Dynamic metastable vortex states in interacting vortex lines

Synchronisation is a phenomenon which can be found in incredibly enormous fields of studies : from a two coupled clocks to a ruby nuclear magnetic resonance laser with a delayed feedback, from atrial pacemaker cells to periodically stimulated fireflies. Historically, Christiaan Huygens first observed that two coupled or forced pendulum clocks can be synchronised. Then it turns out that this effect is very general for systems consist of coupled oscillators, no matter what nature of these oscillators or coupling is.

In our paper, we consider a synchronisation in a superconductivity domain. The usual example of synchronisation in superconductivity is the Josephson junction, which, forced by external microwave, shows typical synchronisation features. For example, the Shapiro steps, regions where while direct current changes, voltage remains the same corresponding to the second Josephson relation. In the non-linear community, they are better known as Arnold’s tongues.). Here we consider another superconducting phenomenon — the Abrikosov vortex, that can also be synchronised and show that it lead not just the same Shapiro steps, to underline that it is an effect of synchronisation, but even to so-called fractional ones.

The first object of our study is a superconducting nanowire with linear defect, which we study numerically by means of Time-Dependent Ginzburg-Landau (TDGL) framework. We do not restrict ourselves to a particular nature of this defect : it can be as natural grain-boundaries in High-Temperature superconductors, or artificially made defects with Force Ion Beam or sputtered ferromagnetic material on top of the nanowire. Applying direct current with alternate microwave component with constant frequency and measuring voltage at some extent of the defect. When current reaches the critical one, Abrikosov vortex and its antivortex enters into the defect under the Lorenz force from the current, and then attracts each other and finally annihilate in the centre. Then the process repeats. When the vortex moves, it dissipates energy and we can calculate instantaneous voltage. Averaging this voltage in time, we can restore current-voltage characteristic (CVC) that usually are measured in experiment. In our simulation, we observed that not only the Shapiro steps are appeared when microwave current component increases, but a so-called fractional Shapiro Steps. These fractional steps are the sign of strongly anharmonic instantaneous voltage that leads to, that is also known as high-order Arnold’s tongues.

In this first model, we discuss one simple vortex-antivortex oscillator, that can mimic Josephson junction with non-sinusoidal current-phase relationship, showing Shapiro steps and fractional Shapiro Steps. But there is still a significant difference. In our next step, we add another parallel linear defect at the distance that vortices in different lines can slightly interact. Now we have two coupled vortex-antivortex oscillators with applying direct current to create vortices and periodic drive of microwave alternate current with fixed frequency as in the first model. Increasing microwave current at first glance, we reproduce the results of the first model. But, starting from some value of microwave current, we observe some abrupt jumps and falls on a CVC. Having a closer look at what happens with vortices motion, we will see that they have different patterns of motion : when vortices enter simultaneously in each line and when they enter sequentially. The first situation appears to have a higher voltage, it is metastable and possible only with sufficient microwave current component because vortices in the neighbouring lines repel each other. This ​​state is responsible for abrupt jumps on the current-voltage characteristic to high Shapiro steps. Then, when the direct current component increases to restore a CVC, a microwave current component is not sufficient and voltage drops as the system transits to a second sequential patter. But both patterns can be synchronous states.

To know more, this link will direct you to the paper.