Manipulation of individual superconducting vortices and stick-slip motion in periodic pinning arrays

We numerically examine the manipulation of superconducting vortices interacting with a moving trap representing a magnetic force tip translating across a superconducting sample containing a periodic array of pinning sites. As a function of the tip velocity and coupling strength, we find five distinct dynamic phases, including a decoupled regime where the vortices are dragged a short distance within a pinning site, an intermediate coupling regime where vortices in neighboring pinning sites exchange places, an intermediate trapping regime where individual vortices are dragged longer distances and exchange modes of vortices occur in the surrounding pins, an intermittent multiple trapping regime where the trap switches between capturing one or two vortices, and a strong coupling regime in which the trap permanently captures and drags two vortices. In some regimes we observe the counterintuitive behavior that slow moving traps couple less strongly to vortices than faster moving traps; however, the fastest moving traps are generally decoupled. The different phases can be characterized by the distances the vortices are displaced and the force fluctuations exerted on the trap. We find different types of stick-slip motion depending on whether vortices are moving into and out of pinning sites, undergoing exchange, or performing correlated motion induced by vortices outside of the trap. Our results are general to the manipulation of other types of particle-based systems interacting with periodic trap arrays, such as colloidal particles or certain types of frictional systems.

Figure 1

(a) Schematic of a superconducting slab containing a square array of artificial pinning sites (yellow) occupied by vortices (red arrows). The number of vortices produced by the magnetic field $\mathbf{B}$ applied perpendicular to the sample plane matches the number of pinning sites. A magnetic force microscope (MFM) tip moves over the sample surface at velocity $v_{tr}$ and is represented by a finite range harmonic trap with a trapping force or strength that can be varied by adjusting the distance between the MFM tip and the sample. (b) Schematic of a $5\lambda \times 5\lambda$ subsection of the system. Open black circles are pinning sites, filled blue circles are the vortices, and the large red circle is the trap which is moving at an angle of $\theta ={30}^{\circ }$ relative to the $x$ axis symmetry direction of the pinning array as indicated by the red arrow.

Figure 2

Vortex positions (filled circles), pinning site locations (open circles), tip trajectory (magenta line), and vortex trajectories (green lines) in an $8\lambda \times 8\lambda$ portion of a system with $F_{tr}=1.0$ and $F_p=0.3$ where the trap moves at an angle of $\theta ={30}^{\circ }$ with respect to the $x$ axis of the pinning array. Red filled circles indicate vortices that were displaced a distance of at least a pin radius due to the motion of the trap. (a) The decoupled phase I at $v_{tr}=0.5$, marked A in Fig. 3, where all the vortices remain pinned. (b) The intermediate coupling phase II at $v_{tr}=0.2$, marked B in Fig. 3, where individual vortices travel a distance $2a$ with the trap before escaping and being replaced by a new trapped vortex. (c) The intermediate trapping phase III at $v_{tr}=0.02$, marked C in Fig. 3, where in addition to translations of the trapped vortex, vortices near but outside the trap move in exchange rings through neighboring pinning sites.

Figure 3

(a) Capture length $C_l$, the average distance a vortex is dragged by the trap, vs trap velocity $v_{tr}$ in the system from Fig. 2 with $F_{tr}=1.0$ and $\theta ={30}^{\circ }$. The points marked A, B, and C correspond to $v_{tr}$ values at which the images in Fig. 2 were obtained. (b) The total displacements $d$ of all the vortices over a time interval during which the trap translates by $D_x=300a$ vs $v_{tr}$. Above $v_{tr}=0.375$, we find the decoupled phase I in which the trap does not drag any vortices. In the intermediate coupling phase II for $0.19<v_{tr}<0.375$, an individual vortex can be dragged by the trap a distance of $2a$ before exchanging places with a pinned vortex. For $v_{tr}<0.19$, the system is in the intermediate trapping phase III where vortices can be dragged a distance of several lattice constants and additional vortices exchange positions among the sites close to the trap. For $v_{tr}<0.012$, vortices are able to escape more easily from the slow trap so $C_l$ drops while $d$ remains large.

Figure 4

(a) $C_l$ vs $v_{tr}$ and (b) $d$ vs $v_{tr}$ for the $\theta ={30}^{\circ }$ system with a decreased trap strength of $F_{tr}=0.5$. The motion is always in the decoupled phase I. (c) $C_l$ vs $v_{tr}$ and (d) $d$ vs $v_{tr}$ in the same system for a strong trap with $F_{tr}=1.8$, where dashed lines indicate the boundaries of phases II, III, IV, and V. In the intermittent multiple trapping phase IV, the trap intermittently captures two vortices, and in the strongly coupled phase V, the trap always captures two vortices. A reentrant window of phase III appears just above phase V.

Figure 5

Vortex positions (filled circles), pinning site locations (open circles), tip trajectory (magenta line), and vortex trajectories (green lines) in an $8\lambda \times 8\lambda$ portion of the $F_{tr}=1.8$ system in Figs. 4 and 4. Red filled circles indicate vortices that were displaced a distance of at least a pin radius due to the motion of the trap. (a) At $v_{tr}=0.02$ in phase V, the trap always contains two vortices and correlated ringlike exchanges of vortices occur in the surrounding regions. As the trap moves, the two trapped vortices dislodge three pinned vortices, and one of these vortices jumps into the empty pinning site immediately behind the moving trap. (b) At $v_{tr}=0.12$ in phase IV, the trap alternates between capturing one and two vortices. (c) Disordered flow in phase III at $v_{tr}=0.25$, with a much weaker perturbation of the surrounding pinned vortices. (d) At $v_{tr}=0.5$ in phase II, a vortex only travels a short distance with the trap before exchanging with a pinned vortex.

Figure 6

Heat map of the total displacements $d$ as a function of $F_{tr}$ vs $v_{tr}$ for driving at (a) $\theta ={30}^{\circ }$, (b) $\theta =0^{\circ }$, (c) $\theta ={15}^{\circ }$, and (d) $\theta ={45}^{\circ }$. Dashed lines are guides to the eye indicating the locations of the different phases: I (decoupled), II (intermediate coupling), III (intermediate trapping), IV (intermittent multiple trapping), and V (strongly coupled).

Figure 7

$d$ vs $v_{tr}$ at $F_{tr}=1.8$. Dashed lines indicate intervals corresponding to the distance $300a$ traveled by the trap during the simulation. (a) At $\theta =0^{\circ }$, there is no II-I transition, but the jumps in $d$ for $v_{tr}<0.075$ indicate the location of the V-IV transition, while the drop in $d$ near $v_{tr}=0.1125$ corresponds to the IV-III transition. (b) At $\theta ={45}^{\circ }$, a clear drop in $d$ occurs at the IV-III transition near $v_{tr}=0.25$. Another drop in $d$ appears at small $v_{tr}$ where the system is in region IV but jumps to region V as the trap velocity increases.

Figure 8

Vortex positions (filled circles), pinning site locations (open circles), tip trajectory (magenta line), and vortex trajectories (green lines) in an $8\lambda \times 8\lambda$ portion of a sample with $F_{tr}=1.8$ and (a)–(c) $\theta =0^{\circ }$. (a) Phase II at $v_{tr}=0.35$, where there is little distortion of the background. (b) Phase III at $v_{tr}=0.1$, where the amount of distortion of the surrounding vortices has increased. (c) Phase IV at $v_{tr}=0.02$, where the multiple dragged vortices induce plastic motion in the surrounding vortices. (d) The phase II motion at $v_{tr}=0.35$ in a sample with $\theta ={45}^{\circ }$.

Figure 9

(a) The shape $U\left(r\right)$ of the pinning potential for parabolic (blue) and Gaussian (red) pinning sites. (b) The corresponding pinning force $F_p\left(r\right)$ showing a sharp cutoff for the parabolic pins and a smooth cutoff for the Gaussian pins.

Figure 10

Heat map of the total displacements $d$ as a function of $F_{tr}$ vs $v_{tr}$ for the system in Fig. 6 with driving at $\theta ={30}^{\circ }$ where the parabolic pinning has been replaced by Gaussian pinning. Dashed lines are guides to the eye indicating the locations of the different phases: I (decoupled), II (intermediate coupling), III (intermediate trapping), IV (intermittent multiple trapping), and V (strongly coupled). Compared to Fig. 6, some small changes in the locations of phases I to V appear, but both the smooth and the parabolic pinning sites exhibit the same dynamic phases.

Figure 11

(a) A representative plot of the time series of the $x$ direction forces $f_x$ experienced by the moving trap in phase I at $\theta ={30}^{\circ }, F_{tr}=1.0$, and $v_{tr}=0.5$. (b) The corresponding distribution function $P\left(f_x\right)$. The time intervals when the trap does not contain a vortex produce the peak at $f_x=0$.

Figure 12

(a) A representative segment of $f_x\left(t\right)$ in phase II at $v_{tr}=0.2$ for a sample with $\theta ={30}^{\circ }$ and $F_{tr}=1.0$. (b) The corresponding $P\left(f_x\right)$. There is no peak at $f_x=0$ since the trap always contains a vortex. (c) $f_x\left(t\right)$ in the same sample in phase III at $v_{tr}=0.05$. (d) The corresponding $P\left(f_x\right)$ contains additional peaks produced by additional modes of motion.

Figure 13

(a) $f_x\left(t\right)$ for a sample with $\theta ={30}^{\circ }$ and $F_{tr}=1.8$ at $v_{tr}=0.12$ in phase IV. The time series has a telegraph noise characteristic in which the two values are produced when the trap alternates between dragging one (higher $f_x$) or two (lower $f_x$) vortices. (b) In phase V at $v_{tr}=0.02$, the trap always captures two vortices and the telegraph noise is lost. (c) In phase IV at $v_{tr}=0.048$, there is a transient signature when the trap initially drags one vortex but then captures a second vortex, producing a clearly visible jump in $f_x$.

Figure 14

(a) $f_x\left(t\right)$ in phase I for a sample with $\theta =0^{\circ }, F_{tr}=1.0$, and $v_{tr}=0.3$. (b) The corresponding $P\left(f_x\right)$ indicates that the fluctuations are periodic.