × 200 μm, and were hole doped to values of p = 0.23 (T c = 56 K), 0.25 (T c = 47 K) and 0.28 (T c = 22 K), or samples A, D and G respectively. Sample G was annealed in pure oxygen at 325 ∘ C. Samples A and D were annealed in 0.1% oxygen in argon at 550 and 500 ∘ C respectively. Samples were annealled for up to 64 h and quenched by quickly removing them from the furnace and placing on a copper block. T c was determined from the mid-point of a.c. susceptibility superconducting transition. Doping values were determined from T c using the relation given in ref. 21 . Tl2201 is tetragonal, with approximate lattice parameters a = b = 3.85 Å and c = 23.1 Å. The samples had mirror-like surfaces as grown and were not cleaved.
Resonant inelastic x-ray scattering. RIXS was carried out at the ID32 beamline at the ESRF, Grenoble, France. Using x-ray absorption spectroscopy (XAS), the incident x-ray energy was tuned to the Cu-L 3 edge at the absorption peak at 931.6 eV. Data were collected with photons polarised linearly, vertical to the scattering plane (LV), in the low resolution/high throughput configuration of the instrument. The energy resolution was further relaxed from 42 to 48 meV by opening the exit slit in order to observe weak elastic scattering. The UB matrix (sample orientation) was determined on ID32 using diffraction from the (002) and (103) Bragg reflections. Scans in the (H, 0, L) plane were made changing sample rotation θ and detector position 2θ in the horizontal plane. Except for those shown in Fig. 3a, data were normalised by the orbital excitations, obtained by integrating RIXS spectra in the energy range 1000 < E < 3000 meV, resulting in a dimensionless ratio of intensities. Error bars on RIXS intensity and the elastic intensity were obtained by taking advantage of single photon counting at ID32, assuming Poisson statistics. Error bars in the fitted parameters were obtained from least square fitting and represent one standard deviation.
The low-energy excitations of the RIXS spectra, where 0 < E < 300 meV were fitted to three features. The elastic peaks were fitted to a Gaussian with a FWHM fixed to the the instrument resolution. This was used to determine zero energy. An inelastic Gaussian peak with FWHM fixed to instrument resolution was fitted to the CuO stretching phonon where present (discussed more in main text). The magnetic paramagnon excitation was fitted to a damped harmonic oscillator that was numerically convoluted with a Gaussian with FWHM set to instrument resolution. To account any higher energy excitations, a linear background was used.
Fermi surface reconstruction. The simulations of Fermi surface reconstruction in Fig. 6 was based on the ARPES derived tight-binding parameterisation of the FS of Tl2201 27 ; εðKÞ ¼ μ þ t 1 2 ðcos k x þ cos k y Þ þ t 2 cos k x cos k y þ t 3 2 ðcos 2k x þ cos 2k y Þþ t 4 2 ðcos 2k x cos k y þ cos k x cos 2k y Þ þ t 5 cos 2k x cos 2k y , with t 1 = − 0.725, t 2 = 0.302, t 3 = 0.0159, t 4 = − 0.0805 and t 5 = 0.0034; for p = 0.25 μ = 0.2382 and p = 0.12 μ = 0.1998. All energies are expressed in eV with lattice parameter a = 1. Small CDW potentials with V 0 = 0.002, Q CDW = (2π/3, 0) and Q CDW = (0, 2π/3) were introduced using the method described in ref. 49 , but with a k-independent potential. This translates copies of the FS by multiples of Q CDW in the a and b directions and introduces small gaps on those parts of the FS connected by Q CDW . In the incommensurate case, which corresponds to our experimental |Q CDW | ≈ 0.31 r.l.u., small gaps will also be present where the other sections of FS cross. However, these gaps will generally be small so the electrons will be able to tunnel across in moderate fields and hence the transport will be largely unaffected 49 .
In the low-field limit, n H ¼ ðB=eÞσ 2 xx =σ xy , where σ xx and σ xy are the longitudinal and Hall conductivities respectively and B is the magnetic field. The sign of n H is therefore the same as σ xy , and which is the sum of positive and negative contributions from the hole and electron pockets respectively which in turn depend predominantly on the mean-free-path on each pocket. The open sections of the Fermi surface will have only a small contribution to σ xy as their contribution is zero if the mean-free-path is isotropic 50 . These open sections will however, make a substantial contribution to σ xx so will affect the magnitude of n H (see Supplementary Note 5 for details of the n H calculations).

Data availability
All relevant data are available from the corresponding authors upon reasonable request and at https://doi.org/10.15151/ESRF-DC-646880761.