We study the graphene band-gap engineering by introducing different defects, namely the defects breaking the inversion symmetry and the ones periodically patterning graphene into superlattice such as the regularly arranged antidots, etc. Comparing to the primitive unit cell of graphene, the pseudo-graphene superlattice, referred to the pristine graphene supercell, modulates the boundary condition accordingly. According to the energy band-folding picture, these superlattices can be categorized into two groups on the basis of the Dirac cone position. In some cases, the Dirac points K and K′ in primitive cell are folded to the Γ point of pseudo-superlattices. The coincidence of Dirac points with Γ point results in the fourfold degeneracy. In these systems, a band gap at Γ point can be opened by introducing periodically arranged defects such as the antidots, which could be easily utilized in experiment, for example, by making the graphene nanomesh through lithography technique. In the other cases, the twofold degenerate Dirac points remain nonequivalent with Γ point in pseudo-superlattice, and the method to break the inversion symmetry could open their band gaps.

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Using both the first-principles method and the tight-binding method, we have carried out detailed studies on the electronic properties of silicene superlattices. According to the band-folding picture, in the hexagonal silicene superlattice (M,N) when both M and N are integer multiples of 3, the simultaneous folding of K and K 0 points to the G point results in the fourfold degeneracy. In the orthogonal one [P,Q], the fourfold degeneracy occurs if Q ¼ 3q (q is an integer) is satisfied. By introducing degenerate perturbation to remove the fourfold degeneracy, the bandgap could be opened no matter whether the inversion symmetry remains or not. For the other cases, the bandgaps could be kept closed by the inversion symmetry preservation. Besides, the studied transport properties confirm the bandgap tuning by applying degenerate perturbation to the silicene superlattice. The new bandgap opening mechanism with degenerate perturbation could benefit further experimental studies of the silicenebased nanostructures for applications in nanoelectronics.

We have performed a detailed density functional theory study on the structural and electronic properties of Na(n)C(60)(-) (n = 1-12) clusters. The calculated vertical detachment energies show good agreement with the experimental data, which confirms the 3p (n = 3p) oscillation rule. The oscillation can be attributed to the combination of the charge depletion distribution induced by removing electrons and the number of the sodium atoms in direct contact with the fullerene. Based on the structural and electronic properties, the Na atoms can be categorized into two groups, one is for the metal atoms directly bonded to the fullerene surface, and the other one is for those without bonding to the fullerene. The Na atoms in group one would donate electrons to both the fullerene and the Na atoms in group two. As the total number of the sodium atoms increases, the number of Na atoms in group one would continue increasing till the size n = 3p - 1 to meet a shoulder from n = 3p - 1 to n = 3p, which accounts for the maximum vertical detachment energy at the size of n = 3p as drawn from the detailed electronic property studies.