Normal modes and specific heat of a lattice of undamped pancake vortices in thin superconducting mulitlayers

Recent high-field (B≤8 T) experiments on single crystals of ${\mathrm{Bi}}_2$ ${\mathrm{Sr}}_2$ ${\mathrm{CaCu}}_2$ ${\mathrm{O}}_{\mathit{x}}$ (BSCCO-2:2:1:2) have found a significant anisotropy in the specific heat, with an excess contribution for perpendicular field (along the c axis) relative to that for parallel or zero field. One possible explanation for these observations is the undamped vibrational modes of the lattice of pancake vortices, which occur only for the perpendicular orientation. The London model of thin superconducting layers provides a simple theoretical picture that allows a direct evaluation of the elastic energy for small deformations of the vortex lattice. For typical temperatures, the resulting vortex contribution ${\mathit{C}}_{\mathit{v}}$ to the heat capacity has a Debye form for two-dimensional phonons, with a Debye temperature ${\mathrm{\Theta }}_{\mathit{v}}$=1/2ħ${\mathrm{\omega }}_{\mathit{c}}$/${\mathit{k}}_{\mathit{B}}$, where ${\mathrm{\omega }}_{\mathit{c}}$ is the electron cyclotron frequency (for B=10 T and free electrons, ${\mathrm{\Theta }}_{\mathit{v}}$≊6.7 K). For T>${\mathrm{\Theta }}_{\mathit{v}}$, the resulting ${\mathit{C}}_{\mathit{v}}$ approaches ${\mathit{k}}_{\mathit{B}}$ per pancake vortex, as appropriate for vortex dynamics and in qualitative agreement with the experimental observations; for T${\mathrm{\Theta }}_{\mathit{v}}$, the theory predicts that ${\mathit{C}}_{\mathit{v}}$∝${\mathit{T}}^2$, but this particular power law remains to be verified experimentally. If an inertial mass ${\mathit{m}}_{\mathit{c}}$ is ascribed to each pancake vortex, an additional set of (Newtonian) normal modes yields a further (Einstein) heat capacity, with ${\mathrm{\Theta }}_{\mathit{E}}$∝${\mathit{m}}_{\mathit{c}}^{\mathrm{-}1}$. The presence (absence) of such a contribution to ${\mathit{C}}_{\mathit{v}}$ could help determine (bound) the value of the currently controversial quantity ${\mathit{m}}_{\mathit{c}}$.