Angular Momentum Transfer via Relativistic Spin-Lattice Coupling from First Principles

The transfer and control of angular momentum is a key aspect for spintronic applications. Only recently, it was shown that it is possible to transfer angular momentum from the spin system to the lattice on ultrashort timescales. To contribute to the understanding of angular momentum transfer between spin and lattice degrees of freedom we present a scheme to calculate fully relativistic spin-lattice coupling parameters from first principles. In addition to the dipole-dipole interactions often discussed in the literature, these parameters give, in particular, access to the spin-lattice effects controlled by spin-orbit coupling. By treating changes in the spin configuration and atomic positions at the same level, closed expressions for the atomic spin-lattice coupling parameters can be derived in a coherent manner up to any order. Analyzing the properties of these parameters, in particular their dependence on spin-orbit coupling, we find that even in bcc Fe the leading term for the angular momentum exchange between the spin system and the lattice is a Dzyaloshiskii-Moriya-type interaction, which is due to the symmetry breaking distortion of the lattice.

Figure 1

SLC parameters versus distortion for bcc Fe: Comparison of the products ${\mathcal{J}}_{ij,i}^{xy,x}·u_i^x$ (solid lines) with the corresponding SLC terms $\mathrm{\Delta }{J}_{ij}^{xy}(u_i^x)=J_{ij}^{xy}(u_i^x)-J_{ij}^{xy}(0)$ calculated for a distorted system with the supercell technique (dotted lines) for one atom $i$ displaced by $u_i^x$ along the $x$ axis. The inset shows the labeling of the nearest neighbor atoms used in the figure.

Figure 2

Magnitude of site-off-diagonal and site-diagonal SLC parameters: DMI $|{\overrightarrow{\mathcal{D}}}_{ij,j}^x|$ and isotropic SLC ${\mathcal{J}}_{ij,j}^{\mathrm{iso},x}$ (top), antisymmetric diagonal components ${\mathcal{J}}_{ij,j}^{\mathrm{dia}-\mathrm{a},x}$ and ${\mathcal{J}}_{ii,k}^{\mathrm{dia}-\mathrm{a},x}$ (middle), and symmetric off-diagonal components ${\mathcal{J}}_{ij,j}^{\mathrm{off}-\mathrm{s},x}$ and ${\mathcal{J}}_{ii,k}^{\mathrm{off}-\mathrm{s},x}$ (bottom).

Figure 3

(a) Longitudinal (top) and perpendicular (bottom) forces (with respect to ${\overrightarrow{r}}_{ij}$) emerging from the isotropic, DMI-type and MCA-type contributions to the SLC parameters. The central spin at $\overrightarrow{r}=(0,0,0)$ is tilted in the $y$ direction while all the others point in the $z$ direction [blue arrows in (b) and (c)]. (b) Directions of the forces with color coding indicating their perpendicular components with red being high and gray being low. (c) Directions of the perpendicular forces.

Figure 4

Imaginary part $\mathrm{Im}{\mathcal{D}}_{\overrightarrow{q}}^{x,\mu }$ (top) and $\mathrm{Im}(J_{\overrightarrow{q}}^{xy,\mu }+J_{\overrightarrow{q}}^{yx,\mu })/2$ (bottom) of the SLC parameters for bcc Fe, with $\mu =x$, $y$, $z$, plotted for $\overrightarrow{q}$ along high-symmetry lines of the Brillouin zone. The real part is zero in all cases.