# Electric Field Dependence of Topological Edge States in One-Bilayer Bi(111): A First-Principles Study

Abstract

We investigate the effect of the electric field on the edge states in one-bilayer Bi(111) by first-principles calculations. We calculate the band structures of armchair and zigzag Bi nanoribbons. With increasing strength of the electric field E > 2.1 V/Å, the armchair nanoribbon shows a topological phase transition from non-trivial metallic edge states to insulating edge states. However, under the same conditions, the zigzag nanoribbon shows a topological phase transition from non-trivial metallic edge states to trivial metallic edge states. We expect that these findings will contribute to the development of, e.g., spin current switches for use in next-generation devices. [DOI: 10.1380/ejssnt.2018.427]
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The study of topological insulators has generally involved search of materials that have this property as an innate quality, distinct from normal insulators. Here we focus on the possibility of converting a normal insulator into a topological one by application of an external electric field that shifts different bands by different energies and induces a specific band inversion, which leads to a topological state. Phosphorene is a two-dimensional (2D) material that can be isolated through mechanical exfoliation from layered black phosphorus, but unlike graphene and silicene, few layers phosphorene has a large band gap (1.5 - 2.2 eV). It was thus unsuspected to exhibit band inversion and the ensuing topological insulator behavior. Using first-principles calculations with applied perpendicular electric field F⊥ we predict a continuous transition from the normal insulator to a topological insulator and eventually to a metal as a function of F⊥. The tuning of topological behavior with electric field would lead to spin-separated, gapless edge states, i.e., quantum spins Hall effect. This finding opens the possibility of converting normal insulating materials into topological ones via electric field, and making a multi-functional "field effect topological transistor" that could manipulate simultaneously both spins and charge carrier. We use our results to formulate some design principles for looking for other 2D materials that could have such an electrical-induced topological transition. - Jan 2010
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M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).- Recent experiments and theories have suggested that strong spin-orbit coupling effects in certain band insulators can give rise to a new phase of quantum matter, the so-called topological insulator, which can show macroscopic quantum-entanglement effects. Such systems feature two-dimensional surface states whose electrodynamic properties are described not by the conventional Maxwell equations but rather by an attached axion field, originally proposed to describe interacting quarks. It has been proposed that a topological insulator with a single Dirac cone interfaced with a superconductor can form the most elementary unit for performing fault-tolerant quantum computation. Here we present an angle-resolved photoemission spectroscopy study that reveals the first observation of such a topological state of matter featuring a single surface Dirac cone realized in the naturally occurring Bi2Se3 class of materials. Our results, supported by our theoretical calculations, demonstrate that undoped Bi2Se3 can serve as the parent matrix compound for the long-sought topological device where in-plane carrier transport would have a purely quantum topological origin. Our study further suggests that the undoped compound reached via n-to-p doping should show topological transport phenomena even at room temperature.
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It is shown that the macroscopic and microscopic dielectric responses to a constant electric field can be directly dealt with by a method which is general, alternative to linear-response theory, and not restricted to linear effects. The basic features of the theory are discussed; explicit calculations of the macroscopic dielectric constant, miccroscopic local fields, and transverse effective charges are presented for Ge and GaAs within the local-density functional theory. - Article
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We study the electronic surface states of the semiconducting alloy BiSb. Using a phenomenological tight binding model we show that the Fermi surface of the 111 surface states encloses an odd number of time reversal invariant momenta (TRIM) in the surface Brillouin zone confirming that the alloy is a strong topological insulator. We then develop general arguments which show that spatial symmetries lead to additional topological structure, and further constrain the surface band structure. Inversion symmetric crystals have 8 Z_2 "parity invariants", which include the 4 Z_2 invariants due to time reversal. The extra invariants determine the "surface fermion parity", which specifies which surface TRIM are enclosed by an odd number of electron or hole pockets. We provide a simple proof of this result, which provides a direct link between the surface states and the bulk parity eigenvalues. We then make specific predictions for the surface state structure for several faces of BiSb. We next show that mirror invariant band structures are characterized by an integer "mirror Chern number", n_M. The sign of n_M in the topological insulator phase of BiSb is related to a previously unexplored Z_2 parameter in the L point k.p theory of pure Bi, which we refer to as the "mirror chirality", \eta. The value of \eta predicted by the tight binding model for Bi disagrees with the value predicted by a more fundamental pseudopotential calculation. This explains a subtle disagreement between our tight binding surface state calculation and previous first principles calculations on Bi. This suggests that the tight binding parameters in the Liu Allen model of Bi need to be reconsidered. Implications for existing and future ARPES experiments and spin polarized ARPES experiments will be discussed. - Topological insulators are new states of quantum matter in which surface states residing in the bulk insulating gap of such systems are protected by time-reversal symmetry. The study of such states was originally inspired by the robustness to scattering of conducting edge states in quantum Hall systems. Recently, such analogies have resulted in the discovery of topologically protected states in two-dimensional and three-dimensional band insulators with large spin–orbit coupling. So far, the only known three-dimensional topological insulator is BixSb1-x, which is an alloy with complex surface states. Here, we present the results of first-principles electronic structure calculations of the layered, stoichiometric crystals Sb2Te3, Sb2Se3, Bi2Te3 and Bi2Se3. Our calculations predict that Sb2Te3, Bi2Te3 and Bi2Se3 are topological insulators, whereas Sb2Se3 is not. These topological insulators have robust and simple surface states consisting of a single Dirac cone at the point. In addition, we predict that Bi2Se3 has a topologically non-trivial energy gap of 0.3 eV, which is larger than the energy scale of room temperature. We further present a simple and unified continuum model that captures the salient topological features of this class of materials.
- We study the electronic states of graphite ribbons with edges of two typical shapes, armchair and zigzag, by performing tight binding band calculations, and find that the graphite ribbons show striking contrast in the electronic states depending on the edge shape. In particular, a zigzag ribbon shows a remarkably sharp peak of density of states at the Fermi level, which does not originate from infinite graphite. We find that the singular electronic states arise from the partly flat bands at the Fermi level, whose wave functions are mainly localized on the zigzag edge. We reveal the puzzle for the emergence of the peculiar edge state by deriving the analytic form in the case of semi-infinite graphite with a zigzag edge. Applying the Hubbard model within the mean-field approximation, we discuss the possible magnetic structure in nanometer-scale micrographite.
- The exact density functional for the ground-state energy is strictly self-interaction-free (i.e., orbitals demonstrably do not self-interact), but many approximations to it, including the local-spin-density (LSD) approximation for exchange and correlation, are not. We present two related methods for the self-interaction correction (SIC) of any density functional for the energy; correction of the self-consistent one-electron potenial follows naturally from the variational principle. Both methods are sanctioned by the Hohenberg-Kohn theorem. Although the first method introduces an orbital-dependent single-particle potential, the second involves a local potential as in the Kohn-Sham scheme. We apply the first method to LSD and show that it properly conserves the number content of the exchange-correlation hole, while substantially improving the description of its shape. We apply this method to a number of physical problems, where the uncorrected LSD approach produces systematic errors. We find systematic improvements, qualitative as well as quantitative, from this simple correction. Benefits of SIC in atomic calculations include (i) improved values for the total energy and for the separate exchange and correlation pieces of it, (ii) accurate binding energies of negative ions, which are wrongly unstable in LSD, (iii) more accurate electron densities, (iv) orbital eigenvalues that closely approximate physical removal energies, including relaxation, and (v) correct longrange behavior of the potential and density. It appears that SIC can also remedy the LSD underestimate of the band gaps in insulators (as shown by numerical calculations for the rare-gas solids and CuCl), and the LSD overestimate of the cohesive energies of transition metals. The LSD spin splitting in atomic Ni and $s$-${}d$ interconfigurational energies of transition elements are almost unchanged by SIC. We also discuss the admissibility of fractional occupation numbers, and present a parametrization of the electron-gas correlation energy at any density, based on the recent results of Ceperley and Alder.
- We report the formation of a bilayer Bi(111) ultrathin film, which is theoretically predicted to be in a two-dimensional quantum spin Hall state, on a Bi(2)Te(3) substrate. From angle-resolved photoemission spectroscopy measurements and ab initio calculations, the electronic structure of the system can be understood as an overlap of the band dispersions of bilayer Bi and Bi(2)Te(3). Our results show that the Dirac cone is actually robust against nonmagnetic perturbations and imply a unique situation where the topologically protected one- and two-dimensional edge states are coexisting at the surface.
- Topological insulators are electronic materials that have a bulk band gap like an ordinary insulator but have protected conducted states on their edge or surface. These states are possible due to the combination of spin-orbit interactions and time-reversal symmetry. The two-dimensional (2D) topological insulator is a quantum spin Hall insulator, which is a close cousin of the integer quantum Hall state. A three-dimensional (3D) topological insulator supports novel spin-polarized 2D Dirac fermions on its surface. In this Colloquium the theoretical foundation for topological insulators and superconductors is reviewed and recent experiments are described in which the signatures of topological insulators have been observed. Transport experiments on HgTe/CdTe quantum wells are described that demonstrate the existence of the edge states predicted for teh quantum spin hall insulator. Experiments on Bi1-xSbx, Bi<2Se3, Bi2Te3 and Sb2Te3 are then discussed that establish these materials as 3D topological insulators and directly probe the topology of their surface states. Exotic states are described that can occur at the surface of a 3D topological insulator due to an induced energy gap. A magnetic gap leads to a novel quantum Hall state that gives rise to a topological magnetoelectric effect. A superconducting energy gap leads to a state that supports Majorana fermions and may provide a new venue for realizing proposals for topological quantum computation. Prospects for observing these exotic states are also discussed, as well as other potential device applications of topological insulators.
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- Aug 2010
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Using first-principles calculations within density functional theory, we explore the feasibility of converting ternary half-Heusler compounds into a new class of three-dimensional topological insulators (3DTI). We demonstrate that the electronic structure of unstrained LaPtBi as a prototype system exhibits a distinct band-inversion feature. The 3DTI phase is realized by applying a uniaxial strain along the [001] direction, which opens a band gap while preserving the inverted band order. A definitive proof of the strained LaPtBi as a 3DTI is provided by directly calculating the topological Z2 invariants in systems without inversion symmetry. We discuss the implications of the present study to other half-Heusler compounds as 3DTI, which, together with the magnetic and superconducting properties of these materials, may provide a rich platform for novel quantum phenomena. - Article
- May 2010
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We theoretically study the generic behavior of the penetration depth of the edge states in two-dimensional quantum spin Hall systems. We found that the momentum-space width of the edge-state dispersion scales with the inverse of the penetration depth. As an example of well-localized edge states, we take the Bi(111) ultrathin film. Its edge states are found to extend almost over the whole Brillouin zone. Correspondingly, the bismuth (111) 1-bilayer system is proposed to have well-localized edge states in contrast to the HgTe quantum well. - 5 pages. Using scanning tunneling spectroscopy and first-principles calculations, we have studied the electronic structure of two different ultrathin bismuth films on a Si(111)-7×7 substrate; a hexagonal film (HEX film) having a bulklike (A7-like) structure and a film having a black-phosphorus-like structure (BP film). The HEX film is metallic because of spin–orbit (SO)-split surface-state bands lying inside the projected bulk band gap near the Fermi level. Another SO-split surface state is also observed inside the SO gap. The BP film exhibits a significant reduction in metallicity in contrast to the HEX film. This is related to the formation of a very stable paired-layer structure, the mechanism of which is similar to that of the stabilization of semiconducting bulk black P. However, unlike bulk black P, a certain extent of metallicity still remains. This slight metallicity can be associated with buckling and strain in the BP film, which is analogous to the fact that shear angle distortion in bulk Bi is responsible for its semimetallicity. Peer reviewed
- Three-dimensional topological insulators are a new state of quantum matter with a bulk gap and odd number of relativistic Dirac fermions on the surface. By investigating the surface state of Bi2Te3 with angle-resolved photoemission spectroscopy, we demonstrate that the surface state consists of a single nondegenerate Dirac cone. Furthermore, with appropriate hole doping, the Fermi level can be tuned to intersect only the surface states, indicating a full energy gap for the bulk states. Our results establish that Bi2Te3 is a simple model system for the three-dimensional topological insulator with a single Dirac cone on the surface. The large bulk gap of Bi2Te3 also points to promising potential for high-temperature spintronics applications.
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- Dec 1986
- Phys Rev B

A new method is proposed for dealing with the dielectric response of a semiconductor to a constant macroscopic electric field. The ``direct'' approach we use is general, an alternative to linear-response theory, and not restricted to linear effects. It consists in incorporating the macroscopic field into the system and in dealing with the ``perturbed'' crystal as with a new system; the potential of the constant field is generated by building in a (periodically repeated) capacitor. We apply this method to Ge and GaAs within the local-density-functional framework and investigate the dielectric response: Macroscopic quantities (ε∞,e*T) are obtained, as well as microscopic ones (variations of local fields). Our approach, self-explanatory in r space, is related here to the conventional linear dielectric theory, which is formulated in terms of dielectric matrices in reciprocal space. Our results shed new light upon the basic mechanisms and trends in electronic dielectric screening; the most popular models for microscopic response are judged in view of the present calculations for real materials. - Article
- Jan 1997
- Phys Rev B

Finite graphite systems having a zigzag edge exhibit a special edge state. The corresponding energy bands are almost flat at the Fermi level and thereby give a sharp peak in the density of states. The charge density in the edge state is strongly localized on the zigzag edge sites. No such localized state appears in graphite systems having an armchair edge. By utilizing the graphene ribbon model, we discuss the effect of the system size and edge shape on the special edge state. By varying the width of the graphene ribbons, we find that the nanometer size effect is crucial for determining the relative importance of the edge state. We also have extended the graphene ribbon to have edges of a general shape, which is defined as a mixture of zigzag and armchair sites. Examining the relative importance of the edge state for graphene ribbons with general edges, we find that a non-negligible edge state survives even in graphene ribbons with less developed zigzag edges. We demonstrate that such an edge shape with three or four zigzag sites per sequence is sufficient to show an edge state, when the system size is on a nanometer scale. The special characteristics of the edge state play a large role in determining the density of states near the Fermi level for graphite networks on a nanometer scale. - Our scanning tunneling microscopy and electron diffraction experiments revealed that a new two-dimensional allotrope of Bi forms on the Si(111)-7x7 surface. This pseudocubic [012]-oriented allotrope is stable up to four atomic layers at room temperature. Above this critical thickness, the entire volume of the film starts to transform into a bulk single-crystal (001) phase, as the bulk contribution in the cohesion becomes dominant. Based on ab initio calculations, we propose that the new allotrope consists of black phosphorus-like puckered layers stabilized by saturating all the p(z) dangling bonds in the film.
- Article
- Oct 2005
- PHYS REV LETT

The quantum spin Hall (QSH) phase is a time reversal invariant electronic state with a bulk electronic band gap that supports the transport of charge and spin in gapless edge states. We show that this phase is associated with a novel Z2 topological invariant, which distinguishes it from an ordinary insulator. The Z2 classification, which is defined for time reversal invariant Hamiltonians, is analogous to the Chern number classification of the quantum Hall effect. We establish the Z2 order of the QSH phase in the two band model of graphene and propose a generalization of the formalism applicable to multiband and interacting systems. - Article
- Jan 2007
- PHYS REV LETT

We show that the spin-Hall conductivity in insulators is related to a magnetic susceptibility representing the strength of the spin-orbit coupling. We use this relationship as a guiding principle to search real materials showing quantum spin-Hall effect. As a result, we theoretically predict that two-dimensional bismuth will show the quantum spin-Hall effect, both by calculating the helical edge states, and by showing the nontriviality of the Z2 topological number, and propose possible experiments. - When electrons are subject to a large external magnetic field, the conventional charge quantum Hall effect dictates that an electronic excitation gap is generated in the sample bulk, but metallic conduction is permitted at the boundary. Recent theoretical models suggest that certain bulk insulators with large spin-orbit interactions may also naturally support conducting topological boundary states in the quantum limit, which opens up the possibility for studying unusual quantum Hall-like phenomena in zero external magnetic fields. Bulk Bi(1-x)Sb(x) single crystals are predicted to be prime candidates for one such unusual Hall phase of matter known as the topological insulator. The hallmark of a topological insulator is the existence of metallic surface states that are higher-dimensional analogues of the edge states that characterize a quantum spin Hall insulator. In addition to its interesting boundary states, the bulk of Bi(1-x)Sb(x) is predicted to exhibit three-dimensional Dirac particles, another topic of heightened current interest following the new findings in two-dimensional graphene and charge quantum Hall fractionalization observed in pure bismuth. However, despite numerous transport and magnetic measurements on the Bi(1-x)Sb(x) family since the 1960s, no direct evidence of either topological Hall states or bulk Dirac particles has been found. Here, using incident-photon-energy-modulated angle-resolved photoemission spectroscopy (IPEM-ARPES), we report the direct observation of massive Dirac particles in the bulk of Bi(0.9)Sb(0.1), locate the Kramers points at the sample's boundary and provide a comprehensive mapping of the Dirac insulator's gapless surface electron bands. These findings taken together suggest that the observed surface state on the boundary of the bulk insulator is a realization of the 'topological metal'. They also suggest that this material has potential application in developing next-generation quantum computing devices that may incorporate 'light-like' bulk carriers and spin-textured surface currents.