Extended Ginzburg-Landau theory of superconductivity from generalized momentum operator and position-dependent mass

Abstract

The concept of generalized momentum operator which is motivated from the generalized uncertainty principle has been extensively investigated in quantum mechanics and various aspects of theoretical and applied physics. In this study, we have applied this formalism to Ginzburg-Landau theory of superconductvity and Abrikosov vortex lattice in type II-superconductors. After deriving the extended Ginzburg-Landau equations, we have discussed several independent structures of the auxiliary function of position operator in the generalized momentum operator and we have analyzed their features in both Ginzburg-Landau and London theories. Comparable properties to the basic formalism have been obtained even without the presence of the cubic nonlinear term in Ginzburg-Landau equations. However, not all structures of the auxiliary function of position operator result on the exclusion of the magnetic field from a superconductor when it's below its critical temperature. But, only specific forms may succeed to eliminate the magnetic field. This approach has been generalized by considered a position-dependent mass of the electric charge. Amazingly, for a position-dependent mass of hyperbolic soliton-like structure, the extended Ginzburg-Landau wave function is identical to the result obtained in the conventional formalism and besides, for specific correlations between the auxiliary function of the position operator and position-dependent mass, the exclusion of the magnetic field is conceivable. We have also discussed the Abrikosov vortex lattice solution based on the extended Ginzburg-Landau formalism with position-dependent mass of the electric charge. It was observed that, for a specific structure of the position-dependent mass and for a quantum number n=0, a transition between a type-II and type-I superconductor takes place if the Ginzburg-Landau parameter is κ=121+257≈2.0634. For large n, the problem depends on the asymptotic form of the Hermite polynomial and periodicity occurs if the electric charge is quantized. Further details are obtained and analyzed.