Effective Continuum Model Research Articles

Electronic properties of graphene/hexagonal-boron-nitride moiré superlattice

We theoretically investigate the electronic structures of moir\'e superlattices arising in monolayer/bilayer graphene stacked on hexagonal boron nitride (hBN) in the presence and absence of magnetic field. We develop an effective continuum model from a microscopic tight-binding lattice Hamiltonian and calculate the electronic structures of graphene-hBN systems with different rotation angles. Using the effective model, we explain the characteristic band properties such as the gap opening at the corners of the superlattice Brillouin zone (mini-Dirac point). We also investigate the energy spectrum and quantum Hall effect of graphene-hBN systems in uniform magnetic field and demonstrate the evolution of the fractal spectrum as a function of the magnetic field. The spectrum generally splits in the valley degrees of freedom ($K$ and ${K}^{\ensuremath{'}}$) due to the lack of the inversion symmetry, and the valley splitting is more significant in bilayer graphene on hBN than in monolayer graphene on hBN because of the stronger inversion-symmetry breaking in bilayer.

VBEFP/PCM: a QM/MM/PCM approach for valence-bond method and its application for the vertical excitations of formaldehyde and acetone in aqueous solution

In this paper, a combined QM/MM/PCM approach, named VBEFP/PCM, is presented for ab initio VB study with a solvent effect incorporated. In VBEFP/PCM, both short-range and long-range solvent effects are taken into account by effective fragment potential (EFP) and polarizable continuum model (PCM), respectively, while the solute molecules are described by valence bond (VB) wave function. Furthermore, VBEFP/PCM, along with VBPCM and VBEFP, is employed for the n→π* vertical excitation of formaldehyde and acetone molecules in aqueous solution. The computational results show that VBEFP/PCM can provide the appropriate solvent shifts, whereas VBPCM underestimates the solvent shifts due to its lack of short-range solvent effect. The VBEFP results strongly rely upon the description of the short-range solvent effect. To explore the role of the solute’s electronic structure in the solvent shift, resonance energy analysis during the excitation is performed. It was found that the solute’s electronic polarization plays the most important role in the solvent shift. The resonance controls the variation of the solute’s wave function during the n→* vertical excitation, which leads to the blue solvent shifts.

Vibration analysis of periodic cellular solids based on an effective couple-stress continuum model

Cellular solids are usually treated as homogeneous continuums with effective properties. Nevertheless, these mechanical properties depend strongly on the ratio of the specimen size to the cell size. These size effects may be accounted for according to preliminary static analysis of effective continuums based on couple-stress theory. In this paper an effective dynamic continuum model, based on couple-stress theory, is proposed to analyze the behavior of free vibrations of periodic cellular solids. In this continuum model, the effective mechanical constants of the effective continuum are deduced by an equivalent energy method. The cellular solid structure is then replaced with the equivalent couple-stress continuum with same overall dimension and shape. Moreover, the finite element formulation of the couple-stress continuum for the generalized eigenvalue analysis is developed to implement the free vibration analysis. The eigenfrequencies of the effective continuum are then obtained via the shear beam theory or the finite element method. A conventional finite element analysis by discretizing each cell of the cellular solids is also carried out to serve as an exact solution. Several structural cases are calculated to demonstrate the accuracy and effectiveness of the proposed continuum model. Good agreement on structural eigenfrequencies between the effective continuum solutions and the exact solutions shows that the proposed continuum model can accurately simulate the dynamic behavior of the cellular solids.

Graphene's morphology and electronic properties from discrete differential geometry

The geometry of two-dimensional crystalline membranes dictates their mechanical, electronic and chemical properties. The local geometry of a surface is determined from the two invariants of the metric and the curvature tensors. Here we discuss those invariants directly from atomic positions in terms of angles, areas, vertex and normal vectors from carbon atoms on the graphene lattice, for arbitrary elastic regimes and atomic conformations, and without recourse to an effective continuum model. The geometrical analysis of graphene membranes under mechanical load is complemented with a study of the local density of states (LDOS), discrete induced gauge potentials, velocity renormalization, and non-trivial electronic effects originating from the scalar deformation potential. The asymmetric LDOS is related to sublattice-specific deformation potential differences, giving rise to the pseudomagnetic field. The results here enable the study of geometrical, mechanical and electronic properties for arbitrarily-shaped graphene membranes in experimentally-relevant regimes without recourse to differential geometry and continuum elasticity.

An effective Cosserat continuum model for waves in a material with microstructure

The effective acoustic properties of a medium can be tailored by appropriately designing its microstructural components. The effective properties of a material may be generally defined as those that describe the limiting case where the wavelength of propagation is infinite compared to the characteristic scale of the microstructure. Generally, the limit of vanishingly small microstructural scale in an elastic medium results in an effective medium that is again elastic. In this contribution, we give an example for which the above limit results not in an effective elastic medium, but rather, an effective Cosserat medium. We briefly review the properties of Cosserat continua, provide a basic derivation of their fundamental equations of motion, and interpret the continuum properties in terms of the microstructural elements.

Electronic transmission throughAB-BAdomain boundary in bilayer graphene

We study the electron transmission through the domain boundary on bilayer graphene separating AB and BA stacking regions. Using the effective continuum model, we calculate the electron transmission probability as a function of the electron energy and the incident angle, for several specific boundary structures. The transmission strongly depends on the crystallographic direction of the boundary and also on the atomic configuration inside. At the low energy, the boundary is either insulating or highly transparent depending on the structure. In insulating cases, the transmission sharply rises when the Fermi energy is increased to a certain level, suggesting that the electric current through the boundary can be controlled by the field effect. The boundary parallel to the zigzag direction generally have different transmission properties between the two different valleys, and this enables to generate the valley polarized current in a certain configuration. We show that those characteristic features can be qualitatively explained by the transverse momentum conservation in the position-dependent band structure in the intermediate region.

Optical absorption in twisted bilayer graphene

We theoretically study the optical absorption property of twisted bilayer graphenes with various stacking geometries, and demonstrate that the spectroscopic characteristics serve as a fingerprint to identify the rotation angle between two layers. We find that the absorption spectrum almost continuously evolves in changing the rotation angle, regardless of the lattice commensurability. The spectrum is characterized by series of peaks associated with the van Hove singularity, and the peak energies systematically shift with the rotation angle. We calculate the optical absorption in two different frameworks; the tight-binding model and the effective continuum model based on the Dirac equation. For small rotation angles less than $10^\circ$, the effective model well reproduces the low-energy band structure and the optical conductivity of the tight-binding model, and also explains the optical selection rule analytically in terms of the symmetry of the effective Hamiltonian.

Continuum Models for the Plastic Deformation of Octet-Truss Lattice Materials Under Multiaxial Loading

The continuum models that take plastic material behavior into consideration are derived to analyze periodic octet-truss lattice materials under multiaxial loading. The main focus of this study is to investigate the basic topology of unit cell structures having the cubic symmetry and to formulate the analytical models for predicting pressure load-dependent stress surfaces accurately. The discrete lattice materials are converted into equivalent model continua that are obtained by physically homogenizing the property of the unit cell structures. The effective continuum models contain information on the mechanical characteristics of internal truss members with respect to axial stiffness, internal stress variable, structural packing, and material density at the microscale level. With the hardening material model introduced in the homogenization process, the plastic flow acting on the microscopic truss members gives rise to extend the elastic domain of the analytical stress function derived from homogenize constitutive equations at the macroscale level. Analytical stress predictions show excellent agreements with the results obtained from finite element (FE) analyses.

Optical properties of nano-multi-layered quantum dot: oscillator strength, absorption coefficient and refractive index

In this paper, the exact solution of Schrodinger equation for multi-layered quantum dot (MLQD) within the effective mass approximation and dielectric continuum model is obtained with finite and infinite confining potential (CP). The MLQD is a nano-structured semiconductor system that consists of a spherical core (GaAs) and a coated spherical shell (Ga $$_{1-x}$$ Al $$_{x}$$ As) as the whole dot is embedded inside a bulk material (Ga $$_{1-y}$$ Al $$_{y}$$ As). Using the obtained energies, wave functions and taking advantage of numeric calculations, the oscillator strength, refractive index and absorbtion coefficient change associated with intersubband electronic transition from the ground state to the first allowed excited state are investigated for different CPs (both finite and infinite) and shell thicknesses. The results show that all values of ground state energy for large core dot radius approach the same value (the energy of bulk material) independent of CPs and shell thicknesses. Also it is shown that the optical properties are strongly affected by the changes in CPs and shell thicknesses.

Linear and nonlinear optical properties of multi-layered spherical nano-systems with donor impurity in the center

In this study, the linear, third-order nonlinear and total absorption coefficients (ACs) of multi-layered quantum dot (MLQD) and multi-layered quantum anti-dot (MLQAD) with a hydrogenic impurity are calculated. The analytical and numerical solutions of Schrodinger equation for both MLQD and MLQAD, within the effective mass approximation and dielectric continuum model, are obtained. As our numerical results indicate, an increase in the optical intensity changes the total AC considerably, but the intensity range that leads to these changes is different for MLQAD and MLQD. It is observed that by changing the incident photon energy, the AC curves corresponding to MLQAD and MLQD are of different shapes and behaviors. The peak heights of AC curves corresponding to MLQAD are strongly affected by changing the core antidot radius and the shell thickness values, however in these cases no considerable changes are observed in peak heights of MLQD. Furthermore, in contrast to MLQAD, the photon energies corresponding to total AC peaks of MLQD are more affected by changing the confining potentials (CPs).

Interaction between injection points during hydraulic fracturing

We study the geomechanical stress interaction between two injection points during hydraulic fracturing (hydrofracking) and how this interaction in combination with disorder influences the fracturing process. To this end, we develop an effective continuum model of the hydrofracking of heterogeneous poroelastic media that captures the coupled dynamics of the fluid pressure and the fractured rock matrix and models both the tensile and shear failure of the rock. For injection points that are separated by less than a critical correlation length, our numerical simulations show that the fracturing process around each point is strongly correlated with the position of the neighboring point. The magnitude of the correlation length depends on the degree of heterogeneity of the rock and is on the order of 30–45 m for rocks with low permeabilities. In the strongly correlated regime, we predict a novel effective fracture force that attracts the fractures toward the neighboring injection point.

Oscillator strengths of the intersubband electronic transitions in the multi-layered nano-antidots with hydrogenic impurity

In this study, we have obtained the exact solutions of the Schr\"{o}dinger equation for a multi-layered quantum antidot (MLQAD) within the effective mass approximation and dielectric continuum model for the spherical symmetry. The MLQAD is nano-structured semiconductor system that consists of a spherical core (e.g. $Ga_{1-x}Al_{x}As $) and a coated spherical shell (e.g. $Ga_{1-y}Al_{y}As $) as the whole anti-dot is embedded inside a bulk material (e.g. $GaAs $). The dependence of the electron energy spectrum and its radial probability density on nano-system radius are studied. The numeric calculations and analysis of oscillator strength of intersubband quantum transition from the ground state into two first allowed excited states at the varying radius, for both the finite and infinite confining potential (CP) as well as constant shell thickness, are performed. It is shown that, in particular, the binding energy and the oscillator strength of the hydrogenic impurity of a MLQAD behave differently from that of a single-layered quantum antidot (SLQAD). For a MLQAD with finite core and shell CPs, the state energies and the oscillator strengths of the impurity are found to be dependent on the shell thickness. At the large core radius and very small shell thickness, our results are closer to respective values for a SLQAD that previously reported.

Majorana Fermions and Z2 Vortices on a Square Lattice

We present a simple model of Majorana fermions on a square lattice, and study zero-energy states due to Z$_2$ vortices. We show the relationship between the Chern number of the ground state and the number of the zero-energy states by numerical calculations for finite systems. We also discuss the relationship for the bulk system by observing the change of the spectral asymmetry. We finally present an effective continuum model with O(2) gauge potential which shows a topological change of the index at a half-flux.

Continuum modeling of a composite material with internal resonators

Two continuum methods were investigated for modeling the dynamic behavior of an acoustic metamaterial in the form of a composite material with internal resonators. First, an effective homogeneous classical continuum model was proposed. The effective elastic constants for this continuum are obtained by taking the static equivalence between the continuum and the composite, while the effective mass density adopts the form of a second order tensor. The second model is also a continuum model that is described by two displacement variables. In addition to the usual displacement vector, a displacement vector for the motion of the resonator mass is included, thus making this multi-displacement model quite different from the classical model for elastic solids. It was shown that the dispersion relations predicted by the proposed two approaches were practically the same. The accuracy of the dispersion curves was verified by finite element analyses. Simplicity is the main advantage of the first approach. However, it has to adopt an unusual frequency-dependent effective mass density which may become negative in certain frequency range. On the other hand, the multi-displacement model can be constructed based on the actual material properties of the composite and, in general, is more versatile for further extensions to complex microstructures.

Vortex and gap generation in gauge models of graphene

Effective quantum field theoretical continuum models for graphene are investigated. The models include a complex scalar field and a vector gauge field. Different gauge theories are considered and their gap patterns for the scalar, vector, and fermion excitations are investigated. Different gauge groups lead to different relations between the gaps, which can be used to experimentally distinguish the gauge theories. In this class of models the fermionic gap is a dynamic quantity. The finite-energy vortex solutions of the gauge models have the flux of the "magnetic field" quantized, making the Bohm-Aharonov effect active even when external electromagnetic fields are absent. The flux comes proportional to the scalar field angular momentum quantum number. The zero modes of the Dirac equation show that the gauge models considered here are compatible with fractionalization.

Size-dependent optimal microstructure design based on couple-stress theory

The purpose of this paper is to propose a size-dependent topology optimization formulation of periodic cellular material microstructures, based on the effective couple-stress continuum model. The present formulation consists of finding the optimal layout of material that minimizes the mean compliance of the macrostructure subject to the constraint of permitted material volume fraction. We determine the effective macroscopic couple-stress constitutive constants by analyzing a unit cell with specified boundary conditions with the representative volume element (RVE) method, based on equivalence of strain energy. The computational model is established by the finite element (FE) method, and the design density and FE stiffness of the RVE are related by the solid isotropic material with penalization power (SIMP) law. The required sensitivity formulation for gradient-based optimization algorithm is also derived. Numerical examples demonstrate that this present formulation can express the size effect during the optimization procedure and provide precise topologies without increase in computational cost.

Interface solitons in quadratic nonlinear photonic lattices

We study the properties of two-color nonlinear localized modes which may exist at the interfaces separating two different periodic photonic lattices in quadratic media, focusing on the impact of phase mismatch of the photonic lattices on the properties, stability, and threshold power requirements for the generation of interface localized modes. We employ both an effective discrete model and continuum model with periodic potential and find good qualitative agreement between both models. Dynamic excitation of interface modes shows that a two-color interface twisted mode splits into two beams with different escaping angles and carrying different energies when entering a uniform medium from the quadratic photonic lattice. The output position and energy contents of each two-color interface solitons can be controlled by judicious tuning of the lattice parameters.

Oscillator strengths of electron quantum transitions in spherical nano-systems with donor impurity in the center

The exact solutions of Schrodinger equation for the electron in Coulomb field of donor impurity are obtained within the effective masses approximation and dielectric continuum model for the spherical quantum dot CdS/SiO 2 and anti-dot ZnS/Cd x Zn 1− x S. The dependences of electron energy spectrum and its probability density on nano-system radius are studied. The numeric calculations and analysis of oscillator strength of intersubband quantum transition from ground into first excited state at the varying radius are performed. It is shown that the oscillator strength for the big quantum dot is close to the respective value for the bulk semiconductor crystal.

Effective couple-stress continuum model of cellular solids and size effects analysis

The purpose of this paper is to develop a homogeneous, orthotropic couple-stress continuum model to take the place of the periodic heterogeneous cellular solids. Through generalizing the definition of the characteristic length for isotropic couple-stress continuum, four characteristic lengths are introduced as material engineering constants for such kind of continuum. In order to determine the effective moduli and the characteristic lengths of the effective couple-stress continuum, a Representative Volume Element (RVE) method is constructed. The effective properties are obtained based on the response of the RVE under prescribed boundary conditions, and our results agree with the analytical solutions in literature. In addition, the influences of the relative density, the topology, the size, and the properties of the solid material of cellular materials on the effective moduli as well as the characteristic lengths are discussed, respectively. Furthermore, the size effects in cellular solid beams are investigated using our effective couple-stress continuum model. The results show that the developed continuum model in this paper can precisely capture the size effects in cellular solids.

Boundary conditions for the moving contact line problem

The physical processes near a moving contact line are investigated systematically using molecular dynamics and continuum mechanics. Constitutive relations for the friction force in the contact line region, the fluid-fluid interfacial force, and the stresses in the fluid-solid interfacial region are studied. Verification of force balance demonstrates the importance of the normal stress jump across the contact line region. Effective boundary conditions are derived using force balance. It is found that in the flow regime studied, the deviation of the wall contact angle from the equilibrium contact angle is proportional to the velocity of the contact line. The effective continuum model is solved numerically and the behavior of the apparent contact angle and the wall contact angle is studied. It is found that the fluid-fluid interface near the wall exhibits a universal behavior. The onset of the nonlinear response for the contact line motion is studied within the framework of Blake’s molecular kinetic theory.