Energy gaps, magnetism, and electric-field effects in bilayer graphene nanoribbons

Using a first-principles density-functional electronic structure method, we study the energy gaps and magnetism in bilayer graphene nanoribbons as a function of the ribbon width and the strength of an external electric field between the layers. We assume AB (Bernal) stacking and consider both armchair and zigzag edges and two edge alignments distinguished by different ways of shifting the top layer with respect to the other. Armchair ribbons exhibit three classes of bilayer gaps which decrease with increasing ribbon width. An external electric field between the layers increases the gap in narrow ribbons and decreases the gap for wide ribbons, a property which can be understood semianalytically using a $\pi$-band tight-binding model and perturbation theory. The magnetic properties of zigzag edge ribbons are different for the two different edge alignments, and not robust for all exchange-correlation approximations considered. Bilayer ribbon gaps are sensitive to the presence or absence of magnetism.

Figure 1

(Color online) Schematic illustration of the two edge alignments we consider in bilayer graphene. (a) $\alpha$ alignment and (b) $\beta$ alignment. The $\beta$ alignment can be obtained from the $\alpha$ alignment by shifting the top layer.

Figure 2

(Color online) Variation of energy gap with ribbon for balanced bilayer armchair ribbons. (a) $\alpha$ alignment and (b) $\beta$ alignment. Three classes of ribbons, denoted as $3p$, $3p+1$, and $3p+2$, are shown. For $\alpha$-aligned ribbons, $p=1–13$ whereas for $\beta$-aligned ribbons, $p=1–7$, 9, 13.

Figure 3

(Color online) Schematic illustration of the TB Hamiltonian for the coupled two-leg ladder system. The labels used here are explained in the text.

Figure 4

(Color online) Variations of energy gap of the three classes of bilayer armchair ribbons with applied electric field and widths between 1–5 nm for $\alpha$-aligned ribbons. For clarity, the ribbon with the width of 4.92 nm is denoted by stars.

Figure 5

Band structure of bilayer zigzag ribbons in $\alpha$ alignment in which the edge atoms are (a) without any magnetic order and (b) with ferromagnetic order between the layers. The width of the ribbon is 1 nm and the Fermi level is set at zero. Clearly, the energy spectrum is gapless in (a) but has a finite gap in (b).

Figure 6

Same as Fig. 5 but for $\beta$-aligned ribbons.

Figure 7

(Color online) Variation of the energy gap with the width of bilayer zigzag ribbons in both edge alignments using both LDA and GGA. For $\alpha$ alignments, LDA predicts a nonmagnetic ground state, and therefore no energy gap in the energy spectrum.

Figure 8

(Color online) Variation of the energy gap with external electric field for zigzag bilayer ribbons with (a) $\alpha$ alignment and (b) $\beta$ alignment.