Theory of tensorial magnetic inertia in terahertz spin dynamics

TL;DR

This work treats magnetic inertia as a tensor $\Delta$ and decomposes it into a scalar $\eta$, a symmetric part $\mathbb{I}$, and a chiral vector $\mathbf{C}$ within the inertial Landau-Lifshitz-Gilbert framework. Using linear response, the authors compute the frequency-dependent susceptibility for ferromagnets, antiferromagnets, and ferrimagnets and extract the precession and nutation resonance poles. They find that precession resonances are largely insensitive to the inertia tensor, while nutation resonances shift to lower frequencies and acquire enhanced damping when the chiral part is present; the symmetric inertia lowers nutation frequencies via $\eta^{\rm eff}=\eta+\mathbb{I}$, and AFM/FiM show additional exchange-enhanced effects. The results provide a tensorial-control knob for THz spin dynamics and predict measurable changes in nutation linewidths and the splitting of nutation modes in AFM/FiM.

Abstract

Magnetic inertia has emerged as a possible way to manipulate ferromagnetic spins at a higher frequency e.g., THz. Theoretical treatments so far have considered the magnetic inertia as a scalar quantity. Here, we explore the magnetic inertial dynamics with a magnetic inertia tensor as macroscopic derivations predicted it to be a tensor. First, the inertia tensor has been decomposed into three terms: (a) scalar and isotropic inertia, (b) anisotropic and symmetric inertia tensor, (c) chiral and antisymmetric tensor. Further, we employ linear response theory to the inertial Landau-Lifshitz-Gilbert equation with the inertia tensor and calculate the effect of chiral and anisotropic inertia on ferromagnets, antiferromagnets, and ferrimagnets. It is established that the precession and nutation resonance frequencies decrease with scalar magnetic inertia. Our results suggest that the nutation resonance frequencies further reduce due to inertia tensor. However, the effective damping of the nutation resonance increases with the chiral and antisymmetric part of the inertia tensor. We show that the precession resonances remain unaffected, while the nutation resonances are modified with the chiral magnetic inertia.

Figures (8)

• Figure 1: Dissipated power has been computed as a function of frequency at different values of inertial relaxation time. The considered parameters are $\gamma/2\pi$ = 28 GHz/T, $M_0 = 2\mu_B$, $\alpha = 0.05$, $K = 10^{-23}$ J, $H_0 = 0$ T.
• Figure 2: The variation of precession and nutation frequencies plotted against $\mathbb{I}_{xx}$. The considered parameters are $\gamma/2\pi$ = 28 GHz/T, $M_0 = 2\mu_B$, $\alpha = 0.05$, $K = 10^{-23}$ J, $H_0 = 0$ T.
• Figure 3: The variation of (a) precession and nutation frequencies and (b) effective damping plotted against $C$. The considered parameters are $\gamma/2\pi$ = 28 GHz/T, $M_0 = 2\mu_B$, $\alpha = 0.05$, $K = 10^{-23}$ J, $H_0 = 0$ T.
• Figure 4: The variation of (a) precession and nutation frequencies and (b) effective damping plotted against $\mathbb{I}_{xx}$. The solid lines represent the approximated expressions derived in Eqs. (\ref{['Eq:19']}) and (\ref{['Eq:20']}), whereas the symbols are calculated data points via the direct solutions of Eq. (\ref{['Eq:12']}). The chiral inertia has been set to $C = 0$. The considered parameters are $M_{0}$ = $2\mu_B$, $\gamma$ = 28 GHz/T, $\alpha$= 0.05, $J = 10^{-21}$ J, $K = 10^{-23}$ J, $D_z = 0$ J, $H_0 = 0$ T.
• Figure 5: The variation of (a) precession and nutation frequencies and (b) effective damping plotted against $C$. The solid lines are for the analytical expressions derived in Eqs. (\ref{['Eq:19']}) and (\ref{['Eq:20']}), where as the symbols are the data from the direct solutions of Eq. (\ref{['Eq:12']}). The considered parameters are $M_{0}$ = $2\mu_B$, $\gamma$ = 28 GHz/T, $\alpha$= 0.05, $J = 10^{-21}$ J, $K = 10^{-23}$ J, $D_z = 0$ J, $H_0 = 0$ T.