One-dimensional Crystal Lattice Research Articles

Analytical spectral design of mechanical metamaterials with inertia amplification

Functional metamaterials offering superior dynamic performances can be conceived by introducing local mechanisms of inertia amplification in the periodic microstructure of cellular composite media. The minimal physical realization of an inertially amplified metamaterial is represented by a one-dimensional crystal lattice, characterized by an intracellular pantograph mechanism. The microstructural dynamics of the periodic tetra-atomic cell is synthetically described by a low-dimension lagrangian model. The lagrangian coordinates account for the longitudinal motion of a pair of elastically coupled principal atoms, rigidly connected by the pantograph arms to a pair of transversely-oscillating secondary atoms, serving as inertial amplifiers. Within the range of small-oscillations, the free propagation of undamped harmonic waves is described by a linear difference equation, governed by a symplectic transfer matrix. First, the band structure of the complex-valued dispersion spectrum is determined analytically, by properly exploiting the formal (time-to-space) analogy with the Floquet Theory for the stability of non-autonomous dynamic systems (direct problem). Second, a spectral design problem is stated and solved by inverting analytically the functions expressing parametrically the boundaries separating attenuation (stop) and propagation (pass) bands in the frequency spectrum (inverse problem). Specifically, the inverse problem solution has the merit of providing simple formulas for parametric design. Such formulas explicitly and exactly identify the mechanical parameters of the inertially amplified metamaterial possessing a desired pass-stop-pass band structure, assigned according to functional design requirements. The existence and uniqueness of the solution in the admissible range of mechanical parameters is discussed. The discussion provides (i) mathematical demonstration for the existence of feasible design solutions or multi-solutions, (ii) complete definition of the physically realizable band structures in the frequency domain, and (iii) alternative criteria to design iso-band structured metamaterials within the admissible domain of mechanical parameters. The extra-customization possibilities offered by iso-band structured metamaterials are analyzed in terms of static and dynamic performances. As feasible limit cases of the analytical spectral design, extreme mechanical metamaterials working as phononic superfilters and/or superpropagators are realizable.

Exact ground state energy of a system with an arbitrary number of dipoles at the sites of a regular one-dimensional crystal lattice

We consider a finite classical system of electric dipoles localized at the sites of a regular one-dimensional crystal lattice. It is shown that the ground state energy of such a system can be calculated exactly for an arbitrary number of dipoles. The expression for the ground state energy can be given in terms of special functions such as Riemann zeta functions and polygamma functions for any arbitrary number of dipoles, and reduces to the correct result in the thermodynamic limit as the number of dipoles tends to infinity. Simpler, but very accurate, approximate expressions for the ground state energy are also introduced.

Splitting Methods for Semi-Classical Hamiltonian Dynamics of Charge Transfer in Nonlinear Lattices

We propose two classes of symplecticity-preserving symmetric splitting methods for semi-classical Hamiltonian dynamics of charge transfer by intrinsic localized modes in nonlinear crystal lattice models. We consider, without loss of generality, one-dimensional crystal lattice models described by classical Hamiltonian dynamics, whereas the charge (electron or hole) is modeled as a quantum particle within the tight-binding approximation. Canonical Hamiltonian equations for coupled lattice-charge dynamics are derived, and a linear analysis of linearized equations with the derivation of the dispersion relations is performed. Structure-preserving splitting methods are constructed by splitting the total Hamiltonian into the sum of Hamiltonians, for which the individual dynamics can be solved exactly. Symmetric methods are obtained with the Strang splitting of exact, symplectic flow maps leading to explicit second-order numerical integrators. Splitting methods that are symplectic and conserve exactly the charge probability are also proposed. Conveniently, they require only one solution of a linear system of equations per time step. The developed methods are computationally efficient and preserve the structure; therefore, they provide new means for qualitative numerical analysis and long-time simulations for charge transfer by nonlinear lattice excitations. The properties of the developed methods are explored and demonstrated numerically considering charge transport by mobile discrete breathers in an example model previously proposed for a layered crystal.

Optical biosensors using plasmonic and photonic crystal band-gap structures for the detection of basal cell cancer

One of the most interesting topics in bio-optics is measuring the refractive index of tissues. Accordingly, two novel optical biosensor configurations for cancer cell detections have been proposed in this paper. These structures are composed of one-dimensional photonic crystal (PC) lattices coupled to two metal–insulator–metal (MIM) plasmonic waveguides. Also, the tapering method is used to improve the matching between the MIM plasmonic waveguides and PC structure in the second proposed topology. The PC lattices at the central part of the structures generate photonic bandgaps (PBGs) with sharp edges in the transmission spectra of the biosensors. These sharp edges are suitable candidates for sensing applications. On the other hand, the long distance between two PBG edges causes that when the low PBG edge is used for sensing mechanism, it does not have an overlapping with the high PBG edge by changing the refractive index of the analyte. Therefore, the proposed biosensors can be used for a wide wavelength range. The maximum obtained sensitivities and FOM values of the designed biosensors are equal to 718.6, 714.3 nm/RIU, and 156.217, 60.1 RIU−1, respectively. The metal and insulator materials which are used in the designed structures are silver, air, and GaAs, respectively. The finite-difference time-domain (FDTD) method is used for the numerical investigation of the proposed structures. Furthermore, the initial structure of the proposed biosensors is analyzed using the transmission line method to verify the FDTD simulations. The attractive and simple topologies of the proposed biosensors and their high sensitivities make them suitable candidates for biosensing applications.

Data-driven intrinsic localized mode detection and classification in one-dimensional crystal lattice model

In this work we propose Support Vector Machine classification algorithms to classify one-dimensional crystal lattice waves from locally sampled data. Different learning datasets of particle displacements, momenta and energy density values are considered. Efficiency of the classification algorithms is further improved by two dimensionality reduction techniques: Principal Component Analysis and Locally Linear Embedding. Robustness of classifiers is investigated and demonstrated. Developed algorithms are successfully applied to detect localized intrinsic modes in three numerical simulations considering a case of two localized stationary breather solutions, a single stationary breather solution in noisy background and two mobile breather collision.

Looking for charge order

Physics At low temperatures, ions in some one-dimensional (1D) crystal lattices are predicted to transition into a chain of dimers. The resulting charge order has been observed in quasi-1D solids consisting of ionic chains with weak interchain interactions. Aiming to find out whether this transition occurs in purely 1D systems, Yang et al. synthesized nanowires of Mo6Se6 from monolayers of the transition metal dichalcogenide MoSe2. This resulted in two types of nanowires: those attached along the edges of MoSe2 and those connecting adjacent MoSe2 flakes. Scanning tunneling spectroscopy revealed coherent charge order in the former, but not the latter, type. The loss of coherence might be caused by the formation of polaronic quasiparticles in the 1D system. Phys. Rev. X 10 , 031061 (2020).

About electron transfer over long distances with tunable sub/supersonic velocities.

Provided in this paper is a theory of long-range electron transfer with near sound (supersonic or subsonic) velocity along one-dimensional crystal lattices. The theory represents the development of an earlier work by introducing Marcus formulation. To illustrate its application to a realistic case, the theory is used to offer an explanation of two puzzling observations made by Donovan and Wilson in transient photoconduction experiments with non-dopable perfectly crystalline polydiacetylene crystals in the presence of an electric field: transport velocity value close to sound velocity being independent of field for four orders of magnitude of field (102 V/m-106 V/m) and, in the low field values, an ultra-high mobility greater than 20 m2/V s. We also study factors eventually leading to lowering of the transport velocity.

Results for the ground state energy of a finite system of dipoles in a one-dimensional crystal lattice

We investigate the ground state energy of a finite classical system consisting of an arbitrary number of electric dipoles localized at the sites of a regular one-dimensional crystal lattice. The ground state energy per dipole can be exactly calculated in the thermodynamic limit but an exact analytical expression for the energy valid for an arbitrary finite number of dipoles is not possible. In this work we obtain an approximate analytical expression for the ground state energy that applies to any given finite number of dipoles. The approximate analytical expression that we report reproduces the exact numerically calculated values of the ground state energy with an astonishing accuracy.

Formulation of an efficient continuum mechanics-based model to study wave propagation in one-dimensional diatomic lattices

In this paper, we formulate an efficient continuum mechanics-based model on the basis of a discrete lattice model. First, the dispersion relation of a lattice wave in a one-dimensional diatomic crystal lattice is derived. Then, the second- and fourth-order continuum models are obtained from the differential-difference equations of motion by using the Padé approximations. The results show that the proposed fourth-order continuum model can predict the dispersion behaviour of the one-dimensional diatomic crystal lattice very well in the first Brillouin zone. Furthermore, the applicability of the present model to the prediction of the dispersion behaviour of the one-dimensional diatomic lattice with internal resonator and inerter is examined. Finally, the vibration frequencies of finite diatomic lattices are calculated by both the discrete and the proposed continuum models.

Continuum model of acoustic metamaterials with diatomic crystal lattice

ABSTRACTThe continuum model of one-dimensional acoustic metamaterial with diatomic crystal lattice is studied in this article. First, the dispersive relation of lattice wave in one-dimensional diatomic crystal lattice of metamaterial is established and compared with that of the classic material. Then, the continuum model of the acoustic metamaterial leads to the classical continuum model, the stain gradient continuum model, and the nonlocal gradient continuum model based on different assumptions. The dispersive curves, which correspond to the three kinds of models, are shown graphically and compared with that of discrete crystal lattice of metamaterial. The disadvantage of the classic continuum model and the strain gradient continuum model are discussed. The nonlocal gradient continuum model is derived based on the nonlocal assumption of continuous displacement field. The stability of dispersive curves is guaranteed. However, the actual prediction effects are still dependent upon the appropriate selection of the nonlocal parameter in the nonlocal gradient continuum model.

Continuum model of a one-dimensional lattice of metamaterials

The continuum model of a one-dimensional crystal lattice of a metamaterial is studied in this paper. First, the dispersive relation of a lattice wave in a one-dimensional crystal lattice of metamaterial is established and compared with that of the classic material. Then, the continuous medium modeling of the metamaterial is studied. It leads to the classical continuum model, the strain gradient continuum model, and the nonlocal gradient continuum model based on different assumptions. The disadvantages of the classic continuum model and the gradient continuum model are discussed. The nonlocal gradient continuum model is derived based on the nonlocal assumption of a continuous displacement field. The stability of dispersive curves is guaranteed, and the conceptions of negative mass and infinite mass are also avoided. The dispersive curves which correspond to the three kinds of models are compared with those of a discrete crystal lattice of metamaterial. The disadvantages of the classic continuum model and the gradient continuum model and the appropriate selection of a nonlocal parameter in the nonlocal gradient continuum model are discussed based on the numerical results.

Design and fabrication of suspended Si3N4 nanobeam cavities

We report on the design, theoretical simulations and fabrication details of photonic crystal nanobeams (PhCNB) in air-bridge silicon nitride structures. The PhC nanobeams consist of a one-dimensional photonic crystal lattice of circular air holes. The design of structures that show band gaps for the waveguide modes will allow the creation of suspended photonic nanobeam cavities in a second step. The silicon nitride layer is deposited on top of a thermally oxidized silicon wafer with LPCVD followed by e-beam lithography and proper wet etching to release the suspended structures.

On the temperature dependence of fast electron transport in crystal lattices

Building upon the findings of Muto et al. [Phys. Lett. A 136, 33 (1989)] and Marchesoni and Lucheroni [Phys. Rev. E 44, 5303 (1991)] about the growth of the number of (anharmonic) lattice solitons with increasing temperature and using a recent transport theory developed by the present authors [A.P. Chetverikov, W. Ebeling, G. Ropke, M.G. Velarde, Eur. Phys. J. B 87, 153 (2014)] here we provide the fractional power law of the temperature dependence of resistivity in a rather general model for one-dimensional crystal lattices as, e.g., conducting polymers. We also show that the determining factor for the transport is the possibility of forming electron-soliton bound states (in short solectrons) with a most significant contribution arising from the (bosonic) bound state of two electrons to a soliton (in short bisolectrons).

Electron pairing and Coulomb repulsion in one-dimensional anharmonic lattices

We show that in anharmonic one-dimensional crystal lattices pairing of electrons or holes in a localized bisolectron state is possible due to the coupling between the charges and the lattice deformation that can overcome the Coulomb repulsion. Such localized states appear as traveling ground singlet states of two extra electrons bound in the potential well created by the local lattice solitonlike deformation. We also find the first excited localized state of two electrons given by a triplet state of two electrons. The results of the analytical study of interacting electrons in a lattice with cubic anharmonicity are compared with the numerical simulations of two electrons in an anharmonic lattice with Morse interactions and taking into account the single-site Hubbard electron-electron repulsion. We find quite a good qualitative agreement between both approaches for a broad range of parameter values. For illustration we give expressions for the bisolectron binding energy with parameter values that are typical for biological macromolecules. We also estimate threshold values of the Coulomb repulsion above which the bisolectron splits into two solectrons.

Effect of anharmonicity on the phonon density of states and specific heat of a monoatomic, one-dimensional crystal lattice

We analyze the lattice equation of motion involving terms up to third order in lattice displacement. The phenomenological arguments suggest that the force constant ${D}_{1}$ of the quadratic term must always be positive and the force constant ${B}_{1}$ of the cubic term may take either positive or negative value. The criterion for stability of the lattice provides constraint on the relative magnitudes of the three force constants. We solve the equation of motion using root mean-square spatial fluctuation approximation and obtain the seminonperturbative dispersion relation both for positive and negative ${B}_{1}$. The nature of phonon density of states curves for positive ${B}_{1}$ show some close resemblance with the experimental observations. At very low temperature, the specific heat of this system to leading order in large positive ${B}_{1}$ varies as square root of temperature and it obeys Debye's $T$ law in one dimension for small negative ${B}_{1}$. At very high temperature, the specific heat may fall below or above its classical value depending on the relative magnitudes of ${B}_{1}$ and ${D}_{1}$ for ${B}_{1}>0$ and it always falls above its classical value for ${B}_{1}<0$. The lattice model with positive ${B}_{1}$ emerges as a good candidate for description of a monoatomic crystal.

Dynamics of martensitic phase boundaries: discreteness, dissipation and inertia

We study a fully inertial model of a martensitic phase transition in a one-dimensional crystal lattice with long-range interactions. The model allows one to represent a broad range of dynamic regimes, from underdamped to overdamped. We systematically compare the discrete model with its various continuum counterparts including elastic, viscoelastic and viscosity-capillarity models. Each of these models generates a particular kinetic relation which links the driving force with the phase boundary velocity. We find that the viscoelastic model provides an upper bound for the critical driving force predicted by the discrete model, while the viscosity-capillarity model delivers a lower bound. We show that at near-sonic velocities, where inertia dominates dispersion, both discrete and continuum models behave qualitatively similarly. At small velocities, and in particular near the depinning threshold, the discreteness prevails and predictions of the continuum models cannot be trusted.

Hyperbolicity, convexity and shock waves in one-dimensional crystalline solids

For a continuum model of one-dimensional anharmonic crystal lattices at finite temperatures, which was derived from a statistical-mechanical model proposed recently, we clarify the classification of its differential system. That is, we determine not only the strict hyperbolicity and convexity regions but also the elliptic and parabolic regions in the space of the state. The melting point is found to be on the boundary of the convexity region. Then we derive the Rankine–Hugoniot relations, and we prove that the admissible shocks are always in the stable region of convexity.

Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice II. Impurities, confinement and Bloch oscillations

We present a computational approach for the WKB approximation of the wavefunction of an electron moving in a periodic one-dimensional crystal lattice by means of a nonstrictly hyperbolic system whose flux function stems from the Bloch spectrum of the Schrödinger operator. This second part focuses on the handling of the source terms which originate from adding a slowly varying exterior potential. Physically, relevant examples are the occurrence of Bloch oscillations in case it is linear, a quadratic one modelling a confining field and the harmonic Coulomb term resulting from the inclusion of a “donor impurity” inside an otherwise perfectly homogeneous lattice.

Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice: I. Homogeneous problems

We present a computational approach for the WKB approximation of the wave function of an electron moving in a periodic one-dimensional crystal lattice. We derive a nonstrictly hyperbolic system for the phase and the intensity where the flux functions originate from the Bloch spectrum of the Schrödinger operator. Relying on the framework of the multibranch entropy solutions introduced by Brenier and Corrias, we compute efficiently multiphase solutions using well adapted and simple numerical schemes. In this first part we present computational results for vanishing exterior potentials which demonstrate the effectiveness of the proposed method.

Reflection of Nonlinear Waves in Crystal Lattices at Finite Temperatures, and Dynamic Fracture

Reflection of nonlinear waves, which are excited by impulsive forces, in one-dimensional crystal lattices at finite temperatures, and dynamic fracture of the lattice induced by the reflection are numerically analyzed using the basic statistical-mechanical equations proposed in the preceding paper. Enhancement of the strength of the reflected waves, and typical fracture patterns are found. Their physical implications are also briefly discussed. The equations are, therefore, confirmed to be valid for analyzing such nonequilibrium phenomena.