Wannier-Function Method Of Energy Bands For Graphene

Graphene’s theoretical research began in1947, so far has a history of more than60years.Actually2D graphene crystals can not exist independently.Until2004,the British astronomy physics professor Andre Geim’s research group from Manchester University prepared it for the frist time,hence,Geim and his partner Konstantin Novoselov won the2010Nobel Prize in physics. Due to the excellent properties of mechanics, thermodynamics, electromagnetism, graphene is expected to have wide application in nano electronics, composite materials, field emission materials, energy storage, and other fields. In recent years,graphene has quickly become one of the hot spot in the field of material science and condensed matter physics.Because graphene is a wide band system and the overlap of electronic wave function of adjacent lattice is larger, the tight-binding approach for calculating the electronic properties of graphene is too rough. In order to reflect the behavior of the electrons in graphene better, tight binding approximation model must be amended.We construct the Wannier functions of graphene and applied it to calculate the energy band.The non-nearest-neighbor hopping terms of electrons are taken into account and the energy spectra of graphene are given analytically. Different energy gap is needed when we put the graphene into application,so we also discuss the electronic structures of the deformed graphene.Wannier functions of graphene are constructed by a linear combination of electronic wave functions. Compare the cell function diagram and the wannier functions diagram, we found the crest of wannier functions is lower, the spacing of neighbor peak of wannier functions is smaller, and the wave of wannier functions is falling faster. If put multiple diagram of the wave function together, the overlap of wave function will be less. We also gives spatial distribution of electric charge and the charge density contour of graphene, the numerical results show that the charge density contour by wannier functions model is more compact, shrinkage on the x axis obviously, which are illustrate the locality of the wannier functions is better.We show the three dimensional spectrum diagram of graphene by wannier functions method.Compared with the three dimensional spectrum diagram of graphene by tight binding approximation method we found that, the graphic of former is more flat and the band of former is more narrow.This is the impact of the locality of wannier functions.In the study of the electronic structures of the deformed graphene by wannier functions method, for considering the only nearest-neighbors we obtain the analytic relations between a tensile force and energy gaps.In the model of graphene with zigzag shaped edges we found the energy gaps broaden as the tensile force increases while the magnitude of broaden is smaller than the numerical result by tight binding approximation method.