Superlattice structures of graphene-based armchair nanoribbons

Based on first-principles calculations we predict that periodically repeated junctions of armchair graphene nanoribbons of different widths form multiple quantum well structures. In these superlattice heterostructures the width as well as the energy-band gap is modulated in real space and specific states are confined in certain segments. Not only the size modulation, but also composition modulation, such as periodically repeated and commensurate heterojunctions of boron nitride and graphene honeycomb nanoribbons, results in a multiple quantum well structure. The geometrical features of the constituent nanoribbons, namely, their widths and lengths, the form of the junction, as well as the symmetry of the resulting superlattice, are the structural parameters available to engineer electronic properties of these quantum structures. We present our analysis regarding the variation of the band gaps and the confined states with these structural parameters. Calculation of transmission coefficient through a double barrier resonant tunneling device formed from a finite segment of such a multiple quantum well structure and placed between metallic electrodes yields resonant peaks which can be identified with electronic states confined in the well. We show that these graphene-based quantum structures can introduce interesting concepts to design nanodevices. Relevance of the quantum structures are discussed in view of the most recent experimental results.

Figure 1

(Color online) Band structure of AGNR(26) calculated with the three different methods used in this work: (a) plane-wave DFT, (b) localized-orbital DFT, (c) and ETB.

Figure 2

(Color online) (a) Bare and hydrogen-terminated AGNR(10). Atomic geometry, electronic band structure, and isosurface charge densities of edge and “uniform” states. The primitive unit cell is delineated with dashed lines and includes $n=10$ carbon atoms. Carbon and hydrogen atoms are shown by large and small balls. (b) Same for AGNR(14). All data in this figure are calculated by first-principles method (Refs. 26, 27).

Figure 3

(Color online) (a) Possible junction angles leading to armchair or zigzag edge shapes. $60°$ angle with the longitudinal axis results in armchair edge at the interface, whereas $30°$ and $90°$ give zigzag edges. Some of the possible superlattice shapes are shown; namely, (b) sharp rectangular, (c) smooth, (d) and sawtoothlike.

Figure 4

(Color online) (a) Atomic structure of AGSL($n_1=10$, $n_2=14$; $s_1=3$, $s_2=3$). The superlattice unit cell and primitive unit cell of each segment are delineated. (b) Band structure with flat bands corresponding to confined states. (c) Isosurface charge density of propagating and confined states. (d) Variation of various superlattice gaps with $s_1=s_2$. (e) Confinement of states versus $s_1=s_2$ calculated by ETB. All data except those in (e) are calculated by using first-principles method (Refs. 26, 27).

Figure 5

(Color online) (a) Variation of band gaps $E_g$, ${\Delta }_1$, and ${\Delta }_2$ of the nanoribbon superlattice AGSL(10,14:3,3). (b) Variation of the total energy $E_T$ with respect to $ϵ$ and the force constant ${\kappa }_{\text{SL}}$ [in $\text{eV}/\text{Å}$] for AGSL(10,14;3,3) and its constituent nanoribbons. All data in this figure are calculated by using first-principles method (Refs. 26, 27).

Figure 6

(Color online) The effect of the variation of narrower region $s_1$ of AGSL $\left(10,14;s_1,3\right)$ from $s_1=3$ to 8. (a) Atomic structure and superlattice unit cell. (b) The variation of band structures. (c) The numerical values for energy gaps $E_g$, ${\Delta }_1$, and ${\Delta }_2$. $E_g$ is the actual band gap of the structure which comes from a dispersive state. ${\Delta }_1$ is the band gap of highest localized state while ${\Delta }_2$ is the band gap for the next dispersive state. The energy of the flat-band states related with ${\Delta }_1$ are confined to $s_2$ and their weights are practically independent of $s_1$. All data in this figure are calculated by using first-principles method (Refs. 26, 27).

Figure 7

(Color online) (a) Band structure of AGSL(10,18;3,3) with flat bands corresponding to confined states $\left({\Delta }_1=E_g\right)$. (b) Isosurface charge density of propagating and confined states. (c) Density of states (DOS) of AGSL(10,18;3,3) with sharp peaks corresponding to confined states. Calculations are performed by using first-principles method (Refs. 26, 27).

Figure 8

(Color online) Energy-band structure of the AGSL(34,70;11,9) superlattice and the charge densities of selected bands. As seen clearly, states associated with flat bands 1, 3, and 4 are confined but the state with dispersive band indicated by 2 is propagating. Calculations have been performed using ETB method.

Figure 9

(Color online) One-dimensional superlattice structure formed from the junction of BN and graphene armchair nanoribbons. (a) Atomic structure and superlattice parameters. (b) Band structures of constituent BN and graphene armchair nanoribbons having 18 atoms in their unit cells and the band structure of the superlattice BN(18)/AGNR(18) each segment having three unit cells $\left(s_1=s_2=3\right)$. (c) Energy-band diagram in real space forming multiple quantum wells (QW) in graphene segments (zones). Isosurface charge densities of states confined to QWs and propagating states are presented for selected bands. All data in this figure are calculated by first-principles method (Refs. 26, 27).

Figure 10

(Color online) (a) Atomic structure of a RTDB device made from a symmetric segment of AGNR(10)/AGNR(18) system and connected to two metallic electrodes. (b) Transmission coefficient $\mathit{T}$ versus energy under zero bias calculated using nonequilibrium Green’s function approach using local-orbital DFT. (c) The energy-level diagram and the charge densities of selected confined and extended states of the decoupled device calculated using plane-wave DFT.