Vacancy-induced bound states in topological insulators

Abstract

We present an exact solution of a modifed Dirac equation for topological insulator in the presence of a hole or vacancy to demonstrate that vacancies may induce bound states in the band gap of topological insulators. They arise due to the Z2 classification of time-reversal invariant insulators, thus are also topologically-protected like the edge states in the quantum spin Hall effect and the surface states in three-dimensional topological insulators. Coexistence of the in-gap bound states and the edge or surface states in topological insulators suggests that imperfections may affect transport properties of topological insulators via additional bound states near the system boundary.PACS numbers: 73.20.Hb, Topological insulators are narrow-band semiconductors with band inversion generated by strong spin-orbit coupling [1]. They are distinguished from the ordinary band insulators according to the Z 2 invariant classification of the gapped band insulators due to the time reversal symmetry. The variation of the Z 2 invariant at their boundaries will lead to the topologically protected edge or surface states with the gapless Dirac energy spectrum [2][3][4][5][6][7]. Imperfections, such as impurity, vacancy, and disorder, are inevitably present in topological insulators. Owing to the time-reversal symmetry, an exciting feature of topological insulator is that its boundary states are expected to be topologically protected against weak non-magnetic impurities or disorders [8,9]. This provoked much interest on the single impurity problem on the surface of a topological insulator, starting with gapless Dirac model [10][11][12][13][14]. However, reminding that the boundary state is only a manifestation of the topological nature of bulk bands, it should also start with the examination of the host bulk to know how the imperfections affect the electronic structure. It is well known that single impurity or defect can induce bound states in many systems, such as in the Yu-Shiba state in s-wave superconductor [15,16] and in d-wave superconductors [17]. Topological defects were discussed in the B-phase of 3 He superfluid[18] and topological insulators and superconductors [19]. Here we report that bound states can form around a single vacancy in the bulk energy gap of topological insulators. These bound states are found to have the same origin as boundary states due to the Z 2 classification, thus are also topologically protected.The formation of the topological bound states can be readily illustrated by reviewing the quantum spin Hall effect in two-dimensional (2D) topological insulators [20][21][22], in which strong spin-orbit coupling twists the bulk conduction and valence bands, leading to a nontrivial Z 2 index. As the Z 2 varies across the edge, edge states arise in the gap with the gapless Dirac dispersion. Unlike the quantum Hall effect in a magnetic field, spin-orbit coupling preserves the time reversal symmetry, so the result-

Subsurface impurities and vacancies in a three-dimensional topological insulator

“…This result does not agree with continuum model results for finite holes in a TI. 16 To expand on this discrepancy we have studied extended bulk vacancy clusters. We find that, while fully-symmetric 5-and 17-site clusters do not have any low-energy resonance states in agreement with continuum results, any asymmetry in the clusters produces E res ≈ 0 resonance peaks.…”

“…(6) showed, the E res = 0 peak is the same as that of a vacancy in an idealized normal insulator, and is thus a very robust result. On the other hand, Shan et al 16 recently used a continuum model to demonstrate the existence of bound states for a finite sized hole in a TI. In that case the bound states are simply a manifestation of the fact that a finite sized hole creates an interior surface in the TI.…”

“…iv) Fully symmetric multiple-site vacancy clusters have no in-gap resonance peaks, in agreement with continuum results. 16 However, any small deviation from full symmetry produces lowlying resonance peaks. Any realistic microscopically created hole in a 3D TI will therefore have a resonance peak around E = 0, mimicking the results of a single vacancy instead of that of a finite size continuum hole.…”

“…In the opposite limit, the properties of deep subsurface impurities ought to be closely connected to those of bulk impurities. Both potential impurities 15 and finite sized holes 16 in the bulk of a 3D TI have previously been treated within a continuum theory focusing on in-gap bound states. In the case of a finite sized hole, it constitutes an interior surface and will thus necessarily host a surface state in a TI.…”

Universal Impurity-Induced Bound State in Topological Superfluids

“…As the gap parameter is fully suppressed at the impurity site, we have a local point defect (i.e., vacuum) that is topologically trivial. Due to the topological nature of the Fermi cloud away from the impurity, there would be an interface between the nontopological and topological components, which can host a gapless Majorana edge state [33]. The observed universal impurity state is precisely such a Majorana edge mode.…”