Theoretical investigation of lindemann's criterion in the melting of two-dimensional vortex-lattice with nano defects

Calculations of the three-dimensional vortex-lattice melting have been mainly based on the use of the Lindemann criterion (u ²) _T = c ² _La ², which estimates the root-mean-square thermal displacement fluctuations ([u ²] _T) ^1/2 of a vortex element at the melting transition as a fraction c _L of the vortexlattice spacing a. Using a series of molecular dynamic simulations, we have tested Lindemann criterion in the melting of two-dimensional vortex lattice systems with periodic square arrays of pinning sites each is 20 nm in diameter. The melting criterion was determined using direct observing of the two-dimensional vortex lattice system and by calculating the Bragg peaks in regions below and above the melting line. Our results suggest the possible applicability of this criterion on the twodimensional vortex lattice systems. We have confirmed the reported values of Lindemann number c _L ~ (0.1-0.2) reported for other systems. We have found an interesting dependence of c _L on the density of vortices, and on the pinning strengths of the superconductor system. No noticeable role of the density of the nano-sized pinning sites was observed.