Renormalization-group analysis of weak collective pinning in type-II superconductors

Abstract

The effect of randomly distributed impurities on vortex lattices in isotropic type-II superconductors is studied within the framework of weak collective pinning theory. Using a renormalization-group (RG) approach, we calculate the size ${R}_{c}$ of collectively pinned vortex bundles in the dispersive regime, ${a}_{0}l{R}_{c}l\ensuremath{\lambda}$ (small bundles), to two-loop accuracy. We assume impurity disorder to be weak and short-range correlated, and neglect thermal effects. Our findings quantitatively refine the lowest-order perturbation result due to A. I. Larkin and Yu. N. Ovchinnikov [J. Low Temp. Phys. $34,$ 409 (1979)]. In particular, we determine the numerical constant in the exponential function and find the algebraic prefactor in ${R}_{c}\ensuremath{\propto}{B}^{\ensuremath{-}1/2}{\ensuremath{\delta}}_{p}^{{\ensuremath{\alpha}}_{2}}\mathrm{exp}({\ensuremath{\alpha}}_{1}/{\ensuremath{\delta}}_{p}),$ where the (dimensionless) parameter ${\ensuremath{\delta}}_{p}\ensuremath{\propto}{B}^{\ensuremath{-}3/2}$ is a measure of the effective disorder strength, B is the magnetic induction, ${\ensuremath{\alpha}}_{1}=16/(9\sqrt{\ensuremath{\pi}})\ensuremath{\approx}1,$ and ${\ensuremath{\alpha}}_{2}=(7\ensuremath{-}5\mathrm{ln}\frac{4}{3})/27\ensuremath{\approx}0.2.$ These refinements lead to an improved description of the activated dynamics of the vortex lattice (creep) and provide us with a more accurate functional dependence of the critical current ${j}_{c}$ on the magnetic field ${j}_{c}\ensuremath{\propto}{B}^{1+3{\ensuremath{\alpha}}_{2}}\mathrm{exp}(\ensuremath{-}\mathrm{const}{B}^{3/2}).$