Optimizing superconductor transport properties through large-scale simulation

Vortex phase diagram in layered superconductors and thermodynamics of vortex lattice melting

Alex Koshelev,
Materials Science Division, Argonne National Laboratory

Thermodynamics and transport properties of Type II superconductors in magnetic field are essentially properties of the “vortex matter.” A peculiarity of high Tc superconductors is a very high amplitude of thermal fluctuations in the vortex state. In very anisotropic superconductors vortex lines consist of very weakly coupled “pancake vortices.” This leads to destruction of the low temperature vortex crystal state well below the mean field upper critical field. The only way to obtain quantitative information about the phase diagram in the vortex state is large scale numerical simulations.


We used CCST resources to study the vortex phase diagram in layered superconductors by Monte-Carlo simulations. We describe superconductor by the Lawrence-Doniack model in the phase approximation, i.e., superconductor is modeled by a system of interacting phases placed in three dimensional rectangular grid. Figure 1 shows an example of the instantaneous phase and vortex distribution. For our parameters one in 36 square cells contains a vortex.

We systematically investigated properties of the melting transitions in superconductors with different anisotropy parameter Gamma.

Fig 1: Typical phase and vortex configuration at low temperatures. Phases at the sites of grid are represented by the tilting angles of arrows


We find that the pancake alignment above the melting transition increases with decreasing of anisotropy. At high anisotropies the melting is accompanied by a significant drop in the coupling energy and the destruction of vortex lines, while at small anisotropies the vortex lines preserve at the transition. The movie shows evolution of vortex line configuration when the temperature is lowered below the melting point for moderately anisotropic superconductor. Figures below compare liquid and crystal configurations for small and large anisotropy parameters.

The system has very important scaling property: thermodynamics of melting transition does not depend separately on field B and anisotropy Gamma but only on combination Gamma. We verified this property by comparing results of different simulations (see Fig. 3). In the linelike melting regime we find several universal properties of the transition and the liquid phase. These properties include the scaling of the melting temperature with anisotropy and magnetic field (see Fig. 3), the effective line tension of vortices in the liquid regime, the latent heat, the entropy jump per entanglement length, and relative jump of Josephson energy at the transition as compared to latent heat. Simulations advanced our understanding of the vortex phase diagram in layered high temperature superconductors.

Fig 2: Triangulation patterns in neighboring layers and density-density correlation functions for competing configurations near the melting temperature for two values of the anisotropy factor Gamma.

Fig 3: Universal phase diagram of layered superconductors. We compare results of different simulation with different filling factors b.


  1. A.E. Koshelev, Pointlike and linelike melting of the vortex lattice in the universal phase diagram of layered superconductors, Phys. Rev. B 56 212 (1997).
  2. A.E. Koshelev and H. Nordborg, Universal properties for linelike melting of the vortex lattice, Phys. Rev. B 59 4358 (1999).