Discrete Frenkel-Kontorova Model Research Articles

The Influence of the Shape and Size of the Nanostructures to the Atoms and Molecules Transport

The discrete Frenkel-Kontorova model for the movement of a foreign atom in a solid, obtained in the local chains approximation, led to the movement of the atom in the Frenkel-Kontorova (F-K) soliton form. This model made it possible to reveal the structure of the F-K soliton, to design the shapes of nanostructures and to take into account their influence on the F-K soliton. The transition to field variables leads the Frenkel-Kontorova equation to the sine-Gordon equation for the displacement field of an atom having solutions also in the form of a soliton. This equation and its solution (soliton) contains coefficients depending on the shape and size of nanostructures. The transition to the sine-Gordon equation allowed us to use the results of Theodorakopoulos 's works related to the consideration of the interaction of elastic vibration modes with a soliton. This made it possible to calculate the diffusion coefficient of the soliton and find the dependence of the diffusion coefficient on the shape and size of nanostructures and temperature.

The ground state of the Frenkel–Kontorova model

The continual approximation of the ground state of the discrete Frenkel–Kontorova model is tested using a symmetric algorithm of numerical simulation. A “kaleidoscope effect” is found, which means that the curves representing the dependences of the relative extension of an N-atom chain vary periodically with increasing N. Stairs of structural transitions for N ≫ 1 are analyzed by the channel selection method with the approximation N = ∞. Images of commensurable and incommensurable structures are constructed. The commensurable–incommensurable phase transitions are stepwise.

Collective transport in the discrete Frenkel-Kontorova model

Through multiscale analysis of the adjoint Fokker-Planck equation, strict bounds are derived for the center of mass diffusivity of an overdamped harmonic chain in a periodic potential, often known as the discrete Frenkel-Kontorova model. Significantly, it is shown that the free energy barrier is a lower bound to the true finite temperature migration barrier for this general and popular system. Numerical simulation confirms the analysis, while effective migration potentials implied by the bounds are employed to give a surprisingly accurate prediction of the nonlinear response.

Structure tuning by Fermi-surfaceshifts in low-dimensional systems

In low-dimensional systems, electron-electron correlation as wellas electron-phonon coupling is enhanced. This tends to linkthe geometry to the Fermi-surface position, for instance viaformation of charge-density waves. The quasi-one-dimensionalc(2×2)-Br/Pt(110) surface system is used here toexplore this effect. NO adsorption gives rise to a considerable Fermi-surface shift and causes the system to go through asuccession of complex phases comprising commensurate,long-range-ordered structures, as well as periodic and chaoticsoliton lattices. The structural evolution is in qualitativeagreement with the predictions of the discrete Frenkel-Kontorova model.

Kink-breather solution in the weakly discrete Frenkel-Kontorova model

The discrete Frenkel-Kontorova model, having the sine-Gordon equation as the continuous analog, was investigated numerically at a small degree of discreteness. Interaction between a kink and a breather in a discrete system was compared with the exact three-soliton solution to the continuous sine-Gordon equation. Nontrivial effects of discreteness were found numerically. One is that a kink and a breather in the discrete system are mutually attractive quasiparticles, so they can be regarded as a three-soliton oscillatory system. The other is the energy exchange between a kink and a breather when their collision takes place in a vicinity of a separatrix solution to the continuous sine-Gordon equation. We have estimated numerically the kink-breather binding energy EB and the maximum possible exchange energy EE for different breather frequencies omega. The results suggest that there is a threshold breather frequency for the "spontaneous" breaking up of the three-soliton oscillatory system into a kink and a breather moving in opposite directions.

Computer Simulation on Kink Dynamics and Kink-Pair Nucleation in the Discrete Frenkel-Kontorova Model

Dynamical properties of kinks in the discrete Frenkel-Kontorova model are investigated with the aid of molecular dynamics method with special attention to kink propagation, kink-pair nucleation and dislocation propagation under an external force. The activation energies for these three processes obtained from our simulations are compared with the theoretical results calculated from our method of constraint(s) and the transition state theory. Simulation results are in good agreement with the theoretical ones.

Static multikink solutions in a discrete Frenkel-Kontorova model with anharmonic interactions.

Solutions representing multikinks or pairs of single kinks and strongly elongated bonds-antikinks-are found in a Frenkel-Kontorova model with Morse interactions and a misfit between the chain and the periodic potential. They are formed by splitting of the existing single kinks at negative misfits (expanded chains) which are greater in absolute value than the limit of metastability of the single kinks. The energy of the multikinks is higher than that of the single kinks and they are always in a metastable state

Accommodation of misfit in epitaxial interfaces: discrete Frenkel-Kontorova model with real interatomic forces

The accommodation of misfit in epitaxial interfaces by homogeneous strain and by misfit dislocations (MD) is studied by means of the 1D model of Frank and van der Merwe in which the elastic interactions are replaced by a real pair-wise potential. A relevant feature of the real potential is its inflection point beyond which the potential is not convex. The latter leads to a significant difference in the physical behaviour in comparison with the system with a convex potential. An alternation of long, weak and short, strong bonds occurs in expanded epilayers. This transition from undistorted to distorted structures is found to be of second-order with respect to both the misfit and the substrate modulation and the critical exponents are equal to 1/2. The commensurate-incommensurate transition is continuous in compressed chains. In expanded chains it is also continuous as long as all the atoms in the ground state sample the convex part of the potential. Above a certain value of the interfacial bonding this condition is violated and the commensurate-incommensurate transition changes into a first-order transition. The energy of MD interaction is considerably smaller in expanded than in compressed chains. It is suppressed additionally by the distortion of the chemical bonds between the MD. The energy of a single MD depends strongly on the absolute value of the misfit; it grows in compressed and decreases in expanded epilayers. At zero value of the misfit the energy of a positive MD (empty potential trough) is smaller than the energy of a negative MD (two atoms in a trough).

Destruction of classical cantori in the quantum Frenkel-Kontorova model

In this paper we investigate the properties of the quantized discrete Frenkel Kontorova model. The structure of the ground state is numerically analyzed by means of the Metropolis algorithm; special attention is given to the effects of quantization on the Cantori structure of the classical ground state. These quantum effects produce a new structure which can be approximately described by using a sawtooth map instead of the Standard map. The dependence of the quantum energy on temperature is also investigated and discussed in connection with these structural modifications.

Low-temperature excitations, specific heat and hierarchical melting of a one-dimensional incommensurate structure

The low-temperature behaviour of an Ising spin model for a one-dimensional incommensurate structure (which is equivalent to the discrete Frenkel-Kontorova model in the non-analytical regime) is analytically investigated. Its specific heat exhibits an infinite series of Schottky anomalies at temperatures Tm going to zero. These anomalies correspond to crossover which can be physically associated with a hierarchical melting of the incommensurate chain. In contrast with early work, the authors show that certain types of phase defect (discommensurations) are the essential low-temperature excitations of the system and in addition that they can be hierarchically classified according to the continued-fraction expansion of the commensurability ratio zeta . Their characteristic energies kBTm and their weights are explicitly calculated at all orders. They present first a physically simple but empirical approach which allows one to predict and to calculate approximately these anomalies. Next, the validity of this approach is confirmed by a more sophisticated and new method of renormalisation with unequal blocks, which becomes exact as T goes to zero. The authors discuss the application of this theory to experimental measurements of the specific heat in K-hollandite.

Mean field phase diagram of the discrete Frenkel-Kontorova model

The mean field phase diagram of the discrete Frenkel-Kontorova model (harmonic chain of particles exposed to a sinusoidal potential) is studied using a fixed irrational value for the ratio of atoms and available sites as a boundary condition. Independent variables are the temperature and the height of the sinusoidal potential. The phase diagram contains two different solid phases, namely an intrinsically pinned incommensurate and an unpinned incommensurate phase. The position of the phase boundary and the nature of the phases are determined. We also investigate the influence of an electric field and the resulting pinning-unpinning transition for fields smaller than the threshold field.

Critical behaviour at the transition by breaking of analyticity and application to the Devil's staircase

The authors investigate numerically the behaviour of the elastic constant of incommensurate ground states of the discrete Frenkel-Kontorova (FK) model below their transition by breaking of analyticity (corresponding to the lattice locking of the incommensurate modulation). They show that the elastic constant is critical and vanishes with an exponent that depends on the irrational commensurability ratio. They propose a scaling relation between this exponent zeta and the exponents of the phonon gap and of the correlation length which are critical from above the transition and which were previously calculated. These results allow one to confirm a previous conjecture which asserts that the Devil's staircase of the discrete FK model is incomplete when the incommensurate configurations are unlocked by the lattice and is complete when they are locked.

The discrete Frenkel-Kontorova model and its extensions: I. Exact results for the ground-states

We present a rigorous study of the classical ground-states under boundary conditions of a class of one-dimensional models generalizing the discrete Frenkel-Kontorova model. The extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic, quasi-periodic and chaotic orbits. For all boundary conditions, we select among all the extremum solutions of the energy of the model, those which correspond to the ground-states of the infinite system. We prove that these ground-states are either periodic (commensurate) or quasi-periodic (incommensurate) but are never chaotic. We also prove the existence of elementary discommensurations which are minimum energy configuration of the model for certain special boundary conditions. The topological structure of the whole set of ground-states is described in details. In addition to physical applications, consequences for twist map homeomorphisms are mentioned. Part II (S. Aubry, P.Y. LeDaeron and G. Andre) will be mostly devoted to exact results on the transition by breaking of analyticity which occurs on the incommensurate ground states when the model parameters vary and on its connection with the stochasticity threshold in the corresponding twist map.

Exact models with a complete Devil's staircase

The author describes two exact models which exhibit a complete Devil's staircase which can both be calculated explicitly with the same method. The first one is a discrete Frenkel-Kontorova model (an elastic chain of atoms submitted to a continuous piecewise parabolic periodic potential). The atomic mean distance of the ground state varies with the model parameters as a complete Devil's staircase. The authors also calculates explicitly the Peierls-Nabarro barrier and the depinning force of the chain which is always locked on the periodic potential. (However, it is noted that the lack of differentiability of this periodic potential always makes the chain locked and the Devil's staircase complete, unlike the case for smooth periodic potentials.) The second model is the Ising chain under an external magnetic field and with long-range antiferromagnetic interactions proposed by Bak and Bruinsma (1982). It is rigorously proved that the ground state of this model is periodically modulated and that its magnetisation varies as a function of the magnetic field which is a complete Devil's staircase. This one is also explicitly calculated.

Critical behaviour at the transition by breaking of analyticity in the discrete Frenkel-Kontorova model

The authors study numerically the transition by breaking of analyticity which occurs in the incommensurate ground state of the Frenkel-Kontorova model (1938) when the amplitude lambda of its periodic-potential V(u) is increased beyond a critical value lambda c. A brief review of the properties of this transition and its connection with the standard map is given. They consider four quantities which are critical when lambda goes to lambda c from upper values: the gap in the phonon spectrum, the coherence length of the ground state, the Peierls-Nabarro barrier and the depinning force. The numerical method is discussed and the authors show in particular that the mapping method is unpracticable for lambda > lambda c. They observe the transition by breaking of analyticity and show that in the stochastic region ( lambda > lambda c) the ground state is never chaotic. Nevertheless in this region metastable chaotic states can be obtained. The numerical calculations, performed with a ratio of the atomic mean distance to the period of the potential V(u), l/2a=(3- square root 5)/2 equivalent to the golden mean, show that critical exponents can be defined for the four critical quantities studied. Their values reveal two scaling laws which are empirically explained.