Locally induced, magnetic order on the surface of a topological insulator nanowire could enable room-temperature topological quantum devices. Here we report on the realization of selective magnetic control over topological surface states on a single facet of a rectangular Bi2Te3 nanowire via a magnetic insulating Fe3O4 substrate. Low-temperature magnetotransport studies provide evidence for local time-reversal symmetry breaking and for enhanced gapping of the interfacial 1D energy spectrum by perpendicular magnetic-field components, leaving the remaining nanowire facets unaffected. Our results open up great opportunities for development of dissipation-less electronics and spintronics.

We present a microscopic theory of the chiral one-dimensional electron gas system localized on the sidewalls of magnetically doped Bi2Se3-family topological insulator nanoribbons in the quantum anomalous Hall effect (QAHE) regime. Our theory is based on a simple continuum model of sidewall states whose parameters are extracted from detailed ribbon and film geometry tight-binding model calculations. In contrast to the familiar case of the quantum Hall effect in semiconductor quantum wells, the number of microscopic chiral channels depends simply and systematically on the ribbon thickness and on the position of the Fermi level within the surface state gap. We use our theory to interpret recent transport experiments that exhibit nonzero longitudinal resistance in samples with accurately quantized Hall conductances.

While spintronics often investigates striking collective spin e ects in large systems, a very important research direction deals with spin-dependent phenomena in nanostructures, reaching the extreme of a single spin conned in a quantum dot, in a molecule, or localized on an impurity or dopant. The issue considered in this chapter involves taking this extreme to the nanoscale and the quest to use rst-principles methods to predict and control the behavior of a few \spins" (down to 1 spin) when they are placed in an interesting environment. Particular interest is on environments for which addressing these systems with external elds and/or electric or spin currents is possible. The realization of such systems, including those that consist of a core of a few transition-metal (TM) atoms carrying a spin, connected and exchanged-coupled through bridging oxo-ligands has been due to work by many experimental researchers at the interface of atomic, molecular and condensed matter physics. This chapter addresses computational problems associated with understanding the behaviors of nanoand molecular-scale spin systems and reports on how the computational complexity increases when such systems are used for elements of electron transport devices. Especially for cases where these elements are attached to substrates with electronegativities that are very di erent than the molecule, or for coulomb blockade systems, or for cases where the spin-ordering within the molecules is weakly antiferromagnetic, the delocalization error in DFT is particularly problematic and one which requires solutions, such as self-interaction corrections, to move forward. We highlight the intersecting elds of spin-ordered nanoscale molecular magnets, electron transport, and coulomb blockade and highlight cases where self-interaction corrected methodologies can improve our predictive power in this emerging field.

We study theoretically the effects of short-range electron-electron interactions on the electronic structure of graphene, in the presence of substitutional impurities. Our computational approach is based on the π orbital tight-binding model for graphene, with the electron-electron interactions treated self-consistently at the level of the mean-field Hubbard model. The finite impurity concentration is modeled using the supercell approach. We compare explicitly noninteracting and interacting cases with varying interaction strength and impurity potential strength. We focus in particular on the interaction-induced modifications in the local density of states around the impurity, which is a quantity that can be directly probed by scanning tunneling spectroscopy of doped graphene. We find that the resonant character of the impurity states near the Fermi level is enhanced by the interactions. Furthermore, the size of the energy gap, which opens up at high-symmetry points of the Brillouin zone of the supercell upon doping, is significantly affected by the interactions. The details of this effect depend subtly on the supercell geometry. We use a perturbative model to explain these features and find quantitative agreement with numerical results.

The response of thin films of Bi$_2$Se$_3$ to a strong perpendicular magnetic field is investigated  by performing magnetic bandstructure calculations for a realistic multi-band tight-binding model.   Several crucial features of Landau quantization in a realistic three-dimensional topological insulator are revealed.  The $n=0$ Landau level is absent in ultra-thin  films, in agreement with experiment.  In films with a crossover thickness of five quintuple layers, there is     a signature of the $n=0$ level, whose overall trend as a function of magnetic field matches the established  low-energy effective-model result.  Importantly, we find a field-dependent splitting and a strong spin-polarization of the $n=0$ level which can be measured experimentally at reasonable field strengths. Our calculations      show  mixing between the surface and bulk Landau levels      which causes the character of levels to evolve with magnetic field.

Combining density functional theory calculations and microscopic tight-binding models, we investigate theoretically theelectronic and magnetic properties of individual substitutional transition-metal impurities (Mn and Fe) positioned in the vicinity of the (110) surface of GaAs. For the case of the [Mn2+](0) plus acceptor-hole (h) complex, the results of a tight-binding model including explicitly the impurity d electrons are in good agreement with approaches that treat the spin ofthe impurity as an effective classical vector. For the case of Fe, where both the neutral isoelectronic [Fe3+](0) and the ionized [Fe2+](-)states are relevant to address scanning tunneling microscopy (STM) experiments, the inclusion of d orbitals is essential. We find that the in-gap electronic structure of Fe impurities is significantly modified by surface effects. For the neutral acceptor state [Fe2+, h](0), the magnetic-anisotropy dependence on the impurity sublayer resembles the case of [Mn2+, h](0). In contrast, for [Fe3+](0) electronic configuration the magnetic anisotropy behaves differently and it is considerably smaller. For this state we predict that it is possible to manipulate the Fe moment, e. g., by an external magnetic field, with detectable consequences in the local density of states probed by STM.

We investigate the properties of a single substitutional Mn impurity and its associated acceptor state on the (111) surface of Bi$_2$Se$_3$ topological insulator. Combining \textit{ab initio} calculations with microscopic tight-binding modeling, we identify the effects of inversion-symmetry and time-reversal-symmetry breaking on the electronic states in the vicinity of the Dirac point. In agreement with experiments, we find evidence that the Mn ion is in ${+2}$ valence state and introduces an acceptor in the bulk band gap. The Mn-acceptor has predominantly $p$--character, and is localized mainly around the Mn impurity and its nearest-neighbor Se atoms. Its electronic structure and spin-polarization are determined by the hybridization between the Mn $d$--levels and the $p$--levels of surrounding Se atoms, which is strongly affected by electronic correlations at the Mn site. The opening of the gap at the Dirac point depends crucially on the quasi-resonant coupling and the strong real-space overlap between the spin-chiral surface states and the mid-gap spin-polarized Mn-acceptor states.

We have investigated the magnetoresistance (MR) in a series of Zn doped (p-type) GaAs nanowires implanted with different Mn concentrations. The nanowires with the lowest Mn concentration (~0.0001%) exhibit a low resistance of a few kΩ at 300K and a 4% positive MR at 1.6K, which can be well described by invoking a spin-split subband model. In contrast, nanowires with the highest Mn concentration (4%) display a large resistance of several MΩ at 300K and a large negative MR of 85% at 1.6K. The large negative MR is interpreted in terms of spin-dependent hopping in a complex magnetic nanowire landscape of magnetic polarons, separated by intermediate regions of Mn impurity spins. Sweeping the magnetic field back and forth for the 4% sample reveals a hysteresis that indicates the presence of a weak ferromagnetic phase. We propose co-doping with Zn to be a promising way to reach the goal of realizing ferromagnetic Ga_1-xMn_xAs nanowires for future nanospintronics.

We report on microscopic tight-binding modeling of surfacestates in Bi$_2$Se$_3$ three-dimensional topological insulator, based on a\textit{sp}$^3$ Slater-Koster Hamiltonian, with parameters calculated fromdensity functional theory. The effect of spin-orbit interaction on theelectronic structure of the bulk and of a slab with finite thickness isinvestigated. In particular, a phenomenological criterion of band inversion isformulated for both bulk and slab, based on the calculated atomic- andorbital-projections of the wavefunctions, associated with valence and conductionband extrema at the center of the Brillouin zone. We carry out athorough analysis of the calculated bandstructures of slabs with varyingthickness, where surface states are identified using a quantitative criterionaccording to their spatial distribution. The thickness-dependent energy gap,attributed to inter-surface interaction, and the emergence of gapless surfacestates for slabs above a critical thickness are investigated. We map out thetransition to the infinite-thickness limit by calculating explicitly themodifications in the spatial distribution and spin-character of the surfacestates wavefunction with increasing the slab thickness. Our numerical analysisshows that the system must be approximately forty quintuple-layers thick toexhibit completely decoupled surface states, localized on the oppositesurfaces. These results have implications on the effect of external perturbationson the surface states near the Dirac point.