Band Gap Scaling Laws In Group Ⅳ Nanotubes：A First-principles Study

Very recently,the new 2D monolayer of tin has been experimentally realized via the epitaxial growth method.That is,since we obtain graphene,we have almost realized all the 2D monolayers in group Ⅳ successively by various approaches excepting for the metal lead element.Like graphene,the other group Ⅳ monolayers,namely,silicene,germanene,and stanene,also behave like the properties of massless Dirac fermions,which give rise to ultimate high charge carrier mobility.Also,these 2D monolayers are promising to support 2D quantum spin Hall states and provide enhanced thermoelectricity,topological superconductivity and the quantum anomalous Hall effect,and so on.Similarly,the 1D nanotubes based on those group Ⅳ elements have also attracted increasing interest with the achievement of these 2D monolayers.In fact,carbon nanotubes(CNTs)have aroused considerable scientific interest due to their novel electronic properties and applications in many fields even earlier than graphene.As we known,one intriguing property of CNTs is that they can exhibit metallic or semiconductor properties depending on their chiral vectors.For silicon nanotubes,For silicon nanotubes,due to their unique gearlike structures caused by partial sp3 hybridization,previous studies have revealed that the armchair nanotubes exhibit semiconductor properties with decreasing band gaps as radii of nanotubes increase.On the other hand,band gaps also oscillate with a period of 3 for zigzag silicon nanotubes.By using the first-principles calculations,the band gap properties of nanotubes formed by group Ⅳ elements have been investigated systemically.we investigate the generality of the band gap properties of a large range of carbon-,silicon-,germaniumand tin-based SWNTs.Armchair nanotubes(n,n)and zigzag nanotubes(n,0)are constructed and their band structures are studied systematically.Our results reveal that for armchair nanotubes,the energy gaps at K points in the Brillouin zone decrease as 1/R scaling law with the radii(r)increasing,while they are scaled by-1/R2+C at Γ points,here,C is a constant.Further studies show that such scaling law of K points is independent of both the chiral vector and the type of elements.Therefore,the band gaps of nanotubes for a given radius can be determined by these scaling laws easily.Interestingly,we also predict the existence of indirect band gap for both germanium and tin nanotubes.Our new findings provide an efficient way to determine the band gaps of group Ⅳ element nanotubes by knowing the radii,as well as to facilitate the design of functional nanodevices.