Figure 1

(Color online) (a) Lattice structure of graphene (or h-BN) constructed by two interpenetrating triangular sublattices, (b) corresponding Brillouin zone and (c) lattice structure of graphene (black points) on h-BN. Boron atoms are denoted as red (darker) circles, nitrogen atoms as blue (lighter) ones.Reuse & Permissions

Figure 2

(Color online) Band structure near $K$ (or $K^{\prime }$) point for (a) graphene bilayer and (e) graphene on h-BN with ${ϵ}^{{\alpha }_1}=-{ϵ}^{{\beta }_1}=2.95\text{eV}$. Graphs (b)–(d) show the band structures for bilayer systems with different values of energy parameters ${ϵ}_{2p_z}^{{\alpha }_1}$ and ${ϵ}_{2p_z}^{{\beta }_1}$ describing the evolution of band structure from graphene bilayer toward graphene on h-BN. The onsite energies sets used for these three graphs: (b) ${ϵ}^{{\alpha }_1}=-{ϵ}^{{\beta }_1}=0.2\text{eV}$, (c) ${ϵ}^{{\alpha }_1}=-{ϵ}^{{\beta }_1}=\left|{\gamma }_1\right|/\sqrt{2}\approx 0.28\text{eV}$, and (d) ${ϵ}^{{\alpha }_1}=-{ϵ}^{{\beta }_1}=0.32\text{eV}$. The hopping parameters of graphite ${\gamma }_1=0.39\text{eV}$ and ${\gamma }_0=3.16\text{eV}$ known from experimental data (Ref. 32) fitted into the Slonczewski-Weiss-McClure model (Refs. 33, 34) and ${\gamma }_0^{\prime }={\gamma }_0$ have been chosen for presented calculations. Blue dotted-dashed line denotes a band with the value $E_1$ [Eq. (19)] at the Dirac points; red dashed line is assigned to the value $E_2$ [Eq. (20)]; and olive dotted to the value $E_4$ [Eq. (22)] and red solid to the value $E_3$ [Eq. (20)].Reuse & Permissions

Figure 3

(Color online) Variation in the energy gap as a function of energy difference between two layers. The insets show the band structures in three characteristic points indicated on the gap variation curve. The TB parameters are taken from Table . Color scheme used in the insets is the same as in Fig. 2.Reuse & Permissions

Figure 4

(Color online) Band structures of graphene on h-BN bilayer for different values of $U$. Blue dotted-dashed line denotes a band with the value $E_1$ [Eq. (24)] at $K$ point; red dashed line is assigned to the value $E_2$ [Eq. (25)]; olive dotted to the value $E_4$ [Eq. (27)] and red solid to the value $E_3$ [Eq. (26)]. The TB parameters are taken from Table .Reuse & Permissions

Figure 5

(Color online) The TB fit along $\Gamma KM$ line in the vicinity of K point for LDA calculations. Red dashed lines denote DFT bands while black dotted lines denote TB fit. The inset show DFT band structure in $\Gamma K$ and $KM$ directions. The energies are measured from the zero level fixed by the condition ${ϵ}_{2p_z}^{{\alpha }_2}=0$.Reuse & Permissions

Figure 6

DFT band structure along $\Gamma KM$ line in momentum space of bilayer graphene/h-BN in the presence of external electric field with the effective value of $\text{U}=1.40\text{V}$ applied in the direction perpendicular to the layers. The inset presents zoom of the band structure in the vicinity of K point, where the band gap is closed.Reuse & Permissions

Figure 7

(Color online) DFT band structure along $\Gamma KM$ line in momentum space of bilayer graphene/h-BN in the presence of external electric field with the effective value of $U=-2.0\text{V}$ applied in the direction perpendicular to the layers. Red arrow points out the edge of conduction band originating from hybridized states comprising $\pi$ carbon band and $\sigma$ boron nitride band.Reuse & Permissions