Energy Dispersions and Minimal Conductivity of Bernal Multilayer Graphene

摘要

In this work, we derive the analytical expression of the energy dispersions and minimal conductivity of the Bernal multilayer graphenes with the trigonal warping γ3 taken into consideration. Under unitary transformation, both the Hamiltonian matrix H and current operator partial deriv H/partial deriv k of an N-layer Bernal graphene can be exactly reduced to block diagonal matrices. As the layer number N is even (odd), the 2N X 2N matrix is decomposed into N/2 four by four blocks (one 2 x 2 and (N-1)/2 4 x 4 blocks). Each of the Hamiltonian blocks represents an independent subsystem, which is an equivalent bilayer or a monolayer graphene. The analytical expression of energy dispersions are then obtained. Most importantly, the minimal conductivity, based on the Kubo formula, of Bernal multilayer graphene is equal to the summation of the direct current conductivity of each subsystem. The analytical formula of minimal conductivity is (N/2) 2A/π (e~2/h) [4/π(e~2/h) + ((N - 1)/ 2)24/π(e~2/h)] for even (odd) N, where 24/π (e~2/h) is the conductivity of a bilayer graphene with trigonal warping [Phys. Rev. Lett. 2007, 99, 066802].