Collective pinning model of the mixed state inYBa2Cu3O7−δ: Critical currents and flux creep

Abstract

Magnetic hysteresis and flux creep measurements in single crystal samples of ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ (YBCO) are presented for a wide range of $B$,$T$ phase space. Some of these samples can be described as weakly or collectively pinned. For these, over a large portion of this phase space, the flux creep can be described in terms of thermally activated single-fluxoid motion. A simple model based on maximizing the pinning energy of a fluxoid segment provides a good, semiquantitative picture of the low-temperature data, where the experimentally measured critical current density $j$ is proportional to $1∕T$ and the activation barrier height is proportional to ${j}^{\ensuremath{-}\ensuremath{\mu}}$, where $\ensuremath{\mu}=1$. In this model individual fluxoids are pinned by stochastic fluctuations in defect concentration, and are driven over the pinning barriers by critical currents and thermal activation. Incorporating flux lattice elasticity into this simple model leads to new predictions for the low-temperature data and allows the simple model to be extended to higher temperature. There are two distinct effects, both of which can be put in the form of effective current densities. One effective current density ${j}_{s}$ arises from direct fluxoid-fluxoid repulsion, and the second effective current density ${j}_{r}$ arises from fluxoid relaxation. In YBCO at $7\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ and $2\phantom{\rule{0.3em}{0ex}}\mathrm{T}$, where the measured critical current density is $j=8.9\ifmmode\times\else\texttimes\fi{}{10}^{9}\phantom{\rule{0.3em}{0ex}}\mathrm{A}∕{\mathrm{m}}^{2}$, we find ${j}_{s}=0.57\ifmmode\times\else\texttimes\fi{}{10}^{9}\phantom{\rule{0.3em}{0ex}}\mathrm{A}∕{\mathrm{m}}^{2}\phantom{\rule{0.3em}{0ex}}(6%)$ and ${j}_{r}=\ensuremath{-}2.1\ifmmode\times\else\texttimes\fi{}{10}^{9}\phantom{\rule{0.3em}{0ex}}\mathrm{A}∕{\mathrm{m}}^{2}$ $(\ensuremath{-}20%)$. We present a discussion of their origin that leads to plausible temperature and field dependences. The model accounts for the rapid drop of $j(T)$ with increasing temperature, the peak effect in $j(B)$ at high temperature, and the temperature and field dependence of the ``critical exponent'' $\ensuremath{\mu}$. Thermal fluxoid vibrations play an important role in the pinning, and we find effects consistent with calculations in the literature. The model postulates that fluxoid motion takes place by hopping in segments on a characteristic length scale ${l}_{\mathrm{model}}$. In the model we find ${l}_{\mathrm{model}}=104\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ at $7\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ and $2\phantom{\rule{0.3em}{0ex}}\mathrm{T}$. A completely independent measurement from the creep-derived four-volume $VX$ yields a length ${l}_{VX}=102\ifmmode\pm\else\textpm\fi{}5\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ at the same temperature and field. Excellent agreement between the two independently determined lengths persists over a wide range of temperatures. A failure of these two lengths to agree marks the boundary for single-fluxoid hopping, and we present a diagram of the pinning regimes in $B$,$T$ phase space. From the measured prefactor of thermally activated creep at $10\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ and $2\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ we infer a value for the attempt frequency ${f}_{a}=8.5\ifmmode\times\else\texttimes\fi{}{10}^{10}\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{\ensuremath{-}1}$. This value is in reasonable agreement with a published theoretical calculation of the relaxation frequency for overdamped fluxoids in an Abrikosov lattice. Finally, based on these data we estimate the mass per unit length of a YBCO fluxoid segment, and compare our result with Suhl's theory to obtain a quasiparticle effective mass of 30 free electron masses.