RKKY interaction mediated by a spin-polarized 2D electron gas with Rashba and altermagnetic coupling

Magnetic interactions between impurity spins play a crucial role in determining magnetic configurations in spintronic systems. Using a Green's function formalism, we investigate Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interactions between two localized spins mediated by a two-dimensional electron gas arising from two spin-polarized bands with Rashba spin-orbit coupling (RSOC) and altermagnetic dispersion. We analyze two distinct regimes: (a) strong out-of-plane ferromagnetic order, and (b) weak in-plane order. For out-of-plane magnetization, the Heisenberg, Ising, and Dzyaloshinskii-Moriya (DM) exchange interaction terms exhibit oscillatory spatial modulations and asymptotically decay as $1/R^{2}$ with impurity separation. Besides, all exchange terms display beating-like patterns that can be tuned via the exchange coupling strength between conduction electrons and ferromagnetic ordering. The DM vector lies within the two-dimensional plane, with the DM interaction being odd in the RSOC strength, oscillatory, and increasing in magnitude with RSOC strength. In contrast, it is even in the out-of-plane Zeeman field strength and oscillatory. Furthermore, the Heisenberg interaction exhibits a non-linear dependence on the altermagnetic band parameter. In the case of weak in-plane order, the Heisenberg exchange interaction shows a non-linear dependence on the in-plane exchange coupling strength.