Skyrmionic configuration and half-quantum vortex-antivortex pair in mesoscopic $p$-wave superconducting noncircular systems

In the framework of the microscopic Bogoliubov-de Gennes theory, we study the topological properties of skyrmionic states under applied magnetic flux in mesoscopic symmetric and asymmetric $p$-wave superconducting noncircular systems. For a perfect square (rectangular) sample, an enclosed square-loop-like (parallelogram-loop-like) chain comprising four spatially separated one-component vortices emerges for a single-skyrmion state with the topological charge $Q=2$. A multiskyrmion state containing two concentric skyrmions can be found in the square case, and more complex skyrmionic structures with $Q>2$ take place when the superconducting pairing interaction becomes stronger. By contrast, the mesoscopic rectangular geometry favors different arrangements of skyrmions. A novel type of multiskyrmion states with two or three separated skyrmions aligning along the long side of the rectangle becomes stable, accompanied with the half-quantum vortex-antivortex (V-Av) pair in one component of the order parameter. Moreover, for the square loops with a small square hole, the singly quantized vortex always traps in the centered or off-centered hole, while the coreless skyrmion located in the superconducting region tries to restore the centrally symmetric character. For some critical displacement of the off-centered hole, the half-quantum V-Av pair can occur in such asymmetric system.

Figure 1

Spatial profiles of the charge current $J_c$ for the vortex-free state (a), the singly quantized vortex state (b), the single-skyrmion state (c), the skyrmion-vortex coexisting state (d), and the multiskyrmion state (e) in a mesoscopic $p$-wave superconducting square with $N_x\times N_y=32\times 32$ when the pairing interaction $V=1$ and the temperature $T=0$. The red and blue colors indicate the current flowing in opposite directions. The current changes sign near the sites with white color.

Figure 2

Spatial profiles of the zero-energy local density of states $\rho \left(0\right)$ for the vortex-free state (a), the singly quantized vortex state (b), the single-skyrmion state (c), the skyrmion-vortex coexisting state (d), and the multiskyrmion state (e) in a mesoscopic $p$-wave superconducting square with $N_x\times N_y=32\times 32$. The zero-energy peaks mainly emerge around the vortex core and near the domain walls, indicating the occurrence of vortex and domain-wall bound states.

Figure 3

Spatial profiles of a singly quantized vortex state in a mesoscopic $p$-wave superconducting square with $N_x\times N_y=32\times 32$ when $\mathrm{\Phi }/{\mathrm{\Phi }}_0=5.5$. Displayed quantities are the amplitude [(a) and (b)] and phase $\varphi$ [(e) and (f)] for the $p$-wave components ${\mathrm{\Delta }}_x$ and ${\mathrm{\Delta }}_y$, the amplitude of the pairing potential ${\mathrm{\Delta }}_{+}$ and ${\mathrm{\Delta }}_{-}$ [(c) and (d)], the relative phase ${\varphi }_{xy}=\cos ({\varphi }_x-{\varphi }_y)$ between ${\mathrm{\Delta }}_x$ and ${\mathrm{\Delta }}_y$ (g), and the total order-parameter amplitude $\left|\mathrm{\Delta }\right|=\sqrt{|{\mathrm{\Delta }}_{+}{|}^2+{\left|{\mathrm{\Delta }}_{-}\right|}^2}$ (h). Note that the winding numbers of ${\mathrm{\Delta }}_x$ and ${\mathrm{\Delta }}_y$ are $L_x=L_y=1$, and the vortex cores in ${\mathrm{\Delta }}_x$ and ${\mathrm{\Delta }}_y$ overlap at the sample center.

Figure 4

Spatial profiles of a single-skyrmion state carrying two flux quanta in a mesoscopic $p$-wave superconducting square with $N_x\times N_y=32\times 32$ when $\mathrm{\Phi }/{\mathrm{\Phi }}_0=6$. Displayed quantities are the same as in Fig. 3. ${\mathrm{\Delta }}_x$ and ${\mathrm{\Delta }}_y$ components contain two half-quantum vortices each, while all four vortex cores are spatially separated on the domain wall, leading to a coreless vortex state carrying two flux quanta with the topological charge $Q=2$.

Figure 5

Spatial profiles of a multiskyrmion state with two $Q=2$ skyrmions in a mesoscopic $p$-wave superconducting square with $N_x\times N_y=32\times 32$ when $\mathrm{\Phi }/{\mathrm{\Phi }}_0=8.5$. Displayed quantities are the same as in Fig. 3. ${\mathrm{\Delta }}_x$ and ${\mathrm{\Delta }}_y$ have the winding numbers $L_x=L_y=4$, and a circular-loop-like domain wall consisting of four outer separated half-quantum vortices as well as a square-loop-like domain wall consisting of four inner separated half-quantum vortices are formed together.

Figure 6

Spatial profiles of a single-skyrmion state carrying five flux quanta in a mesoscopic $p$-wave superconducting square when the pairing interaction $V=1.5$ and $\mathrm{\Phi }/{\mathrm{\Phi }}_0=9.2$ at zero temperature. Displayed quantities are the same as in Fig. 3. Note that $L_x=L_y=5$, and ten separated half-quantum vortices form a coreless skyrmion state with the topological charge $Q=5$.

Figure 7

Spatial profiles of a single-skyrmion state carrying two flux quanta in a mesoscopic $p$-wave superconducting rectangle with $N_x\times N_y=24\times 40$ when $\mathrm{\Phi }/{\mathrm{\Phi }}_0=5.5$. Displayed quantities are the same as in Fig. 4, except (h) shows the zero-energy local density of states $\rho \left(0\right)$. Note that the structure of the domain wall is deformed by the rectangle boundary and an enclosed parallelogram-loop-like wall is formed.

Figure 8

Spatial profiles of a multivortex state carrying two flux quanta in a mesoscopic $p$-wave superconducting rectangle with $N_x\times N_y=24\times 40$ when $\mathrm{\Phi }/{\mathrm{\Phi }}_0=7.5$. Displayed quantities are the same as in Fig. 7. The winding numbers of ${\mathrm{\Delta }}_x$ and ${\mathrm{\Delta }}_y$ are $L_x=L_y=2$, while the vortex cores in ${\mathrm{\Delta }}_x$ and ${\mathrm{\Delta }}_y$ can overlap, resulting in two pointlike topological defects.

Figure 9

Spatial profiles of another type of multiskyrmion states with two skyrmions as compared to the one in Fig. 5 in a mesoscopic $p$-wave superconducting rectangle with $N_x\times N_y=24\times 40$ when $\mathrm{\Phi }/{\mathrm{\Phi }}_0=8.5$. Displayed quantities are the same as in Fig. 7. Note that a half-quantum vortex-antivortex pairing pattern with a centered antivortex (with phase difference $-2\pi$) surrounded by four ordinary vortices takes place in ${\mathrm{\Delta }}_x$.

Figure 10

Spatial profiles of a multiskyrmion state with three skyrmions in a mesoscopic $p$-wave superconducting rectangle with $N_x\times N_y=24\times 40$ when $\mathrm{\Phi }/{\mathrm{\Phi }}_0=9.5$. Displayed quantities are the same as in Fig. 7. Note that two antivortices emerge near the sample center and align orthogonally compared to the other six ordinary vortices in the ${\mathrm{\Delta }}_x$ component.

Figure 11

Spatial profiles of a single-skyrmion state in a mesoscopic $p$-wave superconducting square with $N_x\times N_y=32\times 32$ and a centered $4\times 4$ hole when $\mathrm{\Phi }/{\mathrm{\Phi }}_0=7.5$. Displayed quantities are the same as in Fig. 7. Note that a singly quantized vortex is trapped in the hole.

Figure 12

Spatial profiles of an asymmetric skyrmion state in a mesoscopic $p$-wave superconducting square with $N_x\times N_y=32\times 32$ and a centered $6\times 6$ hole when $\mathrm{\Phi }/{\mathrm{\Phi }}_0=7.75$. Displayed quantities are the same as in Fig. 11. In contrast to the case in Fig. 11, the symmetric skyrmion configuration is broken for a sufficient large hole size, while the skyrmion character keeps topological stable around the hole.

Figure 13

Spatial profiles of a single-skyrmion state in a mesoscopic $p$-wave superconducting square with $N_x\times N_y=32\times 32$ and an off-centered distance $d=2$ of the $4\times 4$ hole when $\mathrm{\Phi }/{\mathrm{\Phi }}_0=7.75$. Displayed quantities are the same as in Fig. 11. A singular vortex still sits in the off-centered hole, while the coreless skyrmion tries to restore the centrally symmetric feature.

Figure 14

Spatial profiles of a single-skyrmion state in a mesoscopic $p$-wave superconducting square with $N_x\times N_y=32\times 32$ and an off-centered distance $d=9$ of the $4\times 4$ hole when $\mathrm{\Phi }/{\mathrm{\Phi }}_0=6.5$. Displayed quantities are the same as in Fig. 11. Note that a single skyrmion appears around the square center, accompanied with the half-quantum vortex-antivortex pair in the ${\mathrm{\Delta }}_x$ component.