Role of intersublattice coupling in current-induced spin torque in Rashba antiferromagnet

We investigate the spin-orbit torque in Rashba antiferromagnet (AFM). The model AFM composes of two square sublattices with magnetizations $m_{1}$ and m
2
(Fig. 1)
^{[1, 2]}
. The intersublattice electron dynamics is described by the hopping energy t, and Rashba coupling $\alpha_{Ri}$ is sublattice-dependent. In linear response theory, the spin-torque acting on $m_{i}$ is derived as $T^{i}= \Delta J_{dd}$ $(m_{i} \times m_{jnei}) \quad + \quad m_{i} \times H_{i}$, where the first term represents the modified AFM exchange coupling between $m_{1}$ and $m_{2}$ with additional strength $\Delta J_{dd}$, the second term is the current induced torque with $H_{i}\alpha_{Ri}(z \times j_{e})$. Firstly, we show that $\Delta J_{dd}$ is induced by the intersubllatice electron dynamics. As shown in Fig. 2a, when the hopping is small, $\Delta J_{dd}$ increases with t and reaches its maximum $\Delta \mathrm{J}_{max}= 2 J_{sd}/ 3 \surd 3 at t=\surd 2 J_{sd}$(red circle), where $J_{sd}$ is the sd coupling. However, when $\mathrm{t}\ge \surd 2 \mathrm{J}_{sd}$, the modulation becomes diminishing. To explain this trend, we can consider the exchange time scale which is $\tau_{sd} \sim1/ \mathrm{J}_{sd}$, and the inter-sublattice hopping time scale $\tau _{hop} \sim1 /$t. When $\mathrm{t}< \mathrm{J}_{sd}$,i.e., $\tau _{sd} < \tau_{hop}$, adiabatic condition can be applied, which allows the angular momentum carried by the (a) Modified AFM coupling as a function of hopping energy t, the electron can be adiabatically transferred between m
1
to m
2
, and $\Delta \mathrm{J}_{dd}$ is scaled with the hopping rate, i.e., $\tau_{sd} > \tau _{hop}$, the angular momentum transfer would be less effective due to the $\Delta $J_{dd} \sim \mathrm{t}$. However, when $\mathrm{t}>$ Jsd, which means mred circle shows the maximum value. (b) Torque fields acting on, the magnitudes diminish at large t. 1 and relaxation, similar to strong spin-flip effect, thus resulting in the diminishing trend as shown. Second, the torque fields acting on m
1
and m
2
are staggered for $\alpha_{R1}=- \alpha_{R2}$ as shown in Fig. 2b. Moreover, the magnitude of the torque fields are reduced as a result of the intersublattice electron dynamics. Indeed, opposite Rashba fields induced opposite spin polarizations in each sublattices, and that can cancel each others through the intersublattice dynamics, thus reduces the torque fields.