Showing papers on "Brillouin zone published in 2021"

Observation of Non-Bloch Parity-Time Symmetry and Exceptional Points.

Abstract: Parity-time (PT)-symmetric Hamiltonians have widespread significance in non-Hermitian physics. A PT-symmetric Hamiltonian can exhibit distinct phases with either real or complex eigenspectrum, while the transition points in between, the so-called exceptional points, give rise to a host of critical behaviors that holds great promise for applications. For spatially periodic non-Hermitian systems, PT symmetries are commonly characterized and observed in line with the Bloch band theory, with exceptional points dwelling in the Brillouin zone. Here, in nonunitary quantum walks of single photons, we uncover a novel family of exceptional points beyond this common wisdom. These "non-Bloch exceptional points" originate from the accumulation of bulk eigenstates near boundaries, known as the non-Hermitian skin effect, and inhabit a generalized Brillouin zone. Our finding opens the avenue toward a generalized PT-symmetry framework, and reveals the intriguing interplay between PT symmetry and non-Hermitian skin effect.

Two-dimensional overdamped fluctuations of the soft perovskite lattice in CsPbBr3

Abstract: Lead halide perovskites exhibit structural instabilities and large atomic fluctuations thought to impact their optical and thermal properties, yet detailed structural and temporal correlations of their atomic motions remain poorly understood. Here, these correlations are resolved in CsPbBr3 crystals using momentum-resolved neutron and X-ray scattering measurements as a function of temperature, complemented with first-principles simulations. We uncover a striking network of diffuse scattering rods, arising from the liquid-like damping of low-energy Br-dominated phonons, reproduced in our simulations of the anharmonic phonon self-energy. These overdamped modes cover a continuum of wave vectors along the edges of the cubic Brillouin zone, corresponding to two-dimensional sheets of correlated rotations in real space, and could represent precursors to proposed two-dimensional polarons. Further, these motions directly impact the electronic gap edge states, linking soft anharmonic lattice dynamics and optoelectronic properties. These results provide insights into the highly unusual atomic dynamics of halide perovskites, relevant to further optimization of their optical and thermal properties. Neutron and X-ray scattering measurements provide further insight into the anharmonic behaviour of lead halide perovskites, revealing that rotations of PbBr6 octahedra in CsPbBr3 crystals occur in a correlated fashion along two-dimensional planes.

Thermodynamically stable skyrmion lattice in a tetragonal frustrated antiferromagnet with dipolar interaction

Abstract: Motivated by recent experimental results for ${\mathrm{GdRu}}_{2}{\mathrm{Si}}_{2}$ [Khanh et al., Nat. Nanotechnol. 15, 444 (2020)], in which a nanometric square skyrmion lattice was observed, we propose a simple analytical mean-field description of the high-temperature part of the phase diagram of centrosymmetric tetragonal frustrated antiferromagnets with dipolar interaction in the external magnetic field. Dipolar forces provide momentum-dependent biaxial anisotropy in reciprocal space. It is shown that in a tetragonal lattice, in the large part of the Brillouin zone, for mutually perpendicular modulation vectors in the $ab$ plane this anisotropy has mutually perpendicular easy axes and collinear middle axes, which leads to double-$Q$ modulated spin structure stabilization. In the large part of its stability region, the latter turns out to be a square skyrmion lattice with a topological charge of $\ifmmode\pm\else\textpm\fi{}1$ per magnetic unit cell, which is determined by the frustrated exchange coupling and thus nanometer sized. Easy and middle axes can be swapped in the presence of additional single-ion easy-axis anisotropy. This results in the different phase diagram. It is argued that the latter case is relevant to ${\mathrm{GdRu}}_{2}{\mathrm{Si}}_{2}$.

Strong-field physics in three-dimensional topological insulators

Abstract: In this work, the authors theoretically investigate high-order harmonic generation from a three-dimensional topological insulator. They find that the surface states enhance the harmonic yield for circularly polarized fields. Such behaviors could be linked to the conical shape of the energy bands near the Dirac point in the Brillouin zone, as well as the associated interband transition dipole moments that form vortexlike features at the cusp of the cone.

Non-Bloch band theory in bosonic Bogoliubov-de Gennes systems

Abstract: Diagonalization of a bosonic Bogoliubov--de Gennes system is intrinsically a non-Hermitian problem, although its Hamiltonian is Hermitian. Here, the authors study the non-Hermitian nature of bosonic Bogoliubov--de Gennes systems by constructing a non-Bloch band theory. Such systems have a unique non-Hermitian Brillouin zone, where Hermitian and non-Hermitian regions coexist. Accordingly, its non-Hermitian skin effect shows instability against infinitesimal instability and reentrant behavior.

Advances in Brillouin–Mandelstam light-scattering spectroscopy

Abstract: Recent years have witnessed a much broader use of Brillouin inelastic light-scattering spectroscopy for the investigation of phonons and magnons in novel materials, nanostructures and devices. Driven by the developments in instrumentation and the strong need for accurate knowledge on the energies of elemental excitations, Brillouin–Mandelstam spectroscopy is rapidly becoming an essential technique that is complementary to Raman inelastic light-scattering spectroscopy. We provide an overview of recent progress in the Brillouin light-scattering technique, focusing on the use of this photonic method for the investigation of confined acoustic phonons, phononic metamaterials and magnon propagation and scattering. This Review emphasizes the emerging applications of Brillouin–Mandelstam spectroscopy for phonon-engineered structures and spintronic devices, and concludes with a perspective on future directions. Nearly 100 years after the prediction of Brillouin light-scattering spectroscopy, or Brillouin–Mandelstam light-scattering spectroscopy, the effect has proved itself a powerful tool for decades. Now its application to probing confined acoustic phonons, phononic metamaterials and magnons is reviewed.

The Interaction between Surface Acoustic Waves and Spin Waves: The Role of Anisotropy and Spatial Profiles of the Modes.

Abstract: The interaction between different types of wave excitation in hybrid systems is usually anisotropic. Magnetoelastic coupling between surface acoustic waves and spin waves strongly depends on the direction of the external magnetic field. However, in the present study we observe that even if the orientation of the field is supportive for the coupling, the magnetoelastic interaction can be significantly reduced for surface acoustic waves with a particular profile in the direction normal to the surface at distances much smaller than the wavelength. We use Brillouin light scattering for the investigation of thermally excited phonons and magnons in a magnetostrictive CoFeB/Au multilayer deposited on a Si substrate. The experimental data are interpreted on the basis of a linearized model of interaction between surface acoustic waves and spin waves.

Symmetry-enforced ideal lanternlike phonons in the ternary nitride Li 6 WN 4

Abstract: Topological physics of phonon spectra has been attracting great interest. Using the first-principle calculations and symmetry analysis, we show the realistic ternary nitride ${\mathrm{Li}}_{6}{\mathrm{WN}}_{4}$ features ideal (nearly flat) nodal-surface and nodal-line structures in its phonon spectra. These nodal degeneracies are shaped like lanterns, and their existence is guaranteed by nonsymmorphic symmetry. The corresponding topological phonon surface state covers exactly half the surface Brillouin zone and can thereby be distinguished from those of conventional nodal-line and nodal-surface semimetals. Our study enriches the classification of topological quantum phases, predicts ideal material candidates, and provides a good platform for investigating the interaction between nodal-line and nodal-surface phonons.

Real-Space Observation of Magnon Interaction with Driven Space-Time Crystals.

Abstract: The concept of space-time crystals (STC), i.e., translational symmetry breaking in time and space, was recently proposed and experimentally demonstrated for quantum systems. Here, we transfer this concept to magnons and experimentally demonstrate a driven STC at room temperature. The STC is realized by strong homogeneous microwave pumping of a micron-sized permalloy (Py) stripe and is directly imaged by scanning transmission x-ray microscopy (STXM). For a fundamental understanding of the formation of the STC, micromagnetic simulations are carefully adapted to model the experimental findings. Beyond the mere generation of a STC, we observe the formation of a magnonic band structure due to back folding of modes at the STC's Brillouin zone boundaries. We show interactions of magnons with the STC that appear as lattice scattering, which results in the generation of ultrashort spin waves (SW) down to 100-nm wavelengths that cannot be described by classical dispersion relations for linear SW excitation. We expect that room-temperature STCs will be useful to investigate nonlinear wave physics, as they can be easily generated and manipulated to control their spatial and temporal band structures.

Magnetic higher-order nodal lines

Abstract: Nodal lines, as one-dimensional band degeneracies in momentum space, usually feature a linear energy splitting Here we propose the concept of magnetic higher-order nodal lines, which are nodal lines with higher-order energy splitting and realized in magnetic systems with broken time-reversal symmetry We provide sufficient symmetry conditions for stabilizing magnetic quadratic and cubic nodal lines, based on which concrete lattice models are constructed to demonstrate their existence Unlike its counterpart in nonmagnetic systems, the magnetic quadratic nodal line can exist as the only band degeneracy at the Fermi level We show that these nodal lines can be accompanied by torus surface states, which form a surface band that span over the whole surface Brillouin zone Under symmetry breaking, these magnetic nodal lines can be transformed into a variety of interesting topological states, such as three-dimensional quantum anomalous Hall insulator, multiple linear nodal lines, and magnetic triple-Weyl semimetal The three-dimensional quantum anomalous Hall insulator features a Hall conductivity ${\ensuremath{\sigma}}_{xy}$ quantized in units of ${e}^{2}/(hd)$, where $d$ is the lattice constant normal to the $x\text{\ensuremath{-}}y$ plane Our work reveals previously unknown topological states and offers guidance to search for them in realistic material systems

Nonequilibrium Lattice Dynamics in Monolayer MoS2.

Abstract: The coupled nonequilibrium dynamics of electrons and phonons in monolayer MoS2 is investigated by combining first-principles calculations of the electron-phonon and phonon-phonon interactions with the time-dependent Boltzmann equation. Strict phase-space constraints in the electron-phonon scattering are found to influence profoundly the decay path of excited electrons and holes, restricting the emission of phonons to crystal momenta close to a few high-symmetry points in the Brillouin zone. As a result of momentum selectivity in the phonon emission, the nonequilibrium lattice dynamics is characterized by the emergence of a highly anisotropic population of phonons in reciprocal space, which persists for up to 10 ps until thermal equilibrium is restored by phonon-phonon scattering. Achieving control of the nonequilibrium dynamics of the lattice may provide unexplored opportunities to selectively enhance the phonon population of two-dimensional crystals and, thereby, transiently tailor electron-phonon interactions over subpicosecond time scales.

Ideal topological nodal-surface phonons in RbTeAu-family materials

Abstract: By using first-principles calculations and symmetry analysis, we uncover that the rubidium-catena-telluridoaurate-family materials in space group (SG) 51 can exhibit ideal nodal-surface phonons separately in the three-dimensional Brillouin zone, characterized by nearly flat phonon bands in their phonon spectra. The nodal surfaces are protected both by the screw rotation symmetry and the time reversal symmetry. Moreover, our theoretical calculations demonstrate that the nodal surfaces are composed by twofold degenerate nodal points, which can form a straight singularity nodal line in the (001) surface, and that the (001) surface states display Dirac-type crossings and surface flat-band phononic states, confirming the topologically nontrivial nature of nodal-surface phonons. Our findings not only support that another class of nodal phononic states, i.e., one-node surface phonons, can exist in realistic materials in 230 SGs, but also put forward an effective way to search for nodal-surface phonons.

Brillouin scattering-theory and experiment: tutorial

Abstract: Brillouin scattering is an important and interesting nonlinear effect involving the interaction between optical and acoustic fields in optical waveguides. It is increasingly useful in the field of photonics, where it supplies a tunable ultra-narrow linewidth response that can be used for applications including sensing, filtering, and lasing, as well as the acoustic storage of optical pulses. This tutorial gives an overview of the fundamentals of Brillouin scattering aimed at newcomers to the field, and covers the physics underlying the interaction, the mathematical theory, and setup details of foundational Brillouin experiments.

Mapping mechanical properties of biological materials via an add-on Brillouin module to confocal microscopes

Abstract: Several techniques have been developed over the past few decades to assess the mechanical properties of biological samples, which has fueled a rapid growth in the fields of biophysics, bioengineering, and mechanobiology. In this context, Brillouin optical spectroscopy has long been known as an intriguing modality for noncontact material characterization. However, limited by speed and sample damage, it had not translated into a viable imaging modality for biomedically relevant materials. Recently, based on a novel spectroscopy strategy that substantially improves the speed of Brillouin measurement, confocal Brillouin microscopy has emerged as a unique complementary tool to traditional methods as it allows noncontact, nonperturbative, label-free measurements of material mechanical properties. The feasibility and potential of this innovative technique at both the cell and tissue level have been extensively demonstrated over the past decade. As Brillouin technology is rapidly recognized, a standard approach for building and operating Brillouin microscopes is required to facilitate the widespread adoption of this technology. In this protocol, we aim to establish a robust approach for instrumentation, and data acquisition and analysis. By carefully following this protocol, we expect that a Brillouin instrument can be built in 5–9 days by a person with basic optics knowledge and alignment experience; the data acquisition as well as postprocessing can be accomplished within 2–8 h. A protocol for implementing Brillouin microscopy to study biological materials. The procedure contains instructions for integrating an add-on Brillouin module with an existing confocal microscope as well as for its calibration and use in data collection and processing.

Hidden wave function of twisted bilayer graphene: The flat band as a Landau level

Abstract: We study zero-energy states of the chirally symmetric continuum model (CS-CM) of twisted bilayer graphene. The zero-energy state obeys the Dirac equation on a torus in the external non-Abelian magnetic field. These zero-energy states could form a flat band---a band where the energy is constant across the Brillouin zone. We prove that the existence of the flat band implies that the wave function of any state from the flat band has a zero and vice versa. We found a hidden flat band of unphysical states in the CS-CM that has a pole instead of a zero. Our main result is that in the basis of the flat band and hidden wave functions the flat band could be interpreted as a Landau level in the external magnetic field. From that interpretation we show the existence of extra flat bands in the magnetic field.

Topological charge density waves at half-integer filling of a moiré superlattice

Abstract: At partial filling of a flat band, strong electronic interactions may favor gapped states harboring emergent topology with quantized Hall conductivity. Emergent topological states have been found in partially filled Landau levels and Hofstadter bands; in both cases, a large magnetic field is required to engineer the underlying flat band. The recent observation of quantum anomalous Hall effects (QAH) in narrow band moire systems has led to the theoretical prediction that such phases may be realized even at zero magnetic field. Here we report the experimental observation of insulators with Chern number $C=1$ in the zero magnetic field limit at $ u=3/2$ and $7/2$ filling of the moire superlattice unit cell in twisted monolayer-bilayer graphene (tMBG). Our observation of Chern insulators at half-integer values of $ u$ suggests spontaneous doubling of the superlattice unit cell, in addition to spin- and valley-ferromagnetism. This is confirmed by Hartree-Fock calculations, which find a topological charge density wave ground state at half filling of the underlying $C=2$ band, in which the Berry curvature is evenly partitioned between occupied and unoccupied states. We find the translation symmetry breaking order parameter is evenly distributed across the entire folded superlattice Brillouin zone, suggesting that the system is in the flat band, strongly correlated limit. Our findings show that the interplay of quantum geometry and Coulomb interactions in moire bands allows for topological phases at fractional superlattice filling that spontaneously break time-reversal symmetry, a prerequisite in pursuit of zero magnetic field phases harboring fractional statistics as elementary excitations or bound to lattice dislocations.

High-Performance Distributed Brillouin Optical Fiber Sensing

Abstract: This paper reviews the recent advances on the high-performance distributed Brillouin optical fiber sensing, which include the conventional distributed Brillouin optical fiber sensing based on backward stimulated Brillouin scattering and two other novel distributed sensing mechanisms based on Brillouin dynamic grating and forward stimulated Brillouin scattering, respectively. As for the conventional distributed Brillouin optical fiber sensing, the spatial resolution has been improved from meter to centimeter in the time-domain scheme and to millimeter in the correlation-domain scheme, respectively; the measurement time has been reduced from minute to millisecond and even to microsecond; the sensing range has reached more than 100 km. Brillouin dynamic grating can be used to measure the birefringence of a polarization-maintaining fiber, which has been explored to realize distributed measurement of temperature, strain, salinity, static pressure, and transverse pressure. More recently, forward stimulated Brillouin scattering has gained considerable interest because of its capacity to detect mechanical features of materials surrounding the optical fiber, and remarkable works using ingenious schemes have managed to realize distributed measurement, which opens a brand-new way to achieve position-resolved substance identification.

Nonperturbative phonon scatterings and the two-channel thermal transport in Tl 3 VSe 4

Abstract: We study the role of nonperturbative phonon scattering in strongly anharmonic materials having ultralow lattice thermal conductivity with unusual temperature dependence. We take ${\mathrm{Tl}}_{3}{\mathrm{VSe}}_{4}$ as an example and investigate its lattice dynamics using perturbation theory (PT) up to the fourth order and molecular dynamics (MD) with a machine-learning potential. We find distinct differences of phonon linewidth between PT and MD in the whole Brillouin zone. The comparison between the theoretical phonon linewidths and experiments suggests that PT severely underestimates the phonon scatterings, even when the fourth-order anharmonicity is included. Moreover, we extend our calculations to higher temperatures and evaluate the two-channel thermal conductivity based on the unified theory developed by Simoncelli et al. [Nat. Phys. 15, 809 (2019)]. We find a crucial coherence contribution to the total thermal conductivity at high temperatures. Our results pave the path for future studies of phonon properties and lattice thermal conductivities of strongly anharmonic crystals beyond the conventional PT realm.

Modeling the optical properties of twisted bilayer photonic crystals

Abstract: We demonstrate a photonic analog of twisted bilayer graphene that has ultra-flat photonic bands and exhibits extreme slow-light behavior. Our twisted bilayer photonic device, which has an operating wavelength in the C-band of the telecom window, uses two crystalline silicon photonic crystal slabs separated by a methyl methacrylate tunneling layer. We numerically determine the magic angle using a finite-element method and the corresponding photonic band structure, which exhibits a flat band over the entire Brillouin zone. This flat band causes the group velocity to approach zero and introduces light localization, which enhances the electromagnetic field at the expense of bandwidth. Using our original plane-wave continuum model, we find that the photonic system has a larger band asymmetry. The band structure can easily be engineered by adjusting the device geometry, giving significant freedom in the design of devices. Our work provides a fundamental understanding of the photonic properties of twisted bilayer photonic crystals and opens the door to the nanoscale-based enhancement of nonlinear effects. The twisted bilayer photonic crystals have near-zero group velocity over the Brillouin zone and provide access to light localization that is far out of the reach of conventional photonic crystals.

Valley Orientation of Electrons and Excitons in Atomically Thin Transition Metal Dichalcogenide Monolayers (Brief Review)

Abstract: The main aspects of physical phenomena associated with the optical orientation of the spin and valley degrees of freedom in transition metal dichalcogenide monolayers and in van der Waals heterostructures based on them have been briefly reviewed. Owing to features of the band structure and spin–orbit coupling in such systems, circularly polarized light induces optical transitions in different valleys K+ and K– of the Brillouin zone; consequently, the optical orientation of charge carriers and excitons is accompanied by their valley polarization. The main features of the band structure of transition metal dichalcogenide monolayers, excitonic effects, and results of theoretical studies of the valley orientation of excitons and electrons at one-photon absorption have been reported. The linear–circular dichroism and valley orientation of free charge carriers and excitons at multiphoton absorption have been studied. Effects associated with the trigonal symmetry of monolayers, including the inversion of valley polarization at two-photon transitions and the second harmonic generation, have been discussed. The considered theoretical models have been illustrated by experimental data.

Measuring Topological Invariants in a Polaritonic Analog of Graphene.

Abstract: Topological materials rely on engineering global properties of their bulk energy bands called topological invariants. These invariants, usually defined over the entire Brillouin zone, are related to the existence of protected edge states. However, for an important class of Hamiltonians corresponding to 2D lattices with time-reversal and chiral symmetry (e.g., graphene), the existence of edge states is linked to invariants that are not defined over the full 2D Brillouin zone, but on reduced 1D subspaces. Here, we demonstrate a novel scheme based on a combined real- and momentum-space measurement to directly access these 1D topological invariants in lattices of semiconductor microcavities confining exciton polaritons. We extract these invariants in arrays emulating the physics of regular and critically compressed graphene where Dirac cones have merged. Our scheme provides a direct evidence of the bulk-edge correspondence in these systems and opens the door to the exploration of more complex topological effects, e.g., involving disorder and interactions.

Topological slow light via coupling chiral edge modes with flatbands

Abstract: Chiral edge modes in photonic topological insulators host great potential to realize slow-light waveguides with topological protection. Increasing the winding of the chiral edge mode around the Brillouin zone can lead to broadband topological slow light with ultra-low group velocity. However, this effect usually requires careful modifications on a relatively large area around the lattice edge. Here, we present a simple and general scheme to achieve broadband topological slow light through coupling the chiral edge modes with flatbands. In this approach, modifications inside the topological lattice are not required. Instead, only several additional resonators that support the flatbands need to be attached at the lattice edge. We demonstrate our idea numerically using a gyromagnetic photonic crystal, which is ready to be tested at microwave frequencies.

Forward stimulated Brillouin scattering and opto-mechanical non-reciprocity in standard polarization maintaining fibres.

Abstract: Opto-mechanical interactions in guided wave media are drawing great interest in fundamental research and applications. Forward stimulated Brillouin scattering, in particular, is widely investigated in optical fibres and photonic integrated circuits. In this work, we report a comprehensive study of forward stimulated Brillouin scattering over standard, panda-type polarization maintaining fibres. We distinguish between intra-polarization scattering, in which two pump tones are co-polarized along one principal axis, and inter-polarization processes driven by orthogonally polarized pump waves. Both processes are quantified in analysis, calculations and experiment. Inter-modal scattering, in particular, introduces cross-polarization switching of probe waves that is non-reciprocal. Switching takes place in multiple wavelength windows. The results provide a first demonstration of opto-mechanical non-reciprocity of forward scatter in standard fibre. The inter-polarization process is applicable to distributed sensors of media outside the cladding and coating boundaries, where light cannot reach. The process may be scaled towards forward Brillouin lasers, optical isolators and circulators and narrowband microwave-photonic filters over longer sections of off-the-shelf polarization maintaining fibres.

Double Dirac nodal line semimetal with a torus surface state

Abstract: We propose a class of nodal line semimetals that host an eightfold-degenerate double Dirac nodal line (DDNL) with negligible spin-orbit coupling. We find only 5 of the 230 space groups host the DDNL. The DDNL can be considered as a combination of two Dirac nodal lines, and has a trivial Berry phase. This leads to two possible but completely different surface states, namely, a torus surface state covering the whole surface Brillouin zone and no surface state at all. Based on first-principles calculations, we predict that the hydrogen storage material LiBH is an ideal DDNL semimetal, where the line resides at Fermi level, is relatively flat in energy, and exhibits a large linear energy range. Interestingly, both the two novel surface states of DDNL can be realized in LiBH. Further, we predict that with a magnetic field parallel to DDNL, the Landau levels of DDNL are doubly degenerate due to a Kramers-like degeneracy and have a doubly degenerate zero mode.

Interorbital nematicity and the origin of a single electron Fermi pocket in FeSe

Abstract: The electronic structure of the enigmatic iron-based superconductor FeSe has puzzled researchers since spectroscopic probes failed to observe the expected electron pocket at the $Y$ point in the 1-Fe Brillouin zone. It has been speculated that this pocket, essential for an understanding of the superconducting state, is either absent or incoherent. Here, we perform a theoretical study of the preferred nematic order originating from nearest-neighbor Coulomb interactions in an electronic model relevant for FeSe. We find that at low temperatures the dominating nematic components are of interorbital ${d}_{xz}\text{\ensuremath{-}}{d}_{xy}$ and ${d}_{yz}\text{\ensuremath{-}}{d}_{xy}$ character, with spontaneously broken amplitudes for these two components. This interorbital nematic order naturally leads to distinct hybridization gaps at the $X$ and $Y$ points of the 1-Fe Brillouin zone, and may thereby produce highly anisotropic Fermi surfaces with only a single electron pocket at one of these momentum-space locations. The associated superconducting gap structure obtained with the generated low-energy electronic band structure from spin-fluctuation mediated pairing agrees well with that measured experimentally. Finally, from a comparison of the computed spin susceptibility to available neutron scattering data, we discuss the necessity of additional self-energy effects, and explore the role of orbital-dependent quasiparticle weights as a minimal means to include them.

Explicit Brillouin Flow Solutions in Magnetrons, Magnetically Insulated Line Oscillators, and Radial Magnetically Insulated Transmission Lines

Abstract: This article re-examines the Brillouin flow solutions in crossed-field diodes, with applications to magnetrons, magnetically insulated line oscillators (MILOs), and magnetically insulated transmission lines (MITLs). The Brillouin flow solutions are constructed for various geometries, including planar magnetrons, MILOs, and MITLs, cylindrical magnetrons with electrons flowing in the azimuthal direction, cylindrical MITLs and MILOs with electrons flowing in the axial direction, and radial MITLs and MILOs with electrons flowing in the radial direction. A common theme of this analysis is that two main external parameters are used to characterize the Brillouin flow: the anode–cathode voltage ( $V_{a}$ ) and the total magnetic flux within the crossed-field diodes ( $A_{a}$ ). These two parameters are equivalent to the gap voltage and a specification of the degree of magnetic insulation, which is approximately equal to the ratio of the magnetic field to the Hull cutoff (HC) magnetic field. The magnetic flux may be provided externally by a magnet (as in a magnetron) or by the wall currents without an external magnet (as in a MILO or MITL), or by some combination of the two, as in the intermediate case of a magnetron–MILO hybrid. Once these two parameters are specified, the electron flow speed at the top of the Brillouin hub is uniquely determined. This immediately yields the Buneman–Hartree (BH) condition according to the Brillouin flow model, whether it be a planar magnetron or a cylindrical MILO. In so doing, we have obtained, for the first time using the Brillouin flow model, the BH condition for a cylindrical MILO, and we show that the same condition is obtained from the single-particle orbit model. We also found that, in general, the electron current within the Brillouin hub contributes only to a very small fraction of the magnetic flux $A_{a}$ , regardless of the gap voltage $V_{a}$ , thereby correcting an erroneous notion that the electron flow within the crossed-field gap could be responsible for the magnetic insulation. Another counter-intuitive finding is that, for a given degree of magnetic insulation, the Brillouin hub height decreases as the gap voltage $V_{a}$ increases. These conclusions, and other results, were based on the simple, explicit analytic expressions that we have obtained for the Brillouin flow profiles, including the velocity, electron density, as well as the self-magnetic field and the self-electric field profiles due to the Brillouin hub electrons. From these analytical expressions, we deduce useful scaling laws that are applicable to the prevailing cases where the magnetic field exceeds 1.5 times the HC magnetic field, and they are valid in both relativistic and non-relativistic regimes. Thus, these scaling laws show the contrast between magnetrons and MILOs, and a ready assessment of the viability of building a moderate-current MILO, a low-voltage MILO, and a magnetron–MILO hybrid which might combine the advantages of a magnetron and a MILO. The Brillouin flow profiles in a radial MITL are explicitly calculated. Additional issues are addressed.

Topological polarization, dual invariants, and surface flat bands in crystalline insulators

Abstract: We describe a crystalline topological insulator (TI) phase of matter that exhibits spontaneous polarization in arbitrary dimensions. The bulk polarization response is constructed by coupling the system to geometric deformations of the underlying crystalline order, represented by local lattice vectors---the elasticity tetrads. This polarization results from the presence of (approximately) flat bands on the surface of such TIs. These flat bands are a consequence of the bulk-boundary correspondence of polarized topological media, and contrary to related nodal line semimetal phases also containing surface flat bands, they span the entire surface Brillouin zone. We also present an example Hamiltonian exhibiting a Lifshitz transition from the nodal line phase to the TI phase with polarization. In addition, we discuss a general classification of three-dimensional (3D) crystalline TI phases and invariants in terms of the elasticity tetrads. The phase with polarization naturally arises from this classification as a dual to the previously better-known 3D TI phase exhibiting quantum (spin) Hall effect. Besides polarization, another implication of the large surface flat band is the susceptibility to interaction effects such as superconductivity: The mean-field critical temperature is proportional to the size of the flat bands, and this type of system may hence exhibit superconductivity with a very high critical temperature.