Formation of quantum spin Hall state on Si surface and energy gap scaling with strength of spin orbit coupling

Prediction of Nontrivial Topological Phases and Rashba Spin-Splitting in BaABTe4 Janus Monolayers (A, B = Al, Ga, In, or Tl)

Two-dimensional (2D) materials have emerged as a significant focus in materials research due to their tunable properties on thermoelectricity, spin-splitting, and nontrivial topology. Specifically, Janus-type 2D materials are interesting due to their additional breaking of inversion or mirror symmetry in the atomic structure. Based on the recently synthesized monolayer MoSi2N4 and previously studied BaIn2Te4 with the chemical formula of MA2Z4, we derive a family of 2D Janus compounds, namely BaABTe4. Using first-principles calculations, a total of six Janus BaABTe4 monolayers (BaAlGaTe4, BaAlInTe4, BaAlTlTe4, BaGaInTe4, BaGaTlTe4, and BaInTlTe4) were investigated for their dynamical stability, electronic, and topological properties. Notably, the Z2 topological invariant calculated using HSE06 hybrid functional reveals that three out of the six monolayers (BaAlGaTe4, BaAlTlTe4, and BaInTlTe4) have nontrivial topological phases, with BaInTlTe4 exhibiting the largest positive system band gap of 17 meV. These three topological monolayers were further confirmed to be dynamically stable based on phonon dispersion and formation energy calculations. Subsequent orbital analysis of BaInTlTe4 showed that the spin-orbit coupling effect drives the topological phase transition, resulting in the band inversion between the s-orbital of In + Tl and p x + p y -orbitals of Te around Γ. Also, the presence of the gapless edge states confirmed the nontrivial topological property. The Janus monolayers were found to exhibit significant Rashba spin-splitting except BaAlInTe4. The topologically nontrivial BaAlTlTe4 has the strongest Rashba strength of αK-Γ= αΓ-M = 1.03 eVÅ. Our results show that the coexisting nature of the nontrivial phase and Rashba-type splitting within the BaABTe4 Janus monolayers might apply to spintronics.

Topological quantum devices: a review

The introduction of the concept of topology into condensed matter physics has greatly deepened our fundamental understanding of transport properties of electrons as well as all other forms of quasi particles in solid materials. It has also fostered a paradigm shift from conventional electronic/optoelectronic devices to novel quantum devices based on topology-enabled quantum device functionalities that transfer energy and information with unprecedented precision, robustness, and efficiency. In this article, the recent research progress in topological quantum devices is reviewed. We first outline the topological spintronic devices underlined by the spin-momentum locking property of topology. We then highlight the topological electronic devices based on quantized electron and dissipationless spin conductivity protected by topology. Finally, we discuss quantum optoelectronic devices with topology-redefined photoexcitation and emission. The field of topological quantum devices is only in its infancy, we envision many significant advances in the near future.

High-Yield Production of Quantum Corrals in a Surface Reconstruction Pattern

The power of surface chemistry to create atomically precise nanoarchitectures offers intriguing opportunities to advance the field of quantum technology. Strategies for building artificial electronic lattices by individually positioning atoms or molecules result in precisely tailored structures but lack structural robustness. Here, taking the advantage of strong bonding of Br atoms on noble metal surfaces, we report the production of stable quantum corrals by dehalogenation of hexabromobenzene molecules on a preheated Au(111) surface. The byproducts, Br adatoms, are confined within a new surface reconstruction pattern and aggregate into nanopores with an average size of 3.7 ± 0.1 nm, which create atomic orbital-like quantum resonance states inside each corral due to the interference of scattered electron waves. Remarkably, the atomic orbitals can be hybridized into molecular-like orbitals with distinct bonding and antibonding states. Our study opens up an avenue to fabricate quantum structures with high yield and superior robustness.

Tunable Mott Dirac and Kagome Bands Engineered on 1T-TaS2

Strongly interacting electrons in hexagonal and kagome lattices exhibit rich phase diagrams of exotic quantum states, including superconductivity and correlated topological orders. However, material realizations of these electronic states have been scarce in nature or by design. Here, we theoretically propose an approach to realize artificial lattices by metal adsorption on a 2D Mott insulator 1T-TaS₂. Alkali, alkaline-earth, and group 13 metal atoms are deposited in (√3 × √3)R30° and 2 × 2 TaS₂ superstructures of honeycomb- and kagome-lattice symmetries exhibiting Dirac and kagome bands, respectively. The strong electron correlation of 1T-TaS₂ drives the honeycomb and kagome systems into correlated topological phases described by Kane-Mele-Hubbard and kagome-Hubbard models. We further show that the 2/3 or 3/4 band filling of Mott Dirac and flat bands can be achieved with a proper concentration of Mg adsorbates. Our proposal may be readily implemented in experiments, offering an attractive condensed-matter platform to exploit the interplay of correlated topological order and superconductivity.

Robust Tunable Large-Gap Quantum Spin Hall States in Monolayer Cu2S on Insulating Substrates

Quantum spin Hall (QSH) insulators with large band gaps and dissipationless edge states are of both technological and scientific interest. Although numerous two-dimensional (2D) systems have been predicted to host the QSH phase, very few of them harbor large band gaps and retain their nontrivial band topology when they are deposited on substrates. Here, based on a first-principles analysis with hybrid functional calculations, we investigated the electronic and topological properties of inversion-asymmetric monolayer copper sulfide (Cu₂S). Interestingly, we found that monolayer Cu₂S possesses an intrinsic QSH phase, Rashba spin splitting, and a large band gap of 220 meV that is suitable for room-temperature applications. Most importantly, we constructed heterostructures of a Cu₂S film on PtTe₂, h-BN, and Cu(111) substrates and found that the topological properties remain preserved upon an interface with these substrates. Our findings suggest Cu₂S as a possible platform to realize inversion-asymmetric QSH insulators with potential applications in low-dissipation electronic devices.

Epitaxial Growth of Ultraflat Bismuthene with Large Topological Band Inversion Enabled by Substrate-Orbital-Filtering Effect

Quantum spin Hall (QSH) systems hold promises of low-power-consuming spintronic devices, yet their practical applications are extremely impeded by the small energy gaps. Fabricating QSH materials with large gaps, especially under the guidance of design principles, is essential for both scientific research and practical applications. Here, we demonstrate that large on-site atomic spin-orbit coupling can be directly exploited via the intriguing substrate-orbital-filtering effect to generate large-gap QSH systems and experimentally realized on the epitaxially synthesized ultraflat bismuthene on Ag(111). Theoretical calculations reveal that the underlying substrate selectively filters Bi p_z orbitals away from the Fermi level, leading p_xy orbitals with nonzero magnetic quantum numbers, resulting in large topological gap of ∼1 eV at the K point. The corresponding topological edge states are identified through scanning tunneling spectroscopy combined with density functional theory calculations. Our findings provide general strategies to design large-gap QSH systems and further explore their topology-related physics.

Engineering of flat bands and Dirac bands in two-dimensional covalent organic frameworks (COFs): relationships among molecular orbital symmetry, lattice symmetry, and electronic-structure characteristics

Two-dimensional covalent organic frameworks (2D-COFs), also referred to as 2D polymer networks, display unusual electronic-structure characteristics, which can significantly enrich and broaden the fields of electronics and spintronics. In this Focus article, our objective is to lay the groundwork for the conceptual description of the fundamental relationships among the COF electronic structures, the symmetries of their 2D lattices, and the frontier molecular orbitals (MOs) of their core and linker components. We focus on monolayers of hexagonal COFs and use tight-binding model analyses to highlight the critical role of the frontier-MO symmetry, in addition to lattice symmetry, in determining the nature of the electronic bands near the Fermi level. We rationalize the intriguing feature that, when the core unit has degenerate highest occupied MOs [or lowest unoccupied MOs], the COF highest valence band [or lowest conduction band] is flat but degenerate with a dispersive band at a high-symmetry point of the Brillouin zone; the consequences of having such band characteristics are briefly described. Multi-layer and bulk 2D COFs are found to maintain the salient features of the monolayer electronic structures albeit with a reduced bandgap due to the interlayer coupling. This Focus article is thus meant to provide an effective framework for the engineering of flat and Dirac bands in 2D polymer networks.

Selective Substrate-Orbital-Filtering Effect to Realize the Large-Gap Quantum Spin Hall Effect

Although Pb harbors a strong spin-orbit coupling effect, pristine plumbene (the last group-IV cousin of graphene) hosts topologically trivial states. Based on first-principles calculations, we demonstrate that epitaxial growth of plumbene on the BaTe(111) surface converts the trivial Pb lattice into a quantum spin Hall (QSH) phase with a large gap of ∼0.3 eV via a selective substrate-orbital-filtering effect. Tight-binding model analyses show the p_z orbital in half of the Pb overlayer is selectively removed by the BaTe substrate, leaving behind a p_z-p_x,y band inversion. Based on the same working principle, the gap can be further increased to ∼0.5-0.6 eV by surface adsorption of H or halogen atoms that filters out the other half of the Pb p_z orbitals. The mechanism of selective substrate-orbital-filtering is general, opening an avenue to explore large-gap QSH insulators in heavy-metal-based materials. It is worth noting that plumbene has already been widely grown on various substrates experimentally.

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.

Functionalization induced quantum spin Hall to quantum anomalous Hall phase transition in monolayer jacutingaite

As novel states of quantum matter, quantum spin Hall (QSH) and quantum anomalous Hall (QAH) states have attracted considerable interest in condensed matter and materials science communities. Recently, a monolayer of the naturally occurring mineral jacutingaite (Pt2HgSe3), was theoretically proposed to be a large-gap QSH insulator and experimentally confirmed. Here, based on first-principles calculations and tight-binding modeling, we demonstrate QSH to QAH phase transition in jacutingaite by chemical functionalization with chalogen. We show that two-dimensional (2D) chalogenated jacutingaite, Pt2HgSe3-X (X = S, Se, Te), is ferromagnetic with Curie temperature up to 316 K, and it exhibits QAH effect with chiral edge states inside a sizeable topological gap. The physical mechanism lies in the adsorption induced transformation of the original Kane-Mele model into an effective four-band model involving (px, py) orbitals on a hexagonal lattice, so that the topological gap size can be controlled by spin-orbit coupling strength of the chalogen (0.28 eV for Pt2HgSe3-Te). These results not only show the promise of functionalization in orbital-engineering of 2D functional structures, but also provide an ideal and practical platform for achieving exotic topological phases for dissipationless transport and quantum computing.

Exotic Topological Bands and Quantum States in Metal-Organic and Covalent-Organic Frameworks

ConspectusMetal-organic and covalent-organic frameworks (MOFs/COFs) have been extensively studied for fundamental interests and their promising applications, taking advantage of their unique structural properties, i.e., high porosity and large surface-to-volume ratio. However, their electronic and magnetic properties have been somewhat overlooked because of their relatively poor performance as conductive and/or magnetic materials. Recent experimental breakthroughs in synthesizing two-dimensional (2D) π-conjugated MOFs/COFs with high conductivity and robust magnetism through doping have generated renewed and increasing interest in their electronic properties. Meanwhile, comprehensive theoretical studies of the underlying physical principles have led to discovery of many exotic quantum states, such as topological insulating states, which were only observed in inorganic systems. Especially, the diversity and high tunability of MOFs/COFs have provided a playground to explore novel quantum physics and quantum chemistry as well as promising applications.The band theory has empowered us to understand the most fundamental electronic properties of inorganic crystalline materials, which can also be used to better understand MOFs/COFs. The first obvious difference between the two is that instead of atomic orbitals residing at lattice sites of inorganic crystals, molecular orbitals of organic ligands are predominant in MOFs/COFs. The second key difference is that usually all atomic orbitals in an inorganic crystal are subject to one common group of lattice symmetry, while atomic orbitals of metal ion and molecular orbitals of different organic ligands in MOFs/COFs belong to different subgroups of lattice symmetries. Both these differences will impact the band structure of MOFs/COFs, in particular making it more complex. Consequently, which subset of bands are of most importance depends strongly on the location of Fermi level, i.e., electron counting and charge doping. Furthermore, there are usually two types of characteristic electrons coupled in MOFs, i.e., strongly correlated localized d and f electrons and diffusive s and p electrons, which interplay with lattice, orbital, and spin degrees of freedom, leading to more exotic topological and magnetic band structures.In this Account, we present an up-to-date review of recent theoretical developments to better understand the exotic band structures of MOFs/COFs. Starting from three fundamental 2D lattice models, i.e., honeycomb, Kagome, and Lieb lattices, exotic Dirac and flat bands as well as the intriguing topological quantum states they host, e.g., quantum spin Hall and quantum anomalous Hall states, are outlined. In addition to the single-lattice models, we further elaborate on combined lattice model Hamiltonians, which give rise to overlapping bands hosting novel quantum states, such as nodal-line Dirac semimetal and unconventional superconducting states. Also, first-principles predictions of candidate MOFs/COFs that host these exotic bands and hence quantum phases are reviewed, which greatly extends the pool of materials beyond inorganic crystals for hosting exotic band structures.

Atomic, electronic and transport properties of In-Au 2D compound on Si(1 0 0)

Two-dimensional (In, Au)/Si(1 0 0)c(2 [Formula: see text] 2) compound was synthesized and its atomic arrangement, electron band structure and low-temperature transport properties were characterized using scanning tunneling microscopy, angle-resolved photoelectron spectroscopy and four-point-probe resistivity measurements assisted with first-principles density-functional-theory calculations. The present results are compared to those obtained earlier for the parent (Tl, Au)/Si(1 0 0)c(2 [Formula: see text] 2) system.

π-Orbital Yin–Yang Kagome bands in anilato-based metal–organic frameworks

π-Orbital Yin–Yang Kagome bands consisting of two flat bands with opposite Chern numbers have been disclosed in anilato-based metal–organic frameworks.

Topological phase transition induced by px,y and pz band inversion in a honeycomb lattice

The search for more types of band inversion-induced topological states is of great scientific and experimental interest. Here, we proposed that the band inversion between p_x,y and p_z orbitals can produce a topological phase transition in honeycomb lattices based on tight-binding model analyses. The corresponding topological phase diagram was mapped out in the parameter space of orbital energy and spin-orbit coupling. Specifically, the quantum anomalous Hall (QAH) effect could be achieved when ferromagnetism was introduced. Moreover, our first-principles calculations demonstrated that the two systems of half-iodinated silicene (Si₂I) and one-third monolayer of bismuth epitaxially grown on the Si(111)-√3 ×√3 surface are ideal candidates for realizing the QAH effect with Curie temperatures of ∼101 K and 118 K, respectively. The underlying physical mechanism of this scheme is generally applicable, offering broader opportunities for the exploration of novel topological states and high-temperature QAH effect systems.

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.

Electronic properties of the two-dimensional (Tl, Rb)/Si(1 1 1)[Formula: see text] compound having a honeycomb-like structure

Heavy metal layers having a honeycomb structure on the Si(1 1 1) surface were theoretically predicted to show prospects for possessing properties of the quantum spin Hall (QSH) insulators. The (Tl, Rb)/Si(1 1 1)[Formula: see text] atomic-layer compound synthesized in the present work is the first real system of such type, where atoms of heavy metal Tl are arranged into the honeycomb structure stabilized by Rb atoms occupying the centers of the honeycomb units. Electronic properties of the (Tl, Rb)/Si(1 1 1)[Formula: see text] compound has been fully characterized experimentally and theoretically and compared with those of the hypothetical (Tl, H)/Si(1 1 1)[Formula: see text] prototype system. It is concluded that the QSH-insulator properties of the Tl-honeycomb layers on Si(1 1 1) surface are dictated by the stable adsorption sites occupied by Tl atoms which, in turn, are controlled by the atom species centering the Tl honeycombs. As a result, the real (Tl, Rb)/Si(1 1 1)[Formula: see text] compound where Tl atoms occupy the T₄ sites does not possess QSH-insulator properties in contrast to the hypothetical (Tl, H)/Si(1 1 1)[Formula: see text] system where Tl atoms reside in the T₁ (on-top) sites and it shows up as a QSH material.

Quantum Spin Hall Effect and Spin Bott Index in a Quasicrystal Lattice

Despite the rapid progress in the field of the quantum spin Hall (QSH) effect, most of the QSH systems studied up to now are based on crystalline materials. Here we propose that the QSH effect can be realized in quasicrystal lattices (QLs). We show that the electronic topology of aperiodic and amorphous insulators can be characterized by a spin Bott index B_{s}. The nontrivial QSH state in a QL is identified by a nonzero spin Bott index B_{s}=1, associated with robust edge states and quantized conductance. We also map out a topological phase diagram in which the QSH state lies in between a normal insulator and a weak metal phase due to the unique wave functions of QLs. Our findings not only provide a better understanding of electronic properties of quasicrystals but also extend the search of the QSH phase to aperiodic and amorphous materials that are experimentally feasible.

Strain-Induced Quantum Spin Hall Effect in Two-Dimensional Methyl-Functionalized Silicene SiCH₃

Quantum Spin Hall (QSH) has potential applications in low energy consuming spintronic devices and has become a researching hotspot recently. It benefits from insulators feature edge states, topologically protected from backscattering by time-reversal symmetry. The properties of methyl functionalized silicene (SiCH₃) have been investigated using first-principles calculations, which show QSH effect under reasonable strain. The origin of the topological characteristic of SiCH₃, is mainly associated with the s-pxy orbitals band inversion at Γ point, whilst the band gap appears under the effect of spin-orbital coupling (SOC). The QSH phase of SiCH₃ is confirmed by the topological invariant Z₂ = 1, as well as helical edge states. The SiCH₃ supported by hexagonal boron nitride (BN) film makes it possible to observe its non-trivial topological phase experimentally, due to the weak interlayer interaction. The results of this work provide a new potential candidate for two-dimensional honeycomb lattice spintronic devices in spintronics.

Two-Dimensional In-Sb Compound on Silicon as a Quantum Spin Hall Insulator

Two-dimensional (2D) topological insulator is a promising quantum phase for achieving dissipationless transport due to the robustness of the gapless edge states resided in the insulating gap providing realization of the quantum spin Hall effect. Searching for two-dimensional realistic materials that are able to provide the quantum spin Hall effect and possessing the feasibility of their experimental preparation is a growing field. Here we report on the two-dimensional (In, Sb)2[Formula: see text]2[Formula: see text] compound synthesized on Si(111) substrate and its comprehensive experimental and theoretical investigations based on an atomic-scale characterization by using scanning tunneling microscopy and angle-resolved photoelectron spectroscopy as well as ab initio density functional theory calculations identifying the synthesized 2D compound as a suitable system for realization of the quantum spin Hall effect without additional functionalization like chemical adsorption, applying strain, or gating.

Quantum anomalous/valley Hall effect and tunable quantum state in hydrogenated arsenene decorated with a transition metal

The quantum anomalous Hall (QAH) effect is superior to the quantum spin Hall (QSH) effect, which can avoid the inelastic scattering of two edge electrons located on one side of a topological nontrivial material, and thus it has attracted both theoretical and experimental interest. Here, we systematically investigate the lattice structures, and electronic and magnetic properties of hydrogenated arsenene decorated with certain transition metals (Cr, Mo and Cu) based on density-functional theory. A unique QAH effect in Mo@AsH is predicted, whose Chern number (C = 1) indicates only one chiral edge channel located on its one side. Then, we prove that this QAH effect realization is closely related with band inversion, which is the competitive result between its spin-orbit coupling (SOC) strength and exchange field. The quantum state of Mo@AsH can also be tuned by an external strain, similar to SOC, and it is noted that its increased topological gap of about 35 meV under 5.0% tensile strain, is large enough to realize the QAH effect at room-temperature. Additionally, the quantum valley Hall effect in Cu@AsH contributed by the inequality of AB sublattices is also found. Our results reveal the physical mechanism to realize the QAH effect in TM@AsH and provide a platform for electrically controllable topological states, which are highly desirable for nanoelectronics and spintronics.

Formation of a large gap quantum spin Hall phase in a 2D trigonal lattice with three p-orbitals

The quantum spin Hall (QSH) phase in a trigonal lattice requires typically a minimal basis of three orbitals with one even parity s and two odd parity p orbitals. Here, based on first-principles calculations combined with tight-binding model analyses and calculations, we demonstrate that depositing 1/3 monolayer Bi or Te atom layers on an existing experimental Ag/Si(111) surface can produce a QSH phase readily but with three p-orbitals (p_x, p_y and p_z). The essential mechanism can be understood by the fact while in 3D, the p_z orbital has an odd parity, its parity becomes even when it is projected onto a 2D surface so as to act in place of the s orbital in the original minimum basis. Furthermore, non-trivial large gaps, i.e., 275.0 meV for Bi and 162.5 meV for Te systems, arise from a spin-orbit coupling induced quadratic p_x-p_y band opening at the Γ point. Our findings will significantly expand the search for a substrate supported QSH phase with a large gap, especially in the Si surface, to new orbital combinations and hence new elements.

First-principles prediction of a giant-gap quantum spin Hall insulator in Pb thin film

The quantum spin Hall (QSH) effect is promising for achieving dissipationless transport devices due to the robust gapless states inside the insulating bulk gap. However, QSH insulators currently suffer from requiring extremely high vacuums or low temperatures. Here, using first-principles calculations, we predict cyanogen-decorated plumbene (PbCN) to be a new QSH phase, with a large gap of 0.92 eV, that is robust and tunable under external strain. The band topology mainly stems from s-p_xy band inversion related to the lattice symmetry, while the strong spin-orbit coupling (SOC) of the Pb atoms only opens a large gap. When halogen atoms are incorporated into PbCN, the resulting inversion-asymmetric PbF_x(CN)_1-x can host the QSH effect, accompanied by the presence of a sizable Rashba spin splitting at the top of the valence band. Furthermore, the Te(111)-terminated BaTe surface is proposed to be an ideal substrate for experimental realization of these monolayers, without destroying their nontrivial topology. These findings provide an ideal platform to enrich topological quantum phenomena and expand the potential applications in high-temperature spintronics.

Nontrivial topology and topological phase transition in two-dimensional monolayer Tl

Topological insulating material with dissipationless edge states is a rising star in spintronics.

Evolution of the topological properties of two-dimensional group IVA materials and device design

Two-dimensional group IVA materials (graphene, silicene, germanene, stanene, and plumbene) are promising candidates for realization of the quantum spin Hall effect and for future device applications.

An sd2 hybridized transition-metal monolayer with a hexagonal lattice: reconstruction between the Dirac and kagome bands

Graphene-like two-dimensional materials have garnered tremendous interest as emerging device materials due to their remarkable properties. However, their applications in spintronics have been limited by the lack of intrinsic magnetism. Here, we perform an ab initio simulation on the structural and electronic properties of several transition-metal (TM) monolayers (TM = Cr, Mo and W) with a honeycomb lattice on a 1/3 monolayer Cl-covered Si(111) surface. Due to the template effect from the halogenated Si substrate, the TM-layers will be maintained in an expanded lattice which is nearly 60% larger than that of the freestanding case. All these isolated TM-layers exhibit ferromagnetic coupling with kagome band structures related to sd² hybridization and a strong interfacial interaction may destroy the topological bands. Interestingly, the W-monolayer on the Cl-covered Si substrate shows a half-metallic behavior. A Dirac point formed at the K point in the spin-down channel is located exactly at the Fermi level which is crucial for the realization of a quantum spin Hall state. Moreover, the reconstruction process between the Dirac and kagome bands is discussed in detail, providing an interesting platform to study the interplay between massless Dirac fermions and heavy fermions.

Quantum spin Hall phase in 2D trigonal lattice

The quantum spin Hall (QSH) phase is an exotic phenomena in condensed-matter physics. Here we show that a minimal basis of three orbitals (s, p_x, p_y) is required to produce a QSH phase via nearest-neighbour hopping in a two-dimensional trigonal lattice. Tight-binding model analyses and calculations show that the QSH phase arises from a spin–orbit coupling (SOC)-induced s–p band inversion or p–p bandgap opening at Brillouin zone centre (Γ point), whose topological phase diagram is mapped out in the parameter space of orbital energy and SOC. Remarkably, based on first-principles calculations, this exact model of QSH phase is shown to be realizable in an experimental system of Au/GaAs(111) surface with an SOC gap of ∼73 meV, facilitating the possible room-temperature measurement. Our results will extend the search for substrate supported QSH materials to new lattice and orbital types.

Whilst different models describing the two-dimensional quantum spin Hall effect exist, very few experimental systems have been realized in which to test theory. Here, the authors present a discrete trigonal lattice model for the quantum spin Hall effect and predict its realization in Au/GaAs(111).

Exploring Ag(111) Substrate for Epitaxially Growing Monolayer Stanene: A First-Principles Study

Stanene, a two-dimensional topological insulator composed of Sn atoms in a hexagonal lattice, is a promising contender to Si in nanoelectronics. Currently it is still a significant challenge to achieve large-area, high-quality monolayer stanene. We explore the potential of Ag(111) surface as an ideal substrate for the epitaxial growth of monolayer stanene. Using first-principles calculations, we study the stability of the structure of stanene in different epitaxial relations with respect to Ag(111) surface, and also the diffusion behavior of Sn adatom on Ag(111) surface. Our study reveals that: (1) the hexagonal structure of stanene monolayer is well reserved on Ag(111) surface; (2) the height of epitaxial stanene monolayer is comparable to the step height of the substrate, enabling the growth to cross the surface step and achieve a large-area stanene; (3) the perfect lattice structure of free-standing stanene can be achieved once the epitaxial stanene monolayer is detached from Ag(111) surface; and finally (4) the diffusion barrier of Sn adatom on Ag(111) surface is found to be only 0.041 eV, allowing the epitaxial growth of stanene monolayer even at low temperatures. Our above revelations strongly suggest that Ag(111) surface is an ideal candidate for growing large-area, high-quality monolayer stanene.

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.

Robust Room-Temperature Quantum Spin Hall Effect in Methyl-functionalized InBi honeycomb film

Two-dimensional (2D) group-III-V honeycomb films have attracted significant interest for their potential application in fields of quantum computing and nanoelectronics. Searching for 2D III-V films with high structural stability and large-gap are crucial for the realizations of dissipationless transport edge states using quantum spin Hall (QSH) effect. Based on first-principles calculations, we predict that the methyl-functionalized InBi monolayer (InBiCH₃) has no dynamic instability, and hosts QSH state with a band gap as large as 0.29 eV, exhibiting an interesting electronic behavior viable for room-temperature applications. The topological characteristic is confirmed by s-p_xy band inversion, topological invariant Z₂ number, and the time-reversal symmetry protected helical edge states. Noticeably, the QSH states are tunable and robust against the mechanical strain, electric field and different levels of methyl coverages. We also find that InBiCH₃ supported on h-BN substrate maintains a nontrivial QSH state, which harbors the edge states lying within the band gap of substrate. These findings demonstrate that the methyl-functionalized III-V films may be a good QSH platform for device design and fabrication in spintronics.

Formation of a quantum spin Hall state on a Ge(111) surface

Using first-principles density functional theory (DFT) hybrid functional calculations, we demonstrate the formation of a quantum spin Hall (QSH) state on a Ge(111) surface. We show that a 1/3 monolayer (ML) Cl-covered Ge(111) surface offers an ideal template for metal, such as Bi, deposition into a stable hexagonal overlayer 2D lattice, which we refer to as Bi@Cl-Ge(111). The band structure and band topology of Bi@Cl-Ge(111) are analyzed with respect to the effect of spin-orbit coupling (SOC). The Bi@Cl-Ge(111) exhibits a QSH state with a band gap of 0.54 eV. In contrast, the Au@Cl-Ge(111) is found to be a trivial semiconducting surface. The Ge(111) substrate acts as an orbital filter to critically select the orbital composition around the Fermi level. Our findings offer another possible system for experimental exploration of the growth of 2D topological materials on conventional semiconductor substrates, where the 2D overlayer is atomically bonded to, but electronically decoupled from, the underlying substrate, exhibiting an isolated topological quantum state inside the substrate band gap.

Controllable band structure and topological phase transition in two-dimensional hydrogenated arsenene

Discovery of two-dimensional (2D) topological insulator such as group-V films initiates challenges in exploring exotic quantum states in low dimensions. Here, we perform first-principles calculations to study the geometric and electronic properties in 2D arsenene monolayer with hydrogenation (HAsH). We predict a new σ-type Dirac cone related to the px,y orbitals of As atoms in HAsH, dependent on in-plane tensile strain. Noticeably, the spin-orbit coupling (SOC) opens a quantum spin Hall (QSH) gap of 193 meV at the Dirac cone. A single pair of topologically protected helical edge states is established for the edges, and its QSH phase is confirmed with topological invariant Z2 = 1. We also propose a 2D quantum well (QW) encapsulating HAsH with the h-BN sheet on each side, which harbors a nontrivial QSH state with the Dirac cone lying within the band gap of cladding BN substrate. These findings provide a promising innovative platform for QSH device design and fabrication operating at room temperature.

Unexpected Giant-Gap Quantum Spin Hall Insulator in Chemically Decorated Plumbene Monolayer

Quantum spin Hall (QSH) effect of two-dimensional (2D) materials features edge states that are topologically protected from backscattering by time-reversal symmetry. However, the major obstacles to the application for QSH effect are the lack of suitable QSH insulators with a large bulk gap. Here, we predict a novel class of 2D QSH insulators in X-decorated plumbene monolayers (PbX; X = H, F, Cl, Br, I) with extraordinarily giant bulk gaps from 1.03 eV to a record value of 1.34 eV. The topological characteristic of PbX mainly originates from s-p(x,y) band inversion related to the lattice symmetry, while the effect of spin-orbital coupling (SOC) is only to open up a giant gap. Their QSH states are identified by nontrivial topological invariant Z2 = 1, as well as a single pair of topologically protected helical edge states locating inside the bulk gap. Noticeably, the QSH gaps of PbX are tunable and robust via external strain. We also propose high-dielectric-constant BN as an ideal substrate for the experimental realization of PbX, maintaining its nontrivial topology. These novel QSH insulators with giant gaps are a promising platform to enrich topological phenomena and expand potential applications at high temperature.

Room Temperature Quantum Spin Hall Insulator in Ethynyl-Derivative Functionalized Stanene Films

Quantum spin Hall (QSH) insulators feature edge states that topologically protected from backscattering. However, the major obstacles to application for QSH effect are the lack of suitable QSH insulators with a large bulk gap. Based on first-principles calculations, we predict a class of large-gap QSH insulators in ethynyl-derivative functionalized stanene (SnC2X; X = H, F, Cl, Br, I), allowing for viable applications at room temperature. Noticeably, the SnC2Cl, SnC2Br, and SnC2I are QSH insulators with a bulk gap of ~0.2 eV, while the SnC2H and SnC2F can be transformed into QSH insulator under the tensile strains. A single pair of topologically protected helical edge states is established for the edge of these systems with the Dirac point locating at the bulk gap, and their QSH states are confirmed with topological invariant Z2 = 1. The films on BN substrate also maintain a nontrivial large-gap QSH effect, which harbors a Dirac cone lying within the band gap. These findings may shed new light in future design and fabrication of large-gap QSH insulators based on two-dimensional honeycomb lattices in spintronics.

Quantum Spin Hall States in Stanene/Ge(111)

For topological insulators to be implemented in practical applications, it is a prerequisite to select suitable substrates that are required to leave insulators’ nontrivial properties and sizable opened band gaps (due to spin-orbital couplings) unaltered. Using ab initio calculations, we predict that Ge(111) surface qualified as a candidate to support stanene sheets, because the band structure of √3 × √3 stanene/Ge(111) (2 × 2) surface displays a typical Dirac cone at Γ point in the vicinity of the Fermi level. Aided with the result of Z₂ invariant calculations, a √3 × √3 stanene/Ge(111) (2 × 2) system has been proved to sustain the nontrivial topological phase, with the prove being confirmed by the edge state calculations of stanene ribbons. This finding can serve as guidance for epitaxial growth of stanene on substrate and render stanene feasible for practical use as a topological insulator.

Interplay between spin-orbit coupling and crystal-field effect in topological insulators

Band inversion, one of the key signatures of time-reversal invariant topological insulators (TIs), arises mostly due to the spin-orbit (SO) coupling. Here, based on ab initio density-functional calculations, we report a theoretical investigation of the SO-driven band inversion in isostructural bismuth and antimony chalcogenide TIs from the viewpoint of its interplay with the crystal-field effect. We calculate the SO-induced energy shift of states in the top valence and bottom conduction manifolds and reproduce this behavior using a simple one-atom model adjusted to incorporate the crystal-field effect. The crystal-field splitting is shown to compete with the SO coupling, that is, stronger crystal-field splitting leads to weaker SO band shift. We further show how both these effects can be controlled by changing the chemical composition, whereas the crystal-field splitting can be tuned by means of uniaxial strain. These results provide a practical guidance to the rational design of novel TIs as well as to controlling the properties of existing materials.

sd(2) Graphene: Kagome band in a hexagonal lattice

Graphene, made of sp^{2} hybridized carbon, is characterized with a Dirac band, representative of its underlying 2D hexagonal lattice. The fundamental understanding of graphene has recently spurred a surge in the search for 2D topological quantum phases in solid-state materials. Here, we propose a new form of 2D material, consisting of sd^{2} hybridized transition metal atoms in hexagonal lattice, called sd^{2} "graphene." The sd^{2} graphene is characterized by bond-centered electronic hopping, which transforms the apparent atomic hexagonal lattice into the physics of a kagome lattice that may exhibit a wide range of topological quantum phases. Based on first-principles calculations, room-temperature quantum anomalous Hall states with an energy gap of ∼0.1  eV are demonstrated for one such lattice made of W, which can be epitaxially grown on a semiconductor surface of 1/3 monolayer Cl-covered Si(111), with high thermodynamic and kinetic stability.