Band structure and optical absorption in multilayer armchair graphene nanoribbons: A Pariser-Parr-Pople model study

Using the tight-binding and Pariser-Parr-Pople model Hamiltonians, we study the electronic structure and optical response of multilayer armchair graphene nanoribbons (AGNRs), both with and without a gate bias. In particular, the influence of the number of layers ($n$), and the strength of the electric field applied perpendicularly to layers, for different types of edge alignments is explored via their electro-optical properties. As a function of increasing $n$, the energy gap initially decreases, eventually saturating for large $n$. The intensity of the linear optical absorption in these systems also increases with increasing $n$ and depends crucially on the polarization direction of the incident light and the type of the edge alignment. This provides an efficient way of determining the nature of the edge alignment, and $n$, in the experiments. In the presence of a gate bias, the intensity of optical absorption behaves in a nontrivial way. The absorption becomes more intense for the large fields in narrow ribbons exhibiting a redshift of the band gap with the increasing field strength, while in broad ribbons exhibiting a blueshift, the absorption becomes weaker. However, for smaller electric fields, the absorption intensity exhibits more complicated behavior with respect to the field strength. Thus, the effect of the gate bias on optical absorption intensity in multilayer AGNRs is in sharp contrast to bilayer graphene, which exhibits only enhancement of the absorption intensity with the increasing electric field.

Figure 1

The structures of bilayer AGNR (a) $\alpha$ alignment and (b) $\beta$ alignment.

Figure 2

Variation of the energy gap with increasing width for bilayer AGNRs, obtained by the tight-binding method for (a) $\alpha$ alignment and (b) $\beta$ alignment.

Figure 3

Variation of the energy gap with increasing width for the AGNRs with eight layers, obtained by the tight-binding method for (a) $\alpha$ alignment and (b) $\beta$ alignment.

Figure 4

Variation of energy gap with number of layers obtained by the tight-binding method for multilayer AGNRs in (a) $\alpha$ alignment and (b) $\beta$ alignment.

Figure 5

Variation of energy gap with number of layers obtained using the PPP model, within the RHF approximation, for multilayer AGNRs with $N_A=8$, in $\alpha$ and $\beta$ alignments.

Figure 6

Band structure of (a) 2-AGNR-11-$\alpha$ and (b) 2-AGNR-11-$\beta$, obtained using the PPP-RHF approach. The inset contains the magnified band structure near the Fermi energy ($E_F=0$) of (a) 2-AGNR-11 (black lines) and (b) monolayer AGNR-11 (dashed (red) lines).

Figure 7

Band structure of (a) 3-AGNR-8-$\alpha$, (b) 4-AGNR-8-$\alpha$, and (c) 10-AGNR-8-$\alpha$ near $E_F$ obtained using the PPP-RHF approach.

Figure 8

Variation of energy gap with $E_z$ for (a) 2-AGNR-8-$\alpha$ and 2-AGNR-24-$\alpha$ and (b) 3-AGNR-8-$\alpha$ and 3-AGNR-24-$\alpha$, obtained using the PPP-RHF approach.

Figure 9

PPP-RHF band structure of 3-AGNR-24-$\alpha$ near $E_F$, for several values of external electric field $E_z$.

Figure 10

Optical absorption spectra of 2-AGNR-11-$\alpha$ calculated using the PPP-RHF approach for the light polarized along the $x$ axis with Coulomb parameters (a) $U=6.0$ eV and ${\kappa }_{i,j}=2.0$ ($i\ne j)$ and ${\kappa }_{i,i}=1$ and (b) $U=8.0$ eV and ${\kappa }_{i,j}=2.0$ ($i\ne j)$ and ${\kappa }_{i,i}=1$. In both cases the black line corresponds to hopping parameters (in eV) $t=2.7$, $t^{\prime }=0.27$, $t_{\perp }=0.4$ eV, and $t_{\perp }^{\prime }=0.3$, the dashed (red) line corresponds to $t=3.16$, $t^{\prime }=0.316$, $t_{\perp }=0.39$ eV, and $t_{\perp }^{\prime }=0.315$.

Figure 11

Optical absorption spectra of 2-AGNR-12-$\alpha$ computed using the PPP-RHF approach for the light polarized along the $x$ axis, with various sets of PPP parameters as explained in the caption of Fig. 10.

Figure 12

Optical absorption spectra of 2-AGNR-8 calculated using the PPP-RHF approach for the light polarized along the (a) $x$ axis, (b) $y$ axis, and (c) $z$ axis. The black lines represent the $\alpha$ alignment case, while the dashed (red) lines denote the $\beta$ alignment. A linewidth of 0.05 eV was assumed throughout.

Figure 13

Optical absorption spectra of 1-AGNR-11-$\alpha$, 2-AGNR-11-$\alpha$, and 3-AGNR-11-$\alpha$ calculated using the PPP-RHF approach, for the light polarized along the $x$ axis. A linewidth of 0.05 eV was assumed throughout.

Figure 14

PPP-RHF absorption spectra at several values of the static electric field $E_z$, for the incident light polarized along the $x$ axis for (a) 2-AGNR-12-$\alpha$ and (b) 2-AGNR-24-$\alpha$. A linewidth of 0.05 eV is assumed throughout.

Figure 15

Variation of peak strength of ${\Sigma }_{11}$ transition of ${\epsilon }_{xx}$ with static electric field $E_z$, computed using the PPP-RHF method, for (a) 2-AGNR-12-$\alpha$ and (b) 2-AGNR-24-$\alpha$.