Vortex Pinning

In a perfect superconducting crystal, the vortices make a vortex lattice. The position of each of the vortices is defined by the repulsive force between them. When a current driven by external source is applied to this crystal, the vortices will be subjected to Lorentz force, because they carry a magnetic moment. Because of this, they will move through the crystal, exiting it on one side, whereas new vortices will be entering it on the other side, so that the total number of vortices in the crystal is always the same. The average Lorentz force per unit volume of the crystal is [15]

where J and B are spatially averaged transport current density and magnetic field inside the crystal, respectively.

Movement of the vortices is associated with a change of magnetic flux in and around the vortices [16]. This creates electromagnetic force that drives the supercurrents into the resistive core of the vortices. In effect, some of the Cooper pairs are broken down and the charge carriers interact with the crystal lattice, resulting in dissipation of their energy and electrical resistance. According to Bardeen and Stephen [16], about half of the dissipation occurs within the vortex core and half in the transition region just outside the core. This means that a perfect superconductor in the mixed state

Figure 2. Spatial variation of B around a magnetic vortex. B is constant inside the nonsuperconducting core of radius of the coherence length It decreases approximately exponentially with distance outside the core, with the decay length equal to London penetration depth A.

Figure 1. Schematic drawing of the field entering the type II superconductor through magnetic vortices (shaded).

Figure 2. Spatial variation of B around a magnetic vortex. B is constant inside the nonsuperconducting core of radius of the coherence length It decreases approximately exponentially with distance outside the core, with the decay length equal to London penetration depth A.

will exhibit electrical resistance regardless of the value of the transport current.

If, however, the superconducting crystal has defects in its structure, there will be local variations in the superconducting properties. This will result in energetically favorable sites for magnetic vortices, where they will be pinned down. Such sites are called the pinning centers. With the vortices pinned, there is again no electrical resistance in the superconductor. However, when applying too large a magnetic field or transport current, Lorentz force on the vortices will exceed the pinning force of the pinning centers. This will set vortices to motion again, and resistance will occur. The current at which the resistance just starts occurring is called the critical current, Ic. Additionally, if the thermal energy is higher than the pinning energy, vortices can also escape from the pinning centers. The thermal energy is proportional to temperature T,

where kB is the Boltzman constant. Because of this, Ic will become zero for certain combinations of magnetic field and temperature, for which vortices are just released from the pinning centers. They are called irreversibility field (Hirr) and irreversibility temperature (Tirr), respectively. Because Tirr < Tc and Hirr < Hc2, electrical resistance in type II superconductors can occur even when they are still in the superconducting state. Values of Hirr, Tirr, and critical current density (Jc) are defined by the vortex pinning.

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