Spin currents in Rashba altermagnets: From equilibrium to nonlinear regimes

We investigate equilibrium (background), linear, and nonlinear spin currents in two-dimensional Rashba spin-orbit coupled altermagnet systems, using a modified spin current operator that includes anomalous velocity arising from nonzero Berry curvature. The background spin current, stemming from spin-orbit coupling and modulated by the altermagnet term (${t}_{j}$), exhibits in-plane polarization, increases linearly with Fermi energy (${\ensuremath{\epsilon}}_{F}$), and is enhanced by both the altermagnet (${t}_{j}$) and the Rashba parameter ($\ensuremath{\lambda}$). Linear spin current is always transverse with out-of-plane polarization and can be viewed as a spin Hall current, primarily driven by band velocity, with ${t}_{j}$ enabling a band-induced contribution [previously absent in simple Rashba systems (${t}_{j}=0$)]. This highlights the altermagnet system as a promising source of spin Hall current generation. For linear spin Hall current, its band contribution's magnitude increases linearly with ${\ensuremath{\epsilon}}_{F}$, while the magnitude of the anomalous component saturates at higher ${\ensuremath{\epsilon}}_{F}$. Further, the magnitude of the spin Hall current is enhanced by ${t}_{j}$ but reduced by $\ensuremath{\lambda}$. Nonlinear spin currents feature both longitudinal and transverse components with in-plane polarization. Both the nonlinear longitudinal spin current from band velocity and the nonlinear transverse spin current from anomalous velocity initially decrease with ${\ensuremath{\epsilon}}_{F}$ before saturating at higher ${\ensuremath{\epsilon}}_{F}$. Importantly, ${t}_{j}$ reduces these currents while $\ensuremath{\lambda}$ enhances them. Meanwhile, the nonlinear transverse current from band velocity increases and then saturates with ${\ensuremath{\epsilon}}_{F}$, enhanced by $\ensuremath{\lambda}$ and showing nonmonotonic variation with ${t}_{j}$. These findings highlight the tunability of spin current behavior through Rashba and altermagnet parameters, offering insights for spintronic applications.