Rotationally invariant formulation of spin-lattice coupling in multiscale modeling

In the spirit of multiscale modeling, we develop a theoretical framework for spin-lattice coupling that connects, on the one hand, to ab initio calculations of spin-lattice coupling parameters and, on the other hand, to the magnetoelastic continuum theory. The derived Hamiltonian describes a closed system of spin and lattice degrees of freedom and explicitly conserves the total momentum, angular momentum, and energy. Using a numerical implementation that corrects earlier Suzuki-Trotter decompositions we perform simulations on the basis of the resulting equations of motion to investigate the combined magnetic and mechanical motion of a ferromagnetic nanoparticle, thereby validating our developed method. In addition to the ferromagnetic resonance mode of the spin system, we find another low-frequency mechanical response and a rotation of the particle according to the Einstein–de Haas effect. The framework developed herein will enable the use of multiscale modeling for investigating and understanding a broad range of magnetomechanical phenomena from slow to ultrafast timescales.

Figure 1

Rotation and translation of a magnetized sample. The reference configuration at $t=0$ is denoted by ${\mathit{R}}_i$ (left), and for $t>0$ by ${\mathit{r}}_i$ (right). During its motion, the easy axis for an atom $i$ at position ${\mathit{r}}_i$ or ${\mathit{R}}_i$ can be defined via its upper and lower neighbors at position $r_i^{z\pm }$.

Figure 2

Coherent magnetization dynamics of a free cubic nanoparticle obtained from SLD in comparison with SD simulations (light curves) for toy model parameters. Top: Magnetization vector components $m_{\alpha }$ vs time. Bottom: Fourier transform of $m_y$.

Figure 3

Mechanical motion of a cubic nanoparticle excited by coherent precession of the magnetization around ${\mathit{n}}_3$ (easy axis). (a) Spiraling magnetization dynamics and easy-axis precession in the time interval $[0,{10}^3]{\mu }_{\mathrm{s}}/\gamma J$ for toy model parameters. (b) Sketch of the two characteristic mechanical modes. (c) Characteristic frequencies of an FePt nanoparticle vs cube size $l$. Dotted lines correspond to theory curves for ${\omega }_{\mathrm{FMR}}$ and ${\omega }_{{\mathit{n}}_3}$ as explained in the text and a guide to the eye for ${\omega }_{\mathrm{EdH}}$.