Magnetization dynamics in layered systems with coexisting bilinear and biquadratic interlayer exchange coupling

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Important aspects of exchange-coupled magnetic layered structures are related to the noncollinear arrangement of sublayer magnetizations which can arise from competition between bilinear (BL) and biquadratic (BQ) interlayer exchange coupling (IEC). In this work, the influence of coexisting BL and BQ IEC of different strengths on magnetization precession in layered systems is investigated both experimentally and theoretically. Laser-induced magnetization precession has been studied in the $\text{Fe/Si}({d}_{\text{Si}}$) multilayers (MLS) as a function of the amplitude ($H$) and orientation angle (${\ensuremath{\theta}}_{H}$) of external magnetic field using time-resolved magneto-optical Kerr (TRMOKE) effect. Strongly changing characters of precession frequency dependencies $\ensuremath{\omega}(H,{\ensuremath{\theta}}_{H})$ for Fe sublayer thickness ${d}_{\text{Fe}}=3$ nm and Si spacer-layer thicknesses (${d}_{\text{Si}}$) varying in the range of 0.9--2.4 nm have been observed. Analytical formulas for acoustic and optic mode dispersion relations with coexisting BL and BQ IEC, scaled by ${J}_{1}$ and ${J}_{2}$ parameters, respectively, for the in-plane effective magnetic anisotropy and arbitrary magnetic field direction were derived, and very good agreement with the experimentally observed frequency dependencies has been obtained. It is shown that BQ coexisting with BL IEC significantly influences on the magnitude and form of dispersion relations. From analytical formula derived, it follows that zero-field optical mode frequency tends to zero as $|{J}_{1}|$ approaches $2{J}_{2}$. The acoustic and optic mode-crossing effect has been observed and it is found that values of crossing fields and frequency gaps strongly increase as ${\ensuremath{\theta}}_{H}$ angles decrease and depend on relative BL and BQ IEC strengths. The BL IEC is of ferromagnetic type with ${J}_{1}\ensuremath{\approx}1.6$ mJ/${\mathrm{m}}^{2}$ for the MLS with ${d}_{\text{Si}}=0.9$ nm, and changes to antiferromagnetic one with ${J}_{1}\ensuremath{\approx}\ensuremath{-}0.9$ mJ/${\mathrm{m}}^{2}$ for the MLS with ${d}_{\text{Si}}=1.4$ nm, while the ${J}_{2}$ parameter of BQ IEC decreases from 1.8 to 1.0 mJ/${\mathrm{m}}^{2}$. The coupling strengths decrease by one to two orders of magnitude for the sample with ${d}_{\text{Si}}=2.4$ nm, but both mode frequencies are still observed and well reproduced by the theory. It is shown that ${J}_{1}$ and ${J}_{2}$ parameters obtained in the TRMOKE experiment coincide within the estimated error bars with the determined from independent measurements of magnetization processes in the static magneto-optical Kerr effect and interpreted with the use of analytical formulas derived. Numerical solutions of coupled Landau-Lifshitz-Gilbert (LLG) equations for acoustic and optic modes, with inclusion of BL IEC, intrinsic Gilbert damping, and spin-pumping damping terms, and extended to include BQ IEC, were performed and fitted to experimental data. It is shown that determined effective damping coefficients on $H$ and ${\ensuremath{\theta}}_{H}$ dependencies for acoustic and optic modes are very well simulated with the use of LLG equation solutions with Gilbert damping, spin-pumping-damping--related effective spin-mixing conductance, and spin-diffusion length parameters included. The dependencies of the parameters on ${d}_{\text{Si}}$ spacer-layer thickness are discussed and compared with available data for other systems.