High-symmetry Points Research Articles

Systematic study for two-dimensional Z2 topological phase transitions at high-symmetry points in all layer groups

We construct a general theory of $Z_2$ topological phase transitions in two-dimensional systems with time-reversal symmetry. We investigate the possibilities of $Z_2$ topological phase transitions at band inversions at all high-symmetry points in $k$-space in all the 80 layer groups. We exclude the layer groups with inversion symmetry because the $Z_2$ topological phase transition is known to be associated with band inversions with an exchange of parities. Among the other layer groups, we find 21 layer groups with insulator-to-insulator transitions with band inversion, and this problem is finally reduced to five point groups $C_3, C_4, C_6, S_4$, and $C_{3h}$. We show how the change of the $Z_2$ topological invariant at a band inversion is entirely determined by the irreps of occupied and unoccupied bands at the high-symmetry point. For example, in the case of $C_3$, we show that the $Z_2$ topological invariants change whenever the band inversion occurs between two Kramers pairs whose $C_3$ eigenvalues are $\{e^{\pi i / 3}, e^{-\pi i / 3}\}$ and $\{-1, -1\}$. These results are not included in the theory of symmetry-based indicators or topological quantum chemistry.

Symmetry-resolved entanglement of C2 -symmetric topological insulators

For a many-body system of arbitrary dimension, we consider fermionic ground states of noninteracting Hamiltonians invariant under a ${C}_{2}$ cyclic group. The absolute difference $\mathrm{\ensuremath{\Delta}}$ between the number of occupied symmetric and antisymmetric single-particle states is a topological invariant in this case. We prove lower bounds on the configurational and the number entropy based on this invariant. Entanglement entropies are particularly important for identifying topological phases protected by spatial symmetries because they lack gapless boundary modes in general. In topological crystalline insulators, the topological invariant $\mathrm{\ensuremath{\Delta}}$ and the entropy bounds can be directly determined from high-symmetry points in the Brillouin zone.

Mapping and Manipulation of Topological Singularities: From Photonic Graphene to T-Graphene

Topological singularities (TSs) in momentum space give rise to intriguing fundamental phenomena as well as unusual material properties, attracting a great deal of interest in the past decade. Recently, we demonstrated universal momentum-to-real-space mapping of TSs and pseudospin angular momentum conversion using photonic honeycomb (graphene-like) and Lieb lattices. Such mapping arises from the nontrivial Berry phase winding and is thus of topological origin. In this paper, after a brief review of previous observations of pseudospin–orbital conversion, we present the first theoretical analysis and experimental demonstration of TS mapping in a new type of Dirac-like structure, namely, the T-graphene lattice. Unlike other lattices, there are two coexisting but distinct TSs located at different high-symmetry points in the first Brillouin zone of T-graphene, which enables controlled topological charge conversion in the same lattice. We show active manipulation of the TS mapping, turning the two TSs into vortices of different helicities, or one into a high-order vortex but the other eliminated and emerged as a quadrupole. Such mapping and manipulation of TSs may find applications in optical communications and quantum information, and may bring insight into the study of other Dirac-like structures with multiple TSs beyond the 2D photonic platform.

Engineering axion insulator and other topological phases in superlattices without inversion symmetry

We study theoretically the interplay between magnetism and topology in three-dimensional HgTe/MnTe superlattices stacked along the (001) axis. Our results show the evolution of the magnetic topological phases with respect to the magnetic configurations. An axion insulator phase is observed for the antiferromagnetic order with the out-of-plane N\'eel vector direction below a critical thickness of MnTe, which is the ground state amongst all magnetic configurations. Defining $T$ as the time-reversal symmetry, this axion insulator phase is protected by a magnetic two-fold rotational symmetry $C_2{\cdot}T$. The axion insulator phase evolves into a trivial insulator as we increase the thickness of the magnetic layers. By switching the N\'eel vector direction into the $ab$ plane, the system realizes different antiferromagnetic topological insulators depending on the thickness of MnTe. These phases feature gapless surface Dirac cones shifted away from high-symmetry points on surfaces perpendicular to the N\'eel vector direction of the magnetic layers. In the presence of ferromagnetism, the system realizes a magnetic Weyl semimetal and a ferromagnetic semimetal for out-of-plane and in-plane magnetization directions, respectively. We observe large anomalous Hall conductivity in the presence of ferromagnetism in the three-dimensional superlattice.

The discovery of the effect of compositional disorder on the opto-electronic properties of the deformed InGaAsP quaternary

This article presents a theoretical study of the optoelectronic properties of InGaAsP alloys fabricated on three different substrates: InP, GaA, and ZnSe. Particular attention has been paid to the study of bandgap energy along points of high symmetry in the Brillouin zone, absorption coefficient, and refractive index of materials without compositional disorder and subjected to deformation. Our goal is to expand the range of optoelectronic properties. We found that the absorption coefficient of the deformed InGaAsP/InP material in the absence of disorder varies between 9.66×104cm−1 and 1.35×105cm−1 changing the content of phosphorus and gallium, making it possible to provide a wide range of refractive indices [3.06, 3.84] compared to GaAs substrates [2.67, 3.02] and ZnSe [2.69, 3.05]. This refractive index contrast can be exploited in the design of optical waveguides.

Dirac Fermions in Blue Phosphorene Monolayer

Abstract2D materials beyond graphene and in particular 2D semiconductors have raised interest due to their unprecedented electronic properties, such as high carrier mobility or tunable bandgap. Blue phosphorene is an allotrope of black phosphorene that resembles graphene as it presents a honeycomb structure. However, it is known to have semiconductor character and the crucial point is to determine whether this hexagonal phase of phosphorene presents Dirac fermions as in graphene. Here, the first compelling experimental evidence of Dirac fermions in blue phosphorene layer grown on Cu(111) surface is presented. The results highlight the formation of a highly ordered blue phosphorene sheet with a clear Dirac cone at the high symmetry points of the Brillouin Zone. The charge carriers behave as massless relativistic particles. Therefore, all the expectations held for graphene, such as high‐speed electronic devices based on ballistic transport at room temperature, may also be applied to blue phosphorene.

Metrology of band topology via resonant inelastic x-ray scattering

Topology is a central notion in the classification of band insulators and the characterization of entangled many-body quantum states. In some cases, it manifests as quantized observables such as quantum Hall conductance. However, being inherently a global property depending on the entirety of the system, its direct measurement has remained elusive to local experimental probes in many cases. Here, we demonstrate that various topological band indices can be directly probed by resonant inelastic x-ray scattering. Specifically, we show that the crystalline symmetry eigenvalues at the high-symmetry momentum points, which determine the band topology, lead to distinct scattering intensity for particular momentum and energy. Our approach can be explicitly demonstrated in several examples such as 1D Su-Schrieffer-Heeger chain, 2D quadrupole insulator, 3D topological band insulator, and chiral hinge insulator. Our result establishes an incisive bulk probe for the measurement of band topology.

Manipulating Dirac States in BaNiS2 by Surface Charge Doping.

In the Dirac semimetal BaNiS2, the Dirac nodes are located along the Γ-M symmetry line of the Brillouin zone, instead of being pinned at fixed high-symmetry points. We take advantage of this peculiar feature to demonstrate the possibility of moving the Dirac bands along the Γ-M symmetry line in reciprocal space by varying the concentration of K atoms adsorbed onto the surface of cleaved BaNiS2 single crystals. By means of first-principles calculations, we give a full account of this observation by considering the effect of the electrons donated by the K atom on the charge transfer gap, which establishes a promising tool for engineering Dirac states at surfaces, interfaces, and heterostructures.

Band engineering of monolayer CaI2, a first-principles approach

ABSTRACT Motivated by recent proposal of enhancing performance of materials via band engineering, first principles calculations are performed to investigate the changes in band structure of monolayer CaI2 by applying isotropic biaxial strain. CaI2 monolayer is found to be an indirect band-gap semiconductor with band-gap of 3.77 eV. Phonon dispersion includes only real phonon modes which reveals dynamical stability of the material. Calculations show that there is a decrease in band-gap as well as effective mass of carriers of the material with increase in compressive strain. At strains of −4%, −8% and −12%, the band gap is found to be 3.49 eV, 3.14 eV and 2.74 eV, respectively. Both the maxima of valence band and minima of conduction band move downwards and maxima of conduction band shifts from high symmetry point M to point K. The changes in band structure and density of states are in consistence with one another. The effective mass of carriers also decreases resulting in an increase in mobility. With a final band-gap of 2.74 eV and increased mobility at −12% strain, the material can be used for different applications as a wide band-gap semiconducting material.

Investigations of Optical Functions and Optical Transitions of 2D Semiconductors by Spectroscopic Ellipsometry and DFT.

Optical functions and transitions are essential for a material to reveal the light-matter interactions and promote its applications. Here, we propose a quantitative strategy to systematically identify the critical point (CP) optical transitions of 2D semiconductors by combining the spectroscopic ellipsometry (SE) and DFT calculations. Optical functions and CPs are determined by SE, and connected to DFT band structure and projected density of states via equal-energy and equal-momentum lines. The combination of SE and DFT provides a powerful tool to investigate the CP optical transitions, including the transition energies and positions in Brillouin zone (BZ), and the involved energy bands and carries. As an example, the single-crystal monolayer WS2 is investigated by the proposed method. Results indicate that six excitonic-type CPs can be quantitatively distinguished in optical function of the monolayer WS2 over the spectral range of 245-1000 nm. These CPs are identified as direct optical transitions from three highest valence bands to three lowest conduction bands at high symmetry points in BZ contributed by electrons in S-3p and W-5d orbitals. Results and discussion on the monolayer WS2 demonstrate the effectiveness and advantages of the proposed method, which is general and can be easily extended to other materials.

Stabilizing tetramethylammonium lead iodide perovskite and exploring its electronic and optical absorption for solar cell absorber application.

Modeling perovskites as solar cell absorbers has become popular due to the breakthrough of methylammonium lead iodide (CH3NH3PbI3). In this study, we modeled a tetramethylammonium lead iodide (CH3)4NPbI3 structure. We further confirmed the stability of the structure by determining the phonon dispersion using density functional perturbation theory. We calculated the spin-orbit and non-spin-orbit coupling-based electronic structure using the Perdew-Burke-Ernzerhof exchange-correlation functional within the generalized gradient approximation of the density functional theory and the self-consistent GW quasiparticle methods. Similarly, the absorption spectra were calculated from the real and imaginary parts of the dielectric tensor obtained from solving the Bethe-Salpeter equation using the GW quasiparticle database. The solar cell absorber spectroscopic limited maximum efficiency was calculated at 293.15 K. The self-consistent GW method without spin-orbit coupling reported bandgaps of 2.63 eV and 2.89 eV for GW0 and GW methods, respectively, in agreement with experimental reports. The phonon dispersion showed positive phonon modes across the high symmetry point, which attest to its thermodynamic stability. The absorption coefficient on the order of 105 was reported along the ultraviolet region. The standard limited maximum efficiency between 7% and 12% was recorded at 293.15 K between 0.01 and 100 μm absorber thicknesses. The thermodynamic stability, high absorption coefficient, and low transmittance indicated exciting prospects for a non-transparent (CH3)4NPbI3 solar cell absorber.

Far-field thermal properties of β12 borophene under an external electric field.

Borophene has been reported as the latest very promising 2D material. We theoretically investigate the thermal radiation of β12 borophene. β12 borophene has been composited on Ag substrate. Theoretical research frequently mentions three types of model of β12 borophene. The energy bands of the three models are different. Homogeneous and inversion symmetric models have a gapless Dirac cone near the K(K') point, while they are gapped in the inversion non-symmetric model. The homogeneous model has gapless triplet fermions and three-band touching points at high-symmetry points. The optical conductivity exhibits peaks due to the energy transition of high-symmetry points. The radiation spectrum follows Wien's displacement law in homogeneous and inversion symmetric models, while it is broken in the inversion non-symmetric model. The total energy radiation changes as the voltage increases for the three β12 borophene models. The radiation of energy can be controlled by applying a suitable voltage.

Polarized coherent emission outside high-symmetry points of dye-coupled plasmonic lattices.

Developing intense, coherent and ultra-fast light sources with nanoscale dimensions is a crucial issue for many applications in nanophotonics. To date, plasmonic nanolasers represent one of the most promising nanophotonic devices capable of this remarkable feature. In the present work we report on the emission properties of two-dimensional Au hexagonal nanodome arrays, fabricated by nanosphere lithography, coupled with a dye liquid solution used as the gain medium. Low-threshold stimulated emission at room temperature is demonstrated by spectral and angle-resolved photoluminescence measurements performed as a function of the pump fluence. The emission arises with narrow angular divergence in off-normal direction, out of high-symmetry points of the plasmonic lattice. The polarization properties of the stimulated emission are investigated, revealing a strong linear polarization character controlled by the polarization orientation of the pumping beam, while the first-order temporal coherence properties are measured by using a tilted-mirrors Michelson interferometer. Finally, by comparing the results obtained for the plasmonic Au nanodomes arrays with those of purely dielectric nanoarrays, the role of the plasmonic modes and the photonic lattice modes in the emission process is highlighted.

Dirac phonons in two-dimensional materials

Phonons are an ideal platform for realizing stable spinless two-dimensional (2D) Dirac points because they have a bosonic nature and hard-to-break time-reversal symmetry. It should be noted that the twofold degenerate nodal points in the phonon dispersions of almost all reported 2D materials are misclassified as ``Dirac points'' owing to a historical issue. The correct name for these twofold degenerate nodal points should be ``Weyl'' because 2D phononic systems are essentially spinless and because each twofold degenerate point is described by a Weyl model in two dimensions. To date, reports of fourfold degenerate Dirac point phonons in 2D materials are lacking. In this paper, we searched through all 80 layer groups (LGs) and discovered that Dirac phonons can be realized in 7 of the 80 LGs. Moreover, the Dirac points in the phonon dispersions of 2D materials can be divided into essential and accidental degenerate points, which appear at high-symmetry points and on high-symmetry lines, respectively. Guided by symmetry analysis, we identified the presence of Dirac phonons in several 2D material candidates with six LGs. This work offers a method for identifying Dirac phonons in 2D and proposes 2D material candidates for realizing Dirac phonons.

Local electron correlation effects on the fermiology of the weak itinerant ferromagnet ZrZn2

The Fermi surface topology plays an important role in the macroscopic properties of metals. It can be particularly sensitive to electron correlation, which appears to be especially significant for the weak itinerant ferromagnet ZrZn2. Here, we look at the differences in the predicted Fermi surface sheets of this metallic compound in its paramagnetic phase for both density functional theory (DFT) and the combination of DFT with dynamical mean field theory (DFT + DMFT). The theoretical spectral functions evaluated at the Fermi level were used along with calculations of the electron–positron momentum density (also known as the two-photon momentum density) in k-space to provide insights into the origin of certain features of the Fermi surface topology. We compare this two photon momentum density to that extracted from the positron annihilation experimental data (2004 Phys. Rev. Lett. 92 107003). The DFT + DMFT densities are in better agreement with the experiment than the DFT, particularly with regard to the flat bands around the L and W high symmetry points. The experimental neck around L, which relates to a van Hove singularity, is present in DFT + DMFT but not in the DFT. We find that these flat bands, and as such the Fermi surface topology, are sensitive to the many body electron correlation description, and show that the positron annihilation technique is able to probe this. This description is significant for the observed behavior such as the Lifshiftz transition around the quantum critical point.

Topological charge-2 Dirac phonons in three dimensions: Theory and realization

Topological Dirac semimetals have been attracting considerable attention. Recently, a new class of Dirac points with a topological charge of $\ifmmode\pm\else\textpm\fi{}2$, named charge-2 Dirac points (C-2 DPs), were proposed in spinless systems. In this paper, we conduct an exhaustive search of all 230 space groups (SGs), finding that six SGs host C-2 DP at high-symmetry points (HSPs) and nine SGs host the C-2 DP on their high-symmetry lines (HSLs). Based on the symmetry analysis, we find the lowest symmetry requirements that enforce a C-2 DP and construct the corresponding effective models. Furthermore, we investigate the difference between C-2 DPs at HSPs and on HSLs, finding that the type-II C-2 DP only appears on HSLs. Via the first-principles calculations, we identify the corresponding realistic materials that host C-2 DPs. Therefore, our paper not only offers the guidelines to identify the materials that host C-2 DPs in spinless systems based on symmetry requirements, but also paves the effective way to study the C-2 DPs in spinless systems.

Lattice distortion optimized hybridization and superlubricity of MoS2/MoSe2 heterointerfaces via Moiré patterns

The moiré superlattices (MSL) of twist two-dimensional van der Waals heterointerfaces accompanying with interfacial lattice distortion present abundant attractive physical properties, including fast interlayer charge-transfer, superconductor-insulator transition, emerging topological phase. Attributing to the optimized atomic lateral forces and elimination of interlayer lock-in by moiré superlattice, the superlubricity can be achieved and its theoretical foundations are still under extensive investigations. Here, the contributions of moiré-induced lattice distortions on the superlubricity of twist MoS2/MoSe2 heterointerfaces are comprehensively investigated by first-principles calculations. The localized electronic redistribution and the orbital hybridization by lattice distortions result in the band evolutions, reflecting the features of high symmetry points under different interfacial orientations. Moreover, the characteristic moiré potential periodically modulating the sliding energy barrier is observed in terms of a sinusoid function, which is stressed to correlate the fluctuation of interlayer charge density with variation of displacement by the fitted dominated parameter α relating to the moiré potential. Therefore, the present procedure explores the variation tendency of energy barrier with band gap and twist angle and can be treated as an essential strategy to reveal the essence of moiré and superlubricity and pave a path to accelerate the development of advanced materials with excellent tribology performance.

Link between Zitterbewegung and topological phase transitions

Topological quantum state described by the global invariant has been extensively studied in theory and experiment. In this letter, we investigate the relationship between \emph{Zitterbewegung} and the topology of systems that reflect the properties of the local and whole energy bands, respectively. We generalize the usual two-band effective Hamiltonian to characterize the topological phase transition of the spin-$J$ topological insulator. By studying \emph{Zitterbewegung} dynamics before and after topological phase transition, we find that the direction of quasiparticles' oscillation can well reflect topological properties. Furthermore, we develop a quantitative calculation formula for the topological invariant in the spin-$J$ Chern insulator and give the selection rule of the corresponding dynamics. Finally, we demonstrate that our theory is valid in different topological systems. The topological invariant can be represented by local dynamical properties of the high-symmetry points in the first Brillouin zone, which provides a new measurement method from the dynamical perspective.

Exact results for the orbital angular momentum of magnons on honeycomb lattices

We obtain exact results for the orbital angular momentum (OAM) of magnons at the high symmetry points of ferromagnetic (FM) and antiferromagnetic (AF) honeycomb lattices in the presence of Dzyallonshinskii–Moriya (DM) interactions. For the FM honeycomb lattice in the absence of DM interactions, the values of the OAM at the corners of the Brillouin zone (BZ) (, ) are alternately for both magnon bands. The presence of DM interactions dramatically changes those values by breaking the degeneracy of the two magnon bands. The OAM values are alternately and 0 for the lower magnon band and and 0 for the upper magnon band. For the AF honeycomb lattice, the values of the OAM at the corners of the BZ are on one of the degenerate magnon bands and on the other, where κ measures the anisotropy and the result is independent of the DM interaction.

Topological nodal-link phonons, three-fold, Dirac and six-fold nodal-point phonons in the insulator SiO2

By using first-principles calculations and symmetry analysis, we study the topologically nontrivial features of sextuple nodal-point phonons together with other kinds of topological phonons in realistic materials. The sextuple nodal-point phonons in all 230 space groups, enumerated in this paper, are localized at the boundaries of the three-dimensional Brillouin zone (BZ), and protected both by time-reversal symmetry () and little-group symmetries. Moreover, in a realistic material sample of insulator SiO2, we find that the sextuple nodal-point phonons exist at the high-symmetry point H of the first BZ, and generate four-fold surface states. Interestingly, owing to the special crystal symmetries in SiO2, triple nodal-point phonons and Dirac phonons exist at the point Γ and P, respectively, and exhibit exotic quadruple surface states. More than these, the topological phononic nodal links also appear around the point Γ, and exhibit drumhead like surface states in this material. Our theoretical work not only proposes an effective way to search for multi-fold topological phonons including Dirac phonons and sextuple nodal-point phonons, but also presents a realistic material sample to realize the coexistence of multiple nodal-point and nodal-link phonons.