Figure 1

(Color online) The model: vortices driven by a Lorentz force produced by an applied current in a superconductor with a square array of pinning sites. The period of the pinning array is $a=2{\lambda }_0$. The pinning sites are shown by dark blue (dark gray) parabolic bars and by dark blue (dark gray) dots on the upper surface. The black dashed lines connecting the dots on the top surface are a guide to the eye. The vortices are shown by red-to-yellow (gray-to-light gray) tubes and by red (gray) larger spots on the top surface. The direction of the applied current $j$ is indicated by a wide light blue (light gray) arrow, and the Lorentz force acting on the vortices $F_d$ is shown by small red (gray) arrows. The direction of the external applied magnetic field $B$ is shown by the brown (gray) arrow.Reuse & Permissions

Figure 2

(Color online) The average vortex velocity $\overline{v}\propto E$ versus current $j$ for $B∕B_{\varphi }=1.074$, $r_p=0.21{\lambda }_0$ and $F_{p0}∕F_0=2$ for increasing [shown by red (black) solid circles] and for decreasing (blue open squares) $j$ (see Ref. 21). (a) No heating effect is taken into account. The regions corresponding to different phases are indicated by the roman numerals from I to V (as in Refs. 20, 21). (b) The sample heats up due to vortex motion. For small values of the drive $j$ (up to $j\sim 3$) the effect of heating is negligible. For $j\sim 3.2$, a jump from phase II to phase III occurs. As a result of the heating, a transition occurs abruptly at $j\sim 4.3$, from regime IV to regime V, where the vortex lattice is entirely unpinned and moves as a whole. A hysteresis now appears in region V: when $\mathit{decreasing}$ the current $j$ down to the value at which the jump from phase IV to phase V previously occurred, when the driving was increased, the overheated vortex lattice now keeps moving as a whole. As a result, we obtain a complicated $N$- and $S$-type VI curve. State $A$ is unstable and the sample divides into filaments in states $B$ and $C$. State $C$ is also unstable. The corresponding stable states are on the lower (point $E$) and on the upper (point $D$) VI-curve branches. Point $j\sim 4.5$ corresponds to the normal transition and at $T>T_c$ we have the usual Ohmic conductivity.Reuse & Permissions

Figure 3

(Color online) The average vortex velocity $\overline{v}\left(j\right)$ for $B∕B_{\varphi }=1.074$, $F_{p0}∕F_0=2.0$ for increasing [red (black) solid circles] and for decreasing (blue open squares) $j$ and different radii of the pinning sites: (a) $r_p=0.23{\lambda }_0$; (b) $r_p=0.20{\lambda }_0$; (c) $r_p=0.19{\lambda }_0$. The function $\overline{v}\left(j\right)$ only slightly changes for radii from $r_p=0.20{\lambda }_0–0.25{\lambda }_0$. For smaller $r_p$, phase IV in the reverse branch disappears (c) since the overheated vortex lattice cannot adjust itself to the pinning array.Reuse & Permissions

Figure 4

(Color online) The average vortex velocity $\overline{v}\left(j\right)$ for increasing [red (black) solid circles] and for decreasing (blue open squares) $j$ for large pinning sites radii: (a) $r_p=0.35{\lambda }_0$; (b) $r_p=0.45{\lambda }_0$. Other parameters are the same as in Fig. 2. An increase of $r_p$ increases the disorder in the system because vortices can then move inside the pinning sites. Phase III disappears in (a). However, the hysteresis related to the heating remains. For larger $r_p$ (b), phase II disappears since all the vortices are pinned.Reuse & Permissions

Figure 5

(Color online) The average vortex velocity $\overline{v}$ as a function of the driving current $j$ for $B∕B_{\varphi }=1.074$, $r_p=0.2{\lambda }_0$, and $F_{p0}∕F_0=2.0$ for increasing [red (black) solid circles] and for decreasing (blue open triangles) $j$ for different amounts of disorder (displacement of the centers of the pinning sites from their regular positions) in the system: (a) $d_{\mathrm{ran}}=0.01(a∕2)$; (b) $d_{\mathrm{ran}}=0.05(a∕2)$; (c) $d_{\mathrm{ran}}=0.1(a∕2)$; (d) $d_{\mathrm{ran}}=0.2(a∕2)$; (e) $d_{\mathrm{ran}}=0.5(a∕2)$; (f) $d_{\mathrm{ran}}=a∕2$. The function $\overline{v}\left(j\right)$ does not appreciably change for a small amount of disorder (a). For $d_{\mathrm{ran}}>0.05(a∕2)$ (b), phase IV disappears in the reverse branch. For $d_{\mathrm{ran}}>0.1(a∕2)$ (c), phase IV is lost in both branches; only some reminiscent features remain, which disappear for larger $d_{\mathrm{ran}}$ (d,e). At maximal disorder $d_{\mathrm{ran}}=a∕2$ (f), only phases I, III, and V remain. However, there is a weak hysteresis related to heating [the inset to (f)].Reuse & Permissions

Figure 6

Electrical circuit and superconducting sample. The current filaments are shown schematically by light gray lines (Ref. 21); $l$ is the sample length, $w$ is the sample half-width, $\Delta w$ is the filament width, and the boundary conditions are stated at the sample edges $y=\pm w$.Reuse & Permissions

Figure 7

(Color online) Schematic VI curves [red (black) solid line and green (dark gray) dashed line] of the superconductor for two different values of the hysteresis due to overheating (Ref. 21). The more pronounced hysteresis loop [shown by the green (dark gray) dashed line] corresponds to a larger value of the characteristic heat release.Reuse & Permissions