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2+1 dimensional Fermions on the low-buckled honey-comb structured lattice plane and classical Casimir-Polder force

We have calculated the Casimir-Polder interaction (CPI) of a micro-particle with a sheet on the basis of the Klimchitskaya-Mostepanenko theory. We find the result that for non-trivial susceptibility values of the sheet and micro-particle, there is crossover between attractive and repulsive behavior. The transition depends only on the impedance, involving permeability and permittivity, apart from the ratio of the film thickness and the micro-particle separation (D/d) and temperature. The approach to calculate CPI of a micro-particle with a silicene sheet involves replacing the dielectric constant of the sample by the static dielectric function obtained using the expressions for the polarization function. The silicene is described by the low-energy Liu-Yao-Feng-Ezawa (LYFE)Model Hamiltonian involving the Dirac matrices in the chiral representation obeying the Clifford algebra.We find that the collective charge excitations at zero doping, i.e., intrinsic plasmons, in this system, are absent in the Dirac limit. The valley-spin-split intrinsic plasmons, however, come into being in the case of the massive Dirac particles with characteristic frequency close to 10 THz.Furthermore, there is a longitudinal electric field induced topological insulator(TI) to spin-valley polarized metal (SVPM) transition in silicene, which is also referred to as the topological phase transition (TPT). The low-energy SVP carriers at TPT possess gap-less (mass-less) and gapped (massive) energy spectra close to the two nodal points in the Brillouin zone with maximum spin-polarization. We find that the magnitude of the Casimir-Polder force at a given ratio of the film thickness and the separation between the micro-particle and the film is greater at TPT than at the topological insulator and trivial insulator phases.