Current-Induced Spin-Wave Doppler Shift in Antiferromagnets

We theoretically study the spin dynamics in antiferromagnets (AFs)under the influence of an electric current. We identify two different sources of spin-transfer torques that stem from uniform (${\boldsymbol v}_n$) and staggered (${\boldsymbol v}_\ell$) electron spin densities. While the former is well recognized, the latter is often overlooked. We show that both ${\boldsymbol v}_n$ and ${\boldsymbol v}_\ell$ contribute equally to the spin-wave Doppler shift. Microscopic calculations are presented for electrons on a two-dimensional square lattice with nearest-neighbor ($t$) and next-nearest-neighbor ($t'$) hopping, which interpolate two opposite transport regimes of strongly-coupled AF ($t'/t \ll 1$) and two weakly coupled ferromagnets ($t'/t \gg 1$). In the AF transport regime ($t'/t \ll 1$), ${\boldsymbol v}_n$ and ${\boldsymbol v}_\ell$ have opposite signs, and the sign of the Doppler shift depends on band filling; ${\boldsymbol v}_n$ (${\boldsymbol v}_\ell$) is dominant near the AF gap (near the band bottom or the top). As $t'/t$ is increased, ${\boldsymbol v}_n$ undergoes a sign change whereas ${\boldsymbol v}_\ell$ does not. In the limit of vanishing $t$, ${\boldsymbol v}_n$ and ${\boldsymbol v}_\ell$ coincide and the spin-transfer torque reduces to that of ferromagnets.