Vortices and magnetic impurities

Ginzburg-Landau vortices in superconductors attract or repel depending on whether the value of the coupling constant $\lambda$ is less than 1 or larger than 1. At critical coupling $\lambda =1$, it was previously observed that a strongly localized magnetic impurity behaves very similarly to a vortex. This remains true for axially symmetric configurations away from critical coupling. In particular, a delta function impurity of a suitable strength is related to a vortex configuration without impurity by singular gauge transformation. However, the interaction of vortices and impurities is more subtle and depends not only on the coupling constant $\lambda$ and the impurity strength, but also on how broad the impurity is. Furthermore, the interaction typically depends on the distance and may be attractive at short distances and repulsive at long distances. Numerical simulations confirm moduli space approximation results for the scattering of one and two vortices with an impurity. However, a double vortex will split up when scattering with an impurity, and the direction of the split depends on the sign of the impurity. Head-on collisions of a single vortex with different impurities away from critical coupling are also briefly discussed.

Figure 1

Vacuum solutions obtained by solving equations (15) with $\lambda =0.5$ (in blue), and $\lambda =1.5$ (in red), in the presence of the impurities $\sigma (r)=4e^{-r^2}$ (solid lines), $\sigma ((r)=-4e^{-r^2}$ (dashed lines), and $\sigma (r)=-8e^{-2r^2}$ (dotted lines). In the subfigures we display (a) profile function $\varphi (r)$, (b) profile function $a_{\theta }(r)$, (c) energy density $E(r)$, (d) magnetic field $B(r)$.

Figure 2

Vacuum solutions obtained by solving Eqs. (15) with three different values of $\lambda$ in the presence of the impurity $\sigma (r)=4e^{-r^2}$. In each subfigure, we display (a) profile function $\varphi (r)$, (b) profile function $a_{\theta }(r)$, (c) energy density $E(r)$, (d) magnetic field $B(r)$.

Figure 3

Vacuum profile functions $\varphi (r),a_{\theta }(r)$ for impurities $\sigma (r)=-4d{e}^{-d{r}^2}$ and coupling constants (a) $\lambda =0.5$, (b) $\lambda =1.0$, (c) $\lambda =1.5$. In each subfigure we plot an $N=1$ vortex profile function for that coupling constant as a solid black line. Note that for $a_{\theta }$ the profile function is shifted down to enable comparison with the impurity vacuum solutions.

Figure 4

Energy $E/\pi$ of profile function solutions obtained by solving Eqs. (15). In each subfigure we display energy against $c$ for impurities of the form $\sigma (r)=c{e}^{-r^2}$ with topological charge (a) $N=0$, and (b) $N=1$.

Figure 5

Energy difference $(E_{1,\sigma }-(E_{0,\sigma }+E_1))/\pi$ as a function of $c$ for impurities of the form $\sigma (r)=c{e}^{-r^2}$, and $\lambda =0.5$, 1.0, 1.5. Where the energy difference is positive, the impurity will repel a vortex, and where it is negative the impurity and vortex will attract.

Figure 6

We plot the critical values of $c$ at which the impurity changes from attracting to repelling a vortex against $d$ for impurities of the form $\sigma (r)=c{e}^{-d{r}^2}$, and $\lambda =0.5$, 1.5. The line $c=-4d$, where the impurities approach a delta function of strength $\alpha =1$ as $d$ increases, is shown as a dashed line for comparison.

Figure 7

The potential (28) for a vortex and impurity with $\lambda =0.5$, 1.0, 1.5 and impurities (a) $\sigma (r)=4e^{-r^2}$, (b) $\sigma (r)=-4e^{-r^2}$, and (c) $\sigma (r)=-16e^{-4r^2}$.

Figure 8

Plot of $|\varphi (x,0){|}^2$ against $x$ for one vortex at critical coupling placed at different positions, and two different impurities $\sigma$.

Figure 9

(a)–(c) Energy $E/\pi$ as a function of the separation $s$ between a single vortex and an impurity $\sigma$ for $\lambda =0.5$ (in blue) and $\lambda =1.5$ (in red). (d)–(f) We zoom into the figures (a)–(c) for $s\ge 2$ and compare them with the corresponding interaction potential (28) shifted by the energy of a well-separated vortex and impurity (shown in black).

Figure 10

Diagram to illustrate the definition of the scattering angle $\mathrm{\Theta }$ and impact parameter $b$ in our simulations.

Figure 11

(a) Scattering angle $\mathrm{\Theta }$ in radians against impact parameter $b$ for the scattering of a single vortex at critical coupling with impurities $\sigma (r)=e^{-r^2}$ (in red), and $\sigma (r)=-e^{-r^2}$ (in blue). The initial vortex speed is $v=0.3$. (b) We display the same data, but for $\sigma (r)=-e^{-r^2}$ we plot $-\mathrm{\Theta }$ for easier comparison with the scattering angle for $\sigma (r)=e^{-r^2}$.

Figure 12

Vortex trajectories during the scattering of a single vortex at critical coupling with impurities (a) $\sigma (r)=e^{-r^2}$ and (b) $\sigma (r)=-e^{-r^2}$ for the impact parameter values $b=0$, 1, 2, 3, 4, 5, 6, 7 and initial velocity $v=0.3$. The location of the impurity is indicated with a black dot. Movies for these trajectories will be available as Supplementary Material [32].

Figure 13

Snapshots of the energy density during the scattering of a single vortex at critical coupling with impurities (a)–(c): $\sigma (r)=e^{-r^2}$ and (d)–(f): $\sigma (r)=-e^{-r^2}$ for initial velocity $v=0.3$. The corresponding movies are available as Supplementary Material [32].

Figure 14

Snapshots of the energy density during the scattering of a ring-shaped 2-vortex at critical coupling with impurities (a)–(c): $\sigma (r)=e^{-r^2}$ and (d)–(f): $\sigma (r)=-e^{-r^2}$ for initial velocity $v=0.3$. The corresponding movies are available as Supplementary Material [32].

Figure 15

Scattering angle $\mathrm{\Theta }$ in radians against impact parameter $b$ for the scattering of two vortices at critical coupling in the presence of impurities $\sigma (r)=e^{-r^2}$ (in red) and $\sigma (r)=-e^{-r^2}$ (in blue).

Figure 16

Vortex trajectories during the scattering of two vortices at critical coupling in the presence of impurities (a) $\sigma (r)=e^{-r^2}$ and (b) $\sigma (r)=-e^{-r^2}$ for impact parameter values $b=0$, 1, 2, 3, 4 and initial velocity $v=0.3$. In (a), we also plot the trajectory for $b=2.54$ (where the scattering angle changes sign) as a black dashed line. The location of the impurity is indicated with a black dot, and the vortices are initially located at $(\pm 4,\pm b)$.

Figure 17

Snapshots of the energy density during the scattering of two $N=1$ vortices at critical coupling through impurities (a)–(c): $\sigma (r)=e^{-r^2}$ and (d)–(f): $\sigma (r)=-e^{-r^2}$ for initial velocity $v=0.3$.

Figure 18

Scattering trajectories for $\lambda \ne 1$. The potential energy $\frac{E}{\pi }$ is displayed in (a), (d) and (g). The corresponding vortex trajectories are displayed in the middle row for $\lambda =1.5$ and in the righthand row for $\lambda =0.5$. A detailed discussion can be found in the main text.