Interplay of Rashba and \textit{sp-d} exchange couplings in magnetic 2DEGs

In diluted magnetic semiconductor (DMS) quantum wells the \textit{sp-d} exchange interaction between the itinerant conduction electrons in the well and the localized electrons in the $d$ orbitals of the Mn impurities gives rise to interesting spin-dependent physics [1]. Recently, the interplay of the Rashba spin-orbit and the \textit{sp-d} exchange interactions in Mn-based wells has been recognized via Shubnikov-de-Haas measurements [2]. While the Rashba spin-orbit has been extensively studied in non-magnetic 2DEGs, its role in DMS systems with a competing \textit{sp-d} exchange interaction has not yet been addressed theoretically. In this work we present a\textbf{\textit{ k.p}} derivation of an effective Hamiltonian for a Mn-based quantum well with competing Rashba and \textit{sp-d} interactions, and show numerical results for the magnetoresistance $\rho _{xx}$ of typical magnetic 2DEGs using our effective Hamiltonian model. Our results shows interesting beating patterns of the $\rho _{xx}$ as a function of the temperature and carrier density which suggests a significant interplay between the spin-orbit and \textit{sp-d} exchange interactions, as a recent experiment observes [2]. [1] J. C. Egues, PRL \textbf{78}, 4578 (1998); H. J. P. Freire and J. C. Egues, cond-mat/0412491. [2] Y. S. Gui \textit{et al.} EPL. \textbf{65}, 393 (2004).