Edge Of Brillouin Zone Research Articles

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.

Layer k -projection and unfolding electronic bands at interfaces

The k-projection method provides an approach to separate the contributions from different constituents in heterostructure systems and can act as an aid to connect the results of experiments and calculations. We show that the technique can be used to "unfold" the calculated electronic bands of interfaces and supercells, and provide local band structure by integrating the projected states over specified regions of space, a step that can be implemented efficiently using fast Fourier transforms. We apply the method to investigate the effects of interfaces in heterostructures consisting of a graphene bilayer on H-saturated SiC(0001), BAs monolayer on the ferromagnetic semiconductor CrI3, silicene on Ag(111), and to the Bi2Se3 surface. Our results reveal that the band structure of the graphene bilayer around the Dirac point is strongly dependent on the termination of SiC(0001): on the C face, the graphene is n doped and a gap of ~0.13 eV is opened, whereas, on the Si face, the graphene is essentially unchanged and neutral. We show that for BAs/CrI3, the magnetic proximity effect can effectively induce a spin splitting up to about 50 meV in BAs. For silicene/Ag(111), our calculations reproduce the angle-resolved photoemission spectroscopy results, including linearly dispersing bands at the edge of the first Brillouin zone of Ag(111); although these states result from the interaction between the silicene overlayer and the substrate, we demonstrate that they are not Dirac states.

Lifetime-shortened acoustic phonons and static order at the Brillouin zone boundary in the organic-inorganic perovskite CH3NH3PbCl3.

Lead halide hybrid perovskites consist of an inorganic framework hosting a molecular cation located in the interstitial space. These compounds have been extensively studied as they have been identified as promising materials for photovoltaic applications with the interaction between the molecular cation and the inorganic framework implicated as influential for the electronic properties. CH3NH3PbCl3 undergoes two structural transitions from a high temperature cubic unit cell to a tetragonal phase at 177 K and then a subsequent orthorhombic transition at 170 K. We have measured the low-frequency lattice dynamics using neutron spectroscopy and observe an energy broadening in the acoustic phonon linewidth towards the high-symmetry point when approaching the transitions. Concomitant with these zone boundary anomalies is a hardening of the entire acoustic phonon branch measured in the limit near the (2, 0, 0) Bragg position with decreasing temperature. Measurements of the elastic scattering at the Brillouin zone edges , and show Bragg peaks appearing below these structural transitions. Based on selection rules of neutron scattering, we suggest that the higher 177 K transition is displacive with a distortion of the local octahedral environment and the lower transition is a rigid tilt transition of the octahedra. We do not observe any critical broadening in energy or momentum, beyond resolution, of these peaks near the transitions. We compare these results to the critical properties reported near the structural transitions in other perovskites and particularly CsPbCl3 [Y. Fujii, S. Hoshino, Y. Yamada, and G. Shirane, Phys. Rev. B 9, 4549 (1974)]. We suggest that the simultaneous onset of static resolution-limited Bragg peaks at the zone boundaries and the changes in acoustic phonon energies near the zone center is evidence of a coupling between the inorganic framework and the molecular cation. The results also highlight the importance of displacive transitions in organic-inorganic hybrid perovskites.

Excitons and trions in two-dimensional semiconductors based on transition metal dichalcogenides

Theoretical and experimental results on excitonic effects in monomolecular layers of transition metal dichalcogenides are reviewed. These two-dimensional semiconductors exhibit a direct bandgap of about 2 eV at the Brillouin zone edges, and the binding energies of their neutral and charged excitons are in the range of hundreds and tens of millielectron-volts, respectively. This implies that electron–hole complexes determine the optical properties of transition metal dichalcogenide monolayers. Topics discussed in this review include the band structure details needed to understand the excitonic effects in these materials, the structure and fine structure of exciton and trion energy levels, the features of the spin and valley dynamics of Coulomb complexes, and how neutral and charged excitons manifest themselves in linear and nonlinear optical effects.

Parity-time symmetry in periodically curved optical waveguides.

The coupling strength between two parity-time (PT) symmetric resonators determines whether the PT phase is broken or not. Here we investigate the scenario that two optical waveguides are spatially curved so that they switch periodically between unbroken and broken PT phases. We show that the existence of locally broken PT phase does not necessarily render a broken phase to waves propagating inside. Criteria are proposed to characterize the collective dynamics of wave near the Brillouin zone (BZ) edge, toward the cases of a totally broken phase, a partially broken phase, or a totally unbroken phase. We also discuss the characteristics of two special kinds of exceptional points (EPs) at the BZ edge, and show that their field patterns are displaced by half a period with each other. Full-wave numerical simulation proves our analysis. Potential applications especially these associated with EPs are discussed. This study helps us to understand how the locally PT-symmetric related eigenstate influences the globally collective dynamics of wave in spatially periodic configuration.

Topological plasmons in dimerized chains of nanoparticles: robustness against long-range quasistatic interactions and retardation effects

We present a simple model of collective plasmons in a dimerized chain of spherical metallic nanoparticles, an elementary example of a topologically nontrivial nanoplasmonic system. Taking into account long-range quasistatic dipolar interactions throughout the chain, we provide an exact analytical expression for the full quasistatic bandstructure of the collective plasmons. An explicit calculation of the Zak phase proves the robustness of the topological physics of the system against the inclusion of long-range Coulomb interactions, despite the broken chiral symmetry. Using an open quantum systems approach, which includes retardation through the plasmon-photon coupling, we go on to analytically evaluate the resulting radiative frequency shifts of the plasmonic spectrum. The bright plasmonic bands experience size-dependent radiative shifts, while the dark bands are essentially unaffected by the light-matter coupling. Notably, the upper transverse-polarized band presents a logarithmic singularity where the quasistatic spectrum intersects the light cone. At wavevectors away from this intersection and for subwavelength nanoparticles, the plasmon-photon coupling only leads to a quantitative reconstruction of the bandstructure and the topologically-protected states at the edge of the first Brillouin zone are essentially unaffected.

Substrate-induced topological minibands in graphene

The honeycomb lattice sets the basic arena for numerous ideas to implement electronic, photonic, or phononic topological bands in (meta-)materials. Novel opportunities to manipulate Dirac electrons in graphene through band engineering arise from superlattice potentials as induced by a substrate such as hexagonal boron-nitride. Making use of the general form of a weak substrate potential as dictated by symmetry, we analytically derive the low-energy minibands of the superstructure, including a characteristic 1.5 Dirac cone deriving from a three-band crossing at the Brillouin zone edge. Assuming a large supercell, we focus on a single Dirac cone (or valley) and find all possible arrangements of the low-energy electron and hole bands in a complete six-dimensional parameter space. We identify the various symmetry planes in parameter space inducing gap closures and find the sectors hosting topological minibands, including also complex band crossings that generate a valley Chern number atypically larger than one. Our map provides a starting point for the systematic design of topological bands by substrate engineering.

Generation of Frenkel defects above the Debye temperature by proliferation of phonons near the Brillouin zone edge

A novel, non-radiative mechanism is reported by which Frenkel pairs of vacancies and interstitials are generated in molar concentrations far above thermal equilibrium. This mechanism is demonstrated in molecular dynamics (MD) simulations of an aluminum single crystal with a free surface. They suggest that three conditions must be fulfilled: (i) lattice vibrations near the Brillouin zone edge are being excited, (ii) these vibrations proliferate at a sufficiently high rate, and (iii) the sample temperature is above the Debye temperature (but significantly below the melting point). The simulations employed an EAM potential for Al. We attempt to draw a confluence between our MD simulations and recent experiments on flash sintering of aluminum. The simulation results are also consistent with flash experiments on polycrystals and single crystals of zirconium and titanium oxides where the Debye temperature was discovered to be the lower limit for the onset of the flash.

Composition Relation between Nonlinear Bloch Waves and Gap Solitons in Periodic Fractional Systems.

Evolution of beams in nonlinear optical media with a fractional-order diffraction is currently attracting a growing interest. We address the existence of linear and nonlinear Bloch waves in fractional systems with a periodic potential. Under a defocusing nonlinearity, nonlinear Bloch waves at the centers or edges of the first Brillouin zone bifurcate from the corresponding linear Bloch modes at different band edges. They can be constructed by directly copying a fundamental gap soliton (in one lattice site) or alternatively copying it and its mirror image to infinite lattice channels. The localized truncated-Bloch-wave solitons bridging nonlinear Bloch waves and gap solitons are also revealed. We thus prove that fundamental gap solitons can be used as unit cells to build nonlinear Bloch waves or truncated-Bloch-wave solitons, even in fractional configurations. Our results provide helpful hints for understanding the dynamics of localized and delocalized nonlinear modes and the relation between them in periodic fractional systems with an optical nonlinearity.

Electronic structure-dependent magneto-optical Raman effect in atomically thin WS2

Magneto-optical effect, such as Faraday rotation, magneto-optical Kerr and Zeeman effect, plays a fundamental role and has found extensive applications in condensed-matter physics. Two-dimensional transition metal dichalcogenides provide us an unprecedented platform for exploring abundant magneto-optical phenomena. Here we report such experiments that demonstrate the band structure-dependent magneto-optical Raman effect in atomically thin WS2. Our experiments in conjunction with theoretical calculations uncover that the direct to indirect transition leads to distinct magneto-optical Raman effect between monolayer and bilayer/trilayer. We further quantitatively determine the optical mobility of carriers at different critical points of conduction band (K and Q points). In addition, our measurements reveal that not only the particular single-phonon mode A1g(Γ) but also multiple phonon replica and phonons at the edges of Brillouin zone display electronic structure-dependent gigantic magneto-optical Raman effect. Our work establishes a firm basis for benchmarking a new microscopic quantum Raman scattering theory, and paves a way towards novel magneto-optical applications based on 2D semiconductors.

Layer-dependent second-order Raman intensity of MoS2 and WSe2 : Influence of intervalley scattering

Acoustic-phonon Raman scattering, as a defect-induced second-order Raman scattering process (with incident photon scattered by one acoustic phonon at the Brillouin-zone edge and the momentum conservation fulfilled by defect scattering), is used as a sensitive tool to study the defects of transition-metal dichalcogenides (TMDs). Moreover, second-order Raman scattering processes are closely related to the valley depolarization of single-layer TMDs in potential valleytronic applications. Here, the layer dependence of second-order Raman intensity of $\mathrm{Mo}{\mathrm{S}}_{2}$ and $\mathrm{WS}{\mathrm{e}}_{2}$ is studied. The electronic band structures of $\mathrm{Mo}{\mathrm{S}}_{2}$ and $\mathrm{WS}{\mathrm{e}}_{2}$ are modified by the layer thicknesses; hence, the resonance conditions for both first-order and second-order Raman scattering processes are tuned. In contrast to the first-order Raman scattering, second-order Raman scattering of $\mathrm{Mo}{\mathrm{S}}_{2}$ and $\mathrm{WS}{\mathrm{e}}_{2}$ involves additional intervalley scattering of electrons by phonons with large momenta. As a result, the electron states that contribute most to the second-order Raman intensity are different from that to first-order process. A weaker layer-tuned resonance enhancement of second-order Raman intensity is observed for both $\mathrm{Mo}{\mathrm{S}}_{2}$ and $\mathrm{WS}{\mathrm{e}}_{2}$. Specifically, when the incident laser has photon energy close to the optical band gap and the Raman spectra are normalized by the first-order Raman peaks, single-layer $\mathrm{Mo}{\mathrm{S}}_{2}$ or $\mathrm{WS}{\mathrm{e}}_{2}$ has the strongest second-order Raman intensity. This layer-dependent second-order Raman intensity can be further utilized as an indicator to identify the layer number of $\mathrm{Mo}{\mathrm{S}}_{2}$ and $\mathrm{WS}{\mathrm{e}}_{2}$.

A Topological View on Optical and Phononic Fabry–Perot Microcavities through the Su–Schrieffer–Heeger Model

Advances in nanofabrication technologies have enabled the study of acoustic wave phenomena in the technologically relevant GHz–THz range. First steps towards applying concepts from topology in nanophononics were made with the proposal of a new topological acoustic resonator, based on the concept of band inversion. In topology, the Su–Schrieffer–Heeger (SSH) model is the paradigm that accounts for the topological properties of many one-dimensional structures. Both the classical Fabry–Perot resonator and the reported topological resonators are based on Distributed Bragg Reflectors (DBRs). A clear and detailed relation between the two systems, however, is still lacking. Here, we show how a parallelism between the standard DBR-based acoustic Fabry–Perot type cavity and the SSH model of polyacetylene can be established. We discuss the existence of surface modes in acoustic DBRs and interface modes in concatenated DBRs and show that these modes are equivalent to Fabry–Perot type cavity modes. Although it is not possible to assign topological invariants to both acoustic bands enclosing the considered minigap in the nanophononic Fabry–Perot case, the existence of the confined mode in a Fabry–Perot cavity can nevertheless be interpreted in terms of the symmetry inversion of the Bloch modes at the Brillouin zone edge.

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

Topological insulating material with dissipationless edge states is a rising star in spintronics. While most two-dimensional (2D) topological insulators belong to group-IV or -V elements in a honeycomb lattice, herein, we propose a new topological phase in the 2D hexagonal group-III crystal, h-Tl, based on a tight-binding model and density-functional theory calculation. Analysis of band dispersion reveals a Dirac nodal-ring near the Fermi level, which is attributed to px,y/pz band crossing. Upon inclusion of spin-orbit coupling (SOC), h-Tl turns into a quantum spin Hall insulator under 21% biaxial strain, confirmed by integrating spin Berry curvature in the Brillouin zone and spin-polarized edge states. A prominent feature of its electronic properties is that the effect of SOC plays two essential roles of both topological gap opening and band inversion between px,y/pz orbitals, which is the first observed phenomenon in 2D materials. This study extends the scope of 2D elemental topological insulators and presents a platform to design new 2D topotronics materials.

Quantum Phases and Collective Excitations of a Spin-Orbit-Coupled Bose–Einstein Condensate in a One-Dimensional Optical Lattice

The ground state of a spin-orbit-coupled Bose gas in a one-dimensional optical lattice is known to exhibit a mixed regime, where the condensate wave function is given by a superposition of multiple Bloch-wave components, and an unmixed one, in which the atoms occupy a single Bloch state. The unmixed regime features two unpolarized Bloch-wave phases, having quasimomentum at the center or at the edge of the first Brillouin zone, and a polarized Bloch-wave phase at intermediate quasimomenta. By calculating the critical values of the Raman coupling and of the lattice strength at the transitions among the various phases, we show the existence of a tricritical point where the mixed, the polarized and the edge-quasimomentum phases meet, and whose appearance is a consequence of the spin-dependent interaction. Furthermore, we evaluate the excitation spectrum in the unmixed regime and we characterize the behavior of the phonon and the roton modes, pointing out the instabilities occurring when a phase transition is approached.

Band warping, band non-parabolicity, and Dirac points in electronic and lattice structures

We illustrate at a fundamental level the physical and mathematical origins of band warping and band non-parabolicity in electronic and vibrational structures. We point out a robust presence of pairs of topologically induced Dirac points in a primitive-rectangular lattice using a p-type tight-binding approximation. We analyze two-dimensional primitive-rectangular and square Bravais lattices with implications that are expected to generalize to more complex structures. Band warping is shown to arise at the onset of a singular transition to a crystal lattice with a larger symmetry group, which allows the possibility of irreducible representations of higher dimensions, hence band degeneracy, at special symmetry points in reciprocal space. Band warping is incompatible with a multi-dimensional Taylor series expansion, whereas band non-parabolicities are associated with multi-dimensional Taylor series expansions to all orders. Still band non-parabolicities may merge into band warping at the onset of a larger symmetry group. Remarkably, while still maintaining a clear connection with that merging, band non-parabolicities may produce pairs of conical intersections at relatively low-symmetry points. Apparently, such conical intersections are robustly maintained by global topology requirements, rather than any local symmetry protection. For two p-type tight-binding bands, we find such pairs of conical intersections drifting along the edges of restricted Brillouin zones of primitive-rectangular Bravais lattices as lattice constants vary relatively to each other, until these conical intersections merge into degenerate warped bands at high-symmetry points at the onset of a square lattice. The conical intersections that we found appear to have similar topological characteristics as Dirac points extensively studied in graphene and other topological insulators, even though our conical intersections have none of the symmetry complexity and protection afforded by the latter more complex structures.

Spin waves in periodic antidot waveguide of complex base

We consider the planar magnonic waveguide with a periodic sequence of antidots forming a zig-zag pattern, where two neighboring antidots are shifted towards the opposite edges of the waveguide. This system has a complex base with two antidots in one unit cell. The Brillouin zone is here two-times narrower than the Brillouin zone for the waveguide without displacement of antidots. We have shown that for dispersion relation folded into a narrower Brillouin zone, a new frequency gap can be opened and their width can be controlled by the shift of the antidots. We found that the different strength of spin wave pinning at the edges of the periodic waveguide (and their antidots) determines the dependence of the width of gap on the shift of antidots. For the systems with completely free or ideally pinned magnetization, these dependencies are qualitatively different. We have found an optimum shift of antidot for maximizing the width of the gap for the system with pinned magnetization. More interestingly, we notice that for this kind of geometry of the structure, the majority of the modes are doubly degenerate at the edge of Brillouin zone and have a finite group velocity at the very close vicinity of the edge of Brillouin zone, for larger values of antidot shift. This empowers us to design a magnonic waveguide to steer the spin waves.

Dirac nodal lines and induced spin Hall effect in metallic rutile oxides

We have found Dirac nodal lines (DNLs) in the band structures of metallic rutile oxides IrO$_2$, OsO$_2$, and RuO$_2$ and revealed a large spin Hall conductivity contributed by these nodal lines, which explains a strong spin Hall effect (SHE) of IrO$_2$ discovered recently. Two types of DNLs exist. The first type forms DNL networks that extend in the whole Brillouin zone and appears only in the absence of spin-orbit coupling (SOC), which induces surface states on the boundary. Because of SOC-induced band anti-crossing, a large intrinsic SHE can be realized in these compounds. The second type appears at the Brillouin zone edges and is stable against SOC because of the protection of nonsymmorphic symmetry. Besides reporting new DNL materials, our work reveals the general relationship between DNLs and the SHE, indicating a way to apply Dirac nodal materials for spintronics.

Negative-index dispersion and accidental mode degeneracy in an asymmetric spoof–insulator–spoof waveguide**Project supported by the National Basic Research Program of China (Grant No. 2013CBA01702) and the National Natural Science Foundation of China (Grant Nos. 61377016, 61575055, 10974039, 61307072, 61308017, and 61405056).

It has been recently demonstrated that negative-index dispersion and mode degeneracy can be achieved by manipulating a spoof–insulator–spoof (SIS) waveguide. In this paper, we propose a new SIS waveguide, which is composed of two spoof surface plasmon polaritons (SSPPs) waveguides drilled with periodic rhomboidal grooves. Both the symmetric and asymmetric cases are investigated. Our simulation results show that the asymmetric SIS waveguides are more significant. By tailoring the tilt of the rhomboidal grooves, the negative-index dispersion can be achieved and the microwave band gap (MBG) can be effectively modulated. At a critical tilt, there appears an accidental mode degeneracy at the edge of the first Brillouin zone. The excitation and propagation of the two coupled modes sustained by the asymmetric SIS waveguides are also demonstrated.

Interaction-Driven Shift and Distortion of a Flat Band in an Optical Lieb Lattice.

We report the momentum-resolved measurement of Bloch bands in an optical Lieb lattice for a Bose-Einstein condensate (BEC). A BEC in the lattice is transported to a desired quasimomentum by applying a constant force. The energy dispersion of the lowest band is obtained by integrating measured group velocities. We also measure the gap from the lowest band to the higher bands with the same quasimomentum, which can be extracted from the oscillation of the sublattice populations after preparing a superposition of the band eigenstates. We show that the experimental results agree with a band calculation based on the Bogoliubov approximation. It is revealed that the second band, which should be flat in a single-particle description, is shifted and, in particular, distorted around the Brillouin zone edge as the interaction strength increases.

Band Gap Opening Induced by the Structural Periodicity in Epitaxial Graphene Buffer Layer.

The epitaxial graphene buffer layer on the Si face of hexagonal SiC shows a promising band gap, of which the precise origin remains to be understood. In this work, we correlate the electronic to the atomic structure of the buffer layer by combining angle resolved photoemission spectroscopy (ARPES), scanning tunneling microscopy (STM), and high-resolution scanning transmission electron microscopy (HR-STEM). We show that the band structure in the buffer has an electronic periodicity related to the structural periodicity observed in STM images and published X-ray diffraction. Our HR-STEM measurements show the bonding of the buffer layer to the SiC at specific locations separated by 1.5 nm. This is consistent with the quasi 6 × 6 periodic corrugation observed in the STM images. The distance between buffer C and SiC is 1.9 Å in the bonded regions and up to 2.8 Å in the decoupled regions, corresponding to a 0.9 Å corrugation of the buffer layer. The decoupled regions are sp2 hybridized. Density functional tight binding (DFTB) calculations demonstrate the presence of a gap at the Dirac point everywhere in the buffer layer, even in the decoupled regions where the buffer layer has an atomic structure close to that of graphene. The surface periodicity also promotes band in the superperiodic Brillouin zone edges as seen by photoemission and confirmed by our calculations.