  A finite spin life-time of conduction electrons may dominate Gilbert damping
of two-dimensional metallic anti-ferromagnets or anti-ferromagnet/metal
heterostructures. We investigate the Gilbert damping tensor for a typical
low-energy model of a metallic anti-ferromagnet 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 N\'eel vector. Isotropic Gilbert damping is restored only if
the electron spin-orbit length is larger than the magnon wavelength. Our theory
applies to MnPS3 monolayer on Pt or to similar systems.
