Pattern formation in vortex matter with pinning and frustrated intervortex interactions

We investigate the effects related to vortex-core deformations when vortices approach each other. As a result of these vortex-core deformations, the vortex-vortex interaction effectively acquires an attractive component leading to a variety of vortex patterns typical for systems with nonmonotonic repulsive-attractive interaction, such as stripes and labyrinths. The core deformations are anisotropic and can induce frustration in the vortex-vortex interaction. In turn, this frustration has an impact on the resulting vortex patterns, which are analyzed in the presence of additional random pinning, as a function of the pinning strength. This analysis can be applicable to vortices in multiband superconductors or to vortices in Bose-Einstein condensates.

Figure 1

Schematic plot of the model vortex-vortex interaction. At large distances, vortices characterized by unperturbed circular-symmetric cores (as shown in the right-hand lower inset) are repelled by the repulsive intervortex interaction force (red arrow in the inset). Eventual vortex-core deformations at short distances (as shown in the left-hand lower inset) give rise to the appearance of effectively multiscale vortex-vortex interaction which, in turn, leads to an additional attractive term in the vortex-vortex interaction. The degree of the core deformation and the corresponding strength of the vortex-vortex attraction force is controlled by the parameter $\beta$. In the plot, the curves are shown for $\beta =0$, 0.2, 0.4, 0.6, 0.8, 1.0, and 1.2, $\gamma =14$, and $r_0=0.7$. The left-hand upper inset shows the profile of the vortex-vortex interaction force in the $xy$ plane, for an anisotropic interaction, where the direction of the vector ${\widehat{\mathbf{r}}}_{ik}$ [see the notations of Eq. (3)] is shown by the black arrow. The right-hand upper inset illustrates the effect of frustration in the intervortex interaction when two close-neighbor vortices with elongated cores are equally separated from a given vortex.

Figure 2

Vortex patterns for $\beta =0.6$ in the absence of pinning and for varying number of vortices per simulation cell: $N_v=1600$ (a), $N_v=2500$ (b), $N_v=3025$ (c), $N_v=3600$ (d), $N_v=4225$ (e), and $N_v=5625$ (f).

Figure 3

Vortex patterns for $\beta =1.2$ in the absence of pinning and for varying number of vortices per simulation cell: $N_v=900$ (a), $N_v=2025$ (b), $N_v=3600$ (c), $N_v=4225$ (d), $N_v=5625$ (e), and $N_v=6400$ (f).

Figure 4

Vortex patterns for $\beta =0.6$ and for varying pinning strength and number of vortices per simulation cell: $f_p=0.3$, $N_v=4225$ (a), $f_p=0.6$, $N_v=4225$ (b), $f_p=0.9$, $N_v=4225$ (c), and $f_p=1.2$, $N_v=4900$ (d). Panels (e) and (f) show scanning Hall probe images ($\approx 50\times 50\mu {\mathrm{m}}^2$) of the vortex distribution in a 160 nm thick superconducting ${\mathrm{MgB}}_2$ film at $T=1.7$ K and magnetic fields of 1.7 Oe (e), 5 Oe (f) [38].

Figure 5

Vortex patterns for $\beta =0.6$ and $\beta =1.2$, and for varying pinning strength and number of vortices per simulation cell: $\beta =0.6$, $f_p=0$, $N_v=3600$ (a), $\beta =1.2$, $f_p=0$, $N_v=5625$ (b), $\beta =0.6$, $f_p=0.6$, $N_v=3600$ (c), and $\beta =1.2$, $f_p=0.6$, $N_v=5625$ (d). Panels (e) and (f) show field-cooled images of vortex patterns in ${\mathrm{MgB}}_2$ at 1 Oe (e) and 5 Oe (f) [15].

Figure 6

These diagrams show the various morphologies of the vortex patterns that are encountered when varying the intervortex attraction force strength and the vortex density $N_v$ (upper panel), and when varying the random pinning strength $f_p$ versus the vortex density $N_v$ (lower panel).

Figure 7

Vortex patterns in the case of frustrated vortex-vortex interaction for $\beta =0.6$ and for varying pinning strength and number of vortices per simulation cell: $f_p=0$, $N_v=3600$ (a), $f_p=0$, $N_v=4225$ (b), and $f_p=0.3$, $N_v=3600$ (c). Panel (d) shows scanning Hall probe image ($\approx 50\times 50\mu {\mathrm{m}}^2$) of the vortex distribution in a 160 nm thick superconducting ${\mathrm{MgB}}_2$ film at $T=1.7$ K and magnetic field of 2.8 Oe [38].