High-energy magnetic excitations in overdoped ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ studied by neutron and resonant inelastic x-ray scattering

We have performed neutron inelastic scattering and resonant inelastic x-ray scattering (RIXS) at the Cu-$L_3$ edge to study high-energy magnetic excitations at energy transfers of more than 100 meV for overdoped ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ with $x=0.25$ $(T_c=15$ K) and $x=0.30$ (nonsuperconducting) using identical single-crystal samples for the two techniques. From constant-energy slices of neutron-scattering cross sections, we have identified magnetic excitations up to $\sim 250$ meV for $x=0.25$. Although the width in the momentum direction is large, the peak positions along the $(\pi ,\pi )$ direction agree with the dispersion relation of the spin wave in the nondoped ${\mathrm{La}}_2{\mathrm{CuO}}_4$ (LCO), which is consistent with the previous RIXS results of cuprate superconductors. Using RIXS at the Cu-$L_3$ edge, we have measured the dispersion relations of the so-called paramagnon mode along both $(\pi ,\pi )$ and $(\pi ,0)$ directions. Although in both directions the neutron and RIXS data connect with each other and the paramagnon along $(\pi ,0)$ agrees well with the LCO spin-wave dispersion, the paramagnon in the $(\pi ,\pi )$ direction probed by RIXS appears to be less dispersive and the excitation energy is lower than the spin wave of LCO near $(\pi /2,\pi /2)$. Thus, our results indicate consistency between neutron inelastic scattering and RIXS, and elucidate the entire magnetic excitation in the $(\pi ,\pi )$ direction by the complementary use of two probes. The polarization dependence of the RIXS profiles indicates that appreciable charge excitations exist in the same energy range of magnetic excitations, reflecting the itinerant character of the overdoped sample. A possible anisotropy in the charge excitation intensity might explain the apparent differences in the paramagnon dispersion in the $(\pi ,\pi )$ direction as detected by the x-ray scattering.

Figure 1

Contour maps of neutron-scattering intensity in the $(H,K)$ zone at (a) $115(\pm 15)$-meV, (b) $145(\pm 15)$-meV, and (c) $205(\pm 25)$-meV energy transfer. The antiferromagnetic zone center $(\pi ,\pi )$ corresponds to $(0.5,0.5)$ in this tetragonal notation. Dashed arrows in (a) indicate trajectories of $(H,H)$ and $(H,1+H)$, on which intensity profiles are analyzed by fits. The inset shows the scan area for neutron and RIXS. The gray area indicates the coverage of detector banks of the neutron measurement that covers the antiferromagnetic zone center $(\pi ,\pi )$. Thick dashed lines indicate the trajectories of RIXS measurements by rotating the sample.

Figure 2

Neutron-scattering profiles along trajectories $(H,H)$ and $(H,1+H)$ at selected energies. Red lines are fits to a function containing two Lorentzians symmetric to the AF zone center. Each Lorentzian component is shown by a dashed line.

Figure 3

(a) Peak positions obtained by fits. Vertical bars indicate the energy range of integration for the data analyses, and horizontal bars indicate full widths at half maximum of Lorentzian peaks. (b) Magnetic peak positions of LSCO $x=0.22$ measured by neutron referred from Ref. [11]. In both figures, solid lines are the spin-wave dispersion of nondoped LCO referred from Ref. [1].

Figure 4

Contour maps of RIXS intensity for $x=0.25$ and $0.30$. The high intensity at $\sim 1.8$ eV is due to the $dd$ excitation. Dispersive feature below 0.5 eV is the paramagnon. Solid dispersive lines are the spin-wave dispersion of LCO. Solid circles indicate the peak positions of neutron magnetic peak in Fig. 3.

Figure 5

Cu-$L_3$ RIXS profiles. Data were analyzed by fitting to a function containing two Gaussians: one for magnon and the other for the $E=0$ components, and a squared Lorentzian describing the tail of the $dd$ excitation. The $E=0$ component is assumed to have resolution width 0.35 eV, while the magnon component is convoluted by the resolution. Thick solid lines are results of fits, thinner solid lines are magnon components, and dashed lines represent the $E=0$ and $dd$-tail components.

Figure 6

Paramagnon dispersions along (a) the $(\pi ,\pi )$ direction for $x=0.25$ (circles) and $0.30$ (closed squares), and those along (b) the $(\pi ,0)$ directions for $x=0.25$ (circles) and for $x=0.26$ thin film (closed diamonds) reported in Ref. [16]. Neutron data in Fig. 3 are also shown by open squares. Solid lines are the spin-wave dispersion of LCO. The thick dashed and gray lines in the inset indicate $q$ trajectories of dispersions measured by RIXS and neutron, respectively.

Figure 7

RIXS profiles measured with $\pi$ and $\sigma$ configurations at $(0.26,0.26)$ and $(0.36,0)$ corresponding to a ${45}^{\circ }$ rotation of the sample from the specular orientation.