## Matching

Without the influence of vortex pinning, flux lines in an ideal type-II-superconductor form a triangular lattice due to repulsive Lorentz forces between them (the Abrikosov lattice [1]). The state of the system where the flux line lattice (or the density of vortices) matches the underlying lattice (or density) of pinning sites, is extremely stable due to energetic reasons. Since the density of flux lines can be tuned by an external magnetic field, there is a first matching field, at which, for a hexagonal pinning lattice, each pinning site is occupied by one single flux line.

The formation of vortices and their arrangement in various types of 'vortex phases', ranging from the ordered, triangular Abrikosov lattice to disordered phases has a strong impact on the electric properties. Numerical simulations show a strong dependence of the critical current of a superconductor containing a rectangular or triangular lattice of pinning sites of an external field [2].

Because of the presence of natural pinning sites like crystal defects and caging of flux in interstitial positions (positions between individual pinning sites), V(I)-curves of a superconductor containing a periodic array of pinning sites may show two linear regimes. These two linear slopes correspond to flux flow after depinning, first from natural and/or interstitial pinning positions and, above a certain driving force, from artificial pinning centers.

The flux line lattice in a superconductor containing a periodic array will on one hand try to occupy as many pinning sites as possible to minimize its energy, on the other hand, the vortices excert repulsive forces on each other, possibly (depending on the number of vortices, the number of pinning sites and their distance from each other) hindering this occupation. Depending on the number of flux lines (which is tunable via an external field), the vortex lattice will form configurations with differing stability. In some cases, high stability of the lattice will make depinning occur only at relatively high depinning forces corresponding to large driving currents. This results in relatively sharp peaks in measurements of critical current vs. magnetic field, as shown in Fig. 2.

[1] A. A. Abrikosov,

*Sov. Phys. JETP*5:1147 (1957)

[2] C. Reichhardt and N. Grønbech-Jensen,

*Phys. Rev. B*63:054510