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.

With soft x-rays, (∼ 930 eV at Cu-L 3 ), the accessible reciprocal space is limited due to the large wavelength of the x-rays compared to the unit cell. The detector arm of the ID32 spectrometer can move in the range 50 • < 2θ < 150 • , where 2θ is the scattering angle of the x-rays. The sample rotation Ω is limited by the angle between in the incoming (α) and outgoing (β) x-rays and the surface of the sample (See Fig. 1a). We are able to work with α, β 9 • . Thus, the accessible region of the (H, 0, L) is shown by the green region of Fig. 1b. For H = −0.31, we can access 2 L 2.8.
For our measurements, we chose L = 2.5 to give us a region of H of either side of H = −δ. For comparison, Table 1 summarises the measurement conditions for some other RIXS studies on cuprates where CDWs have been observed for various L values. This compilation is consistent with the CDW scattering being spread out in L and peaked at certain positions.
Tl2201 has the same body-centered-tetragonal crystal structure (I4/mmm) as LSCO which has the strongest CDW peaks near L = integer + 1/2 positions [1]. Hg1201 has a primitive structure (P 4/mmm) with the strongest CDW peaks being found at L = 1 [2].  The fitting model of the RIXS spectra was comprised of two resolution limited Gaussians, accounting for elastic scattering and a phonon, a damped harmonic oscillator (DHO) to fit spin excitations and finally a linear background, all fitted in the range [−100, 300] meV. The damping parameter of the DHO was fixed to half of the DHO frequency and the whole function was numerically convolved over ±4σ with a resolution limited Gaussian. The DHO model used had the form where ω 0 is the DHO frequency, γ is the damping parameter and I is the intensity.
RIXS spectra collected for our samples of Tl2201 had relatively strong elastic scattering. This makes the phonon less pronounced than in some other RIXS studies e.g. [3] where the phonon peak is stronger than the elastic peak at the zone edge. Thus phonon fitting with acceptably small error bars could only be done on a small subset of measurements we took. The data presented in the main text in Fig. 2 was counted for about 17 minutes per spectrum.
The spectra for the p = 0.25 composition at T c = 45 K and the p = 0.23 composition taken at T c = 56 K are plotted in Fig. 2. Note the p = 0.23 measurements were counted for about 4 minutes per spectrum, resulting in noticeably worse statistics. The resulting phonon dispersions are plotted in Fig. 3. The statistics and range of H for the p = 0.23 data means that little can be said about phonon softening. For the p = 0.25 data, the mean phonon energy for |H| > 0.31 = δ is lower than that in the region |H| ≈ 0.15 which is consistent with a softening of ∼ 10%. However, this result is statistically very weak. Thus, we conclude that 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 to clarify this issue.

SUPPLEMENTARY NOTE 4: COMPARISON TO OTHER CUPRATES
It is interesting to compare the H-integrated CDW intensity to other cuprates. We compared to LSCO (p = 0.12) [8], measured at T c = 30 K with a fixed L = 1.5. The area of the spectra in the range [1000, 3000] meV is used to normalise between materials. The elastic intensity is integrated in the range [−45, 45] meV to avoid the strong phonon in LSCO. The RIXS spectra for Tl2201 and LSCO at their respective CDW positions are plotted in Fig. 5a. In Fig. 5b we plot the H-dependent elastic RIXS intensity. The integrated intensity and correlation lengths of the fits to this data are given in Table. Table 2: H-Integrated CDW intensity I CDW and correlation length of the CDW in Tl2201 and LSCO p = 0.12 [8]. Calculations of the Hall coefficient in the weak-field limit were performed using a two dimensional approximation of the Fermi surface constructed from the ARPES derived tight-binding parameters, and reconstructed with a CDW potential Q CDW = (2π/3, 0) and Q CDW = (0, 2π/3) as as outlined in the Methods. Calculations were performed on a grid of 10 6 k-points. The weak-field expressions for longitudinal and Hall conductivities were calculated using the standard expressions Here v x and v y are the components of the velocity (v x = −1 ∂ε/∂k x ), f is the Fermi function, B is the magnetic field, ε(k) is the quasiparticle energy and τ is the scattering lifetime which we took as constant. This assumption on τ is consistent with the low temperature limit of the behaviour of τ (T ) derived from angle dependent magnetoresistance measurements [9], and so we would expect our results to apply in the low temperature limit. The results were only weakly dependent on the temperature in the Fermi function and we used T = 20 K for the calculation (the size of T is chosen so that the finite k point spacing does not cause problems with the integral). The Hall number is finally calculated as n H = (B/e)σ 2 xx /σ xy , and is expressed in units of carriers per unit cell. To model the experimental data we assumed that the CDW potential, V 0 , increases linearly with decreasing doping p as shown in Fig. 6, being zero for p > 0.255 and saturating at V 0 = 60 meV for p = 0.1425. Increasing V 0 causes the electron pocket to shrink and the Fermi velocities on this pocket to be reduced. σ xy is the sum of the positive and negative contributions from the hole and electron pockets respectively, so a reduction in magnitude of the (negative) electron contribution causes σ xy to increase and hence n H to decrease. On the other hand, reducing p with fixed V 0 causes the electron pocket contribution to grow. Hence increasing V 0 while reducing p produces two opposing effects on the size of the electron and hole pockets and their contribution to n H . Fig. 6: a Hall number n H versus doping for Tl2201. Blue squares are experimental data from Ref. [10], and small red squares are the calculated values. The assumed variation of the CDW potential V 0 with doping p is shown with the magenta solid line. b Fermi surface reconstruction in Tl2201 (p = 0.25) calculated using the model described in the Methods Section for V 0 = 10 meV.
With our assumed p dependence of V 0 , the calculated behaviour of n H is in good agreement with the experimental data as shown in Fig. 6. For p > 0.255 n H (1 + p), consistent with the large Fermi surface found in DFT and quantum oscillation measurements. As the CDW begins to open, for p < 0.255, n H falls towards values closer to n H = p. For p < 0.16, n H changes sign because the hole pocket disappears, however whether this occurs or not depends on the gradient, dV 0 /dp. We note that our calculation does not include any modelling of the pseudogap, which would be expected to reduce the contribution of the hole pocket for p inside the pseudogap regime (p 0.19) and hence make a sign change of n H more likely.