Description of Shapiro steps on the potential energy surface of a Frenkel–Kontorova model, Part II: free boundaries of the chain

Abstract

We explain Shapiro steps in a Frenkel–Kontorova (FK) model for a 1D chain of particles with free boundaries. The action of an external alternating force for the oscillating structure of the chain is important here. The different ’floors’ of the potential energy surface (PES) of this model play an important role. They are regions of kinks, double kinks, and so on. We will find out that the preferable movements are the sliding of kinks or antikinks through the chain. The more kinks / antikinks are included the higher is the ’floor’ through the PES. We find the Shapiro steps moving and oscillating anywhere between the floors. They start with a single jump over the highest SP in the global valley through the PES, like in part I of this series. They finish with complicated oscillations in the PES, for excitations directly over the critical depinning force. We use an FK model with free boundary conditions. In contrast to other results in the past, for this model, we obtain Shapiro steps in an unexpected, inverse sequence. We demonstrate Shapiro steps for a case with soft ’springs’ between an 8-particle FK chain. Graphic abstract

An Analysis of Some Properties and the Use of the Twist Map for the Finite Frenkel–Kontorova Model

“…Newton trajectories connect stationary points of an index difference of one [45] on the original potential energy surface. It is a quite interesting property that the special form of the f vector acts as a change in the orginal distance, a 0 , of the atoms of the FK chain, see value (6), and not only as the tilding of the potential of the first and the last atom of the chain. The FK chain is similar to an accordion.…”

“…If one pulls the ends, all of the bellows relax. Newton trajectories with force f can be used to calculate many kinds of solitones of the FK chain and, thus, intermediate minima of the potential energy surface and saddle points with an increasing index [3,6,19,[46][47][48].…”

“…The latter acts on the extracted subsystem by a potential. Of special interest may be electronic applications [2][3][4][5][6][7] for Wigner electrons or Josephson junctions. Further models are chargedensity wave conductors [8][9][10], charge transport in solids and on crystal surfaces [11], magnetic or ferro-and antiferromagnetic domain walls [12], magnetic superlattices [13], superconductivity [14,15], and vortex matter [16][17][18], to name a few.…”