Edge Of Brillouin Zone Research Articles

A framework for solving atomistic phonon-structure scattering problems in the frequency domain using perfectly matched layer boundaries

We present a numerical approach to the solution of elastic phonon-interface and phonon-nanostructure scattering problems based on a frequency-domain decomposition of the atomistic equations of motion and the use of perfectly matched layer (PML) boundaries. Unlike molecular dynamic wavepacket analysis, the current approach provides the ability to simulate scattering from individual phonon modes, including wavevectors in highly dispersive regimes. Like the atomistic Green's function method, the technique reduces scattering problems to a system of linear algebraic equations via a sparse, tightly banded matrix regardless of dimensionality. However, the use of PML boundaries enables rapid absorption of scattered wave energies at the boundaries and provides a simple and inexpensive interpretation of the scattered phonon energy flux calculated from the energy dissipation rate in the PML. The accuracy of the method is demonstrated on connected monoatomic chains, for which an analytic solution is known. The parameters defining the PML are found to affect the performance and guidelines for selecting optimal parameters are given. The method is used to study the energy transmission coefficient for connected diatomic chains over all available wavevectors for both optical and longitudinal phonons; it is found that when there is discontinuity between sublattices, even connected chains of equivalent acoustic impedance have near-zero transmission coefficient for short wavelengths. The phonon scattering cross section of an embedded nanocylinder is calculated in 2D for a wide range of frequencies to demonstrate the extension of the method to high dimensions. The calculations match continuum theory for long-wavelength phonons and large cylinder radii, but otherwise show complex physics associated with discreteness of the lattice. Examples include Mie oscillations which terminate when incident phonon frequencies exceed the maximum available frequency in the embedded nanocylinder, and scattering efficiencies larger than two near the Brillouin zone edge.

Wave propagation in two-dimensional anisotropic acoustic metamaterials of K4 topology

An acoustic metamaterial is envisaged as a synthesised phononic material the mechanical behaviour of which is determined by its unit cell. The present study investigates one aspect of mechanical behaviour, namely the band structure, in two-dimensional (2D) anisotropic acoustic metamaterials encompassing locally resonant mass-in-mass units connected by massless springs in a K4 topology. The 2D lattice problem is formulated in the direct space (r-space) and the equations of motion are derived using the principle of least action (Hamilton’s principle). Only proportional anisotropy and attenuation-free shock wave propagation have been considered. Floquet–Bloch’s principle is applied, therefore a generic unit cell is studied. The unit cell can represent the entire lattice regardless of its position. It is transformed from the direct lattice in r-space onto its reciprocal lattice conjugate in Fourier space (k-space) and point symmetry operations are applied to Wigner–Seitz primitive cell to derive the first irreducible Brillouin Zone (BZ). The edges of the first irreducible Brillouin Zone in the k-space have then been traversed to generate the full band structure. It was found that the phenomenon of frequency filtering exists and the pass and stop bands are extracted. A follow-up parametric study appreciated the degree and direction of influence of each parameter on the band structure.

Universal dependence of the spin wave band structure on the geometrical characteristics of two-dimensional magnonic crystals.

In the emerging field of magnon-spintronics, spin waves are exploited to encode, carry and process information in materials with periodic modulation of their magnetic properties, named magnonic crystals. These enable the redesign of the spin wave dispersion, thanks to its dependence on the geometric and magnetic parameters, resulting in the appearance of allowed and forbidden band gaps for specific propagation directions. In this work, we analyze the spin waves band structure of two-dimensional magnonic crystals consisting of permalloy square antidot lattices with different geometrical parameters. We show that the frequency of the most intense spin-wave modes, measured by Brillouin light scattering, exhibits a universal dependence on the aspect ratio (thickness over width) of the effective nanowire enclosed between adjacent rows of holes. A similar dependence also applies to both the frequency position and the width of the main band gap of the fundamental (dispersive) mode at the edge of the first Brillouin zone. These experimental findings are successfully explained by calculations based on the plane-wave method. Therefore, a unified vision of the spin-waves characteristics in two-dimensional antidot lattices is provided, paving the way to the design of tailored nanoscale devices, such as tunable magnonic filters and phase-shifters, with predicted functionalities.

Dispersion relation and self-collimation frequency of spoof surface plasmon using tight binding model

The analytical dispersion relation of spoof surface plasmon (SSP) is known only in the low-frequency limit and thus cannot be used to describe various practically important characteristics of SSP in the high-frequency limit (such as multimodal nature, anisotropic propagation, self-collimation). In this article, we consider a square lattice of holes made on a perfect electric conductor and derive a closed form expression of the SSP dispersion relation in the high-frequency limit using a tight binding model. Instead of using prior knowledge of the band diagram along the entire first Brillouin zone (BZ) edge, we analytically determine the hopping parameters by using the eigenfrequencies only at the three high-symmetry points of the square lattice. Using this dispersion relation, we derive an expression for the self-collimation frequency of SSP. We show that this analytical formulation is also applicable to dielectric photonic crystals and can be used to predict the frequencies corresponding to centimetre-scale supercollimation and second band self-collimation in these structures. Finally, we show that our analytical results are in agreement with the simulation results for both SSP and photonic crystals.

The effect of atomic collisions on the quantum phase transition of a Bose–Einstein condensate inside an optical cavity

In this paper, we investigate the effect of atomic collisions on the phase transition form the normal to the superradiant phase in a one-dimensional Bose–Einstein condensate (BEC) trapped inside an optical cavity. Specifically, we show that driving the atoms from the side of the cavity leads to the excitation of modes in the edges of the first Brillouin zone of every energy band, which results in the two-mode approximation of the BEC matter field in the limit of weak coupling regime. The nonlinear effect of atom–atom interaction shifts the threshold of the quantum phase transition of the BEC and also affect the power low behavior of quantum fluctuations in the total particle number. Besides, we show the possibility of controlling the quantum phase transition of the system through the s-wave scattering frequency when the strength of the transverse pumping has been fixed.

Dual landscapes in Anderson localization on discrete lattices

The localization subregions of stationary waves in continuous disordered media have been recently demonstrated to be governed by a hidden landscape that is the solution of a Dirichlet problem expressed with the wave operator. In this theory, the strength of Anderson localization confinement is determined by this landscape, and continuously decreases as the energy increases. However, this picture has to be changed in discrete lattices in which the eigenmodes close to the edge of the first Brillouin zone are as localized as the low energy ones. Here we show that in a 1D discrete lattice, the localization of low and high energy modes is governed by two different landscapes, the high energy landscape being the solution of a dual Dirichlet problem deduced from the low energy one using the symmetries of the Hamiltonian. We illustrate this feature using the one-dimensional tight-binding Hamiltonian with random on-site potentials as a prototype model. Moreover we show that, besides unveiling the subregions of Anderson localization, these dual landscapes also provide an accurate overall estimate of the localization length over the energy spectrum, especially in the weak-disorder regime.

Symmetry-resolved surface-derived electronic structure of MoS2(0 0 0 1)

We find a wave vector dependence of the band symmetries for MoS2(0 0 0 1) in angle-resolved photoemission. The band structures are found to be significantly different for states of even and odd reflection parities, despite the absence of true mirror plane symmetry away from , the Brillouin zone center, along the line to the point, at the Brillouin zone edge. Our measurements agree with density functional theory (DFT) calculations for each band symmetry, with the notable exception of the Mo contributions to the valence band structure of MoS2(0 0 0 1). The band structure is indicative of strong S 3p and Mo 4d hybridization. In particular, the top of the valence band is predominantly composed of Mo derived states near , whereas near Mo as well as Mo 4dxy dominate. In contrast, the bottom of the valence band is dominated by Mo 5s and S 3pz contributions.

Electronic band structure of magnetic bilayer graphene superlattices

Electronic band structure of the bilayer graphene superlattices with δ-function magnetic barriers and zero average magnetic flux is studied within the four-band continuum model, using the transfer matrix method. The periodic magnetic potential effects on the zero-energy touching point between the lowest conduction and the highest valence minibands of pristine bilayer graphene are exactly analyzed. Magnetic potential is shown also to generate the finite-energy touching points between higher minibands at the edges of Brillouin zone. The positions of these points and the related dispersions are determined in the case of symmetric potentials.

The interaction of organic adsorbate vibrations with substrate lattice waves in methyl-Si(111)-(1 × 1).

A combined helium atom scattering and density functional perturbation theory study has been performed to elucidate the surface phonon dispersion relations for both the CH3-Si(111)-(1 × 1) and CD3-Si(111)-(1 × 1) surfaces. The combination of experimental and theoretical methods has allowed characterization of the interactions between the low energy vibrations of the adsorbate and the lattice waves of the underlying substrate, as well as characterization of the interactions between neighboring methyl groups, across the entire wavevector resolved vibrational energy spectrum of each system. The Rayleigh wave was found to hybridize with the surface rocking libration near the surface Brillouin zone edge at both the M̄-point and K̄-point. The calculations indicated that the range of possible energies for the potential barrier to the methyl rotation about the Si-C axis is sufficient to prevent the free rotation of the methyl groups at a room temperature interface. The density functional perturbation theory calculations revealed several other surface phonons that experienced mode-splitting arising from the mutual interaction of adjacent methyl groups. The theory identified a Lucas pair that exists just below the silicon optical bands. For both the CH3- and CD3-terminated Si(111) surfaces, the deformations of the methyl groups were examined and compared to previous experimental and theoretical work on the nature of the surface vibrations. The calculations indicated a splitting of the asymmetric deformation of the methyl group near the zone edges due to steric interactions of adjacent methyl groups. The observed shifts in vibrational energies of the -CD3 groups were consistent with the expected effect of isotopic substitution in this system.

First principles explanation of the positive Seebeck coefficient of lithium.

Lithium is one of the simplest metals, with negative charge carriers and a close reproduction of free-electron dispersion. Experimentally, however, Li is one of a handful of elemental solids (along with Cu, Ag, and Au) where the sign of the Seebeck coefficient (S) is opposite to that of the carrier. This counterintuitive behavior still lacks a satisfactory interpretation. We calculate S fully from first principles, within the framework of Allen's formulation of Boltzmann transport theory. Here it is crucial to avoid the constant relaxation time approximation, which gives a sign for S which is necessarily that of the carriers. Our calculated S are in excellent agreement with experimental data, up to the melting point. In comparison with another alkali metal, Na, we demonstrate that within the simplest nontrivial model for the energy dependency of the electron lifetimes, the rapidly increasing density of states (DOS) across the Fermi energy is related to the sign of S in Li. The exceptional energy dependence of the DOS is beyond the free-electron model, as the dispersion is distorted by the Brillouin zone edge; this has a stronger effect in Li than other alkali metals. The electron lifetime dependency on energy is central, but the details of the electron-phonon interaction are found to be less important, contrary to what has been believed for several decades. Band engineering combined with the mechanism exposed here may open the door to new "ambipolar" thermoelectric materials, with a tunable sign for the thermopower even if either n- or p-type doping is impossible.

Analytical solution for the diffraction of an electromagnetic wave by a graphene grating

An analytical method for diffraction of a plane electromagnetic wave at a periodically modulated graphene sheet is presented. Both interface corrugation and a periodic change in the optical conductivity are considered. Explicit expressions for reflection, transmission, absorption and transformation coefficients in arbitrary diffraction orders are presented. The dispersion relation and decay rates for graphene plasmons of the grating are found. Simple analytical expressions for the value of the bandgap in the vicinity of the first Brillouin zone edge are derived. The optimal amplitude and wavelength, guaranteeing the best matching of the incident light with graphene plasmons are found for the conductivity grating. The analytical results are in a good agreement with first-principles numeric simulations.

Current-induced distortion of the band structure and formation of pseudogaps in magnonic crystals

Using numerical simulations, we have studied how electric current, passing along the periodicity direction in a lateral magnetic superlattice with modulated saturation magnetization, affects the propagation of magnetostatic surface spin waves (MSSWs) across it. It is shown that when the current flows against the normal lattice modes excited by a built-in antenna, it mediates excitation of new MSSW modes. These current-assisted modes are found to be co-propagating with the normal lattice ones but travel with negative group velocities and their wave-packet dispersions opposite to those in the normal lattice modes. Surprisingly, their intensity is high enough to effectively interact with the normal lattice modes under realistic parameters of the lattice and current. This intermode interaction gives rise to new frequency bands where the MSSW intensity is lowered but essentially nonzero (pseudogaps). The pseudogap positions can be shifted by several gigahertz either upwards or downwards with respect to the bandgaps occurring at Brillouin zone edges in the absence of current. The pseudogap shifting depends on the strength of the current and on the lattice magnetization and period.

Forward mode coupling and bandgap formation in magnonic crystals

Abstract The bandgap formation for magnetostatic surface spin waves (MSSWʼs) is numerically simulated in a lateral magnetic superlattice. It is found that, while the bandgaps opened at Brillouin zone edges is an intrinsic property of the periodic lattice, the exact mechanism of their formation can be revealed by incorporating a real source of MSSW excitation into the system. The obtained results uncover that the bandgaps arise due to the coupling between the MSSWʼs travelling away from the excitation source (forward modes). This behavior differs from that provided by Bragg diffraction, which is the well-established mechanism for bandgap formation in periodic lattices.

Graphene–dielectric composite metamaterials: evolution from elliptic to hyperbolic wavevector dispersion and the transverse epsilon-near-zero condition

We investigated a multilayer graphene-dielectric composite material, comprising graphene sheets separated by subwavelength-thick dielectric spacer, and found it to exhibit hyperbolic isofrequency wavevector dispersion at far- and mid-infrared frequencies allowing propagation of waves that would be otherwise evanescent in a dielectric. Electrostatic biasing was considered for tunable and controllable transition from hyperbolic to elliptic dispersion. We explored the validity and limitation of the effective medium approximation (EMA) for modeling wave propagation and cutoff of the propagating spatial spectrum due to the Brillouin zone edge. We found that EMA is capable of predicting the transition of the isofrequency dispersion diagram under certain conditions. The graphene-based composite material allows propagation of backward waves under the hyperbolic dispersion regime and of forward waves under the elliptic regime. Transition from hyperbolic to elliptic dispersion regimes is governed by the transverse epsilon-near-zero (TENZ) condition, which implies a flatter and wider propagating spectrum with higher attenuation, when compared to the hyperbolic regime. We also investigate the tunable transparency of the multilayer at that condition in contrast to other materials exhibiting ENZ phenomena.

The effect of dynamical Bloch oscillations on optical-field-induced current in a wide-gap dielectric

We consider the motion of charge carriers in a bulk wide-gap dielectric interacting with a few-cycle laser pulse. A semiclassical model based on Bloch equations is applied to describe the emerging time-dependent macroscopic currents for laser intensities close to the damage threshold. At such laser intensities, electrons can reach edges of the first Brillouin zone even for electron–phonon scattering rates as high as those known for SiO2. We find that, whenever this happens, Bragg-like reflections of electron waves, also known as Bloch oscillations, affect the dependence of the charge displaced by the laser pulse on its carrier–envelope phase.

Tight-binding model for grapheneπ-bands from maximally localized Wannier functions

The electronic properties of graphene sheets are often understood by starting from a simple phenomenological $\ensuremath{\pi}$-band tight-binding model. We provide a perspective on these models that is based on a study of ab initio maximally localized Wannier wave functions centered at carbon sites. Hopping processes in graphene can be separated into intersublattice contributions responsible for band dispersion near the Dirac point, and intrasublattice contributions responsible for electron-hole symmetry breaking. Both types of corrections to the simplest near-neighbor model can be experimentally relevant. We find that distant neighbor hopping parameters increase the ratio of the full $\ensuremath{\pi}$-band width to the Dirac point velocity and flatten bands along the $KM$ Brillouin-zone edge. We propose a five-parameter model which achieves a good compromise between simplicity and accuracy, and an alternate 15-parameter model achieves better accuracy with some loss of simplicity.

Quantum waveguides with small periodic perturbations: gaps and edges of Brillouin zones

We consider small perturbations of the Laplace operator in a multi-dimensional cylindrical domain by second-order differential operators with periodic coefficients. We show that under certain non-degeneracy conditions, such perturbations can open a gap in the continuous spectrum and give the leading asymptotic terms for the gap edges. We also estimate the values of quasi-momentum at which the spectrum edges are attained. The general machinery is illustrated by several new examples in two- and three-dimensional structures.

Frequency comparison of light transmission in a defected quasi-one-dimensional photonic crystal slab

In a new theoretical investigation, we study light transmission through a photonic crystal (PC) slab with limited boundaries at width. By using a tight binding model, photon dispersion relation and photon Green function for a perfect system are obtained. Then, based on the Lippmann-Schwinger formalism, we calculate the effects of disordering on light transmission in the PC channel. We found that the ratio of the electric field for a defected system with respect to a perfect system at a peculiar frequency is maximized for the wave vector corresponding to the first Brillouin zone (BZ) edge showing photon localization. The electric field difference of the first and second neighboring sites with respect to the defect site on the first BZ edges are depicted in several plots indicating frequency dependence which can be applicable in frequency filtering or resonating cavity studies.

Design of Silicon Photonic Crystal Waveguides for High Gain Raman Amplification Using Two Symmetric Transvers-Electric-Like Slow-Light Modes

We designed silicon photonic crystal (PhC) waveguides (WGs) for efficient silicon Raman amplifiers and lasers. We adopted narrow-width WGs to utilize two symmetric transvers-electric-like (TE-like) guided modes, which permit efficient external coupling for both the pump and Stokes waves. Modifying the size and shape of air holes surrounding the line-defect WG structures could tune the frequency difference between these two modes, at the Brillouin-zone edge, to match the Raman shift of silicon. Thus, small group velocities are also available both for pump and Stokes waves simultaneously, which results in a large enhancement of Raman gain. The enhancement factor of the Raman gain in the designed structure is more than 100 times that reported previously.

Superfluidity and stability of a Bose-Einstein condensate with periodically modulated interatomic interaction

We study theoretically the superfluidity and stability of a Bose-Einstein condensate (BEC) whose interatomic scattering length is periodically modulated with optical Feshbach resonance. Our numerical study finds that the properties of this periodic BEC are strongly influenced by the modulation strength. When the modulation strength is small, only the Bloch waves close to the Brillouin zone edge suffer both Landau and dynamical instabilities. When the modulation strength is strong enough, all Bloch waves become dynamically unstable. In other words, the periodic BEC loses its superfluidity completely.