Electronic state of the limited graphene

Abstract

The limited graphene means that two directions of graphene are limited, one is zigzag type boundary and the other is armchair type boundary. Based on the tight-binding model, the electronic state and band of the limited graphene are given analytically. The results show that there are only two kinds of electronic states, i.e., the standing wave state and edge state. For the standing wave state, the wave function is in the form of sine function in two directions; for the edge state, the wave function is in the form of hyperbolic sine function in the direction of armchair boundary and in the form of sine function in the direction of zigzag boundary. The band is composited of total carbon atom number N discrete eigenvalues. The expression of quantitativly calculating the number of eigenvalues of edge state is deduced. Through the density of states of the limited graphene we analyze the existence of the edge state and the consistency in the infinity case. The results from the analitical method are the same as the numerical resullts. When the width of two restricted boundary goes into infinity, the result of the limited graphene tends to that in the infinity case.