High-symmetry Points Research Articles

Spin-polarized sextuple excitations in ferromagnetic materials

The most fundamental symmetry in condensed matter physics is the crystallographic symmetry, such that solids could exhibit unconventional excitations beyond Dirac and Weyl fermions. Among the unconventional fermions, the sextuple excitation has been predicted in many nonmagnetic materials. However, the study of magnetic sextuple excitations has fallen behind because of the complicated magnetic structures of materials. Here we perform a systematic search for ferromagnetic (FM) sextuple points (SPs) in the absence of spin-orbit coupling. Based on 230 space groups (SGs), we find there are 5 SGs that could host SPs at high-symmetry points. Regarding the stability of the sextuple excitation, we reveal that the minimum little point group at where the SP locates is ${\mathrm{T}}_{d}$. To support this result, we construct the effective model based on the $k\ifmmode\cdot\else\textperiodcentered\fi{}p$ method, from which the linear dispersion can be read off. Via first-principles calculations, we also identify 67 FM material candidates, which realize the SPs with completely/incompletely spin polarized. To support our theory, we pick one of the FM materials, ${\mathrm{Rb}}_{4}{\mathrm{O}}_{6}$, as an example in which there is a fully spin-polarized SP. Finally, our work paves the way to study the spin-polarized SPs and provides a platform to realize the spintronic associated with sextuple excitations.

Nontrivial band topology coupled thermoelectrics in VSe2/MoSe2 van der Waals magnetic Weyl semimetal

The correlation between topological and thermoelectrics promotes numerous interesting electronic phenomena and sets the stage for efficient thermopower devices. Herein, we report nontrivial band topology of 1T–VSe2/1H–MoSe2 van der Waals system and also probe its thermoelectric (TE) characteristics on the basis of first-principle calculations. The crossover of bands, which creates a close loop near Fermi level along M–K high symmetry points, gets inverted at former crossing points of bands, under spin–orbit coupling effect. The calculated Chern Number C = 1 supports the nontrivial band topology whereas the broken time reversal symmetry asserts its magnetic Weyl semimetallic behavior. The nontrivial band topology falls under the category of Type-I Weyl band crossing. We delve into the TE characteristics of the proposed topological material by employing constant relaxation time approximation. The heterostructure shows high electrical conductivity of order 106 S m−1 at both 300 K and 1200 K, and a low magnitude of Seebeck coefficient (S) value of 79.3 μV K−1 near room temperature. Such interplay between the topological phase and TE characteristics can lay foundation for next-generation topological-TE devices.

Discriminant indicator with generalized rotational symmetry

Discriminant indicators with generalized inversion symmetry are computed only from data at the high-symmetry points. They allow a systematic search for exceptional points. In this paper, we propose discriminant indicators for two- and three-dimensional systems with generalized $n$-fold rotational symmetry ($n=4$, $6$). As is the case for generalized inversion symmetry, the indicator taking a nontrivial value predicts the emergence of exceptional points and loops without ambiguity of the reference energy. A distinct difference from the case of generalized inversion symmetry is that the indicator with generalized $n$-fold rotational symmetry ($n=4$, $6$) can be computed only from data at two of four high-symmetry points in the two-dimensional Brillouin zone. Such a difference is also observed in three-dimensional systems. Furthermore, we also propose how to fabricate a two-dimensional system with generalized four-fold rotational symmetry for an electrical circuit.

Magnetic proximity induced valley-contrasting quantum anomalous Hall effect in a graphene- CrBr3 van der Waals heterostructure

The magnetic proximity effect is an imperative tool to comprehend the valley contrasting quantum anomalous Hall (QAH) effect in van der Waals (vdW) heterostructure. The introduction of magnetic exchange and spin orbit interaction together enables to realize topological phases, with a particular emphasis on an interface can be envisioned towards dissipationless electronics. Herein, the proximity coupled valley contrasting QAH effect is predicted in vdW heterostructure consisting of graphene and a ferromagnetic (FM) semiconductor ${\mathrm{CrBr}}_{3}$ with the implication of relativistic effect, from the ab initio density functional theory (DFT) simulation. The valley contrasting QAH effect is observed with a nonzero Chern number at a high-symmetry point stemming from Berry curvature and Wannier charge center (WCC), leading to a topologically nontrivial state. The occurrence of strong magnetic proximity coupling between graphene and ${\mathrm{CrBr}}_{3}$ monolayer is realized intrinsically, from shifting of Hall coefficient value near Fermi level. The opening of the global band gap (178 meV) is observed with the inclusion of spin orbit coupling (SOC). The anomalous Hall conductivity (AHC) demonstrates the presence of two maxima peaked at valley ${K}^{\ensuremath{'}}$ and $K$. The observation of AHC is mainly dominated by nonzero surface charge and localized potential at the heterointerface due to proximity interaction. The Fermi level is found to be located exactly inside the nontrivial global band gap, which can be tuned effectively by applying the external electric field or by introducing a staggered sublattice potential. This robustness makes experimental fabrication highly favorable for developing a valley contrasting QAH device prototype.

Unusual magnetic and transport properties in HoMn6Sn6 kagome magnet

With intricate lattice structures, kagome materials are an excellent platform to study various fascinating topological quantum states. In particular, kagome materials, revealing large responses to external stimuli such as pressure or magnetic field, are subject to special investigation. Here we study the kagome-net ${\mathrm{HoMn}}_{6}{\mathrm{Sn}}_{6}$ magnet that undergoes paramagnetic to ferrimagnetic transition (below 376 K) and reveals spin-reorientation transition below 200 K. In this compound, we observe the topological Hall effect and substantial contribution of anomalous Hall effect above 100 K. We unveil the pressure effects on magnetic ordering at a low magnetic field from the pressure tunable magnetization measurement. By utilizing high-resolution angle-resolved photoemission spectroscopy, Dirac-like dispersion at the high-symmetry point K is revealed in the vicinity of the Fermi level, which is well supported by the first-principles calculations. Our investigation will pave the way to understanding the magnetotransport and electronic properties of various rare-earth-based kagome magnets.

Dirac states in an inclined two-dimensional Su-Schrieffer-Heeger model

We propose to realize Dirac states in an inclined two-dimensional Su-Schrieffer-Heeger model on a square lattice. We show that a pair of Dirac points protected by space-time inversion symmetry appear in the semimetal phase. The locations of these Dirac points are not pinned to any high-symmetry points of the Brillouin zone but are tunable through parameter modulations. Interestingly, the merging of two Dirac points undergoes a topological phase transition that leads to either a weak topological insulator or a nodal-line semimetal. We provide a systematic analysis of these topological phases from both bulk and boundary perspectives combined with symmetry arguments. We also discuss feasible experimental platforms to realize our model.

First- and second-order Raman spectroscopy of monoclinic β−Ga2O3

We employ a combined experimental-theoretical study of the first- and second-order Raman modes of monoclinic $\ensuremath{\beta}\ensuremath{-}{\mathrm{Ga}}_{2}{\mathrm{O}}_{3}$. Gallium oxide has attracted considerable interest due to its deep-UV band gap paired with a high critical field strength, offering promising applications in power electronics. A crucial prerequisite for the future development of ${\mathrm{Ga}}_{2}{\mathrm{O}}_{3}$-based devices is a detailed understanding of the lattice vibrations, i.e., phonons, as they govern important material properties such as elasticity, thermal conductivity, temperature-dependence of the band gap, or free-carrier transport. Polarized micro-Raman spectroscopy measurements on the (010) and ($\overline{2}01$) planes of $\ensuremath{\beta}\ensuremath{-}{\mathrm{Ga}}_{2}{\mathrm{O}}_{3}$ single crystals enable the determination of the phonon frequencies of all 15 first-order and more than 40 second-order Raman modes. The experimental results are correlated with calculations of the mode frequencies, the phonon dispersion relation, and the phonon density of states using density functional perturbation theory. By applying a group-theoretical analysis, we are able to distinguish between overtones and combinational modes and identify the high-symmetry points in the Brillouin zone that contribute to the observed second-order modes. Based on this information, we demonstrate the simultaneous determination of Raman, IR, and acoustic phonons in $\ensuremath{\beta}\ensuremath{-}{\mathrm{Ga}}_{2}{\mathrm{O}}_{3}$ by second-order Raman spectroscopy.

Self-localized topological states in three dimensions

Three-dimensional (3D) topological materials exhibit much richer phenomena than their lower-dimensional counterparts. Here, we propose self-localized topological states (i.e., topological solitons) in a 3D nonlinear photonic Chern insulator. Despite being in the bulk and self-localized in all 3D, the topological solitons at high-symmetry points K and K' rotate in the same direction, due to the underlying topology. Specifically, under the saturable nonlinearity the solitons are stable over a broad frequency range. Our results highlight how topology and nonlinearity interact with each other and can be extended to other 3D topological systems.

Polarization-resolved analysis to solid high-order harmonic generation

We propose a quantitative polarization-resolved quantum trajectory approach to analyse the solid high-order harmonic generation (HHG). By using this approach, we can distinguish the ionization channels of different polarization components in HHG. We take the HHG in hBN under two counter-rotating circularly polarized fields as an example. We find that the right and left circular polarized high harmonics are contributed by the electron trajectories with the ionization channel around the high symmetry points K and K′ respectively. Furthermore, the intensity of the right and left circular polarization harmonics is determined by the interference between these electron trajectories. Our work offers a quantitatively polarization-resolved analysis to understand the underlying mechanism in HHG in the view of electron reciprocal space trajectory.

Hybridization bond states and band structure of graphene: a simple approach

This article reviews the basic theoretical aspects of graphene in order to contribute to making teaching easier and more understandable for those who still cannot understand it in an abstract way. This work presents the basic aspects of sp2 hybridization concepts from a quantum approach, which consists in obtaining the states for the sigma bonds from carbon atoms orientation close to another carbon atom that is our reference point in the crystal lattice. Furthermore, we calculate the band structure of graphene by using the tight-binding model, which can give us a good approximation to the energy levels as a function of the wave vector. This is for students interested in solid-state topic and it is approached in a simple and understandable way step by step to finally obtain the bands structure of the primitive cell of graphene monolayer. Additionally, we calculate the high symmetry points of the first Brillouin zone and the analysis around K points to obtain the well-known Dirac cones. The intention of this work is to show that this topic can be less abstract than it seems.

Electronic structure of the homologous series of Ruddlesden–Popper phases SrO(SrTiO3) n , (n = 0–3, ∞)

Abstract The system SrO(SrTiO3) n contains promising compounds for several applications, whose functionalities all depend in particular on the band structure of the respective crystal. While the electronic structure of SrO and SrTiO3 is sufficiently clarified in literature, there is a lack of information concerning the Ruddlesden–Popper (RP) phases. In this work, density functional theory is used to compute the electronic structure for the homologous series with n = 0–3, ∞. The according band structures are presented and effective masses are given for the complete system. In addition, the calculations are consulted to discuss the thermodynamical stability of the RP phases, confirming the gain of formation energy up to n = 3, as reported in recent literature. A promising possibility for applications has been found, analyzing theses band structures: As the optical gaps at distinct high-symmetry points of the Brillouin zone show different dependencies on the lattice parameters, as it is reported for SrO in literature, a similar behavior could be expected in particular for the RP phase with n = 1.

Structure Optimization and Electronic Properties of Co2tix(X=Si, Ge) Heusler Alloys: A DFT Study

A theoretical study of structural, electronic, magnetic, mechanical and vibrational properties of Co-based heusler alloys, namely Co2TiX(X=Si,Ge); have been studied by the first principle calculations with the Generalized Gradient approximation(GGA) based on Density functional theory(DFT). The total spin magnetic moments per unit cell is 2µB, obeying the Slater-Pauling rule. The electronic band structure along the high symmetry points reveal that majority spin states have metallic character and minority spin states have a band gap at the Fermi level. There are no imaginary phonon mode which illustrate the dynamical stability. The Bulk modulus, young’s modulus, shear modulus, Pugh’s ratio, sound velocities, Debye temperature in accordance with the Voigt-Reuss-Hill approximation. The elastic constant results show that these alloys are mechanically stable.

Experimental Realization of Two-Dimensional Weak Topological Insulators.

We report the experimental realization of a two-dimensional (2D) weak topological insulator (WTI) in spinless Su-Schrieffer-Heeger circuits with parity-time and chiral symmetries. Strong and weak topological indexes are adopted to explain the experimental findings that a Dirac semimetal (DSM) phase and four WTI phases emerge in turn when we modulate the centrosymmetric circuit deformations. In the DSM phase, it is found that the Dirac cone is highly anisotropic and that it is not pinned to any high-symmetry points but can widely move within the Brillouin zone, which eventually leads to the phase transition between WTIs. In addition, we observe a pair of flat-band domain wall states by designing spatially inhomogeneous node connections. Our work provides the first experimental evidence for 2D WTIs, which significantly advances our understanding of the strong and weak nature of topological insulators, the robustness of flat bands, and the itinerant and anisotropic features of Dirac cones.

Topological invariants beyond symmetry indicators: Boundary diagnostics for twofold rotationally symmetric superconductors

Topological crystalline superconductors are known to have possible higher-order topology, which results in Majorana modes on $d-2$ or lower-dimensional boundaries. Given the rich possibilities of boundary signatures, it is desirable to have topological invariants that can predict the type of Majorana modes from band structures. Although symmetry indicators, a type of invariant that depends only on the band data at high-symmetry points, have been proposed for certain crystalline superconductors, there exist symmetry classes in which symmetry indicators fail to distinguish superconductors with different Majorana boundaries. Here, we systematically obtain topological invariants for an example of this kind, two-dimensional time-reversal symmetric superconductors with twofold rotational symmetry $C_2$. First, we show that the nontrivial topology is independent of band data on the high-symmetry points by conducting a momentum-space classification study. Then, from the resulting K groups, we derive calculable expressions for four $\mathbb{Z}_2$ invariants defined on high-symmetry lines or general points in the Brillouin zone. Finally, together with a real-space classification study, we establish the bulk-boundary correspondence and show that the four $\mathbb{Z}_2$ invariants can predict Majorana boundary types from band structures. Our proposed invariants can fuel practical material searches for $C_2$-symmetric topological superconductors featuring Majorana edge and corner modes.

Origin of the frequency-sensitive super-collimation phenomenon from the geometry of band dispersion surface for two-dimensional photonic crystals

Frequency-sensitive super-collimation (FSSC) is a novel dispersion phenomenon of photonic crystals (PhCs) that can realize the beam collimating propagation with very high frequency sensitivity. In order to deeply investigate the origin and the stability of FSSC phenomenon in a wide parameter space, we study the geometry of dispersion surface in detail. Four features for the special geometry of dispersion surface with FSSC are found for rectangular PhCs. The special geometry supports the stability of FSSC in a wide range of parameter space. Two-parameter modulation (TPM) method, in which the aspect ratio β and the dielectric constant of rods ɛr of rectangular lattice are chosen as the key parameters, is used to analyze the geometry of dispersion surface from the frequency changes at the high-symmetry points. Step by step, the origin of such geometry is revealed and the evolving process can be explained by the field distribution changes of Bloch modes at the high-symmetry points. Furthermore, we show that the geometry not only can be used to explain the origin and the stability of FSSC, but also can help us to find other FSSC phenomenons. Theoretically, we believe the geometry of dispersion surface and the TPM can be widely used on the studies of complex dispersion properties of PhCs. The FSSCs found in this work with higher sensitivity or higher stability can help us to design new on-chip PhC devices.

Valley-protected topological interface state of the elastic wave: From discrete model to multistable mechanical metamaterials

Topological metamaterials provide a new strategy to guide wave energy and exhibit unprecedented robustness. In this study, a novel design strategy is proposed to realize programmable topological metamaterials with local resonant eigenstates. Firstly, in the discrete spring-mass model, two resonant masses and two base masses are introduced into the hexagonal lattice, and a local resonant eigenstate-induced Dirac cone can be formed at the high symmetry point of the Brillouin zone. By introducing the spring stiffness difference, a topological bandgap is opened near the Dirac degeneracy frequency. Thereafter, the nontrivial nature of the bandgap is verified by the theoretical evaluation of the Berry curvature and Chern number. Secondly, the symmetry-breaking configuration is extended to a continuum plate-resonator model. The numerical results demonstrate that the interface state wave propagates along the interface path at a frequency located at the edge-bulk band. Finally, by using the asymmetric effective stiffness of the bistable structure between tension and compression, a programmable topological interface path is realized. The proposed reconfigurable design based on asymmetry significantly expands the design space of metamaterials.

First principle based investigation of topological insulating phase in half-Heusler family NaYO (Y = Ag, Au, and Cu)

The ternary half-Heusler compounds have shown great potential for realizing new 3D topological insulators. With band gap tuning and spin orbit coupling these compounds may undergo topological phase transitions. In present work, we explore the possibility of realizing a topological insulating phase in half-Heusler family NaYO (Y = Ag, Au, and Cu). We find that for NaAgO, external strain (∼19%) along with spin–orbit coupling (SOC), is required to achieve band-inversion at Γ high-symmetry point and leads to phase transition from trivial to non-trivial topological insulating phase. In case of NaAuO and NaCuO, non-trivial phase appears in their equilibrium lattice constant, hence only SOC is enough to achieve band-inversion leading to non-trivial topology. The non-centrosymmetric nature of crystal geometry leads to the formation of two twofold degenerate point nodes near the Fermi level.

Systematic generation of the cascade of anomalous dynamical first- and higher-order modes in Floquet topological insulators

After extensive investigation on the Floquet second-order topological insulator (FSOTI) in two dimension (2D), here we propose two driving schemes to systematically engineer the hierarchy of Floquet first-order topological insulator, FSOTI, and Floquet third-order topological insulator in three dimension (3D). Our driving protocols allow these Floquet phases to showcase regular $0$, anomalous $\pi$, and hybrid $0$-$\pi$-modes in a unified phase diagram, obtained for both 2D and 3D systems, while staring from the lower order topological or non-topological phases. Both the step drive and the mass kick protocols exhibit the analogous structure of the evolution operator around the high symmetry points. These eventually enable us to understand the Floquet phase diagrams analytically and the Floquet higher order modes numerically based on finite size systems. The number of $0$ and $\pi$-modes can be tuned irrespective of the frequency in the step drive scheme while we observe frequency driven topological phase transitions for the mass kick protocol. We topologically characterize some of these higher order Floquet phases (harboring either $0$ or anomalous $\pi$ mode) by suitable topological invariant in 2D and 3D cases.

Temperature Dependence of the Indirect Gap and the Direct Optical Transitions at the High-Symmetry Point of the Brillouin Zone and Band Nesting in MoS2, MoSe2, MoTe2, WS2, and WSe2 Crystals.

Following the rise of interest in the properties of transition metal dichalcogenides, many experimental techniques were employed to research them. However, the temperature dependencies of optical transitions, especially those related to band nesting, were not analyzed in detail for many of them. Here, we present successful studies utilizing the photoreflectance method, which, due to its derivative and absorption-like character, allows investigating direct optical transitions at the high-symmetry point of the Brillouin zone and band nesting. By studying the mentioned optical transitions with temperature from 20 to 300 K, we tracked changes in the electronic band structure for the common transition metal dichalcogenides (TMDs), namely, MoS2, MoSe2, MoTe2, WS2, and WSe2. Moreover, transmission and photoacoustic spectroscopies were also employed to investigate the indirect gap in these crystals. For all observed optical transitions assigned to specific k-points of the Brillouin zone, their temperature dependencies were analyzed using the Varshni relation and Bose–Einstein expression. It was shown that the temperature energy shift for the transition associated with band nesting is smaller when compared with the one at high-symmetry point, revealing reduced average electron–phonon interaction strength.

Topological States in Two-Dimensional Su-Schrieffer-Heeger Models

We study the topological properties of the generalized two-dimensional (2D) Su-Schrieffer-Heeger (SSH) models. We show that a pair of Dirac points appear in the Brillouin zone (BZ), consisting a semimetallic phase. Interestingly, the locations of these Dirac points are not pinned to any high-symmetry points of the BZ but tunable by model parameters. Moreover, the merging of two Dirac points undergoes a novel topological phase transition, which leads to either a weak topological insulator or a nodal-line metallic phase. We demonstrate these properties by constructing two specific models, which we referred as type-I and type-II 2D SSH models. The feasible experimental platforms to realize our models are also discussed.