Two-dimensional Frenkel Kontorova Research Articles

Critical behavior of a two-dimensional ferromagnetic film on a non-magnetic substrate

In this paper, we investigate the behavior of a ferromagnetic (FM) film on a nonmagnetic substrate near the Curie point by the computer simulation. The influence of the substrate is specified using the two-dimensional Frenkel-Kontorova (FK) potential. The study is carried out for a two-dimensional film described by the Ising model. At the first step, we calculate the positions of the substrate’s atoms in the ground state depending on the parameters. The parameters are (i) the ratio of the substrate periods and the crystal lattice of the film; and (ii) the ratio of the substrate potential amplitude to the elasticity coefficient of interatomic interaction. The period ratio determines the system coverage ratio. Minimization of the system’s total energy determines the ground state. Calculations show that the ground state has a periodic structure that differs from a square lattice with a coverage coefficient not equal to unity. We calculate the displacements of atoms from the equilibrium position for systems with a different linear scale.

Phase Transition and Magnetoelectric Effect in 2D Ferromagnetic Films on a Ferroelectric Substrate

In this paper, we investigate the behavior of 2D ferromagnetic (FM) films on a ferroelectric (FE) substrate with a periodic structure. The two-dimensional Frenkel–Kontorova (FK) potential simulates the substrate effect on the film. The substrate potential corresponds to a cubic crystal lattice. The Ising model and the Wolf cluster algorithm are used to describe the magnetic behavior of a FM film. The effect of an electric field on a FE substrate leads to its deformation, which is uniform and manifests itself in a period change of the substrate potential. Computer simulation shows that substrate deformations lead to a decrease in the FM film Curie temperature. If the substrate deformations exceed 5%, the film deformations become inhomogeneous. In addition, we derive the dependence of film magnetization on the external electric field.

Structure and energetics of the elbows in the Au(111) herringbone reconstruction

We study the structure of the threading edge dislocations, or ``elbows,'' which are an essential component of the well-known herringbone reconstruction of the (111) surface of Au. Previous work had shown that these dislocations can be stabilized by long-range elastic relaxations into the bulk. However, the validity of the harmonic spring model that had been used to estimate the energies of the dislocations is uncertain. To enable a more refined model of the dislocation energetics, we have imaged the atomic structure of these dislocations using scanning tunneling microscopy. We find that the harmonic spring model does not adequately reproduce the observed structure. We are able to reproduce the structure, however, with a two-dimensional Frenkel-Kontorova (FK) model that uses a pairwise Morse potential to describe the interactions between the top layer Au atoms on a rigid substrate. The parameters of the potential were obtained by fitting the energy of uniaxially compressed phases, or ``stripes'', computed with density functional theory, as a function of surface Au density. Within this model, the formation of the threading dislocations remains unfavorable. However, the large forces on the substrate atoms near the threading-dislocation cores, render the assumption of a completely rigid substrate questionable. Indeed, if the FK parameters are modified to account for the relaxation of just one more atomic layer, threading dislocations can, in principle, become favorable, even without bulk elastic relaxations. Additional evidence for a small elbow energy is that our computed change in the Au(111) surface stress tensor caused by the $(\sqrt{3}\ifmmode\times\else\texttimes\fi{}22)$ reconstruction is considerably smaller than previous estimates.

A quantitative rigidity result for a two-dimensional Frenkel–Kontorova model

We consider a Frenkel–Kontorova system of harmonic oscillators in a two-dimensional Euclidean lattice and we obtain a quantitative estimate on the angular function of the equilibria. The proof relies on a PDE method related to a classical conjecture by E. De Giorgi, also in view of an elegant technique based on complex variables that was introduced by A. Farina.In the discrete setting, a careful analysis of the reminders is needed to exploit these types of methodologies inspired by continuum models.

The superlubricity of the special material with hexagonal symmetry in a two-dimensional Frenkel–Kontorova model

Based on a two-dimensional Frenkel–Kontorova model, the interacting atoms arranged on a two-dimensional hexagonal lattice are studied. It is made to slide by the application of an external driving force in the presence of dissipation over a two-dimensional periodic substrate potential with the hexagonal symmetry. A critical depinning force is defined. It depends on the direction and the magnitude of external driving force, the adhesive force from the substrate, the interaction strength between atoms in the upper layer and especially the misfit angle $$\theta $$ between two layers. The dependence of the state transition from the lock to sliding on the driving force is studied. For the special case of zero misfit angle, the analytical expressions for the critical depinning force are obtained which are in a good agreement with numerical ones. It seems that the superlubricity can appear.

Production of screw dislocations by the external dc and ac forces

The effects of both the ac and the dc external driving forces on the production of screw dislocations are studied by employing the two-dimensional Frenkel–Kontorova model. The ac force is introduced in the model as a way to describe external periodic disturbances or artificial periodic forces. We find that the appearance of the screw dislocation is influenced by the frequency and amplitude of the ac force. It is shown that the screw dislocation can be easily produced by either the lower frequency or the larger amplitude of the ac force.

Propagation and Interaction of Edge Dislocation (Kink) in the Square Lattice**Supported by the National Magnetic Confinement Fusion Science Program of China under Grant No 2014GB104002, the Strategic Priority Research Program of Chinese Academy of Sciences under Grant No XDA03030100, the National Natural Science Foundation of China under Grant Nos 11275156 and 11304324, the Open Project Program of State Key Laboratory

The propagation of kink or edge dislocations in the underdamped generalized two-dimensional Frenkel–Kontorova model with harmonic interaction is studied with numerical simulations. The obtained results show that exactly one line of atoms can be inserted into the lattice, which remains at standstill. However, if more than one line of atoms are inserted into the lattice, then they will split into several lines with α = 1, where α presents the atoms inserted. In other words, only the kink with α = 1 is stable, while the other kinks are unstable, and will split into α = 1 kinks, which remain at standstill.

Ground-State Transition in a Two-Dimensional Frenkel—Kontorova Model

The ground state of a generalized Frenkel—Kontorova model with a transversal degree of freedom is studied. When the coupling strength, K, and the frequency of a single-atom vibration in the transversal direction, ω0y, are increased, the ground state of the model undergoes a transition from a two-dimensional configuration to a one-dimensional one. This transition can manifest in different ways. Furthermore, we find that the prerequisite of a two-dimensional ground state is θ ≠ 1/q.

Resonance and Rectification in a Two-Dimensional Frenkel—Kontorova Model with Triangular Symmetry

The mode-locking phenomena in the dc- and ac-driven overdamped two-dimensional Frenkel-Kontorova model with triangular symmetric structures are studied. The obtained results show that the transverse velocity 〈vx〉 increases in a series of quantized steps nμx (n=0, 1, 2, 3 ···). Moreover, the positive or negative rectification of longitudinal velocity 〈vy〉 can occur when n is an odd number. It is also found in our simulations that the critical depinning force oscillates with the amplitude of ac-driven force, i.e., the system is dominated by the ac-driven force. The oscillatory behavior is strongly determined by the initial phase of ac-force.

Anisotropic character of atoms in a two-dimensional Frenkel—Kontorova model

The dynamics of a certain density of interacting atoms arranged on a two-dimensional square lattice, which is made to slide over a two-dimensional periodic substrate potential with also the quare lattice symmetry, in the presence of dissipation, by an externally applied driving force, is studied. By rotating the misfit angle θ, the dynamical behaviour displays two different tribological regimes: one is smooth, the other becomes intermittent. We comment both on the nature of the atomic dynamics in the locked-to-sliding transition, and on the dynamical states displayed during the atom motion at different values of the driving force. In tribological applications, we also investigate how the main model parameters such as the stiffness strength and the magnitude of the adhesive force affect the static friction of the system. In particular, our simulation indicates that the superlubricity will appear.

The effect of ac-driven force on superlubricity in a two-dimensional Frenkel–Kontorova model

By using the molecular dynamic simulation method with a fourth-order Runge–Kutta algorithm, a two-dimensional dc- and ac-driven Frenkel–Kontorova model with a square symmetry substrate potential for a square lattice layer has been investigated in this paper. For this system, the effects of many different parameters on the static friction force have been studied in detail. It was found that not only the amplitude and frequency of the ac-driven force, but also the direction of dc- and ac-driven forces and the misfit angle between two layers have a strong influence on the static friction force. This indicated that the phenomenon of superlubricity appears easily with larger ac amplitude and smaller ac frequency for some special direction of the external driving force and misfit angle.

Application of Two-Dimensional Frenkel–Kontorova Model to Nanotribology

The dry friction force between two contacting surface layers is studied. The upper layer is arranged on a two-dimensional square lattice and driven by an external driving force. The lower layer is approximated by a two-dimensional periodic substrate potential. This model, usually called two-dimensional Frenkel–Kontorova model, is applied to study the friction forces in this paper. The behaviors of different substrate potential strongly affect the static friction force. It is found that the system has strong anisotropic character. The possibility to obtain superlubricity is suggested.

Friction phenomena in a two-dimensional Frenkel–Kontorova model

By using the molecular dynamic simulation method with a fourth-order Runge–Kutta algorithm, a two-dimensional dc- and ac-driven Frenkel–Kontorova (FK) model with a square symmetry substrate potential for a square lattice layer has been investigated in this paper. For this system, the effects of many different parameters on the average velocity and the static friction force have been studied. It is found that not only the amplitude and frequency of ac-driven force, but also the direction of the external driving force and the misfit angle between two layers have some strong influences on the static friction force. It can be concluded that the superlubricity phenomenon appears easily with a larger ac amplitude and lower ac frequency for some special direction of the external force and misfit angle.

Investigation of superlubricity in a two-dimensional Frenkel–Kontorova model with square lattice symmetry

A two-dimensional Frenkel–Kontorova model with a square symmetry substrate potential for a square lattice layer driven by an external driving force with an arbitrary direction α and an arbitrary misfit angle θ between upper and lower layers is presented in this paper. The effects of the system parameters have been investigated. The application of our results to the tribology is discussed and the dependence of the static friction force on the system parameters is studied. How to make the material with superlubricity is suggested.

Numerical Study of Dynamical Frictional Phenomena Based on the Two-Dimensional Frenkel-Kontorova Model with Impurities

We study numerically dynamical frictional phenomena using the Frenkel-Kontorova model with impurities. Even in the presence of randomness sliding states have a quasi long range order, which changes with the sliding velocity. Such an order affects the velocity dependence of the kinetic frictional force and produces peculiar transverse pinning effects. We discuss them in the relation with the moving Bragg glass theory developed for vortex lattices in type II superconductors. Frictional phenomena have common importance in various physical systems such as vortex lattices in type-II superconductors, charge and spin density waves, interfacial friction of solids and so on. 1),2) We investigate here dynamical frictional phenomena in two-dimensional elastic lattice systems based on the Frenkel-Kontorova (FK) model, 3) and focus especially on the kinetic friction and the transverse pinning effect. The FK model employed here consist of a discrete square lattice with harmonic interatomic force and a substrate lattice with impurities.The periodic potential due to the regular part of the substrate lattice has a two-dimensional sinusoidal form, of which principal axes coincide with those of the lattice on it.Impurities are distributed randomly in the substrate and the potential of each impurity has a Gaussian form.The amplitude of the periodic (impurity) potential is denoted by K(W ).We study overdamped dynamics of the FK model driven by an external force along one of the principal axes.Incommensurate systems are investigated here. The periodic potential pins the lattice even in an incommensurate case when the amplitude K is greater than a critical value Kc ≈ 0.23.The number of sites of a lattice is set to be N =8 9 2 (or 144 2 ) and the impurity density is fixed at about 0.21. A periodic boundary condition is imposed. The kinetic frictional force Fkin is plotted against the sliding velocity v in Fig.1. In the low velocity regime, v � 10, the kinetic frictional force is almost constant and not affected by the periodic potential.In the high velocity regime, however, it decreases with increasing velocity and depends on the periodic potential.The effectiveness of the periodic potential depends on the sliding velocity.This behavior will relate to the change of the order of the sliding lattice as mentioned later. We investigate next the response transverse to the sliding direction.According to the theory of the moving Bragg glass (MBG) developed by Giamarchi and Le Doussal, 4) a quasi long range order of the lattice exists in the transverse direction.

Dislocation nucleation rates during submonolayer growth of heteroepitaxial thin films

Dislocation nucleation rates during submonolayer heteroepitaxial island growth are calculated using a two-dimensional Frenkel-Kontorova (FK) model combined with transition-state theory. A method for determining effective FK adatom pair potentials is tested using embedded atom method (EAM) potentials. Calculated prefactors and activation energies for dislocation nucleation are in excellent agreement with values from a full EAM molecular-dynamics simulation demonstrating the validity of this approach.