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
Highlights
Shapiro steps are reported in many experiments, see part I of this series for references [1]
We concentrate here on such steps in calculations of a Langevin equation with the FK model [2], which many workers saw for a good model of real Shapiro steps
We find out that for an FK chain with free boundaries the results of the FK model with the periodic boundary conditions [1] are not transferable
Summary
Shapiro steps are reported in many experiments, see part I of this series for references [1]. Shapiro steps emerge for a combination of direct and alternating external forces where the frequency of the oscillation of the FK-chain is locked, and its sliding velocity is constant, though the direct force is increased. We specialize in the spring force to a soft value, in comparison to the site-up potential, and in contrast to our former references [3–6]. Beside the use of a Langevin equation [2], an important part of the theory is the use of Newton trajectories (NT) for the description of the ’running’ stationary points of the PES under an external force. They are curves where at every point the gradient of the PES points into the same direction, called the search direction, or even the direction of the external force. Some examples of a changed chain are drawn in the
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