Role of material-dependent properties in THz field-derivative-torque-induced nonlinear magnetization dynamics

TL;DR

The paper addresses ultrafast THz-spin dynamics in ferrimagnets and argues that the relativistic field-derivative torque (FDT) must be included alongside the Landau-Lifshitz-Gilbert (LLG) equation to capture nonlinear responses. It develops a two-sublattice ferrimagnet model where the effective field is modified by FDT through $\mathbf{B}^{\rm eff}_i \rightarrow \mathbf{B}^{\rm eff}_i - \frac{\alpha_i a_i^3}{\gamma_i \mu_{\rm B}} \frac{d H_{\rm THz}}{dt}$ and solves coupled LLG equations with a free-energy density containing exchange $\lambda$, uniaxial anisotropies $K_{\rm Fe}$ and $K_{\rm RE}$, Zeeman, and demagnetization terms. Applied to $\mathrm{Gd}_{3/2}\mathrm{Yb}_{1/2}\mathrm{BiFe}_5\mathrm{O}_{12}$, the study shows that FDT enhances THz magnon amplitudes and induces phase shifts that depend on the sublattice volume ratio $a^3_{\rm Fe}/a^3_{\rm RE}$, with near-equality yielding a significant phase and other ratios altering it. In dual THz-pulse experiments, FDT enables clearly identifiable nonlinear signals in 2D spectra, aligning with prior observations and revealing a damping-dependent threshold for nonlinearity. Overall, the work demonstrates that FDT is a crucial ingredient for accurately modeling and controlling ultrafast ferrimagnetic dynamics at THz frequencies, offering a bridge between theory and experiment and guiding future spintronic applications.

Abstract

The traditional Landau-Lifshitz-Gilbert (LLG) equation has often delineated the linear and nonlinear magnetization dynamics, even at ultrashort timescales e.g., femtoseconds. In contrast, several other non-relativistic and relativistic spin torques have been reported as an extension of the LLG spin dynamics. Here, we explore the contribution of the relativistic field-derivative torque (FDT) in the nonlinear THz magnetization dynamics response applied to ferrimagnets with high Gilbert damping and exchange magnon frequency. Our findings suggest that the FDT plays a significant role in magnetization dynamics in both linear and nonlinear regimes, bridging the gap between the traditional LLG spin dynamics and experimental observations. We find that the coherent THz magnon excitation amplitude is enhanced with the field-derivative torque. Furthermore, a phase shift in the magnon oscillation is induced by the FDT term. This phase shift is almost 90 for the antiferromagnet, while it is almost zero for the ferrimagnet under our investigation. Analyzing the dual THz excitation and their FDT, we find that the nonlinear signals can not be distinctly observed without the FDT terms. However, the inclusion of the FDT terms produces distinct nonlinear signals which matches extremely well with the previously reported experimental results.

Figures (6)

• Figure 1: (a) Modelled effective field representing the THz-induced ZT and the total field incorporating FDT along with ZT. (b) The normalized spectra obtained by performing a fast Fourier transform of the fields in (a). The spectra show a clear shift in the frequency with the incorporation of FDT. (c) The exchange dynamics between the rare earth and the iron moments with and without the FDT. (d) The $x$- and (e) the $z$-components of the individual sublattice magnetization dynamics in the presence of FDT.
• Figure 2: (a) THz excitation of the ferrimagnetic system in the absence and in the presence of field derivative torque (FDT) at different volume ratio of the iron and rare-earth sublattices. (b) The zoomed in part of (a) representing the difference in spin dynamics with and without FDT at $\rm a^{3}_{\rm Fe}=\rm a^{3}_{\rm RE}$. (c) The variation of the maxima of spectral amplitude obtained by the Fourier transforming the spectra in (a). (d) The difference in the phase of the magnon oscillations in (a) upon the incorporation of FDT when compared to the scenario without FDT.
• Figure 3: Exchange oscillations triggered by two THz pulses at three different delay times, 0 ps, 1.5 ps, and 9 ps. The exchange dynamics in the presence and absence of FDT are shown by the solid and dashed lines, respectively.
• Figure 4: Normalized contour plot of 2D Fourier transformed spectra of the exchange nonlinear signal as a function of detection frequency and excitation frequency. (a) and (b) represent the spectra obtained by performing the numerical simulation without and with incorporating the FDT term, respectively. The spectra in (b) appear to have two intense pump-probe ($\rm A_{pu}-\rm B_{pr}$ and $\rm B_{pu}-\rm A_{pr}$) and two echo (ABB and BAA) signals. The green and red arrows indicate the frequency vectors corresponding to THz pulses A and B.
• Figure 5: (a-d) The exchange nonlinear spectra of the Kaplan-Kittel mode for different orders of damping values in the ferrimagnetic system. (e) The maximum of the nonlinear signal, $\textbf{B}_{\rm NL}^{\rm max}$ showing a non-monotonic increase with the increase of the Gilbert damping parameter. The red point shows the result that corresponds to our earlier experimental condition dutta2024experimentalobservationrelativisticfieldderivative.