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, $a_0<R_c<\lambda$ (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 $R_c\propto B^{-1/2}{\delta }_p^{{\alpha }_2}\exp ({\alpha }_1/{\delta }_p),$ where the (dimensionless) parameter ${\delta }_p\propto B^{-3/2}$ is a measure of the effective disorder strength, B is the magnetic induction, ${\alpha }_1=16/\left(9\mathrm{}\sqrt{\pi }\right)\approx 1,$ and ${\alpha }_2=\left(7\mathrm{}-5\ln \frac{4}{3}\right)/27\mathrm{}\approx \mathrm{0.2.}$ 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 $j_c\propto B^{1+3{\alpha }_2}\exp \left(-\mathrm{const}{B}^{3/2}\right).\mathrm{}$

• Received 3 September 1998

DOI:https://doi.org/10.1103/PhysRevB.59.11551