We use the bulk Hamiltonian for a three-dimensional topological insulator such as $rm Bi_2 Se_3$ to study the states which appear on its various surfaces and along the edge between two surfaces. We use both analytical methods based on the surface Ham
iltonians (which are derived from the bulk Hamiltonian) and numerical methods based on a lattice discretization of the bulk Hamiltonian. We find that the application of a potential along an edge can give rise to states localized at that edge. These states have an unusual energy-momentum dispersion which can be controlled by applying a potential along the edge; in particular, the velocity of these states can be tuned to zero. The scattering across the edge is studied as a function of the edge potential. We show that a magnetic field in a particular direction can also give rise to zero energy states on certain edges. We point out possible experimental ways of looking for the various edge states.
We study transport across a line junction lying between two orthogonal topological insulator surfaces and a superconductor which can have either s-wave (spin-singlet) or p-wave (spin-triplet) pairing symmetry. We present a formalism for studying the
effect of a general time-reversal invariant barrier at the junction and show that such a barrier can be completely described by three arbitrary parameters. We compute the charge and the spin conductance across such a junction and study their behaviors as a function of the bias voltage applied across the junction and the three parameters used to characterize the barrier. We find that the presence of topological insulators and a superconductor leads to both Dirac and Schrodinger-like features in charge and spin conductances. We discuss the effect of bound states on the superconducting side of the barrier on the conductance; in particular, we show that for triplet p-wave superconductors such a junction may be used to determine the spin state of its Cooper pairs. Our study reveals that there is a non-zero spin conductance for some particular spin states of the triplet Cooper pairs; this is an effect of the topological insulators which break the spin rotation symmetry. Finally, we find an unusual satellite peak (in addition to the usual zero bias peak) in the spin conductance for p-wave symmetry of the superconductor order parameter.
