Effects of sample geometry on the dynamics and configurations of vortices in mesoscopic superconductors

A computational study of mixed states in mesoscopic type-II superconducting cylinders is presented. The dependence of transient behaviors and steady-state configurations on the value of the applied magnetic field is examined as are the effects of sample size and cross-sectional shape on vortex nucleation and penetration. As is well known, more vortices enter the sample as the sample size grows. It is also found that if a small indentation is made on the sample boundary, vortices can be made to enter the system one by one from the tip of the indentation. An efficient scheme to determine, for any applied field, the equilibrium vortex configuration in mesoscopic samples in a way that is not constrained by sample symmetries, is devised and demonstrated.

Figure 1

(Color online) Examples of the triangulations used in the computational studies. Top: a uniform mesh in a square sample. Bottom: a conforming Delaunay triangulation in an equilateral triangle.

Figure 2

The magnetization (◻) and the corresponding number of vortices (°) vs the value of the external magnetic fields for a sample having square cross section of area $200{\xi }^2$ and for $\kappa =2$.

Figure 3

(Color online) Steady-state vortex configurations for cylindrical samples with symmetric cross sections of area $200{\xi }^2$ and for $\kappa =2$.

Figure 4

(Color online) Steady-state vortex configurations in a rectangle (a) and a rectangular sample with the same area but having a concave indentation (b); $H_{\mathrm{ext}}=1.0$ and $\kappa =2$.

Figure 5

(Color online) Examples of the triangulations used for dented and unindented samples.

Figure 6

(Color online) Time history of the vortex nucleation and penetration process for an indented sample; the area is $346{\xi }^2$, $H_{\mathrm{ext}}=0.84$, and $\kappa =4$.

Figure 7

(Color online) Time history of the vortex nucleation and penetration process for an indented sample; the area is $346{\xi }^2$, $H_{\mathrm{ext}}=0.9$, and $\kappa =4$.

Figure 8

(Color online) The vortex nucleation and penetration process in a sample of area $1024{\xi }^2$ for $H_{\mathrm{ext}}=0.84$ and $\kappa =4$.

Figure 9

(Color online) Vortex nucleation and penetration for an indented square sample having area $346{\xi }^2$ and $\kappa =4$ for $H_{\mathrm{ext}}=1.0$. The location of indentation is chosen to break symmetry.

Figure 10

(Color online) Steady-state vortex configurations for a square sample of area $346{\xi }^2$ with $\kappa =4$. Left figures: configurations for an indented sample obtained using different values for the applied field. Right figures: configurations for an unindented sample obtained from the corresponding configurations in the left column using the applied field value $H_{\mathrm{ext}}=1.144$.

Figure 11

(Color online) The steady-state total Gibbs free energies $G$ of the steady-state configurations in the right figure of Fig. 10 vs the number of vortices $n_{\infty }$, in the configurations.

Figure 12

(Color online) Top: initial configuration determined using an indented sample and $H_{\mathrm{ext}}=1.0$. Bottom: equilibrium configuration for the unindented sample for $H_{\mathrm{ext}}=0.84$.