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We analyze the interaction at the surface of a three-dimensional (3D) topological insulator among 2D electron states belonging to the Dirac cone close to the point of the Brillouin zone and the Rayleigh surface phonon mode. The model deals with an elastic continuum in the long-wavelength limit, in Random Phase Approximation (RPA). Screening of the electronic polarization is quite effective at small wave vectors. On the other hand, the absence of backscattering for the Dirac electrons at the Fermi surface is partly responsible for the reduced influence of the electron-phonon interaction in renormalizing the phonon dispersion at finite wave vectors. We infer that softening of the Rayleigh mode appears as unlikely, at least for the case of a clean and defect-free surface to which our approximate treatment applies. The dielectric response to virtual excitation of the Rayleigh phonon could drive the electron-electron interaction attractive at low frequencies, but the average weak coupling pairing interaction is found to be too small to induce a surface superconducting instability. © 2013 American Physical Society. ; This work was done with financial support from FP7/2007-2013 under Grant No. 264098-MAMA (Multifunctioned Advanced Materials and Nanoscale Phenomena), MIUR-Italy through Prin-Project 2009 "Nanowire high critical temperature superconductor field-effect devices," as well as the Helmholtz Virtual Insitute "New States of Matter." V.P. and F.G. acknowledge financial support from MINECO, Spain, through Grant No. FIS2011-23713, and the European Union, through Grant No. 290846. ; Peer Reviewed

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license. ; The scattering of Dirac electrons by topological defects could be one of the most relevant sources of resistance in graphene and at the boundary surfaces of a three-dimensional topological insulator (3D TI). In the long wavelength, continuous limit of the Dirac equation, the topological defect can be described as a distortion of the metric in curved space, which can be accounted for by a rotation of the Gamma matrices and by a spin connection inherited with the curvature. These features modify the scattering properties of the carriers. We discuss the self-energy of defect formation with this approach and the electron cross-section for intra-valley scattering at an edge dislocation in graphene, including corrections coming from the local stress. The cross-section contribution to the resistivity, ρ, is derived within the Boltzmann theory of transport. On the same lines, we discuss the scattering of a screw dislocation in a two-band 3D TI, like Bi1-xSbx, and we present the analytical simplified form of the wavefunction for gapless helical states bound at the defect. When a 3D TI is sandwiched between two even-parity superconductors, Dirac boundary states acquire superconductive correlations by proximity. In the presence of a magnetic vortex piercing the heterostructure, two Majorana states are localized at the two interfaces and bound to the vortex core. They have a half integer total angular momentum each, to match with the unitary orbital angular momentum of the vortex charge. © 2014 by the authors. ; This work was done with financial support from FP7/2007-2013 under the grant no. 264098—MAMA (Multifunctional Advanced Materials and Nanoscale Phenomena), MIUR (Ministero dell' Istruzione, dell' Università e della Ricerca)-Italy through the Prin-Project 2009 "Nanowire high critical temperature superconductor field-effect devices" and Futuro In Ricerca (FIRB)/2013-2015. Vincenzo Parente and Francisco Guinea acknowledge financial support from MINECO (Ministerio Economía y Competitividad), Spain, through grant FIS2011-23713, and the European Union, through grant 290846. ; Peer Reviewed