Surface Green Function Research Articles

Surface Green's functions for an anisotropic viscoelastic half-plane and their application to contact problems

In this paper, the elastic-like surface Green's functions for an anisotropic viscoelastic half-plane are derived using the time-stepping method. Using the elastic-like surface Green's functions as the core analytical solutions, we develop semi-analytical models (SAMs) and apply them to solve two different contact problems with anisotropic viscoelastic materials. As new modeling approaches, the SAMs developed here can provide fast and efficient approaches to solving contact problems. These methods enable us to consider contact problems with generally anisotropic viscoelastic solids, in which the contact surface is frictional and either smooth or rough, and the applied loads and boundaries can be time-variant. The correctness of the derived surface Green's functions is demonstrated by comparing the numerical results obtained by SAMs and those achieved from the analytical solutions or boundary element methods. Using the obtained numerical results, the impacts of time step size, anisotropy, frictional coefficient, roughness, and applied loads on the contact responses are further analyzed and discussed.

Frictional contact analysis between two-dimensional deformable anisotropic magneto-electro-elastic bodies via a semi-analytical method

In this paper, we utilize the surface Green's function of a magneto-electro-elastic (MEE) half-plane as a general analytical kernel to develop a semi-analytical method (SAM). This SAM is designed to solve the two-dimensional contact problems of two dissimilar deformable anisotropic MEE bodies. Using the surface Green's function, the associated influence matrices in SAM can be obtained in a closed mathematical form and then computed analytically, offering significant benefits in terms of computational efficiency and accuracy. This method can be applied to general contact problems where the contact surface is frictional sliding and subjected to four different electric and magnetic conditions. By assuming the elastic and piezoelectric materials as uncoupled MEE materials, the method can also be applied to contact problems in which anisotropic elastic and piezoelectric materials are involved. To demonstrate the accuracy and applicability of the proposed SAM, four different numerical examples are provided, and their results are compared with those obtained by using other approaches, such as analytical solutions and the previous SAM in the literature. Through the numerical results, parametric studies are also performed, allowing us to explore the influences of material properties, loadings, frictional coefficient, and contact surface profile on mechanical/electrical/magnetic contact behaviors.

Open-boundary cluster model with a parameter-free complex absorbing potential.

In quantum chemical calculations of heterogeneous structures in solids, e.g., when an impurity is located on the surface, the conventional cluster model is insufficient to describe the electronic structure of substrates due to its finite size. The open-boundary cluster model (OCM) overcomes this problem by performing cluster calculations under the outgoing-wave boundary condition. In this method, a complex absorbing potential (CAP) is used to impose the boundary condition, but the CAP used in the previous studies required parameter optimization based on the complex variational principle. This study proposes and applies a parameter-free CAP to OCM calculations. This approach makes it possible to uniquely determine the band-specific CAP based on the surface Green's function theory. Using this CAP, we conducted OCM calculations of the tight-binding model of a one-dimensional semi-infinite chain, and we found that the calculated density of states agreed with the exact one. Surface states of the Newns-Anderson-Grimley model were also computed using the CAP, and the projected density of states on the adsorbed atom was successfully reproduced.

Revealing the influence of edge states on the electronic properties of PtSe2

The properties and performance of 2D nanostructures are known to be critically affected by localized edge states. Here, platinum diselenide (PtSe2) films, synthesized by thermally assisted conversion, are characterized by a variety of spectroscopic and microscopic methods, including scanning tunneling microscopy and spectroscopy, and scanning transmission electron microscopy. We report the presence of distinct edge states, and use surface Green's function methodology in atomistic modeling of different edge atomic structures, together with other theoretical tools, to investigate their origin. Our results indicate that the edges, inherent and abundant in polycrystalline films, can be semiconducting or semimetallic in nature depending on the edge configuration. Experimental results in conjunction with first-principles calculations demonstrate a mapped band profile of the atomically-sharp step edges of PtSe2, which form lateral heterojunctions possessing large asymmetric band offsets in the conduction and valence bands. We further explore the viability of a single step monomaterial heterojunction based field-effect transistor, through atomistic quantum transport simulations. This reveals the critical role of the edge states in creating a wider energy band for carrier transport in the ON-state and hence enabling a remarkably large ION/IOFF ratio > 1010 and near-ideal subthreshold slope, fulfilling the requirements for low-power applications.

Nature of quasiparticle interference in three dimensions

Quasiparticle interference (QPI) imaging is a powerful tool for the study of the low-energy electronic structure of quantum materials. However, the measurement of QPI by scanning tunneling microscopy (STM) is restricted to surfaces and is thus inherently constrained to two dimensions. QPI has proved immensely successful for the study of materials that exhibit a quasi-two-dimensional electronic structure, yet it raises questions about how to interpret QPI in materials that have a highly three-dimensional electronic structure. In this paper, we address this question and establish the methodology required to simulate and understand QPI arising from three-dimensional systems as measured by STM. We calculate the continuum surface Green's function in the presence of a defect, which captures the role of the surface and the vacuum decay of the wave functions. We find that defects at different depths from the surface will produce unique sets of scattering vectors for three-dimensional systems, which nevertheless can be related to the three-dimensional electronic structure of the bulk material. We illustrate the consequences that the three-dimensionality of the electronic structure has on the measured QPI for a simple cubic nearest-neighbor tight-binding model, and then demonstrate application to a real material using a realistic model for PbS. Our method unlocks the use of QPI imaging for the study of quantum materials with three-dimensional electronic structures and introduces a framework to generically account for ${k}_{z}$ dispersions within QPI simulations.

Electronic Band Structure and Surface States in Dirac Semimetal LaAgSb2.

LaAgSb is a Dirac semimetal showing charge density wave (CDW) order. Previous angle-resolved photoemission spectroscopy (ARPES) results suggest the existence of the Dirac-cone-like structure in the vicinity of the Fermi level along the –M direction. This paper is devoted to a complex analysis of the electronic band structure of LaAgSb by means of ARPES and theoretical studies within the ab initio method as well as tight binding model formulation. To investigate the possible surface states, we performed the direct DFT slab calculation and the surface Green function calculation for the (001) surface. The appearance of the surface states, which depends strongly on the surface, points to the conclusion that LaSb termination is realized in the cleaved crystals. Moreover, the surface states predicted by our calculations at the and X points are found by ARPES. Nodal lines, which exist along the X–R and M–A paths due to crystal symmetry, are also observed experimentally. The calculations reveal other nodal lines, which originate from the vanishing of spin–orbit splitting and are located at the X–M–A–R plane at the Brillouin zone boundary. In addition, we analyze the band structure along the –M path to verify whether Dirac surface states can be expected. Their appearance in this region is not confirmed.

A Fully Coupled Time-Domain BEM-FEM Method for the Prediction of Symmetric Hydroelastic Responses of Ships with Forward Speed

This paper presents a direct time-domain method for the prediction of symmetric hydroelastic responses of ships progressing with forward speed in small amplitude waves. A transient time-domain free surface Green function is used for the idealisation of the seakeeping problem using an Earth fixed coordinate system. Free surface ship hydrodynamics are idealised in the time domain by a Green function, and forward speed effects are idealised by a space-state model. Modal actions are accounted for by Timoshenko beam structural dynamics. Flexible fluid structure interaction (FFSI) coupling is enabled by a body boundary condition, and a direct integration Newmark- β scheme is used to obtain symmetric dynamic responses. The method is validated against available published numerical and experimental results. A parametric study for different container ship hull forms confirms that (i) forward speed effects should be taken under consideration as far as practically possible and (ii) hull flexibility effects accounting for hull shear deformation and rotary inertia are more notable for slender hull forms.

Study of the free surface Green function at zero forward speed in finite water depth using Newman’s method

This study details the computation of the finite depth Green function at zero forward speed based on Newman’s method. The Green function is evaluated using either a series expression or an integral expression. An asymptotic analysis of the integral expression is performed and enables an accurate power series approximation over the whole range of wave frequencies. The zero and infinite-frequency limits of the Green function are studied and match their expected values leading to a consistent behaviour with the last results of the literature.The computation of the finite depth Green function is then checked with two verification cases.

Elastic diffusion vibrations of an isotropic Kirchhoff-Love plate under an unsteady distributed transverse load

We investigated an unsteady elastic diffusion vibration of a simply supported rectangular isotropic Kirchhoff-Love plate. The plate is under the action of a distributed transverse load. A model that describes coupled elastic diffusion processes in a multicomponent continuum is used for the mathematical problem formulation. The model is taking into account the diffusion fluxes relaxation. The transverse vibration equations of a rectangular isotropic Kirchhoff-Love plate with diffusion were obtained from the model using the d'Alembert variational principle. The initial-boundary value problem of a freely supported isotropic rectangular plate bending is formulated on the basis of the obtained equations. The plate is under the action of elastic diffusion perturbations distributed over the surface. The problem solution of an unsteady elastic diffusion plate vibration is sought in an integral form. The surface Green's functions are the kernels of the integral representations. To find the Green's functions, we used the Laplace transform in time and the expansion into double trigonometric Fourier series in spatial coordinates. Green's functions in the image domain are represented in the form of rational functions and depend on the Laplace transform parameter. The transition to the original domain is done analytically through residues and tables of operational calculus. The surface Green's function analytical expressions are obtained. As a calculation example, we considered a freely supported elastodiffusive plate under the action of suddenly applied unsteady bending moments distributed over the plate surface. By using a three-component continuum, a numerical study of interactions between unsteady mechanical and diffusion fieldsis done for an isotropic plate. The influence of relaxation effects on the kinetics of mass transfer is investigated. The solution is presented in the analytical form, as well as in the graphs of the displacement fields and concentration increments on time and coordinates. At the end of the publication, the main conclusions are given about the fields coupling effect and the relaxation of diffusion fluxes on the stress-strain state and mass transfer in the plate.

FMM-BEM algorithm based on improved free surface Green's function

In this paper, a Green's function satisfying free surface condition, which is more suitable for solving potential flow problems, is improved, proposed and applied to FMM-BEM to solve the problems of large amount of calculation and long calculation time in the traditional boundary element method. The results show that the FMM-BEM method is more efficient than the BEM method in hydrodynamic analysis of large-scale structures.

WannSymm: A symmetry analysis code for Wannier orbitals

We derived explicit expressions of symmetry operators on Wannier basis, and implemented these operators in WannSymm software. Based on this implementation, WannSymm can i) symmetrize the real-space Hamiltonian output from Wannier90 code, ii) generate symmetry operators of the little group at a specific k-point, and iii) perform symmetry analysis for Wannier band structure. In general, symmetrized Hamiltonians yield improved results compared with the original ones when they are employed for nodal structure searching, surface Green's function calculations, and other model calculations. Program summaryProgram Title: WannSymmCPC Library link to program files:https://doi.org/10.17632/zgf4p85pwx.1Licensing provisions: BSD 3-clauseProgramming language: CExternal libraries: spglib; BLAS and LAPACK; MPI libraries (optional)Nature of problem: Generate the symmetry operators in Wannier basis; calculate the symmetry eigenvalues and characters; symmetrize the real-space Hamiltonian according to the crystal structure. It deals with nonmagnetic or magnetic systems with or without spin-orbit coupling. For magnetic systems, it can deal with ferromagnetic, commensurate antiferromagnetic long range ordered collinear or non-collinear systems with spin-orbit coupling.Solution method: The code requires the real-space Hamiltonian from Wannier projection, the definition of Wannier basis and the crystal structure associated with the defined basis.

Surface states and quasiparticle interference in Bernal and rhombohedral graphite with and without trigonal warping

We use an exact analytical technique [Phys. Rev. B \textbf{101}, 115405 (2020), Phys. Rev. B \textbf{102}, 165117 (2020)] to recover the surface Green's functions for Bernal (ABA) and rhombohedral (ABC) graphite. For rhombohedral graphite we recover the predicted surface flat bands. For Bernal graphite we find that the surface state spectral function is similar to the bilayer one, but the trigonal warping effects are enhanced, and the surface quasiparticles have a much shorter lifetime. We subsequently use the T-matrix formalism to study the quasiparticle interference patterns generated on the surface of semi-infinite ABA and ABC graphite in the presence of impurity scattering. We compare our predictions to experimental STM data of impurity-localized states on the surface of Bernal graphite which appear to be in a good agreement with our calculations.

Transient Green's functions of a buried pulse in a constrained transversely isotropic half-space

In this paper, transient surface Green's functions (GFs) excited by a general dynamic pulse buried in a constrained transversely isotropic (TI) half-space are derived in analytical forms. The constrained transverse isotropy is defined by an explicit relation that expresses one of the five elastic constants in terms of the other four independent constants so that no coupled P-SV wave exists. Use is made of the solutions available for general TI materials in Hankel–Laplace transformed domain in order to extract the corresponding response functions for the constrained material. Under this constraint, the coupled fourth-order wave operator, which represents the convolved motion of the quasi-longitudinal and quasi-transverse waves, is decoupled into two separated second-order wave operators describing the motion of the pure dilatational (P) and pure shear (SV) waves. The involved Laplace transform inversion is carried out using the novel Cagniard scheme so that the final displacement GFs are expressed in terms of the elementary line integrals over a finite interval. The GF solution is valid for the entire range of the constrained TI material including material isotropy as its special case, and clearly illustrates the disturbances due to different stress waves including, P, SV, SH, Rayleigh and diffracted SP waves. Moreover, the GFs show that in this constrained TI material, the compressional and horizontally polarized waves, excited by a source located at an arbitrary depth, propagate towards the surface with direction-dependent velocities. The explicit formulas for these velocities of the waves are obtained in terms of the inclination angle of the wave normal with respect to the axis of material symmetry. On the other hand, the vertically polarized wave and the diffracted SP wave propagate with eigen velocity, independent from the direction of the propagation. Two critical epicentral distances, at which mode conversion in ray diagram occurs, are detected for this constrained TI material. One critical distance marks the phenomena of total reflection of SV-waves where the incident SV-wave is totally reflected and the diffracted SP-wave travels along the free surface. The other marks the distance where the SH- and SP-waves become coalescent, and furthermore, before this epicentral distance the SH-wave is faster, and after this distance, the SP-wave is faster. To show the relevance of this constrained TI material in engineering and material modeling applications and its relation to the physics of TI materials, we also demonstrate that, in addition to its mathematical simplification, this reduced anisotropy can well approximate the elastic coefficients of some real TI materials, particularly those encountered in exploration geophysics and seismology.

Green's function method for dynamic contact calculations.

Resolving atomic scale details while capturing long-range elastic deformation is the principal difficulty when solving contact mechanics problems with computer simulations. Fully atomistic simulations must consider large blocks of atoms to support long-wavelength deformation modes, meaning that most atoms are far removed from the region of interest. Building on earlier methods that used elastic surface Green's functions to compute static substrate deformation, we present a numerically efficient dynamic Green's function technique to treat realistic, time-evolving, elastic solids. Our method solves substrate dynamics in reciprocal space and utilizes precomputed Green's functions that exactly reproduce elastic interactions without retaining the atomic degrees of freedom in the bulk. We invoke physical insights to determine the necessary number of explicit substrate layers required to capture the attenuation of subsurface waves as a function of surface wave vector. We observe that truncating substrate dynamics at depths that fall as a power of wave vector allows us to accurately model wave propagation without implementing arbitrary damping. The framework we have developed substantially accelerates molecular dynamics simulations of large elastic substrates. We apply the method to single asperity contact, impact, and sliding friction problems and present our preliminary findings.

Time domain TEBEM method of ship motion in waves with forward speed by using impulse response function formulation

A numerical method for solving 3D unsteady potential flow problem of ship advancing in waves is proposed. The flow field is divided into inner and outer domain by an artificial matching surface. The inner domain includes ship-wetted surface, matching surface and free surface inside. The free-surface condition is formulated by perturbation about the double-body flow (DB model) or uniform incoming flow assumption (NK model). The outer domain contains matching surface, free surface outside and infinite far-field boundary, where only NK model is applied. The simple Green function and transient free surface Green function are used to form the boundary integral equation (BIE) by the impulse response function method for the inner and outer domain. Taylor Expansion Boundary Element Method (TEBEM) is utilized to solve the BIEs. Matching conditions for the inner-domain and outer-domain flow are enforced by the continuity of velocity potential and normal velocity on the matching surface. In this paper, several ships are selected to predict the hydrodynamic coefficients and displacement at different forward speed. The experiment measurements in physical tank is also used to validate the accuracy of numerical results. Good agreement of 6DOF ship motions in various wave headings can be obtained using TEBEM method.

Quantum spin Hall effect in Ta2M3Te5 (M=Pd,Ni)

Quantum spin Hall (QSH) effect with great promise for the potential application in spintronics and quantum computing has attracted extensive research interest from both theoretical and experimental researchers. Here, we predict monolayer Ta$_2$Pd$_3$Te$_5$ can be a QSH insulator based on first-principles calculations. The interlayer binding energy in the layered van der Waals compound Ta$_2$Pd$_3$Te$_5$ is 19.6 meV/A$^2$; thus, its monolayer/thin-film structures could be readily obtained by exfoliation. The band inversion near the Fermi level ($E_F$) is an intrinsic characteristic, which happens between Ta-$5d$ and Pd-$4d$ orbitals without spin-orbit coupling (SOC). The SOC effect opens a global gap and makes the system a QSH insulator. With the $d$-$d$ band-inverted feature, the nontrivial topology in monolayer Ta$_2$Pd$_3$Te$_5$ is characterized by the time-reversal topological invariant $\mathbb Z_2=1$, which is computed by the one-dimensional (1D) Wilson loop method as implemented in our first-principles calculations. The helical edge modes are also obtained using surface Green's function method. Our calculations show that the QSH state in Ta$_2M_3$Te$_5$ ($M=$ Pd, Ni) can be tuned by external strain. These monolayers and thin films provide feasible platforms for realizing QSH effect as well as related devices.

Time-domain TEBEM method for mean drift force and moment of ships with forward speed under the oblique seas

A numerical method for solving 3D unsteady potential flow problem of ship advancing in waves is put forward. The flow field is divided into an inner and an outer domain by introducing an artificial matching surface. The inner domain is surrounded by ship wetted surface and matching surface as well as part of the free surface. The free surface condition for the inner domain is formulated by perturbation about the double-body flow or uniform incoming flow assumption. The outer domain is surrounded by matching surface and the rest free surface as well as infinite far-field radiation boundary. The free surface condition for the outer domain is formulated by perturbation about uniform incoming flow. The simple Green function and transient free surface Green function are used to form the boundary integral equation (BIE) for the inner and outer domains, respectively. Taylor Expansion Boundary Element Method (TEBEM) is utilized to solve the double-body flow and inner domain and outer domain unsteady flow BIE. Matching conditions for the inner domain flow and outer domain flow are enforced by the continuity of velocity potential and normal velocity on the matching surface. Direct pressure integration on ship wetted surface is used to obtain the first-order and second-order wave forces (moments). The numerical predictions on the displacement, added resistance, sway mean drift force and yaw mean drift moment of the modified KVLCC2 ship at different forward speeds are investigated by the proposed TEBEM method. It is also compared with the other numerical results. The physical tank experiment results are also developed to validate the accuracy of numerical tank results. Compared with the experiment solutions, a good agreement can be obtained by TEBEM method.

Hydrodynamic Analysis and Motions of Ship with Forward Speed via a Three-Dimensional Time-Domain Panel Method

A new three-dimensional (3D) time-domain panel method is developed to solve the ship hydrodynamic problem and motions. For an advancing ship with a constant forward speed in regular waves, the ship’s hull can be discretized and processed into a number of quadrilateral panels. Based on Green’s theorem, an analytical expression for Froude–Krylov (F–K) forces evaluation on the quadrilateral panels is derived without accuracy loss. Within the linear potential theory, the transient free surface Green function (TFSGF) is applied to solve the boundary value problem. To improve the efficiency and numerical stability of TFSGF evaluation, a precise integration method with variable parameters setting for extended identity matrix is developed to compute the TFSGF in the computation domain. Then, radiation and diffraction forces can be evaluated by means of the impulse response function method. The Wigley I hull form is taken as a study case, and the computed hydrodynamic coefficients, wave exciting forces, and motions by the present method are compared with previous literature experimental data and prior published results. It manifests that the three-dimensional time-domain panel method proposed in this paper has good accuracy.

Nonlinear ship motion with forward speed in waves based on 3D time domain hybrid Green function method

In this research, based on three-dimensional (3D) time domain hybrid Green function method, the time-domain nonlinear initial boundary value problem of a ship motion in waves with forward speed is solved. The 3D Rankine source and 3D time domain free surface Green function are adopted to solve flow fields of the inner and the outer domains respectively. Meantime, the instantaneous wet surface during ship motion is divided into boundary elements in every time step, to calculate nonlinear incident wave and static restoring forces. Combining with the linear solution of radiation and diffraction problems, the nonlinear time domain numerical method based on 3D hybrid Green function method for ship motion in waves with forward speed is established. Numerical simulation and analysis are carried out for Wigley and S175 ships with different forward speeds and wave angles respectively by developed numerical solver. The comparison with the experimental results show that the proposed nonlinear numerical method has an obvious improvement in predicting accuracy under complicated conditions such as with higher forward speed and large flare, which proves it has good prospect in engineering applications.

Surface Green's functions and quasiparticle interference in Weyl semimetals

We use the exact analytical technique introduced in Phys. Rev. B 101, 115405 to recover the surface Green's functions and the corresponding Fermi-arc surface states for various lattice models of Weyl semimetals. For these models we use the T-matrix formalism to calculate the quasiparticle interference patterns generated in the presence of impurity scattering. In particular, we consider the models introduced in Phys. Rev. B 93, 041109(R) (A) and Phys. Rev. Lett. 119, 076801 (B), and we find that, as opposed to observations previously obtained via joint density of states and spin-dependent scattering probability, the inter-arc scattering in the quasiparticle interference features is fully suppressed in model A, and very small in model B. Our findings indicate that these models may not correctly describe materials such as MoTe2, since for such materials inter-arc scattering is clearly visible experimentally, e.g., in Nature Phys. 12, 1105-1110 (2016). We also focus on the minimal models proposed by McCormick et al. in Phys. Rev. B 95, 075133, which indeed recover significant inter-arc scattering features.