Vortex-antivortex patterns in mesoscopic superconductors

Gerd Teniers^*, Victor V. Moshchalkov^*, Liviu F. Chibotaru⁺, and Arnout Ceulemans⁺

(^*Nanoscale Supercond. & Magnetism Group, ^*Laboratorium voor Vaste-Stoffysica en Magnetisme, ⁺Afdeling Kwantumchemie, Katholieke Universiteit Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium)

Superconductivity appears in some materials when they are in the state below the critical surface determined by three critical parameters: namely the temperature T_c, the magnetic field H_c and the electrical current J_c. However, it has become apparent that these parameters are not only determined by the properties of the bulk material, but also by the geometry of the sample. The influence of the geometry is investigated with the help of modern microfabrication techniques[1], which makes it possible to produce submicron superconducting structures.

Our theoretical research is focused on the following fundamental problem: how do symmetrical boundary conditions change the behaviour of a superconducting condensate confined in individual nanostructures? For example, symmetrical nanostructures (like disks, triangles, square and rectangles) show a T_c(H) phase boundary between the superconducting and the normal phase strongly differing from the one of the bulk material (see fig. 1).

Fig. 1: Calculated superconducting T-H phase boundary for the bulk H_c3 case(green), disk (black), square(red), triangle(blue) and rectangle (cyan, pink, yellow) with superconductor-vacuum boundary conditions. The flux is defined as Φ=HS, with S the surface of the mesoscopic structure and H the applied magnetic field. The rectangle is shown for different aspect ratios. From the top to bottom curve the aspect ratios for the corresponding rectangle are 2(cyan), 16(pink) and 64(yellow). With increasing aspect ratio the phase boundary evolves from a predominantly linear behavior with oscillations to a parabolic behavior without any oscillations. The observed T_c(H) oscillations are clearly the strongest with the disk.

Moreover, in the neighbourhood of the phase boundary it is necessary for the superconducting condensate to show the same symmetry as the sample. However, how do we place for instance three vortices symmetrically into a square? This problem is solved by spontaneously generating an antivortex-vortex pair, and as such it is possible to place an antivortex in the centre and four vortices in the corners (+1+1-1+1+1=3), which is a configuration with the right symmetry (see fig. 2).

Fig. 2: We determined that it is possible that the discrete symmetry of the triangle and square induce a vortex-antivortex pattern in the triangle[2] and square[3] (shown respectively with a zoom factor of 64x and 16x), which is built up from an antivortex in the center and surrounded by three or four vortices, respectively, on the diagonals, giving a total fluxoid of two and three flux quanta (Φ₀). These vortex patterns stand in contrast to the often expected giant 2Φ₀- or 3Φ₀-vortices.

We wanted to study the evolution of these novel vortex patterns under variations of the magnetic field, the temperature, and the symmetry by deforming a square into a rectangle[4]. Out of this systematic research it has become apparent that these vortex molecules are stable for broad ranges of the magnetic field and temperature. Therefore, we can conclude that the symmetry of the superconducting nanostructures influences substantially the properties of the superconductor even deep in the superconducting state. By improving the critical parameters of these superconductors through nanostructuring the perspectives of their practical applications can be improved.

Fig. 3: Schematic T-H plot of the different vortex patterns that can be observed. Φ₀-vortices, giant 2Φ₀-vortices, and antivortices are represented respectively by black dots, large blue dots and red circles (see also the movie below). The letters show the irreducible representations needed to construct the found symmetry. The black lines separate regions with different vortex patterns in the square. Lines with number one divide two pure symmetrical states with a different fluxoid quantum number and corresponds to a phase transition of the first order[5]. Lines with number two separate regions with a pure or broken symmetry and equal fluxoid quantum. The lines with number three divide regions between pure and broken symmetry and with a different fluxoid quantum number. Lines with number two and three correspond to phase transitions of the second order[5].

In figure 3 the T-H phase diagram for the square is shown for a broad range of temperatures and magnetic fields. We also made a movie to illustrate better the evolution of the vortex patterns as a function of temperature and magnetic field. The right panel in the movie shows the order parameter and the vortex patterns corresponding with the position of the circle in the T-H diagram at the left.

1. MPEG movie (lower quality 20.6MB): vortexpatterns.mpg

2. Quicktime movie (high quality 11.5MB): vortexpatterns.mov

First of all we notice that the number of flux quanta Φ₀ in the square is reduced with decreasing temperature (decreasing coherence length ξ) as expected. However, not only the fluxoid quantum number changes. The vortex patterns also change with decreasing temperature, even when the fluxoid quantum number does not necessarily change.

Specifically, we draw your attention to the field range between the third and the fourth cusp at the T_c phase boundary. The vortex pattern here consists of an antivortex in the center and four vortices on the diagonals leading to a fluxoid quantum number three in the sample (see fig. 3). Lowering the temperature close to the third cusp leads to a decrease in the fluxoid quantum number in the sample to two, which is carried by a giant Φ₀-vortex in the center. This giant vortex in its turn splits into two Φ₀-vortices when lowering the temperature even further. However, away from the third cusp quite a different evolution can be observed, since lowering the temperature leads to the annihilation of the antivortex and the conservation of the fluxoid quantum number. Eventually with decreasing temperature the number of vortices decreases to two and no giant vortex state is observed. Moreover, in the first stage near cusp three, fluxoid quantum number three can only be attained by a vortex-antivortex pattern. So there exists a range of magnetic fields where the vortex-antivortex pattern is always preferred over three singly quantized vortices or a giant vortex with vorticity three. We can conclude that there is a considerable region of stability for the vortex-antivortex pattern. Let us consider some realistic sample sizes to illustrate this. For example, if we take the coherence length for a realistic Al thin film to be ξ(0)=140nm and the side of the square a=2μm, the vortex-antivortex patterns still can be found for temperatures above T=0.75T_c0. Furthermore when taking a smaller sample with the side length a=1μm, the vortex-antivortex pattern is stable for all temperatures below T_c(H). Of course near a temperature of 0 K the GL theory is not the appropriate model anymore. Nonetheless, we can conclude that the vortex-antivortex pattern is stable towards relatively large variations of temperature in a mesoscopic Al sample.

As a final point, let us consider the situation where the fluxoid quantum number is eight. When decreasing the temperature we observe the migration of the four central vortices to the sides and eventually a distorted triangular lattice is formed. This tendency towards a triangular lattice is enhanced for higher fields and lower temperatures, since the considered sample will start to behave like a large thin bulk sample under these circumstances.

[1] V. Moshchalkov, L. Gielen, C. Strunk, R. Jonckheere, X. Qiu, C. V. Haesendonck, and Y. Bruynseraede, Nature 373, 319 (1995).
[2] L. F. Chibotaru, A. Ceulemans, V. Bruyndoncx, and V. V. Moshchalkov, Phys. Rev. Lett. 86, 1323 (2001).
[3] L. F. Chibotaru, A. Ceulemans, V. Bruyndoncx, and V. V. Moshchalkov, Nature 408, 833 (2000).
[4] G. Teniers, L. F. Chibotaru, V. V. Moshchalkov, and A. Ceulemans, Europhys. Lett. 63, 296 (2003).
[5] L. F. Chibotaru, G. Teniers, A. Ceulemans, and V. V. Moshchalkov, Phys. Rev. B 70, 094505 (2004).