Vortex states in nanoscale superconducting squares: The influence of quantum confinement

Bogoliubov–de Gennes theory is used to investigate the effect of the size of a superconducting square on the vortex states in the quantum confinement regime. When the superconducting coherence length is comparable to the Fermi wavelength, the shape resonances of the superconducting order parameter have strong influence on the vortex configuration. Several unconventional vortex states, including asymmetric ones, giant-multivortex combinations, and states comprising giant antivortices, were found as ground states and their stability was found to be very sensitive on the value of $k_F{\xi }_0$, the size of the sample $W$, and the magnetic flux $\Phi$. By increasing the temperature and/or enlarging the size of the sample, quantum confinement is suppressed and the conventional mesoscopic vortex states as predicted by the Ginzburg-Laudau (GL) theory are recovered. However, contrary to the GL results we found that the states containing symmetry-induced vortex-antivortex pairs are stable over the whole temperature range. It turns out that the inhomogeneous order parameter induced by quantum confinement favors vortex-antivortex molecules, as well as giant vortices with a rich structure in the vortex core—unattainable in the GL domain.

Figure 1

Error (solid line) and free energy (dashed line) as a function of iteration steps for sample II for $W/{\xi }_0=5$ at $\Phi /{\Phi }_0=5$ and $T/T_c=0$. Three insets show the initial, intermediate, and final vortex configurations. The solid dots and open diamonds in the insets indicate $L=1$ vortex and $L=-2$ giant antivortex, respectively.

Figure 2

Contour plots of the order parameter $\Delta (x,y)$ with the corresponding diagonal profiles in the absence of applied magnetic field. Panels (a)–(c) are for sample I with sizes $W/{\xi }_0=5,8,10$, respectively. Panels (d) and (e) are for sample II with sizes $5{\xi }_0$ and $10{\xi }_0$, respectively. Panel (f) corresponds to sample III with size $5{\xi }_0$.

Figure 3

Size dependence of the order parameter in the center of the sample $|{\Delta }_c|$ and the spatially averaged $|{\Delta }_m|$ for sample I.

Figure 4

Free energy as a function of the magnetic flux through sample I with size $10{\xi }_0\times 10{\xi }_0$. Different colors indicate the different winding numbers $L$ and the shaded area indicates the flux range over which the vortex state with winding number $L$ is the ground state. The vortex configurations of the ground state marked by solid squares are shown separately in Fig. 5 and the corresponding free energy curves are marked by dots. The inset zooms in the region where $L=5$ and 6 are the ground states.

Figure 5

Contour plot of the amplitude of the order parameter for the ground states marked by solid squares shown in Fig. 4. $L=10\left(\alpha \right)$ and 10$\left(\beta \right)$ are for flux $\Phi /{\Phi }_0=18.2$ and $18.7$, respectively. Darkest areas indicate the positions of vortices. Note that at the center an antivortex (for $L=7$) or a giant vortex [for $L=10\left(\beta \right)$] can spontaneously form. Note also that for some vorticities two different ground states are found (at different magnetic fields).

Figure 6

Phase diagram for samples I–III. Different shadings of the background indicate different types of vortex states. Black (blue) dots indicate vortices and diamonds indicate antivortices in the schematic diagrams of vortex configurations (bottom figures). The symbols are larger when vortices contain multiple flux quanta.

Figure 7

Free energy curves of states with $L=2$ in sample I with size $W/{\xi }_0=10$ (top panel) and the corresponding contour plots of the order parameter (bottom panel). Note that the contour plot of the ground state $L=2\left(e\right)$ is shown in Fig. 5. The inset shows the energy difference between the ground state $L=2\left(e\right)$ and the metastable states $L=2\left(f\right)$ and $L=2\left(g\right)$ at higher field.

Figure 8

Free energy curves of states with $L=3$ in sample I with size $W/{\xi }_0=10$ and the corresponding contour plots of the order parameter. The contour plot of the ground state $L=3\left(g\right)$ is shown in Fig. 5.

Figure 9

Free energy curves of vortex states with $L=4$ in sample I with size $W/{\xi }_0=10$ and the contour plots of the order parameter of selected vortex states. Inset shows details of the free energy curve around $\Phi /{\Phi }_0=6.5$. The contour plot of ground state $L=4\left(k\right)$ is shown in Fig. 5.

Figure 10

Free energy as a function of the magnetic flux through the square sample I for size $W/{\xi }_0=10$ and $T/T_c=0.6$.

Figure 11

The diagonal profile of the order parameter for sample I with size $W/{\xi }_0=10$ for $\Phi /{\Phi }_0=4$ and $L=1$ at different temperatures. The corresponding contour plots of the order parameter at $T/T_c=0$ and $T/T_c=0.6$ are also shown.

Figure 12

Temperature dependence of ${({\xi }_0/{\xi }_1)}^2$ for sample I with size $W/{\xi }_0=10$ and $\Phi /{\Phi }_0=4$.

Figure 13

The contour plots of the order parameter for sample I with size $W/{\xi }_0=10$ at $T/T_c=0.6$ for (a) $L=2$ and (b) $L=3$.

Figure 14

Temperature-flux phase diagram for sample I with size $W/{\xi }_0=5$. The vortex configurations of areas (i)–(iii) are shown as insets in the upper right corner.

Figure 15

Diagonal profile of the order parameter for different vortex states in the GL theory. Blue thick curves represent the order parameter. All the phases have been adjusted such that on the diagonal the order parameter is real. The horizontal line indicates the zero of the order parameter and a vortex appears when it intersects the order parameter. Panels (a) and (c) represent multivortex states. Panels (b) and (d) show giant-vortex states, and (e) shows a vortex-antivortex configuration.

Figure 16

Similar to Fig. 15 but now for the BdG theory. Panels (a) and (b) show the symmetric and asymmetric $L=1$ vortex states, respectively. Panels (c) and (d) show the giant-vortex and vortex–giant-antivortex $L=2$ states, respectively. Panels (e) and (f) show the giant-vortex and vortex-antivortex $L=3$ configurations, respectively.

Figure 17

Vortex state for sample III with $W/{\xi }_0=7$ at $\Phi /{\Phi }_0=20$ with winding number $L=12$. Panels (a) and (b) show contour plots of the order parameter and its phase, respectively. Panel (c) shows schematically the vortex configuration. Dark (blue) dots and open diamonds indicate vortices and antivortices, respectively.