Absence of edge states near the 120${}^{\circ }$ corners of zigzag graphene nanoribbons

Based on first-principles total-energy calculations, we studied the energetics and electronic structure of zigzag graphene nanoribbons with nanometer-scale corners. We found that the formation energy of a short zigzag edge with a 120${}^{\circ }$ corner is substantially smaller than that of a straight zigzag edge with an infinite length. We also found that zigzag graphene ribbons with 120${}^{\circ }$ corners are semiconductors for which the band-gap decreases with increasing length between adjacent corners. The edge state is absent for zigzag graphene ribbons with short corner-corner distances, while the edge state emerges for zigzag ribbons that have a corner-corner distance greater than 1.5 nm. The remarkable stability of the 120${}^{\circ }$ corner of the short zigzag edges is in good agreement with the experimental observation of such corners of graphene edges.

Figure 1

Optimized structures of a graphene ribbon with (a) 120${}^{\circ }$ edge corners and (b) 150${}^{\circ }$ edge corners. The white and black spheres denote the hydrogen and carbon atoms, respectively.

Figure 2

Formation energy per zigzag edge atom of a graphene ribbon with 120${}^{\circ }$ corners as a function of width $W$ at various edge lengths.

Figure 3

Formation energy per zigzag edge atom of graphene ribbons with 120${}^{\circ }$ corners and 150${}^{\circ }$ corners as a function of edge length $L$.

Figure 4

(a, c) Electronic energy bands and (b, d) wave-function distribution (red color region) of valence-band maximum (VBM) and conduction-band minimum (CBM) states at the $\Gamma$ point of a graphene ribbon with 120${}^{\circ }$ corners at $L=4.89$ Å, $W=7.336$ Å and at $L=17.12$ Å, $W=7.336$ Å. Energy is measured from the midgap point of the system. The white and black spheres denote hydrogen and carbon atoms, respectively.

Figure 5

Energy gap of a graphene ribbon with 120${}^{\circ }$ corners as a function of edge length $L$.

Figure 6

Bandwidths of the highest branch of the valence band (HBVB) and the lowest branch of the conduction band (LBCB) of a graphene ribbon with 120${}^{\circ }$ corners as a function of edge length $L$. The bandwidth is normalized by the length of the Brillouin zone due to the difference in the lattice parameter.

Figure 7

(a) Electronic energy band and (b) wave-function distribution (red color region) of the valence-band maximum (VBM) and conduction-band minimum (CBM) states at the $\Gamma$ point of a graphene ribbon with 150${}^{\circ }$ corners at $L=7.336$ Å, $W=7.336$ Å. Energy is measured from the midgap point of the system. The white and black spheres denote hydrogen and carbon atoms, respectively. (c) Energy gap of a graphene ribbon with 150${}^{\circ }$ corners as a function of edge length $L$.