Magnetic nature of the 500 meV peak in ${\text{La}}_{2-x}{\text{Sr}}_x{\text{CuO}}_4$ observed with resonant inelastic x-ray scattering at the $\text{Cu}K$-edge

We present a comprehensive study of the temperature and doping dependence of the 500 meV peak observed at $\mathbf{q}=\left(\pi ,0\right)$ in resonant inelastic x-ray scattering (RIXS) experiments on ${\text{La}}_2{\text{CuO}}_4$. The intensity of this peak persists above the Néel temperature $\left(T_N=320\text{K}\right)$, but decreases gradually with increasing temperature, reaching zero at around $T=500\text{K}$. The peak energy decreases with temperature in close quantitative accord with the behavior of the two-magnon $B_{1g}$ Raman peak in ${\text{La}}_2{\text{CuO}}_4$ and, with suitable rescaling, agrees with the Raman peak shifts in ${\text{EuBa}}_2{\text{Cu}}_3{\text{O}}_6$ and ${\text{K}}_2{\text{NiF}}_4$. The overall dispersion of this excitation in the Brillouin zone is found to be in agreement with theoretical calculations for a two-magnon excitation. Upon doping, the peak intensity decreases analogous to the Raman mode intensity and appears to track the doping dependence of the spin-correlation length. Taken together, these observations strongly suggest that the 500 meV mode is magnetic in character and is likely a two-magnon excitation.

Figure 1

(Color online) Scattering configurations used when scattering in the (a) vertical plane (sector 9ID experiments) and (b) horizontal plane (at sector 30ID and BL11XU). Arrows indicate the wave vectors of the incident and scattered beams. The sample is shown in both cases along the $y$ axis of the figure [note that the vertical direction in real space corresponds to perpendicular to the incident beam and parallel to the plane of the page in (a), while it is perpendicular to the plane of the page in (b)]. The crystallographic axes are also indicated. In (a), the incident-beam polarization is directed out of the page, parallel to the $c$ axis. In (b), the incident polarization is in the scattering plane. The scattering angle $2\theta$ is set to nearly $90°$, which suppresses the in-plane polarization of the scattered beam. The angle $\delta$ is about $20°$, so as to have a large $c$-axis component of the incident polarization.

Figure 2

(Color online) (a) Plot of RIXS intensity vs energy loss, measured at various incident energies at the 30ID beamline (horizontal scattering geometry), for $\mathbf{q}=\left(\pi ,0\right)$. Curves are offset along the $y$ axis as indicated by the thick gray lines on the right. Solid lines are guides to the eyes. (b) Background-subtracted intensity is plotted as a function of energy loss, measured at the 30ID beamline at room temperature for incident energy of 8994 eV, for various momentum transfers. (c) same as (b) but in vertical scattering geometry, from Ref. 35 $\left(T=45\text{K}\right)$. Solid lines are results of fits to Lorentzian line shapes.

Figure 3

(Color online) (a) Dispersion relation of the two-magnon excitation obtained from two separate measurements as described in the figure legend and the lines represent theoretical dispersions calculated by Ament et al. (Ref. 59), Donkov and Chubukov (Ref. 56) ($A_{1g}$, with and without final-state magnon-magnon interactions included in the calculation, as shown by the dash-dotted and vertical dashed lines, respectively), Vernay et al. (Ref. 55) $\left(A_{1g}\right)$, and Nagao and Igarashi (Ref. 57) The energy scale is normalized by the magnetic exchange constant $J\left(\approx 145\text{meV}\right)$. Here, we suppose that $3.5J$ corresponds to 500 meV. The horizontal error bars span the $\mathbf{q}$ resolution. (b) Spectral weight of the background-subtracted data, defined as the product of the Lorentzian peak intensity and linewidths; this does not include constant background so that the spectral weight at zone center is zero. The weights are normalized by the weight at $\mathbf{q}=\left(\pi ,0\right)$. Same legend as in (a), two available theoretical calculations by Ament et al. (Ref. 59) and Donkov and Chubukov (Ref. 56) are plotted.

Figure 4

(Color online) (a) Temperature dependence of $\left(\pi ,0\right)$ RIXS spectra (black squares) of ${\text{La}}_2{\text{CuO}}_4$ obtained at 30ID. The background (thin line) was determined by scaling a Lorentzian fit of the 500 K data by the elastic line intensity at each temperature, as discussed in the text. The spectra are displaced along the $y$ axis for clarity. Thick gray ticks on the right side of the graph indicate zero levels. [(b)–(d)] Background-subtracted data and fits for $T=400$, 250, and 35 K, respectively. [(e) and (f)] Spectra measured at 9ID for temperatures of 250, 150, and 45 K, respectively. The background for the 9ID data was obtained from off-resonance data as described in Sec. 2.

Figure 5

(Color online) Temperature dependence of the Lorentzian fit parameters for the RIXS peaks measured at $\mathbf{q}=\left(\pi ,0\right)$. Shown are the fit parameters (a) intensity and (b) excitation energy. Different symbols correspond to data taken with different experimental conditions: horizontal scattering at 30ID (open circles and filled square: the filled square corresponds to a separate beamtime), vertical scattering at 9ID (filled diamonds), and horizontal scattering at BL11XU (filled triangle). The solid line is a result of the fit to Eq. (1) with fit parameters $E_0=0.50\text{eV}$, $\gamma =1.7$, and $\Omega =0.1\text{eV}$. Dotted vertical line indicates the Néel temperature for ${\text{La}}_2{\text{CuO}}_4$. The dashed curve, which is referred to the right-hand scale, is the temperature dependence of the two-magnon Raman peak position of ${\text{EuBa}}_2{\text{Cu}}_3{\text{O}}_6$ obtained from the peaks of the fitted curves in Ref. 53. On the same scale, the crosses are the two-magnon Raman peak position for ${\text{La}}_2{\text{CuO}}_4$ as measured in Ref. 37.

Figure 6

(Color online) Peak intensity of the 500 meV peak vs doping $x$ for ${\text{La}}_{2-x}{\text{Sr}}_x{\text{CuO}}_4$. The peak intensities were determined from Lorentzian fits of the spectra from Ref. 35 and in Fig. 9. Between different samples, the spectra were normalized by the intensity of the elastic line at the same $\mathbf{Q}$ position and incident energy. Dashed line results from a fit to the form $I\left(x\right)=I_0/\left(1+r\sqrt{x}\right)$, with $r\sim 10$.

Figure 7

(Color online) Comparison of scaled peak energy shift vs scaled temperature in ${\text{La}}_2{\text{CuO}}_4$ as measured by RIXS in this study (circles) and two-magnon Raman scattering in ${\text{EuBa}}_2{\text{Cu}}_3{\text{O}}_6$ (triangles) obtained from the peaks of the fit curves in Ref. 53 and ${\text{K}}_2{\text{NiF}}_4$ (unfilled squares) from Ref. 89. (a) The peak energy shift is normalized by the low-temperature energy and further multiplied by spin $S$, as discussed in the text. The temperature scales are normalized by $JSz$. $J$ of 143 meV is used for ${\text{La}}_2{\text{CuO}}_4$, 100 meV for ${\text{EuBa}}_2{\text{Cu}}_3{\text{O}}_6$ (Ref. 53), and 9.7 meV for ${\text{K}}_2{\text{NiF}}_4$ (Ref. 89), respectively. The lines result from fits to Eq. (1). (b) Same as (a), except normalizing the temperature to the Néel temperature $T_N$, which is 320 K for ${\text{La}}_2{\text{CuO}}_4$, 430 K for ${\text{EuBa}}_2{\text{Cu}}_3{\text{O}}_6$ (Ref. 53), and 97 K for ${\text{K}}_2{\text{NiF}}_4$ (Ref. 90) (in Ref. 89, the temperature is already normalized to $T_N$).

Figure 8

(Color online) (a) The two-magnon density of states for noninteracting magnons with a total energy $\omega$ and two different total momenta. (b) Intensity plot showing the single magnon states that contribute to give a total momentum ${\mathbf{q}}_{tot}=\left(0,0\right)$ at $\omega ={\omega }_{peak}=4J$ (where the two-magnon density of states is a maximum). The two individual magnons have the maximum probability of carrying momenta ${\mathbf{q}}_2=-{\mathbf{q}}_1=\mathbf{k}$, where $\mathbf{k}$ lies on the magnetic zone boundary. (c) Same as in (b) but for ${\mathbf{q}}_{\mathbf{t}\mathbf{o}\mathbf{t}}=\left(\pi ,0\right)$ and $\omega ={\omega }_{peak}=2J\sqrt{3}$. In this case, the maximum probability is for individual magnons to have ${\mathbf{q}}_1={\mathbf{q}}_2=\mathbf{k}$, where $\mathbf{k}=\left(\pm \pi /2,0\right)$, $\left(0,\pm \pi /2\right)$.

Figure 9

(Color online) (a) The RIXS intensity vs photon energy loss, for ${\text{La}}_{1.93}{\text{Sr}}_{0.07}{\text{CuO}}_4$ at $\mathbf{q}=\left(\pi ,0\right)$ and $T=20\text{K}$, for incident energies “on resonance” $E_i=8993\text{eV}$ (filled squares) and “off resonance” (empty circles). For improved statistics in the off-resonance spectrum, we combined spectra of $E_i=8981\text{eV}$ with $E_i=8987\text{eV}$, whose low-energy spectra were identical within error bar. The resultant spectrum was then normalized to match the energy-gain side of the on-resonance spectrum. (b) Subtraction of the spectra shown in (a), which has reasonable error bar only above 0.3 eV energy loss. (c) The same on-resonance spectrum plotted as a function of energy loss (filled squares) and energy gain (open circles) and a fit of the energy-gain spectrum to a Lorentzian line shape plus constant (filled diamonds). (d) Subtraction of a fit of the energy-gain spectrum from the original spectrum.