Effects of edge magnetism and external electric field on energy gaps in multilayer graphene nanoribbons

Using first-principles density-functional theory, we study the electronic structure of multilayer graphene nanoribbons as a function of the ribbon width and the external electric field, applied perpendicular to the ribbon layers. We consider two types of edges (armchair and zigzag), each with two edge alignments (referred to as $\alpha$ and $\beta$ alignments). We show that, as in monolayer and bilayer armchair nanoribbons, multilayer armchair nanoribbons exhibit three classes of energy gaps which decrease with increasing width. Nonmagnetic multilayer zigzag nanoribbons have band structures that are sensitive to the edge alignments and the number of layers, indicating different magnetic properties and resulting energy gaps. We find that energy gaps can be induced in $ABC$-stacked ribbons with a perpendicular external electric field while in other stacking sequences, the gaps decrease or remain closed as the external electric field increases.

Figure 1

(Color online) Schematic illustration of (a) three types of stacking arrangements, labeled by A, B, and C, and (b) two types of edge alignments, $\alpha$ alignment and $\beta$ alignment in multilayer graphene nanoribbons. The two edge alignments are distinguished by different ways of shifting the top layer with respect to the other in finite size multilayer graphene stacks. The arrows indicate the direction of edges along which nanoribbons span infinitely.

Figure 2

Energy-band structure of trilayer armchair nanoribbons for (a) $ABA$-$\alpha$, metallic nanoribbon, (b) $ABA$-$\alpha$, semiconducting nanoribbon (c) $ABC$-$\alpha$, metallic nanoribbon, and (d) $ABC$-$\alpha$, semiconducting nanoribbon. The chosen widths for metallic and semiconducting ribbons are 0.86 nm and 1.11 nm which corresponds to $N=8$ and $N=10$, respectively, where $N$ is the number of carbon chains along the width direction.

Figure 3

(Color online) Variation in the energy gap with widths of (a) $ABA$-$\alpha$, (b) $ABA$-$\beta$, (c) $ABC$-$\alpha$, and (d) $ABC$-$\beta$-aligned nanoribbons. Three classes of the nanoribbons are clearly seen in (a)–(d) and specified by $N=3p$, $3p+1$, and $3p+2$ where $N$ is the number of carbon chains along the width direction. Here $p=1,2,\dots ,13$, which translate to nanoribbons with widths up to 5 nm.

Figure 4

(Color online) Variation in the energy gap with widths of tetralayer nanoribbons (a) $ABAB$-$\alpha$, (b) $ABCA$-$\alpha$. Three classes of nanoribbons are clearly seen. These classes are specified by $N=3p$, $3p+1$, and $3p+2$ where $N$ is the number of carbon chains along the width direction. Here $p=1,2,\dots ,13$, which translate to nanoribbons with widths up to 5 nm.

Figure 5

Energy-band structure of $ABA$-stacked trilayer zigzag nanoribbons with $\alpha$ and $\beta$ alignments. Panels (a) and (b) are, respectively, for nonmagnetic and magnetic $\alpha$-aligned ribbons, and panels (c) and (d) are for $\beta$-aligned ribbons. The width is 1.7 nm which corresponds to $N=8$ where $N$ is the number of zigzag chains along the width directions.

Figure 6

(Color online) Variation in energy gap with the widths of (a) $ABA$ and (b) $ABC$-stacked trilayer zigzag nanoribbons.

Figure 7

Energy-band structure of multilayer zigzag nanoribbons for $ABA$-stacked (a) tetralayer, (b) pentalayer, and (c) hexalayer with $\alpha$ alignment. The left panels show the nanoribbons with nonmagnetic edge atoms and the right panels show the nanoribbons with edge atoms ordered antiferromagnetically layerwise. The width is chosen as 1.7 nm.

Figure 8

(Color online) Variation in the energy gap with perpendicular external electric field for $ABA$-stacked and $ABC$-stacked trilayer armchair ribbons. Panels (a) and (b) are for $\alpha$- and $\beta$-aligned $ABA$-stacked ribbon, respectively, whereas panels (c) and (d) are, respectively, for $ABC$-stacked $\alpha$- and $\beta$-aligned ribbons. Both metallic $\left(W=0.86\text{nm}\right)$ and semiconducting $\left(W=1.11\text{nm}\right)$ nanoribbons are considered which corresponds to $N=8$ and $N=10$, respectively, where $N$ is the number of armchair carbon chains along the width direction.

Figure 9

(Color online) Variation in the energy gap with perpendicular external electric field for (a) $ABA$ and (b) $ABC$-stacked trilayer zigzag nanoribbons with $\alpha$-alignments. $\beta$-aligned nanoribbons show similar behavior. Both narrow ($W=1.74$ and 1.85 nm) and wide ($W=5.11$ and 3.75 nm) nanoribbons are considered.