Spin-orbit Coupling Gaps Research Articles

High spin-Chern-number insulator in α-antimonene with a hidden topological phase

For a time-reversal symmetric system, the quantum spin Hall phase is assumed to be the same as the Z2 topological insulator phase in the existing literature. The spin Chern number Cs is presumed to yield the same topological classification as the Z2 invariant. Here, by investigating the electronic structures of monolayer α-phase group V elements, we uncover the presence of a topological phase in α-Sb, which can be characterized by a spin Chern number Cs = 2, even though it is Z2 trivial. Although α-As and Sb would thus be classified as trivial insulators within the classification schemes, we demonstrate the existence of a phase transition between α-As and Sb, which is induced by band inversions at two generic k points. Without spin–orbit coupling (SOC), α-As is a trivial insulator, while α-Sb is a Dirac semimetal with four Dirac points (DPs) located away from the high-symmetry lines. Inclusion of the SOC gaps out the DPs and induces a nontrivial Berry curvature, endowing α-Sb with a high spin Chern number of Cs = 2. We further show that monolayer α-Sb exhibits either a gapless band structure or a gapless spin spectrum on its edges, as expected from topological considerations.

Optical Transitions of a Single Nodal Ring in SrAs_{3}: Radially and Axially Resolved Characterization.

SrAs_{3} is a unique nodal-line semimetal that contains only a single nodal ring in the Brillouin zone, uninterrupted by any trivial bands near the Fermi energy. We performed axis-resolved optical reflection measurements on SrAs_{3} and observed that the optical conductivity exhibits flat absorption up to 129meV in both the radial and axial directions, confirming the robustness of the universal power-law behavior of the nodal ring. The axis-resolved optical conductivity, in combination with theoretical calculations, further reveals fundamental properties beyond the flat absorption, including the overlap energy of the topological bands, the spin-orbit coupling gap along the nodal ring, and the geometric properties of the nodal ring such as the average ring radius, ring ellipticity, and velocity anisotropy. In addition, our temperature-dependent measurements revealed a spectral weight transfer between intraband and interband transitions, indicating a possible violation of the optical sum rule within the measured energy range.

Rashba-like spin splitting around non-time-reversal-invariant momenta

The ubiquitous Rashba spin splitting occurring around time-reversal-invariant momenta (TRIM) reflects the fundamental interplay between spin-orbit coupling (SOC) and crystalline symmetry. Unexpectedly, recent measurements reveal novel Rashba spin splitting around the momenta lacking time-reversal symmetry (non-TRIM), whose mechanism remains elusive. Here, we theoretically elucidate the microscopic origin of Rashba spin splitting around non-TRIM and identify the design principles for its appearance. Conventionally, a Zeeman-like spin splitting dominates around non-TRIM due to the broken time-reversal symmetry and diminishes the Rashba-like features. Our results show that the Bloch wave function with specific orbital representation at certain non-TRIM could eliminate the first-order SOC effect. As a result, the corresponding little group hosts a type of quasisymmetry that commutes with the dominating SOC Hamiltonian, leading to a tiny SOC gap and helical spin texture. Furthermore, such a quasi-symmetry-induced tiny gap could manifest large Berry curvature, which is desirable for various topological applications. Our work expands the concept of Rashba physics beyond the conventional TRIM scenario, paving an avenue for designing novel spin-splitting materials through the cooperation of symmetry and orbital engineering.

Flat band separation and robust spin Berry curvature in bilayer kagome metals

AbstractKagome materials have emerged as a setting for emergent electronic phenomena that encompass different aspects of symmetry and topology. It is debated whether the XV6Sn6 kagome family (where X is a rare-earth element), a recently discovered family of bilayer kagome metals, hosts a topologically non-trivial ground state resulting from the opening of spin–orbit coupling gaps. These states would carry a finite spin Berry curvature, and topological surface states. Here we investigate the spin and electronic structure of the XV6Sn6 kagome family. We obtain evidence for a finite spin Berry curvature contribution at the centre of the Brillouin zone, where the nearly flat band detaches from the dispersing Dirac band because of spin–orbit coupling. In addition, the spin Berry curvature is further investigated in the charge density wave regime of ScV6Sn6 and it is found to be robust against the onset of the temperature-driven ordered phase. Utilizing the sensitivity of angle-resolved photoemission spectroscopy to the spin and orbital angular momentum, our work unveils the spin Berry curvature of topological kagome metals and helps to define its spectroscopic fingerprint.

Visualization of Tunable Weyl Line in A-A Stacking Kagome Magnets.

Kagome magnets provide a fascinating platform for a plethora of topological quantum phenomena, in which the delicate interplay between frustrated crystal structure, magnetization, and spin-orbit coupling (SOC) can engender highly tunable topological states. Here, utilizing angle-resolved photoemission spectroscopy, the Weyl lines are directly visualized with strong out-of-plane dispersion in the A-A stacked kagome magnet GdMn6 Sn6 . Remarkably, the Weyl lines exhibit a strong magnetization-direction-tunable SOC gap and binding energy tunability after substituting Gd with Tb and Li, respectively. These results not only illustrate the magnetization direction and valence counting as efficient tuning knobs for realizing and controlling distinct 3D topological phases, but also demonstrate AMn6 Sn6 (A = rare earth, or Li, Mg, or Ca) as a versatile material family for exploring diverse emergent topological quantum responses.

Polarizability, plasmons, and screening in 1T′-MoS2 with tilted Dirac bands

In the presence of an external vertical electric field and strain, it is evident that 1T′-MoS2 exhibits tilted Dirac bands which are valley-spin-polarized. Additionally, this material experiences a topological phase change between a topological insulator and band insulator for a critical value of the electric field. Using linear response theory, we calculated the polarization function which is in turn employed to obtain the dielectric function. This latter quantity is subsequently utilized in calculations to determine the plasmons dispersion relation, their decay rate and impurity screening corresponding to various levels of doping, the critical applied vertical electric field strengths and the spin-orbit coupling gap in 1T′-MoS2 with tilted Dirac bands.

Twisted nodal wires and three-dimensional quantum spin Hall effect in distorted square-net compounds

Recently, square-net materials have attracted lots of attention for the Dirac semimetal phase with negligible spin-orbit coupling (SOC) gap, e.g., ZrSiS/LaSbTe and ${\mathrm{CaMnSb}}_{2}$. In this paper, we demonstrate that the Jahn-Teller effect enlarges the nontrivial SOC gap in the distorted structure, e.g., LaAsS and ${\mathrm{SrZnSb}}_{2}$. Its distorted $X$ square-net layer ($X=$ P, As, Sb, Bi) resembles a quantum spin Hall (QSH) insulator. Since these QSH layers are simply stacked in the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}$ direction and weakly coupled, three-dimensional QSH effect can be expected in these distorted materials, such as insulating compounds ${\mathrm{CeAs}}_{1+x}{\mathrm{Se}}_{1\ensuremath{-}y}$ and ${\mathrm{EuCdSb}}_{2}$. Our detailed calculations show that it hosts two twisted nodal wires without SOC [each consists of two noncontractible time-reversal symmetry- and inversion symmetry-protected nodal lines touching at a fourfold degenerate point], while with SOC it becomes a topological crystalline insulator with symmetry indicators $(000;2)$ and mirror Chern numbers $(0,0)$. The nontrivial band topology is characterized by a generalized spin Chern number ${C}_{s+}=2$ when there is a gap between two sets of ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{s}}_{x}$ eigenvalues. The nontrivial topology of these materials can be well reproduced by our tight-binding model and the calculated spin Hall conductivity is quantized to ${\ensuremath{\sigma}}_{yz}^{x}=(\frac{\ensuremath{\hbar}}{e})\frac{{G}_{x}{e}^{2}}{\ensuremath{\pi}h}$ with ${G}_{x}$ a reciprocal lattice vector.

Revealing a charge-density-wave gap in the predicted weak topological insulator HoSbTe

HoSbTe was predicted to be a weak topological insulator, whose spin-orbit coupling (SOC) gaps are reported to be as large as hundreds of meV. Utilizing infrared spectroscopy, we find that the compound is of metallic nature from 350 K down to 10 K. Particularly, both of its itinerant carrier density and scattering rate are demonstrated to decrease with temperature cooling, which is responsible for the appearance of a broad hump feature in the temperature dependent resistivity around 200 K. More importantly, we reveal the appearance of a charge density wave (CDW) gap in addition to the SOC related gap. The energy scale of the CDW gap is identified to be 364 meV at 10 K, which shift to 252 meV at 350 K. The coexistence of CDW and SOC gaps in the same compound paves a new avenue to explore more intriguing physics.

Optical conductivity signatures of open Dirac nodal lines

We investigate the optical conductivity and far-infrared magneto-optical response of ${\mathrm{BaNiS}}_{2}$, a simple square-lattice semimetal characterized by Dirac nodal lines that disperse exclusively along the out-of-plane direction. With the magnetic field aligned along the nodal line the in-plane Landau level spectra show a nearly $\sqrt{B}$ behavior, the hallmark of a conical-band dispersion with a small spin-orbit coupling gap. The optical conductivity exhibits an unusual temperature-independent isosbestic line, ending at a Van Hove singularity. First-principles calculations unambiguously assign the isosbestic line to transitions across Dirac nodal states. Our work suggests a universal topology of the electronic structure of Dirac nodal lines.

Anisotropic Berry phase in the Dirac nodal-line semimetal ZrSiS: The effect of spin-orbit coupling

The topological nodal-line semimetals (NLSMs) possess a loop of Dirac nodes in the $k$ space with linear dispersion, different from the point nodes in Dirac/Weyl semimetals. While the quantum transport associated with the topologically nontrivial Dirac fermions has been investigated extensively, features uniquely associated with the extended nodal lines remain to be demonstrated. Here, we investigate the quantum oscillations (QOs) in the nodal-line semimetal ZrSiS, with the electron transport along the $c$ axis and magnetic field rotating in the $ab$ plane. The extremal orbits identified through the field orientation dependence of the QOs interlock with the nodal line, leading to a nonzero Berry phase. Most importantly, the Berry phase shows a significant dependence on the magnetic field orientation, which we argue to be due to the finite spin-orbit coupling gap. Our results demonstrate the importance of the spin-orbit coupling and the nodal-line dispersion in understanding the quantum transport of NLSMs.

Topological insulators in the NaCaBi family with large spin-orbit coupling gaps

By means of first-principles calculations and crystal structure searching techniques, we predict that a new NaCaBi family crystallized into the ZrBeSi-type structure (\ie $P6_{3}/mmc$) are strong topological insulators (STIs). Taking $P6_{3}/mmc$ NaCaBi as an example, the calculated band structure indicates that there is a band inversion between two opposite-parity bands at the $\Gamma$ point. In contrast to the well-known Bi$_2$Se$_3$ family, the band inversion in the NaCaBi family has already occured even without spin-orbit coupling (SOC), giving rise to a nodal ring surrounding $\Gamma$ in the $k_z=0$ plane (protected by $M_z$ symmetry). With time reversal symmetry $(\cal T)$ and inversion symmetry $(\cal I)$, the spinless nodal-line metallic phase protected by $[{\cal TI}]^2=1$ is the weak-SOC limit of the spinful topological insulating phase. Upon including SOC, the nodal ring is gapped, driving the system into a STI. Besides inversion symmetry, the nontrivial topology of NaCaBi can also be indicated by $\bar{6}$ symmetry. More surprisingly, the SOC-induced band gap in NaCaBi is about 0.34 eV, which is larger than the energy scale of room temperature. Four other compounds (KBaBi, KSrBi, RbBaBi and RbSrBi) in the family are stable at ambient pressure, both in thermodynamics and lattice dynamics, even though the gaps of them are smaller than that of NaCaBi. Thus, they provide good platforms to study topological states both in theory and experiments.

Anisotropic longitudinal optical conductivities of tilted Dirac bands in 1T′−MoS2

1T$^\prime$-MoS$_2$ exhibits valley-spin-polarized tilted Dirac bands in the presence of external vertical electric field and undergoes a topological phase transition between the topological insulator and band insulator around the critical value of the electric field. Within the linear response theory, we theoretically investigate the anisotropic longitudinal optical conductivities of tilted Dirac bands in both undoped and doped 1T$^\prime$-MoS$_2$, including the effects of the vertical electric field. The influence of the spin-orbit coupling gap, band tilting, and vertical electric field on the optical conductivities of tilted Dirac bands is revealed. A theoretical scheme for probing the topological phase transition in 1T$^\prime$-MoS$_2$ via exotic behaviors of longitudinal optical conductivities is proposed. The results for 1T$^\prime$-MoS$_2$ are expected to be qualitatively valid for other monolayer tilted gapped Dirac materials, such as $\alpha$-SnS$_2$, TaCoTe$_2$, and TaIrTe$_4$, due to the similarity in their band structures.

Dimensional engineering of a topological insulating phase in Half-Heusler LiMgAs

We propose a novel technique of dimensional engineering to realize low dimensional topological insulator from a trivial three dimensional parent. This is achieved by confining the bulk system to one dimension along a particular crystal direction, thus enhancing the quantum confinement effects in the system. We investigate this mechanism in the Half-Heusler compound LiMgAs with face-centered cubic (FCC) structure. At ambient conditions the bulk FCC structure exhibits a semi-conducting nature. But, under the influence of high volume expansive pressure (VEP) the system undergoes a topological phase transition (TPT) from semi-conducting to semi-metallic forming a Dirac cone. At a critical VEP we observe that, spin-orbit coupling (SOC) effects introduce a gap of sim 1.5 meV in the Dirac cone at high symmetry point Gamma in the Brillouin zone. This phase of bulk LiMgAs exhibits a trivial nature characterized by the {mathbb {Z}}_2 invariants as (0,000). By further performing dimensional engineering, we cleave [111] plane from the bulk FCC structure and confine the system in one dimension. This low-dimensional phase of LiMgAs has structure similar to the two dimensional {text {1T-MoS}}_2 system. Under a relatively lower compressive strain, the low-dimensional system undergoes a TPT and exhibits a non-trivial topological nature characterized by the SOC gap of sim 55 meV and {mathbb {Z}}_2 invariant nu = 1. Although both, the low-dimensional and bulk phase exhibit edge and surface states, the low-dimensional phase is far more superior and exceptional as compared to the bulk parent in terms of the velocity of Fermions ({text {v}}_f) across the surface states. Such a system has promising applications in nano-electronics.

Enhanced transport of spin-orbit-coupled Bose gases in disordered potentials.

Anderson localization is a single-particle localization phenomena in disordered media that is accompanied by an absence of diffusion. Spin-orbit coupling (SOC) describes an interaction between a particle's spin and its momentum that directly affects its energy dispersion, for example, creating dispersion relations with gaps and multiple local minima. We show theoretically that combining one-dimensional spin-orbit coupling with a transverse Zeeman field suppresses the effects of disorder, thereby increasing the localization length and conductivity. This increase results from a suppression of backscattering between states in the gap of the SOC dispersion relation. Here, we focus specifically on the interplay of disorder from an optical speckle potential and SOC generated by two-photon Raman processes in quasi-one-dimensional Bose-Einstein condensates. We first describe backscattering by using a Fermi golden rule approach, and then numerically confirm this picture by solving the time-dependent one-dimensional Gross-Pitaevskii equation for a weakly interacting Bose-Einstein condensate with SOC and disorder. We find that on the tens of millisecond timescale of typical cold atom experiments moving in harmonic traps, initial states with momentum in the zero-momentum SOC gap evolve with negligible backscattering, while without SOC these same states rapidly localize.

Quantum Phase Engineering of Two-Dimensional Post-Transition Metals by Substrates: Toward a Room-Temperature Quantum Anomalous Hall Insulator.

We propose a new strategy to engineer topological and magnetic properties of two-dimensional (2D) hexagonal lattices consisting of post-transition metals. Our first-principles calculations demonstrate that substrates serve as templates to form 2D lattices with high thermodynamic stability, where their topological properties as well as magnetic properties sensitively change as a function of lattice constants, i.e., the system undergoes a first-order phase transition from nonmagnetic to ferromagnetic state above a critical lattice constant. Consequently, substrates can be used to explore versatile magnetic, electronic, and quantum topological properties. We establish phase diagrams of versatile quantum phases in terms of exchange coupling and spin-orbit coupling effectively tuned by the lattice constants. We further reveal the first room-temperature quantum anomalous Hall (QAH) effect, i.e., Sn on 2√3 × 2√3 graphane is a QAH insulator with a large spin-orbit coupling gap of ∼0.2 eV and a Curie temperature of ∼380 K by using the 2D anisotropic Heisenberg model.

Model Hamiltonian for the Quantum Anomalous Hall State in Iron-Halogenide

We examine quantum anomalous Hall (QAH) insulators with intrinsic magnetism displaying quantized Hall conductance at zero magnetic fields. The spin-momentum locking of the topological edge stats promises QAH insulators with great potential in device applications in the field of spintronics. Here, we generalize Haldane’s model on the honeycomb lattice to a more realistic two-orbital case without the artificial real-space complex hopping. Instead, we introduce an intraorbital coupling, stemming directly from the local spin-orbit coupling (SOC). Our dxy/dx2–y2 model may be viewed as a generalization of the bismuthene px/py-model for correlated d-orbitals. It promises a large SOC gap, featuring a high operating temperature. This two-orbital model nicely explains the low-energy excitation and the topology of two-dimensional ferromagnetic iron-halogenides. Furthermore, we find that electronic correlations can drive the QAH states to a c = 0 phase, in which every band carries a nonzero Chern number. Our work not only provides a realistic QAH model, but also generalizes the nontrivial band topology to correlated orbitals, which demonstrates an exciting topological phase transition driven by Coulomb repulsions. Both the model and the material candidates provide excellent platforms for future study of the interplay between electronic correlations and nontrivial band topology.

Saddle-point Van Hove singularity and dual topological state in Pt2HgSe3

Saddle-point van Hove singularities in the topological surface states are interesting because they can provide a new pathway for accessing exotic correlated phenomena in topological materials. Here, based on first-principles calculations combined with a $\mathbf {k \cdot p}$ model Hamiltonian analysis, we show that the layered platinum mineral jacutingaite (Pt$_2$HgSe$_3$) harbours saddle-like topological surface states with associated van Hove singularities. Pt$_2$HgSe$_3$ is shown to host two distinct types of nodal lines without spin-orbit coupling (SOC) which are protected by combined inversion ($I$) and time-reversal ($T$) symmetries. Switching on the SOC gaps out the nodal lines and drives the system into a topological insulator state with nonzero weak topological invariant $Z_2=(0;001)$ and mirror Chern number $n_M=2$. Surface states on the naturally cleaved (001) surface are found to be nontrivial with a unique saddle-like energy dispersion with type II van Hove singularities. We also discuss how modulating the crystal structure can drive Pt$_2$HgSe$_3$ into a Dirac semimetal state with a pair of Dirac points. Our results indicate that Pt$_2$HgSe$_3$ is an ideal candidate material for exploring the properties of topological insulators with saddle-like surface states.

Strong spin-orbit coupling and Dirac nodal lines in the three-dimensional electronic structure of metallic rutile IrO2

Using high-resolution angle-resolved photoemission spectroscopy and ab initio calculation, we have studied the bulk and surface electronic structure of metallic rutile $5d$ transition metal oxide ${\mathrm{IrO}}_{2}$ that harbors both edge and corner sharing Ir-O octahedrons. We observe strong modulation of the band structure by spin-orbit coupling (SOC). The measured band structure is well reproduced by our ab initio calculation without band renormalization, suggesting the absence of the SOC-enhanced correlation effect in ${\mathrm{IrO}}_{2}$. In accordance with the calculation, we visualize two types of Dirac nodal lines (DNLs) protected by mirror symmetry and nonsymmorphic crystal symmetry, respectively. SOC gaps the first type of DNLs, which may contribute largely to the strong spin Hall effect. The second type of DNLs at the edges of Brillouin zone, however, remain intact against SOC. Our results not only provide important insights into the exotic transport properties of ${\mathrm{IrO}}_{2}$, but also shed light on the understanding of the role of SOC in the iridate family.

Resonance Microwave Measurements of an Intrinsic Spin-Orbit Coupling Gap in Graphene: A Possible Indication of a Topological State.

In 2005, Kane and Mele [Phys. Rev. Lett. 95, 226801 (2005)PRLTAO0031-900710.1103/PhysRevLett.95.226801] predicted that at sufficiently low energy, graphene exhibits a topological state of matter with an energy gap generated by the atomic spin-orbit interaction. However, this intrinsic gap has not been measured to this date. In this Letter, we exploit the chirality of the low-energy states to resolve this gap. We probe the spin states experimentally by employing low temperature microwave excitation in a resistively detected electron-spin resonance on graphene. The structure of the topological bands is reflected in our transport experiments, where our numerical models allow us to identify the resonance signatures. We determine the intrinsic spin-orbit bulk gap to be exactly 42.2 μeV. Electron-spin resonance experiments can reveal the competition between the intrinsic spin-orbit coupling and classical Zeeman energy that arises at low magnetic fields and demonstrate that graphene remains to be a material with surprising properties.

Graphene-like Be3X2 (X = C, Si, Ge, Sn): A new family of two-dimensional topological insulators**Project supported by the National Natural Science Foundation of China (Grant Nos. 11674136 and 11564022), the Thousand Talents Plan — the Recruitment Program for Young Professionals, China (Grant No. 1097816002), Yunnan Province for Recruiting High-Caliber Technological Talents, China (Grant No. 1097816002), Reserve Talents for Yunnan Young

Using first-principle calculations, we predict a new family of stable two-dimensional (2D) topological insulators (TI), monolayer Be3X2 (X = C, Si, Ge, Sn) with honeycomb Kagome lattice. Based on the configuration of Be3C2, which has been reported to be a 2D Dirac material, we construct the other three 2D materials and confirm their stability according to their chemical bonding properties and phonon-dispersion relationships. Because of their tiny spin–orbit coupling (SOC) gaps, Be3C2 and Be3Si2 are 2D Dirac materials with high Fermi velocity at the same order of magnitude as that of graphene. For Be3Ge2 and Be3Sn2, the SOC gaps are 1.5 meV and 11.7 meV, and their topological nontrivial properties are also confirmed by their semi-infinite Dirac edge states. Our findings not only extend the family of 2D Dirac materials, but also open an avenue to track new 2DTI.