Effects of magnetism and electric field on the energy gap of bilayer graphene nanoflakes

We study the effect of magnetism and perpendicular external electric field strengths on the energy gap of length confined bilayer graphene nanoribbons (or nanoflakes) as a function of ribbon width and length using a first-principles density-functional electronic structure method and a semilocal exchange-correlation approximation. We assume $AB$ (Bernal) bilayer stacking and consider both armchair and zigzag edges, and for each edge type, we consider the two edge alignments, namely, $\alpha$ and $\beta$ edge alignment. For the armchair nanoflakes we identify three distinct classes of bilayer energy gaps, determined by the number of carbon chains in the width direction ($N=3p$, $3p+1$ and $3p+2$, $p$ is an integer), and the gaps decrease with increasing width except for class $3p+2$ armchair nanoribbons. Metallic-like behavior seen in armchair bilayer nanoribbons are found to be absent in armchair nanoflakes. Class $3p+2$ armchair nanoflakes show significant length dependence. We find that the gaps decrease with the applied electric fields due to large intrinsic gap of the nanoflake. The existence of a critical gap with respect to the applied field, therefore, is not predicted by our calculations. Magnetism between the layers plays a major role in enhancing the gap values resulting from the geometrical confinement, hinting at an interplay of magnetism and geometrical confinement in finite size bilayer graphene.

Figure 1

(Color online) Schematic illustration of the two edge alignments we consider in bilayer graphene nanoflakes. (a) $\alpha$ alignment and (b) $\beta$ alignment.

Figure 2

(Color online) Variation in the energy gap with width of $\alpha$-aligned bilayer armchair nanoflakes for a fixed length $L\approx 3.2\text{nm}$ (open circles). Three classes of the nanoflakes are clearly seen in (a) $3p$, (b) $3p+1$, and (c) $3p+2$ where $p$ is an integer specifying the number $N$ of carbon chains along the width direction. Here $p=1,2,\cdots ,8$, which translate to nanoflakes with widths less than but close to 3 nm. The gaps of the magnetic nanoflakes (open circles) are compared with those of the nonmagnetic nanoflakes (solid circles) and three classes of nonmagnetic nanoribbons (open triangles). It should be noted that armchair nanoribbons are all nonmagnetic and all the gaps are obtained using the same semilocal GGA potential.

Figure 3

(Color online) Same as Fig. 2 but for $\beta$-aligned bilayer armchair nanoflakes.

Figure 4

(Color online) Variations in the energy gap of the (a) $\alpha$-aligned and (b) $\beta$-aligned bilayer zigzag nanoflakes with a fixed length $L\approx 3.3\text{nm}$. For comparison, the gap values of nonmagnetic nanoflakes and magnetic nanoribbons are also shown. All the gap calculations are done with the same semilocal GGA potential and for magnetic calculations an interlayer antiferromagnetic order was considered.

Figure 5

(Color online) Variation in the armchair and zigzag nanoflake gaps with different lengths are shown. In longer nanoflakes, the nonmagnetic gaps are shown to be smaller than their magnetic counterpart.

Figure 6

(Color online) Variation in the energy gap with perpendicular external electric field for bilayer nanoflakes with $\alpha$ alignments for (a) armchair $\left(L\approx 3.2\text{nm}\right)$ and (b) zigzag nanoflakes $\left(L\approx 3.3\text{nm}\right)$. For each class of armchair nanoflakes, three representative widths specified by the interger $p=3$, 6, and 8, are chosen. The gaps are shown at three representative values of the external electric field strength. The maximum electric field strength applied is close to 1 V/nm.