We study the effect of Landau-Zener (LZ) tunneling caused by the varying sweeping rate of external field, formulating and approximately solving the problem with many levels of the LZ tunneling rate. Comparing with the steadily vary about sweeping fie
ld, the LZ tunneling rate will be essentially changed because of the unsteady variation of the sweeping field in time. Thus could help us to make the particles with in lower states transit periodically to upper states within the finite time.
The hybrid orbitals of single-wall carbon nanotubes are given according to the structure of the nanotube. Because the energy levels of these hybrid orbitals are close to each other, the sigma-orbitals will affect the behavior of the pi-electrons, whi
ch is called the scattering of pi- electrons. This scattering effect is taken into account in the nanotube and the local wave function of pi-electrons is constructed, which is called the extended Wannier function. In the Wannier representation, the electronic hopping energies and the energy gap of the tubes (9,0) and (9,9) are calculated. Our results show that the band gap of the tubes increases in direct ratio with the scattering coefficients of sigma-orbitals and this scattering is able to enhance the localization of pi-electrons.
According to the tight-binding approximation, we investigate the electronic structures of graphene ribbons with zigzag shaped edges (ZGRs) and armchair shaped edges (AGRs) drawn by the tensile force, and obtain the analytic relations between the ener
gy bands of pi-electrons in ZGR, AGR and the tensile force based on only considering the nearest-neighbor interaction and the hydrogen-like atomic wave function is considered as pi-electron wave function. Importantly, we find the tensile force can open an energy gap at the K point for ZGR and AGR, and the force perpendicular to the zigzag edges can open energy gap more easily besides the gap values of ZGR and AGR at the K point both increase as the tensile force increases.
