Fermions on the low-buckled honey-comb structured lattice plane and classical Casimir–Polder force

Abstract

We start with the well-known expression for the vacuum polarization and suitably modify it for 2[Formula: see text]1-dimensional spin–orbit coupled (SOC) fermions on the low-buckled honey-comb structured lattice plane described by the low-energy Liu–Yao–Feng–Ezawa (LYFE) model Hamiltonian involving the Dirac matrices in the chiral representation obeying the Clifford algebra. The silicene and germanene fit this description suitably. They have the Dirac cones similar to those of graphene and SOC is much stronger. The system could be normal or ferromagnetic in nature. The silicene turns into the latter type if there is exchange field arising due to the proximity coupling to a ferromagnet (FM) such as depositing Fe atoms to the silicene surface. For the silicene, we find that the many-body effects considerably change the bare Coulomb potential by way of the dependence of the Coulomb propagator on the real-spin, iso-spin and the potential due to an electric field applied perpendicular to the silicene plane. The computation aspect of the Casimir–Polder force (CPF) needs to be investigated in this paper. An important quantity in this process is the dielectric response function (DRF) of the material. The plasmon branch was obtained by finding the zeros of DRF in the long-wavelength limit. This leads to the plasmon frequencies. 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. Our scheme to calculate the Casimir–Polder interaction (CPI) of a micro-particle with a sheet involves replacing the dielectric constant of the sample in the CPI expression obtained on the basis of the Lifshitz theory by the static DRF obtained using the expressions for the polarization function we started with. Though the approach replaces a macroscopic constant by a microscopic quantity, it has the distinct advantage of the many-body effect inclusion seamlessly. We find the result that for the nontrivial susceptibility and polarizability values of the sheet and micro-particle, respectively, there is crossover between attractive and repulsive behavior. The transition depends only on these response functions apart from the ratio of the film thickness and the micro-particle separation ([Formula: see text]/[Formula: see text]) and temperature. 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 gapless (massless) 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 CPF at a given ratio of the film thickness and the separation between the micro-particle and the film are greater at TPT than at the TI and trivial insulator phases.