Surface barrier and lower critical field in ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{7\mathrm{-}\mathrm{\delta }}$ superconductors

The fields for the first vortex entry and last vortex exit, ${\mathit{H}}_1$ and ${\mathit{H}}_2$, and the lower critical field ${\mathit{H}}_{\mathit{c}1}$ for a grain-aligned, sintered ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{7\mathrm{-}\mathrm{\delta }}$ superconductor have been determined from saturated magnetic-hysteresis loops using an extended critical-state model. For fields oriented along the grain c axis, ${\mathit{H}}_1$ increases with decreasing temperature, showing an upturn below 50 K, whereas ${\mathit{H}}_2$ remains small and positive, in general agreement with the theory of Bean-Livingston surface barriers. ${\mathit{H}}_{\mathit{c}1}$ has a Bardeen-Cooper-Schrieffer temperature dependence above 50 K, but it rises at low temperatures. For fields oriented in the ab plane, ${\mathit{H}}_{\mathit{c}1}$ has a similar temperature dependence, but surface barriers are not evident in the magnetization.