Geometrically induced transitions between semimetal and semiconductor in graphene

How the long-range ordering and local defect configurations modify the electronic structure of graphene remains an outstanding problem in nanoscience, which precludes the practical method of patterning graphene from being widely adopted for making graphene-based electronic and optoelectronic devices, because a small variation in supercell geometry could change the patterned graphene from a semimetal to a semiconductor, or vice versa. Based on the effective Hamiltonian formalism, here we reveal that a semimetal-to-semiconductor transition can be induced geometrically without breaking the sublattice symmetry. For the same patterning periodicity, however, breaking the sublattice symmetry increases the gap, while phase cancellation can lead to a semiconductor-to-semimetal transition in non-Bravais lattices. Our theory predicts the analytic relationship between long-range defect ordering and band-gap opening/closure in graphene, which is in excellent agreement with our numerical ab initio calculations of graphene nanomeshes, partially hydrogen passivated and boron-nitride-doped graphene.

Figure 1

Crystal (left) and electronic band (right) structures for (a) H-passivated graphene, (c) GNM, and (e) BN-doped graphene with supercell lattice parameters $(n_1,m_1,n_2,m_2)=(6,-6,4,4)$. (b, d, f) The corresponding structures with $(n_1,m_1,n_2,m_2)=(7,-7,4,4)$. Here C, H, B, and N atoms are denoted by gold, blue, green, and red circles, respectively. The black rectangle in each panel indicates the supercell of defected graphene.

Figure 2

Band gap ($E_g$) as a function of defect percentage for (a) partially H-passivated graphene and GNMs and (b) BN-doped graphene.

Figure 3

Crystal (left) and electronic band (right) structures of defected graphene sheets, which have $(7,1,8,-7)$ supercells in (a) and (b) and $(9,0,0,9)$ supercells in (c)–(h). Except for (a), defects in these graphene structures form non-Bravais lattices. (a–d) H-passivated graphene; (e, f) GNMs; (g, h) BN-doped graphene. Supercells in these graphene structures are indicated by black rhombuses.

Figure 4

Crystal (left) and electronic band (right) structures of BN-doped graphene with supercell lattice parameters of $(6,1,-3,9)$ in (a) and (b) and $(6,3,-2,7)$ in (c) and (d). (a, c) Four doped areas in a supercell are identical, while (b, d) in a supercell two of the four BN-doped areas are different from the other two, so that an overall balance between $A$ and $B$ sublattices is reached. In all panels the supercell is indicated by a black rhomboid.