Torque equilibrium spin wave theory of Raman scattering in an anisotropic triangular lattice antiferromagnet with Dzyaloshinskii-Moriya interaction

We apply torque equilibrium spin wave theory (TESWT) to investigate an anisotropic XXZ antiferromagnetic model with Dzyaloshinskii-Moriya interaction in a triangular lattice. Considering the quasiparticle vacuum as our reference, we provide an accurate analysis of the noncollinear ground state of a frustrated triangular lattice magnet using the TESWT formalism. We elucidate the effects of quantum fluctuations on the ordering wave vector based on model system parameters. We study the single-magnon dispersion, the two-magnon continuum using the spectral function, and the Raman spectrum of bimagnon and trimagnon excitations. We present our results for the $HH,VV$, and the $HV$ polarization Raman geometry dependence of the bimagnon and trimagnon excitation spectra where $H\left(V\right)$ represents horizontal (vertical) polarization. Our calculations show that both the $HH$ and the $HV$ polarization spectra can be used to determine the degree of anisotropy of our system. We calculate the Raman spectra of ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$ and ${\mathrm{Cs}}_2{\mathrm{CuCl}}_4$.

Figure 1

(a) Triangular lattice with anisotropic exchange constants $J$ and $J^{\prime }$ acting along the bonds. (b) Experimental geometry setup to study polarization effects within Raman scattering. ${\mathit{\delta }}_{1,2}$ denotes lattice vectors. $H$ (horizontal) and $V$ (vertical) indicates polarization direction of the incoming and outgoing light.

Figure 2

Ordering wave vector (a) $Q/\pi$ and (b) $(Q-Q_{cl})/\pi$ versus $J^{\prime }$ for spin-$\frac{1}{2}$ system. The black, blue, and red lines show results for parameters $(D,\mathrm{\Delta })$ equal to (0,1), (0.05,1), and (0,0.95), respectively.

Figure 3

Momentum and energy dependence of the spectral function $A(\mathbf{k},\omega )$ for a spin-$\frac{1}{2}$ system. The points in the chosen path are defined as $\mathrm{\Gamma }=(0,0), K=(4\pi /3,0), K^{\prime }=(2\pi /3,2\pi /\sqrt{3})$, and $M=(0,2\pi /\sqrt{3})$. The parameters $(J^{\prime },D,\mathrm{\Delta })$ are (a) (0.5,0,1), (b) (0.5,0.05,1), (c) (0.5,0,0.95), and (d) (1.1,0,1). The solid red lines show the on-shell dispersion ${\omega }_{\mathbf{k}}$ [see Eq. (9)].

Figure 4

Noninteracting bimagnon (solid lines) and trimagnon (dashed lines) Raman spectra of spin-$\frac{1}{2}$ system in (a) $HH$ and (b) $HV$ polarizations. DM interaction and XXZ anisotropy are absent in the system. The red and black lines are calculated with $J^{\prime }=1$ under $C_{3v}$ symmetry. The blue and green lines are calculated with $J^{\prime }=0.5$ under $C_{2v}$ symmetry.

Figure 5

Interacting Raman bimagnon spectra of spin-$\frac{1}{2}$ systems in $HV$ polarization with different parameters $(J^{\prime },D,\mathrm{\Delta },S)$.

Figure 6

Interacting bimagnon and noninteracting trimagnon Raman spectra of ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$ and ${\mathrm{Cs}}_2{\mathrm{CuCl}}_4$. The left (right) column is under $HH$ ($HV$) polarization. The first line is for ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$ with $(J^{\prime },D,\mathrm{\Delta })=(1,0,0.954)$ [50]. The second line is for ${\mathrm{Cs}}_2{\mathrm{CuCl}}_4$ with $(J^{\prime },D,\mathrm{\Delta })=(0.316,0.025,1)$ [47].