We study the electronic structures and the transport properties of graphene sheets having two-dimensional (2D) and one-dimensional (1D) periodic nanoholes using a first-principles method. The symmetries and periodicities of graphene with 2D periodic nanoholes are classified as metallic or semiconducting. The transport properties for 1D periodic nanoholes can be understood by analogy to 2D periodic nanoholes, but the conduction gaps are smaller than those of 2D periodic nanoholes due to the absence of quantum confinement in one direction. However, when sequential zigzags lie on the hole edge, the conduction gaps become significantly wide. We also show the neck width (

w

) dependency of the conduction gap (

E_{g}

) of large-scale 2D periodic irregular holes for comparison with recent experiments, using the tight-binding approximation method. The value of

E_{g}

is generally inversely proportional to

w

, agreeing with the experiment. However, the distribution of

E_{g}

at each

w

is quite wide, which is attributed to the various lengths of the sequential zigzag edges. Our analysis shows that zigzag edges play a critical role in determining the conduction gaps for graphene with both 2D and 1D periodic nanoholes. These conclusions are obtained on the basis of the results for nonmagnetic planar high-symmetry configurations. Although we also investigate the most stable configurations by considering magnetism and nonplanar geometries, it does not significantly affect the transport properties discussed in this paper. DOI: http://dx.doi.org/10.1103/PhysRevB.84.075467 ©2011 American Physical Society

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