Nutation in antiferromagnetic resonance

The effect of inertial spin dynamics is compared between ferromagnetic, antiferromagnetic, and ferrimagnetic systems. The linear response to an oscillating external magnetic field is calculated within the framework of the inertial Landau-Lifshitz-Gilbert equation using analytical theory and computer simulations. Precession and nutation resonance peaks are identified, and it is demonstrated that the precession frequencies are reduced by the spin inertia, while the lifetime of the excitations is enhanced. The interplay between precession and nutation is found to be the most prominent in antiferromagnets, where the timescale of the exchange-driven sublattice dynamics is comparable to inertial relaxation times. Consequently, antiferromagnetic resonance techniques should be better suited for the search for intrinsic inertial spin dynamics on ultrafast timescales than ferromagnetic resonance.

Figure 1

The rate of energy dissipation in the ferromagnet as a function of frequency for several values of the inertial relaxation time, (a) $\eta =1$ fs, (b) $\eta =10$ fs, (c) $\eta =100$ fs, and (d) $\eta =1$ ps. The lines denote the results of the analytical calculations and the symbols of the atomistic simulations for a single macrospin. All curves are compared to the analytical expression obtained without the inertial term. The other parameters are $\gamma =1.76\times {10}^{11}{\mathrm{T}}^{-1}{\mathrm{s}}^{-1}, M_0=2{\mu }_{\text{B}}, H_0=1$ T, $K={10}^{-23}$ J, $\alpha =0.05$, and $\left|h\right|=0.001$ T.

Figure 2

The rate of energy dissipation for the antiferromagnet as a function of frequency for several values of the inertial relaxation time ${\eta }_A={\eta }_B=\eta$, (a) $\eta =1$ fs, (b) $\eta =10$ fs, (c) $\eta =100$ fs, and (d) $\eta =1$ ps. The lines denote the results of the analytical calculations and the symbols of the atomistic spin simulations for two coupled macrospins. All curves are compared to the analytical expression obtained without the inertial term. The other parameters are $M_{A0}=M_{B0}=2{\mu }_{\text{B}}, {\gamma }_A={\gamma }_B=1.76\times {10}^{11}{\mathrm{T}}^{-1}{\mathrm{s}}^{-1}, {\alpha }_A={\alpha }_B=0.05, K_A=K_B={10}^{-23}$ J, $J={10}^{-21}$ J, $H_0=1$ T, and $|h_A|=|h_B|=0.001$ T. The insets show the precession resonances on a smaller frequency and power scale.

Figure 3

Real part of the precession resonance frequencies as a function of inertial relaxation time $\eta$, (a) for AFMs with $M_{A0}=M_{B0}=2{\mu }_{\text{B}}$ and (b) for FiMs with $M_{A0}=5M_{B0}=10{\mu }_{\text{B}}$. The other parameters are ${\gamma }_A={\gamma }_B=1.76\times {10}^{11}{\mathrm{T}}^{-1}{\mathrm{s}}^{-1}, {\alpha }_A={\alpha }_B=0.05, K_A=K_B={10}^{-23}$ J, $J={10}^{-21}$ J, $H_0=1$ T.

Figure 4

Effective damping parameters of the resonance modes as a function of inertial relaxation time $\eta$, for (a) AFMs with $M_{A0}=M_{B0}=2{\mu }_{\text{B}}$ and (b) FiMs with $M_{A0}=5M_{B0}=10{\mu }_{\text{B}}$. The other parameters are ${\gamma }_A={\gamma }_B=1.76\times {10}^{11}{\mathrm{T}}^{-1}{\mathrm{s}}^{-1}, {\alpha }_A={\alpha }_B=0.05, K_A=K_B={10}^{-23}$ J, $J={10}^{-21}$ J, $H_0=1$ T.