Dynamical spin susceptibility in ${\mathrm{La}}_2{\mathrm{CuO}}_4$ studied by resonant inelastic x-ray scattering

Resonant inelastic x-ray scattering (RIXS) is a powerful probe of elementary excitations in solids. It is now widely applied to study magnetic excitations. However, its complex cross section means that RIXS has been more difficult to interpret than inelastic neutron scattering (INS). Here we report $\sim 37$ meV resolution RIXS measurements of the magnetic excitations in ${\mathrm{La}}_2{\mathrm{CuO}}_4$, the antiferromagnetic parent of one system of high-temperature superconductors. At high energies ($\sim 2$ eV), the RIXS spectra show angular-dependent $dd$ orbital excitations in agreement with previous RIXS studies but show new structure. They are interpreted with single-site multiplet calculations. At low energies ($\lesssim 0.3$ eV), we model the wave-vector-dependent single magnon RIXS intensity as the product of the calculated single-ion spin-flip RIXS cross section and the dynamical structure factor $S(\mathbf{Q},\omega )$ of the spin-wave excitations. When $S(\mathbf{Q},\omega )$ is extracted from our data, the wave-vector-dependence of the single-magnon pole intensity shows a similar variation to that observed by INS. Our results confirm that suitably corrected RIXS data can yield the genuine wave-vector and energy dependence of $S(\mathbf{Q},\omega )$ for a cuprate antiferromagnet. In addition to spin waves, our data show structured multimagnon excitations with dispersing peaks in the intensity at energies higher than the single-magnon excitations.

Figure 1

RIXS experimental geometries for the two samples. (a) The LCO001 sample at $\varphi =0^{\mathrm{o}}$, probing $(h,0)$. (b) The LCO001 sample at $\varphi ={45}^{\mathrm{o}}$, probing $(h,h)$. (c) The LCO100 sample probing $(h,h)$. For each orientation, the projection of the momentum transfer and the relationship between $\theta$ and the momentum transfer is shown. (d) shows various excitations resolved in a typical RIXS spectrum.

Figure 2

Orbital excitations in RIXS showing (a) the $dd$ transitions in LCO and (b) a comparison of the calculated $dd$ peaks from ligand field theory (LFT) and RIXS measurements for a typical spectrum at $\theta ={120}^{\circ }$ along the $(h,0)$ direction, LCO001 orientation and with $\sigma$ polarization. (c)–(h) show the intensity of the $dd$ excitations from fitted RIXS data compared with the relative intensities calculated in LFT.

Figure 3

The low-energy excitations in LCO showing (a)–(f) self-absorption corrected RIXS intensity maps and (g)–(l) representative RIXS spectra. Total fits to the data are shown in red. The DHO magnon and multimagnon fit are shown in pink and yellow, respectively. Gaussian fits to the elastic peak (purple) and phonon (blue and green) peaks.

Figure 4

(a)–(d) show the energy dispersion of magnon excitations measured by RIXS. Red symbols indicate ${\omega }_0$ extracted from the damped harmonic oscillator fit to the magnons and data from the LCO001 and LCO100 orientations are indicated with circles and squares, respectively. The dashed line shows the magnon dispersion extracted from INS [16]. (e)–(f) show comparison of the magnon line shape measured with $\pi$ polarized RIXS in pink and INS in cyan, at wave vectors close to $(1/2,0)$ and $(1/4,1/4)$. The INS data are scaled to the RIXS data and a constant offset of 0.5 is added to the INS to compensate for a possible over subtraction of the background in Ref. [16].

Figure 5

Details of the deconvolution procedure for the magnon intensity. (a)–(f) show the self-absorption corrected magnon intensity, $I_{\mathrm{corr}}$ (filled circles), compared to the single-ion spin-flip cross section $R_{\mathrm{spin}}$ (black line). (g)–(l) show the fully deconvoluted intensity, $S{(\mathbf{Q},\omega )}_{\mathrm{RIXS}}$ as a function of Q. The value of $\theta$ is indicated by the color.

Figure 6

Comparison of the spin-wave intensity for RIXS, INS, and the theoretical values from SWT. INS data are shown as blue diamonds and the SWT as a black dashed line. RIXS data from the LCO001 and LCO100 orientation are indicated in pink and white marks, respectively, and $\sigma$ and $\pi$ polarization are represented by circles and squares, respectively. In each case $I$ is the energy integral of $S(\mathbf{Q},\omega )$ over the spin-wave pole.

Figure 7

The projected XAS spectra along the $a$ and $c$ axes in LCO samples with $\sigma$ and $\pi$ polarizations. $f_a\left(E_i\right),f_c\left(E_i\right)$ have fixed intensity due to the fixed incident energy while $f_a\left(E_f\right),f_c\left(E_f\right)$ of emitted x rays are energy dependent.

Figure 8

The calculated self-absorption factor, $C_{\mathrm{SA}}$ for excitations measured in LCO as a function of excitation energy relative to Cu $L_3$ edge, incident angle $\theta$ and polarization.

Figure 9

RIXS intensity before and after applying the self-absorption correction. Measured data are shown in white and corrected data in red. The effect is shown for $dd$ excitations in (a)–(f) and in single magnons in (g)–(l).

Figure 10

Experimental RIXS $dd$ spectra under all configurations.