Rotational transition, domain formation, dislocations, and defects in vortex systems with combined sixfold and twelvefold anisotropic interactions

We introduce a phenomenological model for a pairwise repulsive interaction potential of vortices in a type-II superconductor, consisting of superimposed sixfold and twelvefold anisotropies. Using numerical simulations we study how the vortex lattice configuration varies as the magnitudes of the two anisotropic interaction terms change. A triangular lattice appears for all values, and rotates through ${30}^{\circ }$ as the ratio of the sixfold and twlevefold anisotropy amplitudes is varied, in agreement with experimental results. The transition causes the vortex lattice to split into domains that have rotated clockwise or counterclockwise, with grain boundaries that are “decorated” by dislocations consisting of fivefold and sevenfold coordinated vortices. We also find intradomain dislocations and defects, and characterize them in terms of their energy cost. We discuss how this model could be generalized to other particle-based systems with anisotropic interactions, such as colloids, and consider the limit of very large anisotropy where it is possible to create cluster crystal states.

Figure 1

Equipotential lines for the vortex-vortex interaction in Eq. (2) at different values of the anisotropy ratio: $\kappa =$ (a) $-6$, (b) $-4$, (c) $-2$, (d) 0, (e) 2, (f) 4, and (g) 6.

Figure 2

Coordination number $P_n$ vs temperature $F^T$ during an annealing process with $K_6=0.05$, $K_{12}=0.05$, and a system size of $L=144\lambda .$ We plot only $P_5$, $P_6$, and $P_7$. The temperature decreases during the anneal, meaning that the system moves from right to left along the curves. At $F^T=0$, we find $P_5=0.011$, $P_6=0.978$, and $P_7=0.011$.

Figure 3

Structure factor $S\left(\mathit{k}\right)$ plotted as a height field for (a) the isotropic case with $K_6=K_{12}=0$, (b) intermediate anisotropy with $K_6=0$ and $K_{12}=0.0125$, (c) large anisotropy with $K_6=-0.4$ and $K_{12}=0.1$, and (e) very large anisotropy with $K_6=-1.2$ and $K_{12}=0.2$. The dashed rectangle in (b) indicates the reciprocal space area shown in Figs. 4 to 4. Circle segments in (c) show the the expected radius ($0.71\times 2\pi /\lambda$) for a uniform triangular VL with 14 250 vortices and a system size of $180\lambda \times 180\lambda$. (d) Inverse transform of the innermost contribution (indicated by the circles) to the structure factor in panel (c). (f) Real-space vortex positions for the very large anisotropy system in (e), showing the formation of an ordered lattice of vortex clusters.

Figure 4

Vortex lattice rotation transition for $K_{12}=0.0125$. (a) Azimuthal dependence of the structure factor, averaged over all six ${60}^{\circ }$ segments, vs the anisotropy ratio $\kappa$. The azimuthal angle range is illustrated in panel (b). (b)–(h) $S\left(\mathit{k}\right)$ for select values of $\kappa$ for the region of reciprocal space indicated in Fig. 3. Here $\kappa =$ (b) $-6$, (c) $-4$, (d) $-2.25$, (e) 0, (f) 2.25, (g) 4, and (h) 6. The definition of the VL splitting angle $\mathrm{\Delta }\theta$ for $\kappa \le 0$ and $\kappa \ge 0$ is shown in panels (d) and (f), respectively, while the rotation angle $\theta$ is shown in panel (e).

Figure 5

(a) Peak position $\theta$ and (b) peak splitting $\mathrm{\Delta }\theta$, obtained from the structure factor, plotted vs anisotropy ratio $\kappa$ for different values of $K_{12}$. Lines indicate the locations of the minima in the interaction potential, Eq. (3). The inset in (a) shows the standard deviation $\sigma$ between the simulated value of $\theta$ and the actual location of the potential minimum as a function of $K_{12}$.

Figure 6

Structure factor peak position $\theta$ vs anisotropy ratio $\kappa$ for different system sizes at $K_{12}=0.0125$. The line corresponds to the interaction potential minimum ${\theta }_{\min }$ obtained from Eq. (3).

Figure 7

Edge dislocations observed in (a,c) the sixfold anisotropy regime ($\kappa =-5.25$) and (b,d) the twelveold anisotropy regime ($\kappa =0$). Delaunay triangulations (a,b) indicate the Burgers circuit (dashed black line), nonzero Burgers vector (black arrow), the extra lattice planes originating on the five-coordinated vortex (solid blue lines) and the direction of their bisector (dashed blue line). Five-, six- and seven-coordinated vortices are plotted as blue, white, and red circles, respectively. The corresponding heatmaps (c,d) show the deviation $\mathrm{\Delta }U/U_{\text{av}}$ of the energy $U$ of each vortex, based on its interactions with the surrounding VL, from the system wide average energy $U_{\text{av}}$. White lines indicate high symmetry directions used to determine the range of the dislocations.

Figure 8

Average number of defects occurring per simulation vs the location $|{\theta }_{\min }|$ of the minimum in the interaction potential. Double and triple defects correspond to combinations of two or three 5 to 7 dislocations, respectively.

Figure 9

Vacancy defects. Delaunay triangulations overlaid with heatmaps showing, respectively, the coordination and energy for each vortex. Twofold symmetric crushed: (a) $V_{2b}^{\left(12\right)}$, (b) $V_{2a}^{\left(6\right)}$, (c) $D_{2b}^{\left(6\right)}$. Twofold symmetric split: (d) $S{V}^{\left(12\right)}$, (e) $S{D}_a^{\left(6\right)}$, (f) $ST{r}_a^{\left(6\right)}$. Threefold symmetric: (g) $D_3^{\left(12\right)}$, (h) $T{r}_3^{\left(12\right)}$, (i) $T{e}_3^{\left(12\right)}$. White lines indicate high symmetry directions used to determine the range of vacancy defects.

Figure 10

Interstitial defects. Delaunay triangulations overlaid with heatmaps showing, respectively, the coordination and energy for each vortex. Twofold symmetric crushed: (a) $I_2^{\left(6\right)}$, (b) $I_{2d}^{\left(6\right)}$. Twofold symmetric split: (c) $S{I}_2^{\left(12\right)}$, (d) $SD{I}_2^{\left(12\right)}$, (e) $STr{I}_{2d}^{\left(12\right)}$. Threefold symmetric: (f) $I_3^{\left(12\right)}$. White lines indicate high symmetry directions used to determine the range of interstitial defects.

Figure 11

Range of the energy deviation of dislocations and defects, plotted as $\left|\mathrm{\Delta }U\right|$ vs $|\mathit{r}·\widehat{\mathit{e}}|$, a distance taken along the high symmetry direction indicated by white lines in Figs. 7, 9, and 10. In all cases lines are exponential fits to the data. (a) Edge dislocations. (b) Double vacancy defects. (c) Triple vacancy defects. (d) Interstitial defects.

Figure 12

Grain boundaries plotted as Delaunay triangulations for different values of the anisotropy ratio: (a) $\kappa =-1.75$, $\mathrm{\Delta }\theta =21.4^{\circ }$, (b) $\kappa =-2.00$, $\mathrm{\Delta }\theta =20.0^{\circ }$, and (c) $\kappa =-3.25$, $\mathrm{\Delta }\theta =11.9^{\circ }$. For simplicity, we plot only fivefold and sevenfold coordinated vortices along with their adjacent sixfold coordinated neighbors, and the VL orientation in the bulk is indicated by a single set of lattice planes. Vortices are colored according to their coordination number as in Fig. 7, and the plots are overlaid on a heatmap indicating the relative energy $U/U_{\text{av}}$ for each vortex. In panel (c), the white line indicates the location of the grain boundary.

Figure 13

Heatmap showing the frequency of ${\theta }_{5–7}$, the angle of the line connecting the fivefold and sevenfold coordinated vortices in grain boundary edge dislocations, as a function of $\kappa$. All values of ${\theta }_{5–7}$ are mapped into a single ${60}^{\circ }$ segment. The heavy white curve indicates ${\theta }_{\text{min}}$, and the vertical lines show the values of $\kappa$ corresponding to $\mathrm{\Sigma }=7$.