Quantitative theory of transport in vortex matter of type-II superconductors in the presence of random pinning

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We quantitatively describe the competition between interactions, thermal fluctuations, and random quenched disorder using the dynamical Martin-Siggia-Rose approach [Phys. Rev. A 8, 423 (1973)] to the Ginzburg-Landau model of the vortex matter. The approach first used by Dorsey et al. [Phys. Rev. B 45, 523 (1992)] to describe the linear response far from ${H}_{c1}$ is generalized to include both pinning and finite voltage. It allows one to calculate the non-Ohmic $I\text{\ensuremath{-}}V$ curve, thereby extending the theory beyond the linear response. The static flux line lattice in type-II superconductors undergoes a transition into three disordered phases: the vortex liquid (not pinned), the homogeneous vortex glass (pinned, if one disregards an exponentially small creep at finite temperatures), and the crystalline Bragg glass (pinned) due to both thermal fluctuations and disorder. The location of the glass transition line in the homogeneous phase is determined and compared to experiments. The line is clearly different from both the melting line and the second peak line describing the translational and rotational symmetry breaking at high and low temperatures, respectively. Time correlation and response functions of the order parameter as functions of the time difference are calculated in both the liquid and the amorphous homogeneous phases. They determine the relaxation properties of the vortex matter due to the combined effect of pinning and thermal fluctuation. We calculate the critical current as a function of magnetic field and temperature in the homogeneous phase. The surface in the $J\text{\ensuremath{-}}B\text{\ensuremath{-}}T$ space defined by this function separates between a dissipative moving vortex matter regime and vortex glass. A quantitative theory of the peak effect, qualitatively different from the conventional one due to Pippard [C. Tang, X. Ling, S. Bhattacharya, and P. M. Chaikin, Europhys. Lett. 35, 597 (1996); A. B. Pippard, Philos. Mag. 19, 217 (1969); A. I. Larkin and Yu. N. Ovchinnikov, J. Low Temp. Phys. 34, 409 (1978)], is proposed.