Resonant inelastic x-ray scattering study of the spin and charge excitations in the overdoped superconductor ${\mathrm{La}}_{1.77}{\mathrm{Sr}}_{0.23}{\mathrm{CuO}}_4$

We present a resonant inelastic x-ray scattering (RIXS) study of spin and charge excitations in overdoped ${\mathrm{La}}_{1.77}{\mathrm{Sr}}_{0.23}{\mathrm{CuO}}_4$ along two high-symmetry directions. The line shape of these excitations is analyzed and they are shown to be highly overdamped. Their spectral weight and damping are found to be strongly momentum dependent. Qualitative agreement between these observations and a calculated random-phase approximation susceptibility is obtained for this overdoped compound, implying that a significant contribution to the RIXS signal stems from a continuum of charge excitations. Furthermore, this suggests that the spin excitations in the overdoped regime can be captured qualitatively by an itinerant picture. Our calculations also predict a low-energy spin-excitation branch to exist along the nodal direction near the zone center. With the energy resolution of the present experiment, this branch is not resolvable, but we show that the next generation of high-resolution spectrometers will be able to test this prediction.

Figure 1

(a) RIXS spectrum, recorded on overdoped LSCO $x=0.23$ using $\sigma$-polarized light, displays elastic scattering, a low-energy excitation, and a $dd$ excitation. The inset shows the scattering geometry and reciprocal space $(h,k)$ schematically. (b) Overdamped response function showing how ${\chi }^{\prime \prime }\to 0$ for $\omega \to 0$. (c) Interpolated RIXS intensity, with elastic scattering subtracted, vs momentum $q=(h,0), (h,h)$ and photon energy loss $\omega$. Red ticks indicate the grid of the spectra used for the interpolation.

Figure 2

RIXS spectra recorded on LSCO $x=0.23$, at $T=20$ K, in grazing incidence geometry using $\sigma$-polarized light tuned to the Cu $L_3$ edge (930 eV) line. (a)–(e) Spectra measured with momenta along $q=(h,0)$ as indicated in the inset of (a). (f)–(j) Spectra taken along $q=(h,h)$ as indicated in the inset of (f) and (g). Blue and gray shaded areas are modeled contributions from elastic and low-energy excitations on top of a cubic background (dashed line). Solid black line is the sum of these contributions. See text for further explanation.

Figure 3

Damping $\gamma /2$ (see text) of the low-energy excitations measured by RIXS in LSCO $x=0.23$ (this work) and ${\mathrm{Ba}}_{0.6}{\mathrm{K}}_{0.4}{\mathrm{Fe}}_2{\mathrm{As}}_2$ [20] along the high-symmetry directions $\mathrm{\Gamma }X$ and $\mathrm{\Gamma }M$. Error bars in the bottom panels are set by the applied energy resolution (65 meV—HWHM), which is also indicated by a horizontal dashed line.

Figure 4

Comparison of the spin-excitation dispersions ${\omega }_0$ and ${\omega }_1$ (see text) extracted on ${\mathrm{La}}_2{\mathrm{CuO}}_4$ (LCO) with the low-energy excitation dispersions on ${\mathrm{La}}_{1.77}{\mathrm{Sr}}_{0.23}{\mathrm{CuO}}_4$ along high-symmetry directions as indicated. Data obtained from INS and RIXS are displayed by square and circular points, respectively. For LCO, good agreement between INS ($\square$, Ref. [40]) and RIXS ($\circ$, Ref. [10]) is found along the $\mathrm{\Gamma }X$ direction. No overlap between RIXS ($•$, this work) and INS ($\square$, Ref. [26]) has been reached for overdoped compositions of LSCO. The inset indicates the high-symmetry directions and displays the calculated static Lindhard susceptibility (for $\omega \to 0$) (see text for further explanation).

Figure 5

Integrated intensity ${\chi }_0^{\prime \prime }·\gamma$ of the low-energy excitations measured by RIXS in LSCO $x=0.23$ along the high-symmetry directions $\mathrm{\Gamma }X$ and $\mathrm{\Gamma }M$. The dashed lines are a guide to the eyes.

Figure 6

(a) Calculated RPA susceptibility along the nodal $(h,h)$ and antinodal direction $q=(h,0)$. (b) The RPA susceptibility convoluted by instrument resolution to make a direct comparison to Fig. 1. (c)–(e) A zoom of the low-energy nodal RPA susceptibility. In (d) and (e), a Gaussian convolution with FWHM = 60 and 130 meV has been applied. This demonstrates that a spectrometer with a FWHM = 60 meV energy resolution at the Cu $L_3$ edge is sufficient to test the RPA prediction of low-energy nodal spin excitations.