Abstract:
We found that non-magnetic defects in two-dimensional topological insulators induce bound states of two kinds for each spin orientation: electron- and hole-like states. Depending on the sign of the defect potential these states can be also of two kinds with different distribution of the electron density. The density has a maximum or minimum in the center. A surprising effect caused by the topological order is a singular dependence of the bound-state energy on the defect potential.

Abstract:
Non-invasive local probes are needed to characterize bulk defects in binary and ternary chalcogenides. These defects contribute to the non-ideal behavior of topological insulators. We have studied bulk electronic properties via $^{125}$Te NMR in Bi$_2$Te$_3$, Sb$_2$Te$_3$, Bi$_{0.5}$Sb$_{1.5}$Te$_3$, Bi$_2$Te$_2$Se and Bi$_2$Te$_2$S. A distribution of defects gives rise to asymmetry in the powder lineshapes. We show how the Knight shift, line shape and spin-lattice relaxation report on carrier density, spin-orbit coupling and phase separation in the bulk. The present study confirms that the ordered ternary compound Bi$_2$Te$_2$Se is the best TI candidate material at the present time. Our results, which are in good agreement with transport and ARPES studies, help establish the NMR probe as a valuable method to characterize the bulk properties of these materials.

Abstract:
Controlling the flow of spin and charge currents in topological insulators (TIs) is a crucial requirement for applications in quantum computation and spin electronics. We demonstrate that such control can be established in nanoscopic two-dimensional TIs by breaking their time reversal symmetry via magnetic defects. This allows for the creation of nearly fully spin-polarized charge currents, and the design of highly tunable spin diodes. Similar effects can also be realized in mesoscale hybrid structures in which TIs interface with ferro- or antiferromagnets.

Abstract:
The electronic transport experiments on topological insulators exhibit a dilemma. A negative cusp in magnetoconductivity is widely believed as a quantum transport signature of the topological surface states, which are immune from localization and exhibit the weak antilocalization. However, the measured conductivity drops logarithmically when lowering temperature, showing a typical feature of the weak localization as in ordinary disordered metals. Here, we present a conductivity formula for massless and massive Dirac fermions as a function of magnetic field and temperature, by taking into account the electron-electron interaction and quantum interference simultaneously. The formula reconciles the dilemma by explicitly clarifying that the temperature dependence of the conductivity is dominated by the interaction while the magnetoconductivity is mainly contributed by the quantum interference. The theory paves the road to quantitatively study the transport in topological insulators and other two-dimensional Dirac-like systems, such as graphene, transition metal dichalcogenides, and silicene.

Abstract:
The study of the propagation of electrons with a varying spinor orientability is performed using the coordinate transformation method. Topological Insulators are characterized by an odd number of changes of the orientability in the Brillouin zone. For defects the change in orientability takes place for closed orbits in real space. Both cases are characterized by nontrivial spin connections. Using this method , we derive the form of the spin connections for topological defects in three dimensional Topological Insulators. On the surface of a Topological Insulator, the presence an edge dislocation gives rise to a spin connection controlled by torsion. We find that electrons propagate along two dimensional regions and confined circular contours. We compute for the edge dislocations the tunneling density of states. The edge dislocations violates parity symmetry resulting in a current measured by the in-plane component of the spin on the surface.

Abstract:
We develop a unified framework to classify topological defects in insulators and superconductors described by spatially modulated Bloch and Bogoliubov de Gennes Hamiltonians. We consider Hamiltonians H(k,r) that vary slowly with adiabatic parameters r surrounding the defect and belong to any of the ten symmetry classes defined by time reversal symmetry and particle-hole symmetry. The topological classes for such defects are identified, and explicit formulas for the topological invariants are presented. We introduce a generalization of the bulk-boundary correspondence that relates the topological classes to defect Hamiltonians to the presence of protected gapless modes at the defect. Many examples of line and point defects in three dimensional systems will be discussed. These can host one dimensional chiral Dirac fermions, helical Dirac fermions, chiral Majorana fermions and helical Majorana fermions, as well as zero dimensional chiral and Majorana zero modes. This approach can also be used to classify temporal pumping cycles, such as the Thouless charge pump, as well as a fermion parity pump, which is related to the Ising non-Abelian statistics of defects that support Majorana zero modes.

Abstract:
Topological superconductors may undergo transitions between phases with different topological numbers which, like the case of topological insulators, are related to the presence of gapless (Majorana) edge states. In $\mathbb{Z}$ topological insulators the charge Hall conductivity is quantized, being proportional to the number of gapless states running at the edge. In a superconductor, however, charge is not conserved and, therefore, $\sigma_{xy}$ is not quantized, even in the case of a $\mathbb{Z}$ topological superconductor. Here it is shown that while the $\sigma_{xy}$ evolves continuously between different topological phases of a $\mathbb{Z}$ topological superconductor, its derivatives display sharp features signaling the topological transitions. We consider in detail the case of a triplet superconductor with p-wave symmetry in the presence of Rashba spin-orbit (SO) coupling and externally applied Zeeman spin splitting. Generalization to the cases where the pairing vector is not aligned with that of the SO coupling is given. We generalize also to the cases where the normal system is already topologically non-trivial.

Abstract:
The requirement for large bulk resistivity in topological insulators has led to the design of complex ternary and quaternary phases with balanced donor and acceptor levels. A common feature of the optimized phases is that they lie close to the p to n transition. The tetradymite Bi2Te3_xSex system exhibits minimum bulk conductance at the ordered composition Bi2Te2Se. By combining local and integral measurements of the density of states, we find that the point of minimum electrical conductivity at x=1.0 where carriers change from hole-like to electron-like is characterized by conductivity of the mixed type. Our experimental findings, which are interpreted within the framework of a two band model for the different carrier types, indicate that the mixed state originates from different type of native defects that strongly compensate at the crossover point.

Abstract:
Topological insulators (TIs) have the singular distinction of being electronic insulators while harboring metallic, conductive surfaces. In ordinary materials, defects such as cracks and deformations are barriers to electrical conduction, intuitively making the material more electrically resistive. Peculiarly, 3D TIs should become better conductors when they are cracked because the cracks themselves, which act as conductive topological surfaces, provide additional paths for the electrical current. Significantly, for a TI material, any surface or extended defect harbors such conduction. In this letter, we demonstrate that small subsurface cracks formed within the predicted 3D TI samarium hexaboride (SmB$_{6}$) via systematic scratching or sanding results in such an increase in the electrical conduction. SmB$_{6}$ is in a unique position among TIs to exhibit this effect because its single-crystals are thick enough to harbor cracks, and because it remarkably does not appear to suffer from conduction through bulk impurities. Our results not only strengthen the building case for SmB$_{6}$'s topological nature, but are relevant to all TIs with cracks, including TI films with grain boundaries.

Abstract:
We study in this paper time-reversal $\delta$-impurity scattering effects in the bulk of topological insulators (TI) in two and three dimensions. Specifically we consider how impurity scattering strength is affected by the bulk band structure of topological insulators. An interesting band inversion effect associated with the change of the system from ordinary to topological insulator is pointed out. Experimental consequences of our findings are discussed.