Anisotropy effects on the critical magnetic field in CaC$_{6}$ superconductor

The calcium intercalated graphite (CaC$_{6}$) is considered to be a representative material of the graphite intercalated superconductors, which exhibits sizable anisotropy of the Fermi surface (FS). Herein, the influence of the FS anisotropy on the critical magnetic field ($H_{C}$) in CaC$_{6}$ superconductor is comprehensively analyzed within the Migdal-Eliashberg (M-E) formalism. To precisely account for the mentioned anisotropy effects, the analysis is conducted in the framework of the six-band approximation, hitherto not employed for calculations of the $H_{C}$ function in CaC$_{6}$ material. For convenience, the obtained results are compared with the available one- and three-band estimates reported by using the M-E theory. A notable signatures of the increased number of bands are observed for the temperature dependent $H_{C}$ functions. In particular, the $H_{C}$ function decreases at T=0 K as the number of the considered bands is higher. Moreover, it is argued that the six-band formalism yields the most physically relevant shape of the $H_{C}$ function among all considered approximations. Therefore, the six-band model introduces not only quantitative but also qualitative changes to the results, in comparison to the other discussed FS approximations. This observation supports postulate that the six-band model constitutes minimal structure for FS of CaC$_{6}$, but also magnify importance of anisotropy effects for the critical magnetic field calculations.