Vortex dynamics for low-$\kappa$ type-II superconductors

Based on a model appropriate for “marginally type-II” superconducting system, we study the dynamics of vortices with competing interactions by Langevin dynamics simulation. In addition to pinned states and plastic flow, we find that the moving vortex system forms ordered bubble configurations and stripe structures, depending on pinning strength and driving force. The vortex system exhibits a marked hysteresis in its velocity-force characteristic, which results from a dynamical stripe reordering due to intervortex attraction.

Figure 1

Effect of density of pinning centers (black open circles) $n$${}_{\mathrm{p}}$ on the vortex (magenta solid circles) phase at $f_{\mathrm{pv}}=5f_0$ without an external driving force. The top row shows simulation results for $B=0.5B_{\mathrm{c}}2$ with increasing ratio $n_{\mathrm{p}}$ of number pinning centers to number of vortices: (a) $n_{\mathrm{p}}=0.1$, (b) $n_{\mathrm{p}}=0.2$, (c) $n_{\mathrm{p}}=0.5$, and (d) $n_{\mathrm{p}}=1$. The vortices for $n_{\mathrm{p}}=0$ form an ordered bubble state. The bottom row shows simulation results for $B=0.65B_{\mathrm{c}}20$: (e) $n_{\mathrm{p}}=0.1$, (f) $n_{\mathrm{p}}=0.2$, (g) $n_{\mathrm{p}}=0.5$, and (h) $n_{\mathrm{p}}=1$. The vortices for $n_{\mathrm{p}}=0$ form an ordered stripe state.

Figure 2

Vortex configurations for different driving forces at $B=0.65B_{\mathrm{c}}2$ and $f_{\mathrm{pv}}=28.2f_0$: (a) $F^L=0$, (b) $F^L=20f_0$, (c) $F^L=40f_0$, and (d) $F^L=100f_0$. Solid circles and open circles are, respectively, the vortices and the pinning centers.

Figure 3

Dynamic phase diagram for different pinning strengths and driving forces at $B=0.65B_{\mathrm{c}}2$. The typical structures of these phases are shown in Figs. 2a, 2b, 2c, 2d.

Figure 4

(a) Velocity-force curve for a fixed force scanning rate $d{F}^L/dt=\pm 0.05(f_0/t_0)$, the density of pinning centers (black open circles) $n_{\mathrm{p}}=1$, $f_{\mathrm{pv}}=10$, and $B=0.2B_{\mathrm{c}}2$. The arrows indicate the evolution of the driving force. The inset is the velocity-force curve for the case of pure repulsion ($q=0$), suggesting that the hysteresis arises due to intervortex attraction. (b) The vortex configuration (magenta solid circles) corresponding to labels A in the velocity-force curve shown in (a) at $F^{\mathrm{L}}=8f_0$, and (c) the corresponding snapshot for label B (also at $F^{\mathrm{L}}=8f_0$).