Magnetic excitations in ${\mathrm{La}}_2\mathrm{Cu}{\mathrm{O}}_4$ probed by indirect resonant inelastic x-ray scattering

Recent experiments on ${\mathrm{La}}_2\mathrm{Cu}{\mathrm{O}}_4$ suggest that indirect resonant inelastic x-ray scattering (RIXS) might provide a probe for transversal spin dynamics. We present in detail a systematic expansion of the relevant magnetic RIXS cross section by using the ultrashort core-hole lifetime (UCL) approximation. We compute the scattering intensity and its momentum dependence in leading order of the UCL expansion. The scattering is due to two-magnon processes and is calculated within a linear spin-wave expansion of the Heisenberg spin model for this compound, including longer range and cyclic spin interactions. We observe that the latter terms in the Hamiltonian enhance the first moment of the spectrum if they strengthen the antiferromagnetic ordering. The theoretical spectra agree very well with experimental data, including the observation that scattering intensity vanishes for the transferred momenta $\mathbf{q}=(0,0)$ and $\mathbf{q}=(\pi ,\pi )$. We show that at finite temperature, there is an additional single-magnon contribution to the scattering with a spectral weight proportional to $T^3$. We also compute the leading corrections to the UCL approximation and find them to be small, setting the UCL results on a solid basis. All this univocally points to the conclusion that the observed low temperature RIXS intensity in ${\mathrm{La}}_2\mathrm{Cu}{\mathrm{O}}_4$ is due to two-magnon scattering.

Figure 1

(Color online) In RIXS, a photon of momentum ${\mathbf{q}}_{\mathrm{in}}$ and energy tuned to the $K$ edge of a transition metal ion $({\omega }_{\mathrm{in}}^0={\omega }_{\mathrm{res}})$ creates a core hole at a certain site. The superexchange interaction between this site and a neighboring other site is modified because the energy of the virtual intermediate states is changed. The same site Coulomb repulsion $U$ is lowered by $U_c$ if the core-hole site contains no holes and is raised by $U_c$ if there are two holes present. Summing the amplitudes for both processes, we obtain the modified superexchange interaction [see Eq. (9)].

Figure 2

(Color online) Modification of the superexchange interaction in the well- and poorly screened intermediate states. In the poorly screened state, the core-hole potential $U_c$ modifies the superexchange. For the well-screened state, however, the copper $3d$ hole on the core-hole site is transfered to a neighboring oxygen, and superexchange is only of order $\mathcal{O}\left(t_{pd}^2\right)$, independent of $U_c$.

Figure 3

(Color online) (a) RIXS spectrum and (b) two-magnon DOS for a nearest neighbor Heisenberg antiferromagnet with exchange interaction $J$ as a function of transferred momentum $\mathbf{q}$ for a cut through the (c) Brillouin zone. The dashed line indicates the magnetic BZ boundary.

Figure 4

(a) First moment and (b) total spectral weight of the RIXS spectrum. The solid lines are obtained by using interaction strengths determined from neutron data (next neighbor coupling $J=146.3\mathrm{meV}$, second and third neighbor couplings $J^{\prime }=J^{″}=2\mathrm{meV}$, and ring exchange $J_c=61\mathrm{meV}$) (Ref. 23). The dashed lines have only nearest neighbor interaction.

Figure 5

RIXS intensity for various points in the BZ. Each figure contains the bare theoretical data (dashed line), the convolution with experimental resolution (solid line), and the experimental data from Ref. 17. For these figures, we used $J=146.3\mathrm{meV}$, second and third neighbor couplings $J^{\prime }=J^{″}=2\mathrm{meV}$, and ring exchange $J_c=61\mathrm{meV}$. The latter contribution is evaluated theoretically using a mean field approximation. These values were found in neutron scattering experiments (Ref. 23). These experiments were analyzed using the wave-vector independent renormalization factor $Z_c=1.18$, which is also used to generate the theoretical curves. The theoretical intensity is scaled independently in each figure to match the experiment. The overall scale factors differ at most by a factor 2.5, which is comparable to experimental uncertainty in absolute intensities (Ref. 38).

Figure 6

Comparison between spectral weight for single-magnon scattering $W_1$ (dashed lines) for various temperatures and zero temperature two-magnon scattering $W_2$ (solid line), all obtained numerically. The figure displays the $T^3$ behavior from Eq. (32) for the single-magnon intensity. For ${\mathrm{La}}_2\mathrm{Cu}{\mathrm{O}}_4$, $J\approx 146\mathrm{meV}$, and at room temperature, we have ${\beta }_{\mathrm{RT}}J\approx 5.8$.

Figure 7

(Color online) The leading order correction to the scattering amplitude does not interfere with the first order. (a) shows the contribution to the cross section from Eq. (36). The full, corrected cross section is shown in (b). There is an appreciable correction only at $\mathbf{q}=\mathbf{0}$.