invariant under inversion of both magnetic field and wavevector [@DAmico2019]. Observation of chiral spin waves in DMS QWs is possible when Rashba SO coupling is of the order of spin-splitting energy. This condition may be fulfilled by adjusting the QW width, carrier density, and $x_{\text{eff}}$. Indeed, under this condition, chiral spin waves were observed in CdMnTe QWs [@Perez2016]. It was found that the dispersions of energy and damping rate are both shifted by a wave vector [**q**]{}$_0$, which depends on the relative strength of the SO field. Thus, a change of a SO coupling strength (e.g., through gating or PPC effect) can alter the group velocity of spin waves in a system. Control over the propagation direction of spin waves can find its application in magnonic devices. Spin waves observed in CdMnTe QWs were also found to be a perspective source of spin-based THz radiation [@Rungsawang2013].

In Refs. [@Korenev2015; @Akimov2017] it was found that interplay between localized moments, phonons, and carriers can even lead to long-range, ferromagnet-induced proximity effect in a ferromagnet-semiconductor hybrid system. In the studied system, a cobalt layer was deposited on the cap layer of CdTe/CdMgTe QW which did not contain any magnetic doping and spin polarization 