Lower critical field of a Josephson-coupled layer model of high-Tcsuperconductors

We calculate the lower critical field ${\mathit{H}}_{\mathit{c}1}$ for a magnetic field applied parallel to the layers of a Josephson-coupled layer model of high-${\mathit{T}}_{\mathit{c}}$ superconductors. This result is obtained by using an expression for the vortex line energy that explicitly involves the gauge-invariant phase difference between superconducting layers. We examine the behavior of ${\mathit{H}}_{\mathit{c}1}$ as a function of temperature and discuss the manifestation of the superconductor discreteness below a crossover temperature where the coherence length ${\xi }_{\mathit{c}}$(T) becomes comparable to the lattice constant c. This temperature dependence is contrasted with the standard result for ${\mathit{H}}_{\mathit{c}1}$ for a continuous anisotropic type-II superconductor.