Tight binding Hamiltonian and energy dispersion relations in monolayer graphene

Abstract

The crystal structure of monolayer graphene is a honeycomb structure of carbon atoms with each unit cell containing two carbon atoms; one π electron is associated with each carbon atom as a conduction electron. The first Brillouin zone is derived as a hexagon containing one electron per atom. The wave function in the tight binding approximation is written as the sum of two component wave functions through an undetermined coefficient. Each component wave function is given by the linear combination of the normalized carbon orbitals centered at each of the same type of carbon atom. The elements of the Hamiltonian matrix are defined with the overlap integral or the exchange integral by component wave functions. With the elimination of the undetermined coefficient, and since only nearest neighbor hopping is allowed in the tight binding approximation, the final Hamiltonian matrix elements become zero for diagonal elements and hopping energies ℏv_F (κ_x−iκ_y) and ℏv_F (κ_x+iκ_y), respectively, for off-diagonal elements. κ is the wave vector measured from the corner points of the first Brillouin zone, v_F=3a₀γ₀/2ℏ=0.874×10⁶ [m·s⁻¹]∼c/300 is the Fermi velocity, i.e. the velocity of the massless particle after the supposed π electron conversion; a₀ is the distance between nearest neighbor carbon atoms; −γ₀ is the hopping integral; ℏ is the Planck constant divided by 2π; and c is the velocity of light. The Hamiltonian matrix is obtained as ℏv_Fσ̂·κ; and the energy dispersion relation as ε=±ℏv_Fκ. σ̂ is the Pauli matrix.