Technology where a solitary dopant acts as the active component of an opto-electronic device is an emerging  field known as solotronics, and bears the promise to revolutionize the way in which information is stored, processed and transmitted. Magnetic doped semiconductors and in particular (Ga, Mn)As, the archetype of dilute magnetic semiconductors, and topological insulators (TIs), a new phase of quantum matter with unconventional characteristics, are two classes of quantum materials that have the potential to advance spin-electronics technology. The quest to understand and control, at the atomic level, how a few magnetic atoms precisely positioned in a complex environment respond to external stimuli, is the red thread that connects these two quantum materials in the research presented here.

The goal of the thesis is in part to elucidate the properties of transition metal (TM) impurities near the surface of GaAs semiconductors with focus on their response to local magnetic and electric fields, as well as to investigate the real-time dynamics of their localized spins. Our theoretical analysis, based on density functional theory (DFT) and using tight-binding (TB) models, addresses the mid-gap electronic structure, the local density of states (LDOS) and the magnetic anisotropy energy of individual Mn and Fe impurities near the (110) surface of GaAs. We investigate the effect of a magnetic field on the Mn acceptor LDOS measured in cross-sectional scanning tunneling microscopy, and provide an explanation of why the experimental LDOS images depend weakly on the field direction despite the strongly anisotropic nature of the Mn acceptor wavefunction. We also investigate the effects of a local electrostatic field generated by nearby charged As vacancies, on individual and pairs of ferromagnetically coupled magnetic dopants near the surface of GaAs, providing a means to control electrically the exchange interaction of Mn pairs. Finally, using the mixed quantum-classical scheme for spin dynamics, we calculate explicitly the time evolution of the Mn spin and its bound acceptor, and analyze the dynamic interaction between pairs of ferromagnetically coupled magnetic impurities in a nanoscaled semiconductor.

The second part of the thesis deals with the theoretical investigation of a single substitutional Mn impurity and its associated acceptor state on the (111) surface of Bi₂Se₃ TI, using an approach that combines DFT and TB calculations. Our analysis clarifies the crucial role played by the spatial overlap and the quasi-resonant coupling between the Mn-acceptor and the topological surface states inside the Bi₂Se₃ band gap, in the opening of a gap at the Dirac point. Strong electronic correlations are also found to contribute significantly to the mechanism leading to the gap, since they control the hybridization between the p orbitals of nearest-neighbor Se atoms and the acceptor spin-polarization. Our results explain the effects of inversion-symmetry and time-reversal symmetry breaking on the electronic states in the vicinity of the Dirac point, and contribute to clarifying the origin of surface-ferromagnetism in TIs. The promising potential of magnetic-doped TIs accentuates the importance of our contribution to the understanding of the interplay between magnetic order and topological protected surface states.