Two-magnon excitations in resonant inelastic x-ray scattering studied within spin density wave formalism

We study two-magnon excitations in resonant inelastic x-ray scattering (RIXS) at the transition-metal $K$ edge. Instead of working with effective Heisenberg spin models, we work with a Hubbard-type model ($d\text{−}p$ model) for a typical insulating cuprate ${\mathrm{La}}_2{\mathrm{CuO}}_4$. For the antiferromagnetic ground state within the spin density wave (SDW) mean-field formalism, we calculate the dynamical correlation function within the random-phase approximation (RPA), and then obtain two-magnon excitation spectra by calculating the convolution of it. Coupling between the $K$-shell hole and the magnons in the intermediate state is calculated by means of diagrammatic perturbation expansion in the Coulomb interaction. The calculated momentum dependence of RIXS spectra agrees well with that of experiments. A notable difference from previous calculations based on the Heisenberg spin models is that RIXS spectra have a large two-magnon weight near the zone center, which may be confirmed by further careful high-resolution experiments.

Figure 1

Schematic picture of typical processes of two-magnon creation. (a) An electron-hole pair is excited on the $d$ bands to screen the $1s$ core hole (only the correlated $d$-electron states are drawn, and the excited $1s$ hole and $p$ electron are not shown explicitly). $\sigma$ represents a spin state. (b) An electron below $E_F$ goes into the created hole state, by changing its spin (${\sigma }^{\prime }\to \sigma$) and exciting a magnon ${\mathrm{m}}_1$. Wavy line represents a magnon. (c) Finally, the electron excited initially above $E_F$ goes into the hole state below $E_F$ by changing its spin ($\sigma \to {\sigma }^{\prime }$) and exciting another magnon ${\mathrm{m}}_2$. Another typical process of two-magnon creation is represented by (a) $\to$ (b') $\to$ (c').

Figure 2

(a) Summation of diagrams for RPA. Thick wavy line and oriented solid lines represent the RPA susceptibility (i.e., the single-magnon propagator) and the Green's function for the $3d$ electrons, respectively. The empty circle represents the $3d$ Coulomb interaction matrix ${\mathrm{\Gamma }}^{\left(0\right)}$. (b) Diagrammatic expression of ${\mathrm{\Pi }}_{{\sigma }_1^{\prime }{\sigma }_2^{\prime },{\sigma }_2{\sigma }_1}(\mathbf{q},t^{\prime }-t)$, where the momenta absorbed to and emitted from the electron system are $\mathbf{q}-{\mathbf{q}}_{{\sigma }_1{\sigma }_2}$ and $\mathbf{q}-{\mathbf{q}}_{{\sigma }_1^{\prime }{\sigma }_2^{\prime }}$, respectively. (c) Typical two contributions from two-magnon excitations to RIXS spectra. External oriented broken wavy lines represent x rays. Oriented thick, intermediate, and thin solid lines represent the propagation of Cu $1s$, $3d$, and $4p$ electrons, respectively. Filled circular vertex represents the Coulomb interaction $V_{sd}$ between the Cu $1s$ and $3d$ electrons at each Cu site. The thick solid wavy lines, c, ${\mathrm{m}}_1$, and ${\mathrm{m}}_2$, represent the RPA propagators, ${\chi }_{{\tau }_1{\tau }_2,\sigma \sigma }^C(\mathbf{Q},\mathrm{\Omega })$, ${\mathrm{\Pi }}_{{\sigma }_1^{\prime }{\sigma }_2^{\prime },{\sigma }_2{\sigma }_1}(\mathbf{p},\zeta )$, and ${\mathrm{\Pi }}_{{\sigma }_4^{\prime }{\sigma }_3^{\prime },{\sigma }_3{\sigma }_4}(\mathbf{Q}-\mathbf{p},\mathrm{\Omega }-\zeta )$, respectively. (d) Extracted bare part of two-magnon propagation. The total carried momentum is $\mathbf{Q}$.

Figure 3

(a) Calculated dynamical spin correlation function $S_{zz}(\mathbf{q},\omega )$ along symmetry lines is represented by the gray-level map in a logarithmic scale. (b) Integrated intensity along symmetry lines is represented by a curve, where $S_{zz}(\mathbf{q},\omega )$ is integrated in $\omega$ up to 1 eV. In both panels, the plots are the neutron-scattering data read from Ref. [48]: $T=10$ (empty symbols) and 295 K (filled symbols). Squares are obtained for $E_i=250$ meV, circles for $E_i=600$ meV, and triangles for $E_i=750$ meV, where $E_i$ is the incident neutron energy [48].

Figure 4

Comparison with experimental data at various momentum transfers. Solid and broken curves are the calculated results of $W(q\mathbf{e};q^{\prime }{\mathbf{e}}^{\prime })$ and $\tilde{W}(\mathbf{Q},\mathrm{\Omega })$, respectively, where $Q\equiv (\mathbf{Q},\mathrm{\Omega })\equiv q-q^{\prime }\equiv (\mathbf{q}-{\mathbf{q}}^{\prime },\omega -{\omega }^{\prime })$, $\mathrm{\Omega }$ is the energy loss of x rays. Plots are the experimental data read from Ref. [19]. X-ray polarization direction is fixed to $E\parallel c$. The calculated intensities are scaled to match the experimental plots, using a scale factor common to all the panels.

Figure 5

(a) Calculated RIXS intensity $W(q\mathbf{e};q^{\prime }{\mathbf{e}}^{\prime })$ and experimental peak positions along symmetry lines. Plots are experimental data read from Fig. 3(a) of Ref. [20]. (b) Integrated weights: solid and broken curves are the calculated results of $I(\mathbf{e},{\mathbf{e}}^{\prime };\mathbf{Q})$ and $\tilde{I}\left(\mathbf{Q}\right)$, respectively, and plots are the experimental data read from Fig. 3(b) of Ref. [20]. The weights are normalized by the value at $\mathbf{Q}=(\pi ,0)$. For the experimental data, the background is subtracted so that the spectral weight at the zone center is zero [20]. In (a) and (b), two symbols, circle and square, represent two different data sets at $T=300$ K and $T=45$ K from two different beamlines [20]. Horizontal bars of plots represent the Q resolution [20]. In the calculated results of (a) and (b), x-ray polarization direction is always fixed to $E\parallel c$, and the incident x-ray energy for calculation is fixed to $\omega =8992$ eV. (c) Bare two-magnon part, $\tilde{W}(\mathbf{Q},\mathrm{\Omega })$, calculated by Eq. (51). (d) $|{\mathrm{\Lambda }}_{\tau \tau }(\mathbf{Q},\mathrm{\Omega })|$ calculated by Eq. (46) along symmetry lines.

Figure 6

Incident x-ray energy dependence for two polarization directions (a) $E\parallel ab$ and (b) $E\parallel c$. X-ray momentum transfer is fixed to $\mathbf{Q}=(\pi ,0)$.

Figure 7

Resonant x-ray absorption spectra for $E\parallel ab$ and $E\parallel c$ geometries. Solid and broken curves are the calculated XAS and the Cu-$p_{\mu }$ partial DOS ${\rho }_{4p\mu }(\omega +{\varepsilon }_{1s})$, respectively. Plots are the experimental data read from Ref. [50]. The experimental intensity is rescaled so that the peak intensity matches the calculated one.