Research Paper: First Order Energy Near Dirac Points for Graphene and Carbon Nanotubes with Arbitrary Chirality

The effective Hamiltonian describing indirect exchange interactions of type Ruderman- Kittel-Kasuya-Yosida [RKKY] between the magnetic impurities near the Dirac Points is based on the first-order energy. Besides, the points where the impurities are located are important in identifying the oscillatory behaviour of these interactions. Then in this paper, we first intend to obtain the effective Hamiltonian elements near the Dirac points for 2D graphene structures using the tight-binding approximation and then the phase factors between neighbor Dirac points are noticed. The obtained results are extended to obtain the effective Hamiltonian and then the first-order energy for carbon nanotubes. Using the quantized wave vector in the circumferential direction of nanotubes at Dirac Points, we examine the condition for nanotubes to be metallic or semiconductors. The obtained results based on the tight binding could be used to study the magnetic interactions at low energy for graphene structures as well as carbon nanotubes