Fig. 3: Magnetophononic wave-mixing.

a Schematic of layered magnetic ordering in MnBi₂Te₄, with blue and purple denoting opposite in-plane spin orientations, and gray denoting disordered spins. The antiferromagnetic (AFM) wavevector is shown schematically, labeled “q_AFM”. b The dispersion relations of the A_1g⁽¹⁾ and A_1g⁽²⁾ modes along the c-axis, calculated using density functional theory. The experimental zone-center and zone-boundary phonon frequencies are denoted using colored and empty circles respectively. c Schematic of wave-mixing for zone-center and zone-boundary modes. The wavevectors of the photon (i = incident, r = reflected), phonon, and AFM spin-wave are shown using gray, green, and purple arrows (not drawn to scale). d Modulation of the interlayer exchange coupling \({J}^{\perp }\) by Raman phonons. Inset shows the eigendisplacements of two representative phonons. e Comparison between the calculated magnetophononic scattering cross-section \(\left|\frac{d{J}^{\perp }}{{du}}\right|\) and the experimental zone-boundary ratio σ (see text for definition). Error bars are standard deviations in fit values. f Schematic of superexchange (SE) and super-superexchange (SSE), with Δ denoting the interlayer distance, θ denoting the Mn–Te–Mn bond angle, and pink and blue clouds denoting SE and SSE pathways, respectively.