Spin dynamics with inertia in metallic ferromagnets

The nonadiabatic contribution of environmental degrees of freedom yields an effective inertia of spin in the effective spin dynamics. In this paper, we study several aspects of the inertia of spin in metallic ferromagnets: (i) a concrete expression of the spin inertia $m_s$: $m_s=\hslash S_c/\left(2g_{\mathrm{sd}}\right)$, where $S_c$ is the spin polarization of conduction electrons and $g_{\mathrm{sd}}$ is the $sd$ coupling constant; (ii) a dynamical behavior of spin with inertia, discussed from the viewpoints of a spinning top and of a particle on a sphere; (iii) the behavior of spin waves and domain walls in the presence of inertia and the behavior of spin with inertia under a time-dependent magnetic field.

Figure 1

The dynamics of spin with inertia, a symmetric spinning top, and a massive charged particle on a sphere subject to a monopole magnetic field ${\mathcal{B}}_{\mathrm{m}}$ are classically equivalent. They undergo a precession motion accompanied by nutation under applied fields, which are a magnetic field $B$ for spin, a gravitational field $g$ for the top, and an electric field $\mathcal{E}$ for the particle, respectively. See the main text for details.

Figure 2

(a) Without any magnetic field, the general motion of spin with finite inertia is a free precession motion, just like a spinning top. The spin $S\mathit{n}$ precesses around the total angular momentum ${\mathit{j}}^0$. (b) When a constant magnetic field $\mathit{B}$ is applied, the total angular momentum ${\mathit{j}}^0$ precesses around $\mathit{B}$. Therefore the “free precession cone” in (a) precesses around $\mathit{B}$ as a whole, which corresponds to the Larmor precession in the absence of the inertia. What was called the free precession in (a) is now called the nutation.

Figure 3

(a) The trajectory of a spin, which fluctuates around $S_z=S$ point (this fluctuation propagates in space as a spin wave). In the figure, two fluctuation modes are superimposed. One is the usual gapless mode rotating clockwise seen from the positive $z$ direction (denoted by s1), while the other is a gapful mode rotating counterclockwise (denoted by s2). (b) The dispersion relations for the two modes of a spin wave. Besides the usual gapless spin wave mode (in green color and denoted by s1), a spin with inertia has a gapful spin wave mode (in red color and denoted by s2), which is the nutation of a spin propagating in space by the spin-spin exchange. See Eq. (30).

Figure 4

The collective coordinates $(X,\lambda {\varphi }_0)$ of a domain wall can be regarded as coordinates of a particle on a cylinder. This particle is subject to a magnetic field flux $\mathcal{B}=S{N}_{\mathrm{w}}/{\lambda }^2$ penetrating the cylinder perpendicularly. The inertia of spin introduces the mass $M_{\mathrm{w}}=m_s{N}_{\mathrm{w}}/{\lambda }^2$ into this particle.

Figure 5

Response of a spin with finite inertia to a time-dependent magnetic field $B\left(t\right)=B_{0}tanh\left(f(t-t_0)\right)$ in $z$ direction with $B_0=100\mathrm{THz}$ and $f=2\pi \mathrm{THz}=6.3\mathrm{THz}$. We take the spin amplitude as $S=\hslash$ and the inertia as $m_s/\hslash =0.1\mathrm{ps}/\left(2\pi \right)=0.016\mathrm{ps}$. (a) The time profiles of $S_z/S(=n_z)$ of the spin and of the applied magnetic field $B$. (b) The trajectory of the spin. The time is divided into three domains I–III. In I, the magnetic field points in the negative $z$ direction and the spin precesses counterclockwise seen from the positive $z$ direction. In II, the magnetic field switches to positive $z$ direction and the $z$ component of spin, $S_z/S$, also changes accordingly. In III, the magnetic field points in the positive $z$ direction and the spin precesses clockwise.

Figure 6

Trajectories of spin $S\mathit{n}$ and angular momentum ${\mathit{j}}^0=S\mathit{n}+m_s\mathit{n}\times \dot{\mathit{n}}$. (a) When $B\ll S/m_s$, spin nutation forms a circle shape. The additional component $m_s\mathit{n}\times \dot{\mathit{n}}$ points inward to the center of the nutation, and the trajectory of the angular momentum sits within that of spin. (b) When $B\sim S/m_s$, spin nutation forms a ripple shape. The additional component $m_s\mathit{n}\times \dot{\mathit{n}}$ points to one side of the spin trajectory. As a result, the trajectories of spin and the angular momentum deviate.

Figure 7

Replot of the trajectory of the spin in Fig. 5 (from slightly a different view angle), together with that of the angular momentum. The magnetic field is $B\sim 100\mathrm{THz}$ and $S/m_s\sim 20\pi \mathrm{THz}=62.8\mathrm{THz}$, corresponding to the case (b) in Fig. 6. Although the $z$-component of spin changes in time, that of the angular momentum is constant in motion. The axes are in unit of $S$.