One-dimensional edge states with giant spin splitting in a bismuth thin film

Giant Rashba-splitting of one-dimensional metallic states in Bi dimer lines on InAs(100)

Bismuth produces different types of ordered superstructures on the InAs(100) surface, depending on the growth procedure and coverage. The (2 × 1) phase forms at completion of one Bi monolayer and consists of a uniformly oriented array of parallel lines of Bi dimers. Scanning tunneling and core level spectroscopies demonstrate its metallic character, in contrast with the semiconducting properties expected on the basis of the electron counting principle. The weak electronic coupling among neighboring lines gives rise to quasi one-dimensional Bi-derived bands with open contours at the Fermi level. Spin- and angle-resolved photoelectron spectroscopy reveals a giant Rashba splitting of these bands, in good agreement with ab initio electronic structure calculations. The very high density of the dimer lines, the metallic and quasi one-dimensional band dispersion and the Rashba-like spin texture make the Bi/InAs(100)-(2 × 1) phase an intriguing system, where novel transport regimes can be studied.

Cluster Formation of Self-Assembled Triarylbismuthanes and Charge Transport Characterizations of Gold-Triarylbismuthane-Gold Junctions

Organometallic molecules are promising for molecular electronic devices due to their potential to improve electrical conductance through access to complex orbital covalency that is not available to light-element organic molecules. However, studies of the formation of organometallic monolayers and their charge transport properties are scarce. Here, we report the cluster formation and charge transport properties of gold-triarylbismuthane-gold molecular junctions. We found that triarylbismuthane molecules with -CN anchoring groups form clusters during the creation of self-assembled submonolayers. This clustering is attributed to strong interactions between the bismuth (Bi) center and the nitrogen atom in the -CN group of adjacent molecules. Examination of the influence of -NH2 and -CN anchoring groups on junction conductance revealed that, despite a stronger binding energy between the -NH2 group and gold, the conductance per molecular unit (i.e., molecule for the -NH2 group and cluster for the -CN group) is higher with the -CN anchoring group. Further analysis showed that an increase in the number of -CN groups from one to three within the junctions leads to a decrease in conductance while increasing the size of the cluster. This demonstrates the significant effects of different anchoring groups and the impact of varying the number of -CN groups on both the charge transport and cluster formation. This study highlights the importance of selecting the appropriate anchoring group in the design of molecular junctions. Additionally, controlling the size and formation of clusters can be a strategic approach to engineering charge transport in molecular junctions.

Observation of edge states derived from topological helix chains

Introducing the concept of topology has revolutionized materials classification, leading to the discovery of topological insulators and Dirac-Weyl semimetals1-3. One of the most fundamental theories underpinning topological materials is the Su-Schrieffer-Heeger (SSH) model4,5, which was developed in 1979-decades before the recognition of topological insulators-to describe conducting polymers. Distinct from the vast majority of known topological insulators with two and three dimensions1-3, the SSH model predicts a one-dimensional analogue of topological insulators, which hosts topological bound states at the endpoints of a chain4-8. To establish this unique and pivotal state, it is crucial to identify the low-energy excitations stemming from bound states, but this has remained unknown in solids because of the absence of suitable platforms. Here we report unusual electronic states that support the emergent bound states in elemental tellurium, the single helix of which was recently proposed to realize an extended version of the SSH chain9,10. Using spin- and angle-resolved photoemission spectroscopy with a micro-focused beam, we have shown spin-polarized in-gap states confined to the edges of the (0001) surface. Our density functional theory calculations indicate that these states are attributed to the interacting bound states originating from the one-dimensional array of SSH tellurium chains. Helices in solids offer a promising experimental platform for investigating exotic properties associated with the SSH chain and exploring topological phases through dimensionality control.

Photoinduced Electronic and Spin Topological Phase Transitions in Monolayer Bismuth

Ultrathin bismuth exhibits rich physics including strong spin-orbit coupling, ferroelectricity, nontrivial topology, and light-induced structural dynamics. We use ab initio calculations to show that light can induce structural transitions to four transient phases in bismuth monolayers. These light-induced phases exhibit nontrivial topological character, which we illustrate using the recently introduced concept of spin bands and spin-resolved Wilson loops. Specifically, we find that the topology changes via the closing of the electron and spin band gaps during photoinduced structural phase transitions, leading to distinct edge states. Our study provides strategies to tailor electronic and spin topology via ultrafast control of photoexcited carriers and associated structural dynamics.

Phase-Selective Epitaxy of Trigonal and Orthorhombic Bismuth Thin Films on Si (111)

Over the past three decades, the growth of Bi thin films has been extensively explored due to their potential applications in various fields such as thermoelectrics, ferroelectrics, and recently for topological and neuromorphic applications, too. Despite significant research efforts in these areas, achieving reliable and controllable growth of high-quality Bi thin-film allotropes has remained a challenge. Previous studies have reported the growth of trigonal and orthorhombic phases on various substrates yielding low-quality epilayers characterized by surface morphology. In this study, we present a systematic growth investigation, enabling the high-quality growth of Bi epilayers on Bi-terminated Si (111) 1 × 1 surfaces using molecular beam epitaxy. Our work yields a phase map that demonstrates the realization of trigonal, orthorhombic, and pseudocubic thin-film allotropes of Bi. In-depth characterization through X-ray diffraction (XRD) techniques and scanning transmission electron microscopy (STEM) analysis provides a comprehensive understanding of phase segregation, phase stability, phase transformation, and phase-dependent thickness limitations in various Bi thin-film allotropes. Our study provides recipes for the realization of high-quality Bi thin films with desired phases, offering opportunities for the scalable refinement of Bi into quantum and neuromorphic devices and for revisiting technological proposals for this versatile material platform from the past 30 years.

Spin-charge separation and quantum spin Hall effect of [Formula: see text]-bismuthene

Multiple works suggest the possibility of classification of quantum spin Hall effect with magnetic flux tubes, which cause separation of spin and charge degrees of freedom and pumping of spin or Kramers-pair. However, the proof of principle demonstration of spin-charge separation is yet to be accomplished for realistic, ab initio band structures of spin-orbit-coupled materials, lacking spin-conservation law. In this work, we perform thought experiments with magnetic flux tubes on [Formula: see text]-bismuthene, and demonstrate spin-charge separation, and quantized pumping of spin for three insulating states, that can be accessed by tuning filling fractions. With a combined analysis of momentum-space topology and real-space response, we identify important role of bands supporting even integer invariants, which cannot be addressed with symmetry-based indicators. Our work sets a new standard for the computational diagnosis of two-dimensional, quantum spin-Hall materials by going beyond the [Formula: see text] paradigm and providing an avenue for precise determination of the bulk invariant through computation of quantized, real-space response.

Totally Spin-Polarized Currents in an Interferometer with Spin-Orbit Coupling and the Absence of Magnetic Field Effects

The paper studies the electronic current in a one-dimensional lead under the effect of spin-orbit coupling and its injection into a metallic conductor through two contacts, forming a closed loop. When an external potential is applied, the time reversal symmetry is broken and the wave vector k of the circulating electrons that contribute to the current is spin-dependent. As the wave function phase depends upon the vector k, the closed path in the circuit produces spin-dependent current interference. This creates a physical scenario in which a spin-polarized current emerges, even in the absence of external magnetic fields or magnetic materials. It is possible to find points in the system's parameter space and, depending upon its geometry, the value of the Fermi energy and the spin-orbit intensities, for which the electronic states participating in the current have only one spin, creating a high and totally spin-polarized conductance. For a potential of a few tens of meV, it is possible to obtain a spin-polarized current of the order of μA. The properties of the obtained electronic current qualify the proposed device as a potentially important tool for spintronics applications.

Phase Transformation-Induced Quantum Dot States on the Bi/Si(111) Surface

Nanopatterns at near atomic dimensions with controllable quantum dot states (QDSs) are promising candidates for the continued downscaling of electronic devices. Herein, we report a phase transition-induced QD system achieved on the √3 × √3-Bi/Si(111) surface reconstruction, which points the way to a novel strategy on QDS implementation. Combining scanning tunneling microscopy, scanning tunneling spectroscopy, and density functional theory (DFT) calculations, the structure, energy dispersion, and size effect on band gap of the QDs are measured and verified. As-created QDs can be manipulated with a dot size down to 2 nm via Bi phase transformation, which, in turn, is triggered by thermal annealing at 700 K. The transition mechanism is also supported by our DFT calculations, and an empirical analytical model is developed to predict the transformation kinetics.

Novel optimization perspectives for thermoelectric properties based on Rashba spin splitting: a mini review

The energy problem has recently become increasingly more serious, therefore the rational use of heat energy and conversion into electrical energy is particularly important. The thermoelectric (TE) field is closely related to human life, as heat from automobiles, heat dissipation from high-power electrical appliances, or other electrical products that produce a lot of heat, can all be transformed with TE materials. The search for TE materials with an excellent performance and effective TE optimization strategies (STs) has attracted significant attention owing to the fact that thermal energy can be directly converted into electric energy. In contrast to the common TE-optimized STs, such as constructing point defects or reducing dimensionality, spin-related optimization STs have emerged from previous published research, such as the spin Seebeck effect or the Rashba effect, in which the Rashba effect shows an effective method to break through the bottleneck of ZT optimization. In this review, typical high ZT materials, common traditional optimized STs, Rashba-type TE materials and their corresponding ZT values are comprehensively discussed. The TE performance of Rashba-type materials is analysed, such as BiTeX (X = I, Br), GeTe, BiSbSeTe₂, and the BiSb monolayer. Moreover, the TE optimization mechanisms (band engineering, phonon engineering, and Rashba spin-split engineering) are summarised. Finally, the development and challenges of Rashba spin-split combined with TE in breaking the bottleneck in ZT optimization are highlighted.

Quantum well states and sizable Rashba splitting on Pb induced α-phase Bi/Si(111) surface reconstruction

Quantum well states (QWSs) with sizable Rashba splitting are a promising quantum phase to achieve spin-split current for quantum computing and spintronics due to their controllable band structures. However, most QWSs were achieved upon metallic substrates with strong bulk electron transport. Developing semiconductor-based QWSs is preferable to minimize substrate interference. Here we report a Pb induced surface reconstruction on Bi/Si(111) α phase. Combining scanning tunneling microscopy (STM) and density functional theory (DFT) the atomic structure has been determined. QWSs and a sizable Rashba band splitting are predicted, with the latter comparable to what is found in other semiconductor heterostructures and an order of magnitude higher than that in Pb/Si(111) QWSs.

Quantum Size Effects, Multiple Dirac Cones, and Edge States in Ultrathin Bi(110) Films

The presence of inherently strong spin-orbit coupling in bismuth, its unique layer-dependent band topology and high carrier mobility make it an interesting system for both fundamental studies and applications. Theoretically, it has been suggested that strong quantum size effects should be present in the Bi(110) films, with the possibility of Dirac Fermion states in the odd-bilayer (BL) films, originating from dangling p_z orbitals and quantum-spin hall (QSH) states in the even-bilayer films. However, the experimental verification of these claims has been lacking. Here, we study the electronic structure of Bi(110) films grown on a high-T_c superconductor, Bi₂Sr₂CaCu₂O_8+δ (Bi2212) using angle-resolved photoemission spectroscopy (ARPES). We observe an oscillatory behavior of electronic structure with the film thickness and identify the Dirac-states in the odd-bilayer films, consistent with the theoretical predictions. In the even-bilayer films, we find another Dirac state that was predicted to play a crucial role in the QSH effect. In the low thickness limit, we observe several extremely one-dimensional states, probably originating from the edge-states of Bi(110) islands. Our results provide a much needed experimental insight into the electronic and structural properties of Bi(110) films.

Evidence for higher order topology in Bi and Bi0.92Sb0.08

Higher order topological insulators (HOTIs) are a new class of topological materials which host protected states at the corners or hinges of a crystal. HOTIs provide an intriguing alternative platform for helical and chiral edge states and Majorana modes, but there are very few known materials in this class. Recent studies have proposed Bi as a potential HOTI, however, its topological classification is not yet well accepted. In this work, we show that the (110) facets of Bi and BiSb alloys can be used to unequivocally establish the topology of these systems. Bi and Bi_0.92Sb_0.08 (110) films were grown on silicon substrates using molecular beam epitaxy and studied by scanning tunneling spectroscopy. The surfaces manifest rectangular islands which show localized hinge states on three out of the four edges, consistent with the theory for the HOTI phase. This establishes Bi and Bi_0.92Sb_0.08 as HOTIs, and raises questions about the topological classification of the full family of Bi_xSb_1−x alloys.

The experimental realization of higher order topological insulator (HOTI) in solid state materials remains elusive. Here, Aggarwal et al. reveal hinge states on three edges of both Bi and Bi_0.92Sb_0.08 (110) islands, supporting them as a class of HOTI.

Atomically Thin Quantum Spin Hall Insulators

Atomically thin topological materials are attracting growing attention for their potential to radically transform classical and quantum electronic device concepts. Among them is the quantum spin Hall (QSH) insulator-a 2D state of matter that arises from interplay of topological band inversion and strong spin-orbit coupling, with large tunable bulk bandgaps up to 800 meV and gapless, 1D edge states. Reviewing recent advances in materials science and engineering alongside theoretical description, the QSH materials library is surveyed with focus on the prospects for QSH-based device applications. In particular, theoretical predictions of nontrivial superconducting pairing in the QSH state toward Majorana-based topological quantum computing are discussed, which are the next frontier in QSH materials research.

Scanning tunneling spectroscopy studies of topological materials

Topological materials have become promising materials for next-generation devices by utilizing their exotic electronic states. Their exotic states caused by spin-orbital coupling usually locate on the surfaces or at the edges. Scanning tunneling spectroscopy (STS) is a powerful tool to reveal the local electronic structures of condensed matters. Therefore, STS provides us with an almost perfect method to access the exotic states of topological materials. In this topical review, we report the current investigations by several methods based on the STS technique for layered topological material from transition metal dichalcogenide Weyl semimetals (WTe₂ and MoTe₂) to two dimensional topological insulators (layered bismuth and silicene). The electronic characteristics of these layered topological materials are experimentally identified.

Giant Rashba splitting in one-dimensional atomic tellurium chains

The search for a one-dimensional (1D) system with purely 1D bands and strong Rashba spin splitting is essential for the realization of Majorana fermions and spin transport but presents a fundamental challenge to date. Herein, using first-principles calculations, we demonstrated that atomic Tellurium (Te) chains exhibit purely 1D bands and giant Rashba spin splitting, and their splitting parameters depend strongly on strain and structure distortion. This phenomenon stems from the helical structure of atomic Te chains, which can not only sustain significant strain but also realize the synergy of orbital angular momentum and in-chain potential gradient in enhancing spin splitting. The structure distortion of stretched helical Te chains is critical to execute this synergy, generating a large Rashba spin splitting among the known systems. Our findings proposed a potential 1D giant Rashba splitting system for exploring spintronics and Majorana fermions, and provide routes for engineering spin splitting in other materials.

Resolving the topological classification of bismuth with topological defects

The growing diversity of topological classes leads to ambiguity between classes that share similar boundary phenomenology. This is the status of bulk bismuth. Recent studies have classified it as either a strong or a higher-order topological insulator, both of which host helical modes on their boundaries. We resolve the topological classification of bismuth by spectroscopically mapping the response of its boundary modes to a screw-dislocation. We find that the one-dimensional mode, on step-edges, extends over a wide energy range and does not open a gap near the screw-dislocations. This signifies that this mode binds to the screw-dislocation, as expected for a material with nonzero weak indices. We argue that the small energy gap, at the time reversal invariant momentum L, positions bismuth within the critical region of a topological phase transition between a higher-order topological insulator and a strong topological insulator with nonzero weak indices.

Band Topology of Bismuth Quantum Films

Bismuth has been the key element in the discovery and development of topological insulator materials. Previous theoretical studies indicated that Bi is topologically trivial and it can transform into the topological phase by alloying with Sb. However, recent high-resolution angle-resolved photoemission spectroscopy (ARPES) measurements strongly suggested a topological band structure in pure Bi, conflicting with the theoretical results. To address this issue, we studied the band structure of Bi and Sb films by ARPES and first-principles calculations. The quantum confinement effectively enlarges the energy gap in the band structure of Bi films and enables a direct visualization of the Z 2 topological invariant of Bi. We find that Bi quantum films in topologically trivial and nontrivial phases respond differently to surface perturbations. This way, we establish experimental criteria for detecting the band topology of Bi by spectroscopic methods.

Topological electronic structure and Rashba effect in Bi thin layers: theoretical predictions and experiments

The goal of the present review is to cross-compare theoretical predictions with selected experimental results on bismuth thin films exhibiting topological properties and a strong Rashba effect. The theoretical prediction that a single free-standing Bi(1 1 1) bilayer is a topological insulator has triggered a large series of studies of ultrathin Bi(1 1 1) films grown on various substrates. Using selected examples we review theoretical predictions of atomic and electronic structure of Bi thin films exhibiting topological properties due to interaction with a substrate. We also survey experimental signatures of topological surface states and Rashba effect, as obtained mostly by angle- and spin-resolved photoelectron spectroscopy.

Higher-Order Topology in Bismuth

The mathematical field of topology has become a framework to describe the low-energy electronic structure of crystalline solids. A typical feature of a bulk insulating three-dimensional topological crystal are conducting two-dimensional surface states. This constitutes the topological bulk-boundary correspondence. Here, we establish that the electronic structure of bismuth, an element consistently described as bulk topologically trivial, is in fact topological and follows a generalized bulk-boundary correspondence of higher-order: not the surfaces of the crystal, but its hinges host topologically protected conducting modes. These hinge modes are protected against localization by time-reversal symmetry locally, and globally by the three-fold rotational symmetry and inversion symmetry of the bismuth crystal. We support our claim theoretically and experimentally. Our theoretical analysis is based on symmetry arguments, topological indices, first-principle calculations, and the recently introduced framework of topological quantum chemistry. We provide supporting evidence from two complementary experimental techniques. With scanning-tunneling spectroscopy, we probe the unique signatures of the rotational symmetry of the one-dimensional states located at step edges of the crystal surface. With Josephson interferometry, we demonstrate their universal topological contribution to the electronic transport. Our work establishes bismuth as a higher-order topological insulator.

Ultrathin Bismuth Film on 1T-TaS2: Structural Transition and Charge-Density-Wave Proximity Effect

We have fabricated bismuth (Bi) ultrathin films on a charge-density-wave (CDW) compound 1T-TaS₂ and elucidated electronic states by angle-resolved photoemission spectroscopy and first-principles band-structure calculations. We found that the Bi film on 1T-TaS₂ undergoes a structural transition from (111) to (110) upon reducing the film thickness, accompanied by a drastic change in the energy band structure. We also revealed that while two-bilayer-thick Bi(110) film on Si(111) is characterized by a dispersive band touching the Fermi level ( E_F), the energy band of the same film on 1T-TaS₂ exhibits holelike dispersion with a finite energy gap at E_F. We discuss the origin of such intriguing differences in terms of the CDW proximity effect.

Ballistic edge states in Bismuth nanowires revealed by SQUID interferometry

The protection against backscattering provided by topology is a striking property. In two-dimensional insulators, a consequence of this topological protection is the ballistic nature of the one-dimensional helical edge states. One demonstration of ballisticity is the quantized Hall conductance. Here we provide another demonstration of ballistic transport, in the way the edge states carry a supercurrent. The system we have investigated is a micrometre-long monocrystalline bismuth nanowire with topological surfaces, that we connect to two superconducting electrodes. We have measured the relation between the Josephson current flowing through the nanowire and the superconducting phase difference at its ends, the current–phase relation. The sharp sawtooth-shaped phase-modulated current–phase relation we find demonstrates that transport occurs selectively along two ballistic edges of the nanowire. In addition, we show that a magnetic field induces 0–π transitions and φ₀-junction behaviour, providing a way to manipulate the phase of the supercurrent-carrying edge states and generate spin supercurrents.

Demonstration of ballistic conduction and spin polarization of edge state currents in two dimensional topological insulators remains a challenge. Here, Murani et al. report a direct signature of ballistic one dimensional transport along the topological surfaces of a bismuth nanowire connected to superconducting electrodes.

Bismuthene on a SiC substrate: A candidate for a high-temperature quantum spin Hall material

Quantum spin Hall materials hold the promise of revolutionary devices with dissipationless spin currents but have required cryogenic temperatures owing to small energy gaps. Here we show theoretically that a room-temperature regime with a large energy gap may be achievable within a paradigm that exploits the atomic spin-orbit coupling. The concept is based on a substrate-supported monolayer of a high-atomic number element and is experimentally realized as a bismuth honeycomb lattice on top of the insulating silicon carbide substrate SiC(0001). Using scanning tunneling spectroscopy, we detect a gap of ~0.8 electron volt and conductive edge states consistent with theory. Our combined theoretical and experimental results demonstrate a concept for a quantum spin Hall wide-gap scenario, where the chemical potential resides in the global system gap, ensuring robust edge conductance.

Orbital angular momentum analysis for giant spin splitting in solids and nanostructures

Giant spin splitting (GSS) of electronic bands, which is several orders of magnitude greater than the standard Rashba effect has been observed in various systems including noble-metal surfaces and thin films of transition-metal dichalcogenides. Previous studies reported that orbital angular momentum (OAM) is not quenched in some GSS materials and that the atomic spin-orbit interaction (SOI) generates spin splitting in some solid states via the interorbital hopping. Although the unquenched OAM may be closely related to the interorbital hopping, their relationship is hardly studied in the aspect of using the unquenched OAM as a control parameter of GSS. Here, we analyze OAM in GSS materials by using the interorbital-hopping mechanism and first-principles calculations. We report that the interatomic hopping between different-parity orbitals, which is generated by specific broken mirror symmetry, produces k-dependent OAM, resulting in valley-dependent GSS in WSe₂ monolayer, Rashba-type GSS in Au (111) surface, and Dresselhaus-type GSS in bulk HgTe. We also demonstrate systematic control of OAM by pressure, external fields, and substrates, thereby controlling the spin splitting, and discuss the temperature dependence of OAM. Our results provide a simplified picture for systematic design and control of GSS materials.

Resolving the one-dimensional singularity edge states of Bi(1 1 1) thin films

We report our investigation on the electronic properties of the step edges on a Bi(1 1 1) surface in epitiaxially grown thin films, using scanning tunneling microscopy and spectroscopy. Our results show three differential conductance peaks including the previously reported peak in the spectra recorded at the step edges. Our analysis indicates that all of the three peaks can be ascribed to the van Hove singularities and thus to the band extrema of the one-dimensional edge bands, according to the quasiparticle interference and the Fourier transform patterns. These edge states show an overall penetration length of about 5 nm, but they also show different spatial distributions perpendicular to the edge. The well-determined band extrema may provide information for establishing a better model to describe the electronic topology of the step edge in the Bi(1 1 1) films.

Decay length of surface-state wave functions on Bi(1 1 1)

We calculate the decay length in surface normal direction of the surface-state wave functions on a clean Bi(1 1 1) surface as a function of two-dimensional (2D) wave vector [Formula: see text] along the [Formula: see text] line. For this purpose, we perform a first-principles density functional theory (DFT) calculation for semi-infinite Bi(1 1 1) by employing the surface embedded Green's function technique. The decay length of the two surface bands is found to be  ∼24 Bi bilayers at [Formula: see text], while it remains less than 5 Bi bilayers when [Formula: see text] is away from [Formula: see text] and [Formula: see text]. At [Formula: see text], the degenerate surface bands are split from the upper boundary energy of the projected bulk valence bands only by 5 meV. In spite of this, the decay length of these bands at [Formula: see text] is less than 10 Bi bilayers due to the large effective mass (small curvature) of the highest valence band in the surface normal direction.

Proving Nontrivial Topology of Pure Bismuth by Quantum Confinement

The topology of pure Bi is controversial because of its very small (∼10  meV) band gap. Here we perform high-resolution angle-resolved photoelectron spectroscopy measurements systematically on 14-202 bilayer Bi films. Using high-quality films, we succeed in observing quantized bulk bands with energy separations down to ∼10  meV. Detailed analyses on the phase shift of the confined wave functions precisely determine the surface and bulk electronic structures, which unambiguously show nontrivial topology. The present results not only prove the fundamental property of Bi but also introduce a capability of the quantum-confinement approach.

Topological phase transition and quantum spin Hall edge states of antimony few layers

While two-dimensional (2D) topological insulators (TI’s) initiated the field of topological materials, only very few materials were discovered to date and the direct access to their quantum spin Hall edge states has been challenging due to material issues. Here, we introduce a new 2D TI material, Sb few layer films. Electronic structures of ultrathin Sb islands grown on Bi₂Te₂Se are investigated by scanning tunneling microscopy. The maps of local density of states clearly identify robust edge electronic states over the thickness of three bilayers in clear contrast to thinner islands. This indicates that topological edge states emerge through a 2D topological phase transition predicted between three and four bilayer films in recent theory. The non-trivial phase transition and edge states are confirmed for epitaxial films by extensive density-functional-theory calculations. This work provides an important material platform to exploit microscopic aspects of the quantum spin Hall phase and its quantum phase transition.

Topological phases in two-dimensional materials: a review

Topological phases with insulating bulk and gapless surface or edge modes have attracted intensive attention because of their fundamental physics implications and potential applications in dissipationless electronics and spintronics. In this review, we mainly focus on recent progress in the engineering of topologically nontrivial phases (such as [Formula: see text] topological insulators, quantum anomalous Hall effects, quantum valley Hall effects etc) in two-dimensional systems, including quantum wells, atomic crystal layers of elements from group III to group VII, and the transition metal compounds.

Prediction of topological phase transition in X2–SiGe monolayers

Quantum spin Hall (QSH) insulators exhibit a bulk insulting gap and metallic edge states characterized by nontrivial topology.

Role of Quantum and Surface-State Effects in the Bulk Fermi-Level Position of Ultrathin Bi Films

We performed high-resolution photon-energy and polarization-dependent ARPES measurements on ultrathin Bi(111) films [6-180 bilayers (BL), 2.5-70 nm thick] formed on Si(111). In addition to the extensively studied surface states (SSs), the edge of the bulk valence band was clearly measured by using S-polarized light. We found direct evidence that this valence band edge, which forms a hole pocket in the bulk Bi crystal, does not cross the Fermi level for the 180 BL thick film. This is consistent with the predicted semimetal-to-semiconductor transition due to the quantum-size effect [V.B. Sandomirskii, Sov. Phys. JETP 25, 101 (1967)]. However, it became metallic again when the film thickness was decreased (below 30 BL). A plausible explanation for this phenomenon is the modification of the charge neutrality condition due to the size effect of the SSs.