Stability of degenerate vortex states and multi-quanta confinement effects in a nanostructured superconductor with Kagome lattice of elongated antidots

In the present work, we directly visualize the multi-quanta cages (MQCs) consisting of the giant vortices pinned by the elongated antidots using low-temperature scanning Hall probe microscopy. The periodic but sufficiently isolated MQCs, observed at various magnetic fields, are in a good agreement with the simulated vortex states based on the time-dependent Ginzburg–Landau (tdGL) equations. Due to the competition between the interstitial vortices and the pinned giant vortices, the formation and collapse of the MQCs can be tuned by varying magnetic field. The experimental statistics of the interstitial vortices confined in the MQCs show that the interstitial vortex patterns become more disordered at higher magnetic fields. The stability of the degenerate vortex states and the multi-quanta confinement effects under an external current are also investigated by using the tdGL simulations. The splitting of the free energy of the degenerate vortex states indicates that applying external current can eliminate parts of the degenerate vortex states.


Introduction
The magnetic flux penetrates into type-II superconductors in the mixed state in the form of quantized vortex with flux Φ 0 [1]. The vortices in macroscopic pinning-free superconductors arrange themselves into triangular Abrikosov lattice due to the repulsive vortex-vortex (V-V) interactions [2]. However, in the presence of the interplay between the inter-vortex interactions and the confinement, vortex matter in mesoscopic superconductors with nanostructures can display complex and unique features [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17], which have attracted a lot of interest over the past decades. The vortices in mesoscopic superconductors (e.g. triangular, square, disk, etc.) can form vortex-antivortex configurations [3][4][5][6], vortex fusion and giant vortex states [9][10][11][12][13][14][15][16][17], which strongly depend on the shape and the size of the samples [18,19] (i.e., shape and finite-size effects). Moreover, the theory and experiments revealed that the interactions between vortices and pinning sites should also be considered [20][21][22][23][24][25]. For example, the vortex clusters and the giant vortices can be observed in the presence of the strong disorder [21] and the void structures [23]. In addition, very recent experiments have indicated that weak pinning has a clear impact on the vortex distributions in nanostructured superconductors [22,24,25]. The confinement of vortices, studied in these literatures, results from the screening current flowing along the sample edges [22], which is a continuous function with magnetic field. However, the vortex states in mesoscopic superconductors with quantized confinement are still quite far from being well understood.
The pinning sites can prevent the vortex motion and thus enhance the critical current, which is a great benefit for the applications of superconductors. The strong commensurability effect was generally observed in the dependence of the critical currents on the magnetic field. However, at high magnetic fields, the repulsive potentials of the pinning centers occupied by vortices lead to the excess vortices preferentially locating in the interstitial sites between the pinning centers [26][27][28]. The critical currents in this case are mainly determined by the confinement effects on the interstitial vortices at higher magnetic fields. However, the interstitial vortices are highly mobile and their confinement is much weaker than the pinning effect, which leads to a strong reduction of the critical current [29]. The vortex configurations rely on the competitions between two types of vortices, the pinned vortices and the interstitial vortices, which vary with the pinning landscapes. It has attracted considerable academic attention to explore the vortex configurations in nanostructured superconductors with periodic and nonuniform pinning sites [29][30][31][32][33][34][35][36][37][38][39][40][41][42] (square, honeycomb, etc). Although the quantized confinement effects can be observed in the superconducting films with antidots, the confinement areas are not isolated and the strong interactions between the interstitial vortices in different confinement areas are still observed.
In this work, we explore the multi-quanta confinement effects in a nanostructured superconducting film that is divided into periodic but sufficiently isolated hexagonal areas. The issue is motivated from the recent works, where some unique physical properties have been observed in superconductors with complex pinning landscapes, such as honeycomb and Kagome patterns [43][44][45]. The well-designed structures provide opportunities to explore complex but fascinating physical phenomena. Numerical simulations first predicted that the artificial square and Kagome vortex ices, analogous to spin ice, can be realized in nanostructured superconductors with double-well pinning sites [46]. The formation of the square vortex ice states was confirmed by the transport measurements and then was directly visualized by using scanning Hall probe microscope (SHPM) [47][48][49]. Very recently, we directly observed some specific patterns of a vortex ice system in a Kagome lattice of paired antidots: an additional degree of freedom leading to even more robust vortex ice states [50]. By reducing the Kagome lattice spacing we found that the ordered vortex ice states cannot be further stabilized when the inter-vortex interaction becomes stronger. Furthermore, our recent work demonstrated that the anisotropy of the pinned vortices and the anisotropy of a superconducting film with intentionally designed the Kagome lattice of elongated antidots results in a much more complex V-V interaction than in the conventional antidot-structures [51]. As a consequence, many distinct degenerate vortex states and large configuration entropy were observed using SHPM at several fractional matching fields and simple local rules were found to characterize the degenerate vortex states based on the experiments and numerical simulations. However, the stability of the degenerate vortex states under applied external currents and the competition between the elongated vortices pinned by the antidots and the interstitial vortices at higher magnetic fields has not been studied yet.
In the present paper, we investigate the attractive features of the sample with a Kagome lattice of elongated antidots to study the confinement of vortices. (i) The multi-quanta confinement results from the quantized giant vortices pinned by the elongated antidots. (ii) The superconducting film is divided into many sufficiently isolated hexagonal areas of micron-size and the vortices in a hexagonal area have little impact on the vortices in next one. (iii) One can study the stability of the degenerate vortex states and the confinement just by introducing current. As such, in this paper, we explore the confinement effects exerted on the interstitial vortices by the multi-quanta cages (MQCs) consisting of the elongated giant vortices. By varying magnetic field, the MQCs formation, collapse and competition between the MQCs and the interstitial vortices are observed using SHPM. Furthermore, the stability of different degenerate vortex states and the confinement effects of the MQCs under an external current are also studied based on the time-dependent Ginzburg-Landau (tdGL) simulations. Our numerical results demonstrate that the external current provides an effective way to eliminate parts of the degenerate vortex states due to the splitting of the free energy.

Experiments
As shown in figure 1(a), the Kagome lattice consisting of antidots has been created to explore the degenerate vortex states at different fractional matching fields in our recent work [51]. In the present paper, we directly probe the confinement effects at higher magnetic fields using SHPM in the same sample. Two types of vortices are observed simultaneously at higher magnetic fields, i.e., the quantized giant vortices pinned by the elongated antidots and the interstitial vortices confined in the hexagonal areas. In this case, the confinement of the interstitial vortices in the MQCs of giant elongated vortices are mainly determined by the competitions between such two types of vortices.
The Kagome lattice is prepared with the conventional electron-beam lithography. The size of each elongated antidot is 1 μm×3 μm (width×length). Six elongated antidots surround a hexagonal area with the edge of 4.73 μm and three nearest-neighboring antidots form a Y-shape at a vertex of the hexagon. An 85 nm-thick Pb film with the Kagome lattice of elongated antidots is deposited on a Si/SiO 2 substrate with ultra-high vacuum (3×10 −8 Torr) electron-beam evaporator calibrated with a quartz monitor. The background temperature for Pb deposition is 77 K provided by cooling with liquid nitrogen to ensure a homogeneous growth. A Ge layer with a thickness of 10 nm is also deposited on top of the Pb layer to prevent oxidation. The source materials are 99.999%-pure Pb and 99.9999%-pure Ge. The recent experiments showed that the coherence length ξ(0) of plane Pb film with the same thickness is 52 nm, and the magnetic penetration depth λ(0)=94 nm [52]. By taking into account the influence of the perforation [53], one can estimate the effective penetration length of the present sample with elongated antidots 0 0 where ν is the ratio of antidots area to total area.
In all our field cooling (FC) measurements, we first apply a magnetic field perpendicularly to the sample surface above the critical temperature T c (7.3 K), and then the temperature is reduced down to the working temperature T=4.25 K. Then the vortex patterns can be visualized directly by using low-temperature SHPM with a magnetic field resolution of 0.01 mT and a temperature stability better than 1 mK [52,54,55]. All the SHPM images of vortex patterns are obtained by lifting the Hall cross about 400 nm after approaching the sample surface by a scanning tunneling microscope tip assembled together with the Hall cross in our measurements.

Theoretical formalism
In order to understand the vortex configurations in the Kagome lattice observed at higher magnetic fields, we carried out numerical simulations by using the tdGL equations. Additionally, the tdGL equations also provide a useful tool to explore the stability of different degenerate vortex states and the confinement effects by calculating the critical currents. The tdGL equations can be written as [56][57][58], where A , y and J s are the order parameter, the vector potential and the supercurrent density, respectively. The physical quantities are normalized as follows. The distance is scaled by the coherence length ξ, time by ξ 2 /D where D is the diffusion coefficient, the order parameter by its equilibrium value in the absence of the magnetic field, the vector potential by c e 2  x, the magnetic field by H c x pl = [59]. To compare with the magnetic field profiles of the vortex patterns obtained in the SHPM experiments, the magnetic field can be derived by using the Biot-Savart law, where A 0 is the vector potential at a uniform magnetic field.
In order to compare with experimental results, the same elongated antidots size and Kagome lattice constant as the sample (see section 2) are used in the simulations. Considering the fact that our sample is sufficiently large Figure 1. (a) The AFM image of a nanostructured superconducting film with the Kagome lattice of elongated antidots [51]. The size of the Kagome lattice (blue dashed lines) is 4.10 μm. (b) Three-dimensional SHPM magnetic field profiles of vortex state observed in the field cooling measurement at the first matching field H 1 . and the scanning area in our FC measurements is far away from the sample edge, the sample is assumed to be infinite. As such, the periodic boundary conditions for ψ and A in both x and y directions are used in the simulations (see detail in [59]). Because the thickness of the sample is sufficiently small, the variations of the order parameter and the currents along the thickness can be neglected. To reproduce the vortex patterns observed in the FC experiments, we start the simulations from different randomly generated initial conditions. where H 1 is the first matching field. More and more vortices are trapped by the antidots with increasing magnetic field until each elongated antidot is occupied by one vortex at H 1 . As shown in threedimensional SHPM image of figure 1(b), the vortex configuration provides a vortex cage, which is expected to confine the interstitial vortices. In this subsection, we pay attention to the confinement effects in the Kagome lattice of elongated antidots. The upper panels of figure 2 show the SHPM images of vortex patterns observed in different FC measurements by varying magnetic field. Indeed, one can find that the vortex pattern forms a single-quantum cage (Φ 0 -cage, the vortex configuration with only one vortex pinned by each antidot) at 0.107 mT and each hexagonal cage can confine at most one interstitial vortex (see the left two panels in figure 2). The interstitial vortices are located at the center of the Φ 0 -cages since the repulsive interactions between the elongated vortices and the interstitial vortices.

Results and discussions
With increasing magnetic field, the incoming interstitial vortices can collapse the Φ 0 -cages. The 2Φ 0 -cages with two vortices pinned by each antidot (giant 2Φ 0 -vortex) are formed to confine more interstitial vortices. Figure 2 shows that each 2Φ 0 -cages can confine 2-5 interstitial vortices. The interstitial vortices act as vertices and form polygonal patterns (triangle, square, pentagon) in each 2Φ 0 -cage. Owing to the weak pinning nature of the superconducting film, the confined vortices cannot form ideal configurations. In order to compare with the experimental results, we add random weak pinning sites in the simulations. One can see that the vortex ground states in the presence of weak pinning sites based on the tdGL simulations (lower panels) are in a good agreement with the experimental vortex configurations (upper panels). At even higher magnetic fields, figure 3 shows that each elongated antidot is occupied by a giant 3Φ 0 -vortex to form 3Φ 0 -cages. Both experimental and theoretical results indicate that such 3Φ 0 -cages can confine 7, 8, 9 and even more vortices.
In order to understand the formation and collapse of the MQCs, we calculated the free energy of the system based on the tdGL simulations in the absence of pinning sites. Note that the difference between the free energy with and without random pinning sites is less than 2% and is not shown in the figure. As shown in figure 4(a), the total free energy of the system (G t ) keeps increasing with magnetic field H a , and it is mainly contributed by the condensation energy (G p ). Nevertheless, the magnetic energy G h is not monotonous with magnetic field. One can see that G h increases during the formation of the MQCs with increasing magnetic field. However, it keeps decreasing when MQCs confine the interstitial vortices. Moreover, the magnetic energy decreases rapidly at the beginning of MQCs confining vortices, and it decreases more and more slowly until the MQCs are collapsed by the interstitial vortices.
Due to the weak pinning sites, the experimental and simulated vortex patterns show that the interstitial vortices become more disordered at higher magnetic fields than that at low magnetic fields (compare figure 2 with figure 3). Thus we make the statistics of the interstitial vortex system at various magnetic fields. Figure 4(b) shows the experimental distances of the nearest-neighbor interstitial vortices and the average distance (d ). For comparison, the nearest-neighbor interstitial vortices are identified based on the tdGL simulations. For example, there are 5 pairs of nearest-neighbor interstitial vortices (end-to-end) in each hexagon at H a =0.392 mT because simulations show that the interstitial vortices form a pentagon in each cage without pinning sites. In order to see the order-disorder of interstitial vortices confined in cages, figure 4(b) where N is the counts of the nearest-neighbor distances) at various fields. Notwithstanding the disordered interstitial vortex patterns at high fields, one can see that d in the experiments agrees quite well with the theoretical results based on the tdGL simulations. Both the experimental and simulated results illustrate that the vortex distance at H a =0.285 mT is even smaller than that at H a =0.321 mT. This can be explained by the V-V interactions. Regardless of the elongated vortices, each interstitial vortex is subjected to the repulsive forces from other two interstitial vortices at 0.321 mT. The total repulsive force on  each interstitial vortex becomes 3 times stronger than that at 0.285 mT. As a result, the vortex distance will become larger to keep balance in the vortex cage.
The normalized variance of distances reflects the disorder of the interstitial vortices confined in the cages. One can see from figure 4(b) that the normalized variance of distances at high fields is much larger than that at low fields, which indicates that the interstitial vortex configurations are relatively ordered at low magnetic fields but they become quite disordered at high fields. For example, as shown in vortex configurations at H a =0.356 mT and 0.606 mT (points 1 and 2), the upper and lower panels of figure 4(c) show the histograms of the nearestneighbor distances and the normal fittings of the two vortex states, respectively. One can see that most of the nearest-neighboring distances of the interstitial vortices observed at 0.356 mT are distributed around the average value d (point 1, upper panel). In contrast, one can see a much wider distribution with additional peaks for the disordered vortex state at H a =0.606 mT (point 2, lower panel). As a matter of fact, the similar order-disorder phenomenon has also been observed in other vortex systems [60,61] and it has been recognised that the random environments are usually unaviodable in physical systems.
The vortex states are very important to explore the properties of superconductors with the pinning sites, such as the critical current (see next subsection). As shown above, the vortex states in the Kagome lattice of elongated antidots at certain magnetic fields strongly depend on the competition of the elongated vortices and the interstitial vortices. Figure 5(a) shows the MQCs formation/collapse regions obtained by varying magnetic field. In order to make the results clear, it is divided into 6 regions based on the magnetic field dependence of the interstitial vortices and the pinned vortices at elongated antidots. The average number of the elongated vortices per antidots (N p ) and the interstitial vortices per cage (N c ) increase separately with magnetic field in regions I and II, respectively, which indicates that all vortices first form the complete Φ 0 -cages, and then Φ 0 -cages confine the interstitial vortices (Panels A and B). With increasing magnetic field, N p and N c increase at the same time in region III until the 2Φ 0 -cages are formed completely at point C. As shown in the inset, it is observed that N c increases rapidly to 2 around N p =4/3. This is because the appearance of much stronger interaction between two nearest-neighboring giant elongated vortices at the hexagonal vertices prevents the rapid formations of the elongated vortices. Region IV shows that 2Φ 0 -cages are powerful enough to confine 2-5 vortices and the decreasing rate of the magnetic energy is nearly zero at H a =0.392 mT (see figure 4(a)). N p and N c in region V increase linearly with magnetic field simultaneously, which is in contrast to the nonlinear increase in region III. Meanwhile, the magnetic energy G h increases much more slowly than that in region III. In our measurements, the SHPM images of the vortex configurations at even higher magnetic fields show that the 3Φ 0 -cages can confine at most 11 interstitial vortices. The columnar bars with different colors in the inset show the maximum confinement ability of the MQCs. The panels of figure 5(b) represent three-dimensional magnetic field profiles of the vortex states in a Φ 0 -cage, 2Φ 0 -cage and 3Φ 0 -cage, respectively.

Stability of degenerate vortex states and confinement effects under external current
In this subsection, we first investigate the effect of an external current on the degenerate vortex states at several fractional matching fields. Then we explore the stability of the degenerate vortex states and the confinement effects in the Kagome lattice in the presence of the external current by using the tdGL simulations.
Due to the particular design of the Kagome lattice of elongated antidots, various degenerate vortex states, with nearly the same free energy, are observed in the FC measurements. As shown in figures 6(a) and 7(a), the vortex configurations are mainly ruled by the specific local constraints, i.e., all vertices comply with oneoccupied/two-empty at H 1 /3 (two-occupied/one-empty at 2H 1 /3); half vertices comply with the rule of oneoccupied/two-empty and the remaining vertices comply with two-occupied/one-empty at H 1 /2 [51]. The middle and right column panels show the distinct developments of the degenerate vortex states with external currents j ax /j 0 =0.1 and j ay /j 0 =0.1, respectively. It can be seen that the variation of the magnetic field profiles depends on the orientations of elongated antidots with respect to the external current. For example, in the vortex state S A at H 1 /3, the magnetic field profiles at the elongated antidots parallel to the current j ax are shifted from the center of antidots and asymmetrical magnetic dipoles can be observed at the empty antidots. In the case of j ay , the magnetic field profiles of vortices at antidots perpendicular to the currents are deformed and more or less like the magnetic field profiles of the pearl vortices. Additionally, some raindrop-like magnetic field profiles can be observed in other cases of external currents. One should note that the deformed magnetic field profiles are the combined effects of vortex current around antidots and external current crowding at antidots. The similar phenomena were also observed at topological defects due to the Meissner current or external current [62][63][64][65][66]. By comparing the evolutions of magnetic field of S A and S B , one can see that the equivalence of the unit cells of the observed ice-like states are broken by the external current (symmetry break) although the different vortexoccupations in unit cells contribute the same free energy to the vortex system in the case of no applied current. Figure 6(b) shows the evolutions of the free energy of the degenerate vortex states (S A, S B and S I , S II ) with increasing j ax and j ay . Although the free energy of different degenerate vortex states is the same in the absence of external currents, it is interesting to note that the free energy of the degenerate vortex states splits into two different curves by the external currents subjected along the y-axis. The difference value becomes larger and larger with increasing j ay . Such phenomenon is reminiscent of the Zeeman and Stark effects, in which the magnetic field or electric field can lead to the splitting of spectral lines. If the sample is subjected to the external current along the x-axis ( j ax ), however, it is surprising that the free energy of the vortex states is nearly the same with increasing currents. But it seems to be an accident since the splitting of the free energy can be observed in both cases of j ax and j ay at H 1 /2 (see figure 7(b)). Additionally, we should note that the critical depinning currents for two degenerate vortex states are different although j ay cannot lead to the free energy splitting. Since the vortices favor forming the states with the minimum free energy, the splitting of the free energy implies a possible effective method to remove parts of the degenerate vortex states by the external current during FC. As such, only parts of the degenerate vortex states can be observed under the external current due to the high energy barrier for the vortex state transition.    Figure 8 shows the developments of vortex patterns just before the vortex cages are collapsed under external currents j ax and j ay in the absence of pinning sites. One can see that the shape of the vortices pinned by the elongated antidots changes by the external currents. However, the interstitial vortices move close to the wall of the MQCs and rearrange themselves in the MQCs. For example, the external currents j ax and j ay cause the rotation of the polygons consisting of interstitial vortices in the 2Φ 0 -cages. Additionally, it is worth noting that the interstitial vortices are driven into the antidots under j ay rather than getting through the gap between two nearest-neighboring antidots at vertex.
The capsule vortices can be driven out from the antidots and the interstitial vortices break the MQCs when the Lorentz force is strong enough. Figures 9(a) and (b), respectively, show the critical currents j cx and j cy of the vortex states at various magnetic fields based on the tdGL simulations. The main feature is that both the critical currents j cx and j cy are divided into two regions, i.e., the critical depinning currents in region I at H a H 1 and the critical confining currents in region II at H a > H 1 . As mentioned above, the critical currents in region I arise from the pinning effect of the antidots while the the critical currents in region II are mainly determined by the confinement effects of the MQCs. One can see that the critical depinning currents at lower magnetic fields are much larger than the critical confining currents at higher magnetic fields, which indicates that the pinning effect is much stronger than the confinement effects. In region I, no significant matching effect can be observed because the V-V interactions are much weaker than the pinning effect of elongated antidots. It is interesting to note that the critical current at H a =0.285 mT is about twice larger than that at 0.143 mT. This can be explained by the fact that the confinement effects of the 2Φ 0 -cages is twice stronger than the Φ 0 -cages and there is no superposition of driving force on the interstitial vortices due to the parallel arrangement of two vortices to the external current (see fourth columnar panels in figure 8). As shown in figures 6 and 7, even for the same vortex state, the anisotropic Kagome lattice leads to different developments of the free energy by external currents j ax and j ay .
The tdGL simulations predict that the degenerate vortex states at several fractional matching fields have various developments of the free energy under the external currents j ax and j ay (figures 6 and 7). As shown in figure 9, different critical depinning currents should be seen due to the different degenerate vortex states after every quenching. Nevertheless, one may just observe one average value of the critical current in the experimental measurements due to the following possible reasons. (i) It is very difficult to observe a long-range ordered vortex states and the vortex configurations are generally combined with different local vortex arrangements. (ii) The critical currents measured in experiments may correspond to more stable vortex states rearranged by the external currents. The currents may lead to the vortex state transitions within very short a time before the critical currents are measured.

Conclusions
In summary, we investigate periodic MQCs to confine vortices by dividing a nanostructured superconducting film into many isolated areas of micron-size with a Kagome lattice of elongated antidots. The vortex configurations in the presence of the competition between the pinned elongated vortices at various magnetic fields are directly observed by using SHPM. Due to the presence of the random pinnings, it is found that the interstitial vortex states become more disordered at high magnetic fields. Furthermore, by subjecting external Figure 8. The simulated vortex states with the interstitial vortices confined in the hexagonal cages in the absence of the pinning sites and the vortex patterns just before the MQCs are collapsed by the interstitial vortices under the external currents j ax and j ay , respectively. currents to superconductors, the stability of different degenerate vortex states at several fractional matching fields and the confinement effects of MQCs at higher magnetic fields are studied based on the tdGL simulations. The free energy splitting of the degenerate vortex states is observed with increasing the external currents and we predict that parts of degenerate vortex states can be eliminated by applying external currents to superconductors during FC. The experimental and theoretical results presented in this paper provide a deeper insight to explore the stability of the degenerate vortex states, the V-V interactions and the vortex state transitions in superconductors with complex pinning landscapes.