Peak Effect in the Critical Current of Type II Superconductors with Strong Magnetic Vortex Pinning

We perform 2D Langevin simulations studying the peak effect (PE) of the critical current taking into account the temperature dependence of the competing forces. We observe and report that the PE results from the competition of vortex-vortex interactions and vortex-pin interactions which have different temperature dependencies. The simulations reveal that the PE can take place only for certain pinning strengths, densities of pinning centers, and driving forces, which is in good agreement with experiments. No apparent vortex order-disorder transition is observed across the PE regime. In addition, the PE is a dynamical phenomenon, and thermal fluctuations can speed up the process for the formation of the PE.

Figure 1

(a) The dip effect in the velocity $V_x$ (solid triangles) and the number of pinned vortices $N_v^p/N_v$ (open circles) as a function of temperature. Inset: Schematic diagram of elastic force $f_{vv}(T)$ and pinning force $f_{pv}(T)$ as a function of temperature. (b)–(e) Vortex (solid circles) distribution at temperatures marked in (a). (b) $T=T_1$, where almost all vortices are unpinned. (c) $T=T_{ps}$, where almost all of the vortices are pinned, and each pin (open circles) can trap several vortices, shown in the enlarged graph. (d) $T_{pe}>T\ge T_2$, where part of the pinned vortices are unpinned and the vortex channels are formed. (e) $T=T_3$, where most of vortices are unpinned and only a smaller number of vortices are trapped, shown in the enlarged graph.

Figure 2

Effect of pinning strength $f_0$ on the PE at a fixed density of pinning centers. The Lorentz force $f_{pv0}$ varies from 4.5 to $90f_0$ (4.5, 6, 6.6, 7.5, 9, 10.5, 90)$f_0$. Inset: Effect of density of the pinning center on the PE at a fixed pinning strength and Lorentz force.

Figure 3

Effect of temperature ramping rates $d{T}_c/d{t}_0$ on the PE, $d{T}_c/d{t}_0$ ranging from $-0.002$ to $0.1T_c/t_0$ ($-0.002$, $-0.02$, $-0.04$, $-0.06$, $-0.08$, $-0.1$)$T_c/t_0$. Inset: Effect of the driving force on the PE.

Figure 4

$V_x-T/T_c$ characteristics for various thermal fluctuation forces. Inset: $V_x-T/T_c$ characteristics without the thermal fluctuations for different temperature ramping rates.