This paper presents scaling rules for the band gaps of graphene nanoribbons (GNRs) based on first-principles calculations. The GNRs considered have either armchair or zigzag shaped edges with hydrogen passivation. Both types of ribbons are shown to have band gaps, which differs from results of simple tight-binding calculations or solutions of the Dirac equation. The origin of energy gaps for GNRs with armchair edges arises from quantum confinement and edge effects. For GNRs with zigzag edges, gaps arise from a staggered sublattice potential due to edge magnetization. The rich gap structure for armchair edge GNRs is obtained analytically including edge effects. These results reproduce ab initio calculations well. The electronic structure of nanoscale carbon materials, such as fullerenes and carbon nanotubes, has been studied extensively due to their potential applications in nanoelectronics. Among carbon nanostructures, graphene ribbons with nanometer-sized widths have been studied extensively. GNRs can be realized by cutting or patterning epitaxially grown graphenes. The electronic structures of GNRs have been modeled using boundary conditions on the Schrödinger equation with simple tight-binding approximations or a 2D free massless particle Dirac equation. In previous models, it was predicted that GNRs with armchair edges can be either metallic or semiconducting depending on their widths, while GNRs with zigzag edges are metallic with edge states. However, edge effects are crucial in determining the bandgaps of GNRs. The spin degree of freedom is also important because GNRs with zigzag edges have narrow-band edge states at the Fermi energy, implying possible magnetization at the edges. The authors performed first-principles calculations and theoretical analysis to explore the relation between the bandgap and the geometries of GNRs. They show that GNRs with hydrogen passivated armchair or zigzag edges both have nonzero and direct bandgaps. The origins of the bandgaps for different types of edges vary. For armchair edge GNRs, the bandgaps originate from quantum confinement and edge effects. For zigzag edge GNRs, the bandgaps arise from a staggered sublattice potential due to spin ordered states at the edges. The authors used first-principles self-consistent pseudopotential method with local density approximation. They found that the bandgaps of GNRs with armchair edges decrease as the width increases. The energy gaps exhibit three distinct family behaviors. The TB results using a constant nearest neighbor hopping integral are quite different from those by first-principles calculations. The first-principles calculations show that the gaps are well separated into three different categories. The gaps for armchair edge GNRs are inversely proportional to their width. The authors also found that the bandgaps of zigzag edge GNRs decreaseThis paper presents scaling rules for the band gaps of graphene nanoribbons (GNRs) based on first-principles calculations. The GNRs considered have either armchair or zigzag shaped edges with hydrogen passivation. Both types of ribbons are shown to have band gaps, which differs from results of simple tight-binding calculations or solutions of the Dirac equation. The origin of energy gaps for GNRs with armchair edges arises from quantum confinement and edge effects. For GNRs with zigzag edges, gaps arise from a staggered sublattice potential due to edge magnetization. The rich gap structure for armchair edge GNRs is obtained analytically including edge effects. These results reproduce ab initio calculations well. The electronic structure of nanoscale carbon materials, such as fullerenes and carbon nanotubes, has been studied extensively due to their potential applications in nanoelectronics. Among carbon nanostructures, graphene ribbons with nanometer-sized widths have been studied extensively. GNRs can be realized by cutting or patterning epitaxially grown graphenes. The electronic structures of GNRs have been modeled using boundary conditions on the Schrödinger equation with simple tight-binding approximations or a 2D free massless particle Dirac equation. In previous models, it was predicted that GNRs with armchair edges can be either metallic or semiconducting depending on their widths, while GNRs with zigzag edges are metallic with edge states. However, edge effects are crucial in determining the bandgaps of GNRs. The spin degree of freedom is also important because GNRs with zigzag edges have narrow-band edge states at the Fermi energy, implying possible magnetization at the edges. The authors performed first-principles calculations and theoretical analysis to explore the relation between the bandgap and the geometries of GNRs. They show that GNRs with hydrogen passivated armchair or zigzag edges both have nonzero and direct bandgaps. The origins of the bandgaps for different types of edges vary. For armchair edge GNRs, the bandgaps originate from quantum confinement and edge effects. For zigzag edge GNRs, the bandgaps arise from a staggered sublattice potential due to spin ordered states at the edges. The authors used first-principles self-consistent pseudopotential method with local density approximation. They found that the bandgaps of GNRs with armchair edges decrease as the width increases. The energy gaps exhibit three distinct family behaviors. The TB results using a constant nearest neighbor hopping integral are quite different from those by first-principles calculations. The first-principles calculations show that the gaps are well separated into three different categories. The gaps for armchair edge GNRs are inversely proportional to their width. The authors also found that the bandgaps of zigzag edge GNRs decrease