Gilbert damping in two-dimensional metallic antiferromagnets

A finite spin life-time of conduction electrons may dominate Gilbert damping of two-dimensional metallic antiferromagnets or antiferromagnet/metal heterostructures. We investigate the Gilbert damping tensor for a typical low-energy model of a metallic antiferromagnet system with honeycomb magnetic lattice and Rashba spin-orbit coupling for conduction electrons. We distinguish three regimes of spin relaxation: exchange-dominated relaxation for weak spin-orbit coupling strength, Elliot-Yafet relaxation for moderate spin-orbit coupling, and Dyakonov-Perel relaxation for strong spin-orbit coupling. We show, however, that the latter regime takes place only for the in-plane Gilbert damping component. We also show that anisotropy of Gilbert damping persists for any finite spin-orbit interaction strength provided we consider no spatial variation of the Néel vector. Isotropic Gilbert damping is restored only if the electron spin-orbit length is larger than the magnon wavelength. Our theory applies to ${\mathrm{MnPS}}_3$ monolayer on Pt or to similar systems.

Figure 1

Three antiferromagnetic phases commonly found among van der Waals magnets. Left to right: Néel, zig-zag, and stripy.

Figure 2

Gilbert damping in the limit ${\mathrm{\Delta }}_{\text{sd}}=0$. Dotted (green) lines correspond to the results of the numerical evaluation of ${\overline{\alpha }}_{m,\perp ,\parallel }^{\left(l\right)}$ for $l=0,1,2$ as a function of the parameter $\lambda \tau$. The dashed (orange) line corresponds to the diffusive (fully vertex corrected) results for ${\overline{\alpha }}_m^{\perp ,\parallel .}$.

Figure 3

Numerical results for the Gilbert damping components in the diffusive limit (vertex corrected) as the function of the spin-orbit coupling strength $\lambda$. The results correspond to $\varepsilon \tau =50$ and ${\mathrm{\Delta }}_{\text{sd}}\tau =0.1$ and agree with the asymptotic expressions of Eq. (2). Three different regimes can be distinguished for ${\overline{\alpha }}_m^{\parallel }$: (i) spin-orbit-independent damping ${\overline{\alpha }}_m^{\parallel }\propto {\varepsilon }^3\tau /{\mathrm{\Delta }}_{\text{sd}}^2$ for the exchange dominated regime, $\lambda \tau \ll {\mathrm{\Delta }}_{\text{sd}}/\varepsilon$, (ii) the damping ${\overline{\alpha }}_m^{\parallel }\propto \varepsilon /{\lambda }^2\tau$ for Elliot-Yafet relaxation regime, ${\mathrm{\Delta }}_{\text{sd}}/\varepsilon \ll \lambda \tau \ll 1$, and (iii) the damping ${\overline{\alpha }}_m^{\parallel }\propto \varepsilon \tau$ for the Dyakonov-Perel relaxation regime, $\lambda \tau \gg 1$. The latter regime is manifestly absent for ${\overline{\alpha }}_m^{\perp }$ in accordance with Eqs. (12) and (13).

Figure 4

Numerical evaluation of Gilbert damping anisotropy in the limit $\lambda \to 0$. Isotropic damping tensor is restored only if $\lambda =0$ with ultimate numerical precision. The factor of 2 anisotropy emerges at any finite $\lambda$, no matter how small it is, and only depends on the numerical precision $n$, i.e., the number of digits contained in each variable during computation. The crossover from isotropic to anisotropic damping can be understood only by considering finite, though vanishingly small, magnon $q$ vectors.

Figure 5

Band structure for the effective model of Eq. (5) in a vicinity of $\mathbf{K}$ valley assuming $n_z=1$. Electron bands touch for $\lambda =2{\mathrm{\Delta }}_{\text{sd}}$. The regime $\lambda \le 2{\mathrm{\Delta }}_{\text{sd}}$ corresponds to spin-orbit band inversion. The band structure in the valley ${\mathbf{K}}^{\prime }$ is inverted. Our microscopic analysis is performed in the electron-doped regime for the Fermi energy above the gap as illustrated by the top dashed line. The bottom dashed line denotes zero energy (half-filling).

Figure 6

Numerical evaluation of the inverse Gilbert damping $1/{\overline{\alpha }}_m^{\parallel }$ as a function of the momentum relaxation time $\tau$. The inverse damping is peaked at $\tau \propto 1/\lambda$, which also corresponds to the maximum of the antiferromagnetic resonance quality factor in accordance with Eq. (21).