Screened hydrogen model of excitons in semiconducting nanoribbons

The optical response of quasi-one-dimensional systems is often dominated by tightly bound excitons that significantly influence their basic electronic properties. Despite their importance for device performance, accurately predicting their excitonic effects typically requires computationally demanding many-body approaches. Here, we present a simplified model to describe the static macroscopic dielectric function, which depends only on the width of the quasi-one-dimensional system and its polarizability per unit length. We show that at certain interaction distances, the screened Coulomb potential is greater than its bare counterpart, which results from the enhanced repulsive electron-electron interactions. As a test case, we study 14 different nanoribbons, 12 of them armchair graphene nanoribbons of different families. Initially, we devised a simplified equation to estimate the exciton binding energy and extension that provides results comparable to those from the full Bethe-Salpeter equation, albeit for a specific nanoribbon family. Then, we used our proposed screening potential to solve the 1D Wannier-Mott equation, which turns out to be a broad approach that is able to predict binding energies that match quite well the ones obtained with the Bethe-Salpeter equation, irrespective of the nanoribbon family.

Figure 1

(a) Schematic representation of an armchair ribbon indicating the number of dimmer lines across the ribbon width. The ribbon is periodic along the $x$ direction. (b) RPA ab initio static dielectric response function for a 4-AGNR (solid blue line). The green line represents the fit to the function ${\varepsilon }_{\text{1D}}\left(q_x\right)\approx 1-8{\alpha }_{\text{1D}}{q}_x^{2}ln(q_{x}L/4)$. The ribbon width and polarizability per unit length are $L=5.54\AA$ and ${\alpha }_{\text{1D}}=2.59{\AA }^2$, respectively. The inset shows the dependence of ${\alpha }_{\text{1D}}$ with the quasiparticle band gap of 14 different armchair ribbons. The dot, triangle, and diamond symbols differentiate the different nanoribbon families, while the indices next to the symbols indicate the nanoribbon index. The orange symbols correspond to selected armchair boron nitride nanoribbons (ABNRs) [29]. The gray dashed line represents a fitted reciprocal function used as guide to the eyes.

Figure 2

Comparison between the screened and bare energy potential energy for a 4-AGNR. The bare energy potential is obtained by setting ${\alpha }_{\text{1D}}=0$ in Eq. (4). The insets shows the induced charge density distribution around a point charge at $x=0$. The polarizabilities per unit length for the 5- and 14-AGNRs are ${\alpha }_{\text{1D}}=32.4{\AA }^2$ and ${\alpha }_{\text{1D}}=183.6{\AA }^2$, respectively.

Figure 3

(a) $n$-dependent static dielectric function versus the exciton state number $n$ for the 3-, 4-, and 5-AGNRs. The dielectric function is computed by solving Eq. (8). The inset shows the average exciton extension varying with the exciton state number for the 4-AGNR, which is obtained with Eq. (7). (b) Binding energy for the lowest-energy bright excitons in 12 different AGNRs as a function of the expectation value of their exciton extension. The labels represent the index of each AGNR. The symbols represent the different AGNR families. The horizontal (vertical) axis is normalized to results obtained by solving the full Bethe-Salpeter equation binding energy (width of the N-AGNR). The inset shows the dependence of the logarithmic factor in Eq. (8) with the reduced mass.

Figure 4

(a) Comparison of the lowest-energy bright exciton binding energy calculated with the Bethe-Salpeter equation ($x$ axis) and the 1D-Schrödinger equation model given by Eq. (9) ($y$ axis), for 14 nanoribbons. The symbols highlight the family of each AGNR while the labels $N$=3,4,..12 indicate the width of the ribbon. The inset shows a comparison of the variation of the binding energy with the exciton state number for the BSE (black dots) and Schrödinger (red dots) approaches for the 4-AGNR. The right-handed figure presents a comparison of the normalized electron distribution along the ribbon axis for (b) the first, (c) second, and (d) third bright exciton states of a 4-AGNR.

Figure 5

Schematic representation of the quasi-one-dimensional system for modeling the screening. Here, we consider an electron ($-$) and hole ($+$) in a ribbon, extending along the $x$ direction, of width $L$ and thickness $b$.