Prevailing Charge Order in Overdoped ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ beyond the Superconducting Dome

The extremely overdoped cuprates are generally considered to be Fermi liquid metals without exotic orders, whereas the underdoped cuprates harbor intertwined states. Contrary to this conventional wisdom, using Cu $L_3$-edge and O $K$-edge resonant x-ray scattering, we reveal a charge order (CO) correlation in overdoped ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ ($0.35\le x\le 0.6$) beyond the superconducting dome. This CO has a periodicity of $\sim 6$ lattice units with correlation lengths of $\sim 20$ lattice units. It shows similar in-plane momentum and polarization dependence and dispersive excitations as the CO of underdoped cuprates, but its maximum intensity differs along the $c$ direction and persists up to 300 K. This CO correlation cannot be explained by the Fermi surface instability and its origin remains to be understood. Our results suggest that CO is prevailing in the overdoped metallic regime and requires a reassessment of the picture of overdoped cuprates as weakly correlated Fermi liquids.

Figure 1

Observation of charge order correlation by RIXS in overdoped metallic ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ ($x=0.45$). The UHB and LHB refer to upper Hubbard band and lower Hubbard band, respectively. (a) Schematic plot of RIXS process at O $K$ edge. (b) A typical RIXS spectrum at ${\mathbf{q}}_{||}=$ (0.12, 0), displaying the elastic peak, phonon, charge, and orbital excitations. Inset: resistance curve displaying the metallic nature of the sample. (c) Integrated intensity of elastic peaks for positive and negative $(H,0)$ and $(H,H)$ directions, using $\sigma$ polarization. Red and blue curves are Lorentzian peak fits to the data with a polynomial background (gray dashed lines). (d) Polarization measurements with $\sigma$- and $\pi$-polarized light, collected at 33 and 250 K. (e) Reciprocal-space image. The blue shaded region is the accessible momentum-transfer range of O $K$ edge. The green and blue lines indicate the momentum cuts, and red dots indicate the observed CO peaks. (f) XAS spectra near the ZRS absorption peak with $\sigma$ polarization at normal incidence. Incident energy dependence of the integrated intensity of elastic peak and orbital excitation, normalized to the value at the ZRS peak.

Figure 2

Cu $L_3$ edge and O $K$ edge REIXS studies of charge order correlation in ${\mathrm{La}}_{1.55}{\mathrm{Sr}}_{0.45}{\mathrm{CuO}}_4$. (a) Observation of CO correlation along $(0,-K)$ direction at both Cu $L_3$ edge and O $K$ edge, offset is applied for clarity. (b) Polarization dependence of the CO peak, collected at 930 eV and $L=1.1$ r.l.u. (c) Detuning measurements near the Cu $L_3$ edge after self-absorption correction (see Supplemental Material) [18]. (d) $L$ dependence of CO correlation within the accessible range of [1.1, 1.8] r.l.u. at 930 eV, collected with $\sigma$ polarization.

Figure 3

Anisotropic momentum dependence of phonon intensity and dispersive CO excitations in ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ ($x=0.45$). (a),(c) The elastic intensity map and (b),(d) the inelastic RIXS intensity map for visualizing the CO correlation and phonon branches along $(H,0)$ direction and $(H,H)$ direction, respectively. (e)–(g) Integrated intensity for buckling phonon, acoustic phonon, and CO correlation, respectively, along both directions. Details of the fitting are presented in the Supplemental Material [18].

Figure 4

Doping dependence of charge order correlation in overdoped ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ (LSCO) and the extended phase diagram. (a)–(c) CO peak profiles measured by Cu $L$ edge REIXS in LSCO with $x=0.35$, 0.45, 0.6, respectively. The offset is applied for clarity. $L$ is fixed at 1.1 r.l.u. The peak is nearly temperature independent up to 300 K. (d) The extended CO phase diagram of cuprates. It shows superconducting dome defined by $T_c$, antiferromagnetism (AFM) defined by $T_N$ [44], pseudogap determined from the Nernst coefficient [45], underdoped charge order and charge fluctuation [16, 35, 46], and overdoped charge order. (e) The doping dependence of the CO wave vector in LSCO and nesting vector obtained from Lindhard function.