Electric field controlled energy gap and Landau levels in silicene

We investigate the electric field controlled energy gap and the Landau levels in silicene in detail. The energy gap at different Dirac points has different closing and reopening conditions and the 2-fold degeneracy induced by the K-K' symmetry is resolved. An externally applied electric field gives rise to two Rashba spin-orbit-couplings between the nearest neighbour and the next nearest neighbour in silicene. Both these couplings can resolve the spin degeneracy at some isolated values of the electric field, where the crossover of the successive Landau levels become anti-crossover. Except some special values of the electric field, the 4-fold degeneracy of energy levels associated with the K-K' symmetry and spin symmetry is completely resolved in silicene, each level has a definite spin polarization, which correspond to the quantum Hall plateaux with filling factor ν=0, ±1, ±2,….