Spin-wave approach to the two-magnon Raman scattering in a $J_{1x}\text{−}{J}_{1y}\text{−}{J}_2\text{−}{J}_c$ antiferromagnetic Heisenberg model

We study the two-magnon nonresonant Raman scattering in the $(\pi ,\pi )$ and $(\pi ,0)$ in-plane ordered antiferromagnetic phases of a spin-$S{J}_{1x}\text{−}{J}_{1y}\text{−}{J}_2\text{−}{J}_c$ Heisenberg model on the tetragonal lattice within the framework of the spin-wave theory. We show that magnon-magnon interactions may significantly change the shape of spectra even in large-$S$ cases. We discuss the effects of various tuning factors to the two-magnon Raman spectra. We find that both the magnetic frustration $J_2/J_1$ and the interlayer exchange coupling $J_c$ may significantly affect the spectra in both the $B_{1g}$ and the $A_{1g}^{\prime }$ channels in the $(\pi ,\pi )$ Néel ordered phase. Moreover, we find a splitting of the two-magnon peak in the $(\pi ,0)$ collinear antiferromagnetic phase. We further discuss the implications of our results to the ${\mathrm{BaMnBi}}_2$ and iron pnictide systems.

Figure 1

Ground-state phase diagram of the classical $J_{1x}\text{−}{J}_{1y}\text{−}{J}_2$ model in the regime $J_{1x}>0,J_{1y}/J_{1x}\ge -1$ and $J_{1y}+2J_2\ge 0$, where the ground state stabilizes in-plane AFM orders with wave vectors $(\pi ,0),(0,\pi )$, and $(\pi ,\pi )$, respectively. The solid lines mark the phase boundaries.

Figure 2

Ladder diagrams for two-magnon propagator (15). A line with a single arrow represents $\alpha$ magnon propagator $G_{\alpha \alpha }(\mathbf{k},\omega )$. A line with a double arrow represents $\beta$ magnon propagator $G_{\beta \beta }(\mathbf{k},\omega )$.

Figure 3

Two-magnon Raman spectra for the $J_1\text{−}{J}_2$ model with $S{J}_1=1,S{J}_2=0.3$. (a), (b) Spectra for $S=1$ highlighting the effects of Oguchi and m-m interaction terms. (c) Dependence of $S$ in $B_{1g}$ (solid lines) and $A_{1g}^{\prime }$ (dashed lines) channels, both are calculated with the Oguchi and m-m interaction terms taken into account.

Figure 4

(a), (b) Two-magnon Raman spectra for the $J_1\text{−}{J}_2$ model $S{J}_1=1,S=2$ with different $J_2$; (c) magnon dispersion and (d) one-magnon DoS for $S{J}_1=1S{J}_2=\pm 0.2,S=2$.

Figure 5

Two-magnon Raman spectra for frustrated anisotropic $J_{1x}\text{−}{J}_{1y}\text{−}{J}_2$ model with $S{J}_{1x}=1,S{J}_{1y}=0.9,S{J}_2=0.3,S=2$. (b) Reference isotropic model with the same $S,J_{1x}$, and $J_2$.

Figure 6

$J_c$ dependence of two-magnon Raman spectra for frustrated $J_1\text{−}{J}_2\text{−}{J}_c$ model $S{J}_1=1,S{J}_2=0.3,S=2$. Solid lines represent $B_{1g}$ channel, and dashed lines represent $A_{1g}^{\prime }$ channel.

Figure 7

$J_{1y}$ dependence of two-magnon Raman spectra for anisotropic $J_{1x}\text{−}{J}_{1y}\text{−}{J}_2$ model with $S{J}_{1x}=1,S{J}_2=0.4,S=1$.

Figure 8

$S$ dependence of two-magnon Raman spectra for an anisotropic $J_{1x}\text{−}{J}_{1y}\text{−}{J}_2$ model. (a), (b) Unfrustrated model $S{J}_{1x}=1,S{J}_{1y}=-0.1,S{J}_2=0.4$; (c), (d) frustrated model $S{J}_{1x}=1,S{J}_{1y}=0.6,S{J}_2=0.4$.

Figure 9

Measured and calculated two-magnon Raman spectra for ${\mathrm{BaMn}}_2{\mathrm{Bi}}_2$. The measured two-magnon Raman spectra at 10 K in the $B_{1g}$ and $A_{1g}^{\prime }$ channels (black curves) are compared to the calculated ones with exchange parameters from an inelastic neutron scattering measurement (“Para1,” red curves) and with further modified exchange parameters (“Para2,” blue curves), respectively. The magnon-magnon interaction has been taken into account in the calculation.

Figure 10

Comparison of theoretical (ED [31] and spin-wave approaches) two-magnon Raman spectra for ${\mathrm{CaFe}}_2{\mathrm{As}}_2$. In calculations we have taken the anisotropic $J_{1x}\text{−}{J}_{1y}\text{−}{J}_2$ model with $S{J}_{1x}=1,S{J}_{1y}=-0.1,S{J}_2=0.4,S=1/2$. (a) $B_{1g}$ channel; (b) $A_{1g}^{\prime }$ channel.