Charge density wave fluctuations in ${\text{La}}_{2-x}$${\text{Sr}}_x$${\text{CuO}}_4$ and their competition with superconductivity

We report hard (14 keV) x-ray diffraction measurements on three compositions ($x=0.11,0.12,0.13)$ of the high-temperature superconductor ${\text{La}}_{2-x}$${\text{Sr}}_x$${\text{CuO}}_4$. All samples show charge density wave (CDW) order with onset temperatures in the range 51–80 K and ordering wave vectors close to (0.23,0,0.5). The CDW is strongest with the longest in-plane correlation length near 1/8 doping. On entering the superconducting state the CDW is suppressed, demonstrating the strong competition between the charge order and superconductivity. CDW order coexists with incommensurate magnetic order and the wave vectors of the two modulations have the simple relationship ${\mathit{\delta }}_{\mathrm{charge}}=2{\mathit{\delta }}_{\mathrm{spin}}$. The intensity of the CDW Bragg peak tracks the intensity of the low-energy (quasielastic) spin fluctuations. We present a phase diagram of ${\text{La}}_{2-x}$${\text{Sr}}_x$${\text{CuO}}_4$ including the pseudogap phase, CDW, and magnetic order.

Figure 1

(a) Schematic drawing of an intensity-modulated rod of scattering in reciprocal space due to the CDW. Trajectories of scans in other panels are shown. (b)–(f) CDW peaks for various $h$ and $\ell$ values for ${\text{La}}_{1.88}$${\text{Sr}}_{0.12}$${\text{CuO}}_4$.

Figure 2

Intensities of CDW satellite peaks in the $(h,0,\ell )$ plane of reciprocal space. The area of the filled circles are proportional to the CDW peak intensities. Crosses ($\times$) are positions investigated where no CDW peak was detected.

Figure 3

(a) $\ell$ dependence of the scattering along the line (4+$\delta$,0,$\ell$) for $T=30$ and 8 K for ${\text{La}}_{1.87}$${\text{Sr}}_{0.13}$${\text{CuO}}_4$. (b) Points show $\ell$ dependence of CDW scattering. The signal has been isolated by subtracting 8 K data, where the CDW signal is weak, from 30-K data where the signal is strongest. The solid line shows equivalent data collected on ${\text{La}}_{1.875}$${\text{Sr}}_{0.125}$${\text{CuO}}_4$ from Ref. [30].

Figure 4

(a)–(c) Scans through the CDW satellite peak at $T=30\text{K}\gtrsim T_c$ and at 150 K showing the doping dependence of the incommensurate wave vector and approximate strength of the peak. The high-temperature scans have been offset for clarity. Solid line in (b) shows the instrumental resolution.

Figure 5

Spin and charge incommensurability vs doping for ${\text{La}}_{2-x}$${\text{Sr}}_x$${\text{CuO}}_4$ (LSCO) [12, 15], ${\text{La}}_{2-x}$${\text{Ba}}_x$${\text{CuO}}_4$ (LBCO) [17], ${\text{YBa}}_2$${\text{Cu}}_3$${\text{O}}_{6+x}$ [35], and ${\text{Bi}}_2$${\text{Sr}}_{2-x}$${\text{La}}_x$${\text{CuO}}_{6+x}$ (Bi2201) [4]. Open symbols represent spin order or the strong spin fluctuations with in-plane wave vector $(1/2+{\delta }_{\mathrm{spin}},1/2)$. Closed symbols are CDW peaks at $({\delta }_{\mathrm{charge}},0)$.

Figure 6

(a) $h$-dependent scans through the $(\delta ,0,12.5)$ CDW peak for various temperatures for ${\text{La}}_{1.88}$${\text{Sr}}_{0.12}$${\text{CuO}}_4$. Solid lines are fits to a Gaussian line shape. All scans, except for $T=8$ K, have been offset for clarity. (b) and (c) Peak heights and widths of the CDW peak extracted from the fits in (a). The inset to (b) shows 30 and 8 K data from (a) plotted together with linear backgrounds subtracted. This illustrates the suppression of the CDW in the superconducting state.

Figure 7

Temperature dependence of the peak intensity of the $(4+\delta ,0,12.5)$ CDW peak for Sr dopings $x=0.11$,0.12, and 0.13. Peak heights are determined from fitting data such as that displayed in Fig. 6. Heights are plotted with respect to average height in the range 90–150 K. The solid lines are power law fits described in the main text. The open circles in (b) show inelastic neutron scattering (INS) measurements of spin fluctuations at the spin ordering wave vector ${\mathbf{q}}_{\mathrm{SDW}}=(1/2+{\delta }_{\mathrm{spin}},1/2)=(0.625,0.5)$ and energy transfer $E=0.3$ meV for ${\text{La}}_{1.88}$${\text{Ba}}_{0.12}$${\text{CuO}}_4$ from Ref. [36]. The intensities of the INS data have been multiplied by 1200.

Figure 8

Temperature vs doping phase diagram of ${\text{La}}_{2-x}$${\text{Sr}}_x$${\text{CuO}}_4$. $T_{\mathrm{CDW}}$ is the onset temperature of charge density wave order determined from the present x-ray experiment. $T_{\mathrm{SDW}}$ is the onset temperature of the incommensurate magnetic order observed with neutron scattering [10, 12, 13, 14, 15], nuclear magnetic resonance (NMR), and muon spin resonance ($\mu$SR) [16]. $T_c$ is the superconducting transition temperature from Ref. [11]. $T^{☆}$ is the pseudogap onset temperature determined from the upturn in the Nernst coefficient [37, 38].