First-principles study of electronic and optical properties of small edge-functionalized penta-graphene quantum dots Open Access

Low-dimensional materials with controllable optoelectronic properties from the bottom-up approach have rapidly received attention from the physical community, thanks to their unique mechanical and optoelectronic properties and their potential applications in the semiconductor industry.^1–5 Graphene quantum dots,⁶ a new kind of carbon nanometer material, exhibit rich electronic, optical, and magnetic behaviors when being doped by heteroatoms, such as boron, nitrogen, oxygen, and phosphorous,⁷ or being passivated by selective edge-functionalized groups.^8–14 For instance, doping of nitrogen to hexagonal graphene quantum dots allows us to tune the photoluminescent spectra of these quantum dots effectively.¹⁵ Besides the potential application on the spin qubit,¹⁶ quantum dots may play as an essential platform to improve the electronic as well as the optical properties of other low-dimensional materials, such as to enhance the solar cell energy-conversion efficiency.^17,18

Penta-graphene (PG), a two-dimensional (2D) pentagonal carbon allotrope, has received considerable attention owing to its interesting sp³-like and sp²-like hybridization of carbon atoms.^19–23 The intrinsic in-plane anisotropy of PG thin films could lead to complex phase transitions,²⁴ although the stability of the PG monolayer is still controversial. However, a recent attempt to grow large-area PG domains on Cu foils by chemical vapor deposition anchors the possibility to form PG monolayers experimentally.²⁵ Such low-symmetrical 2D pentagonal configurations could bring a new level of flexibility to tune their electronic as well as the optical properties effectively. In addition, monolayer amorphous carbon with five-, six-, seven-, and eight-member rings was successfully synthesized by laser-assisted chemical vapor deposition.²⁶ In addition, in this work, the properties of monolayer amorphous carbon investigated by density-functional-theory calculations are consistent with experimental measurements. Very recently, the successful synthesis of pentagonal PdSe₂ via the method of mechanical exfoliation, chemical vapor deposition, and so on, having a bandgap between 1.2 and 1.9 eV, high mobility (electron/hole), and high responsivity, shows the potential to synthesize similar types of pentagonal materials.²⁷ Alternatively, the PG sheets can be cut along different crystallographic orientations to realize quantum confinement, such as PG nanoribbons or PG quantum dots (PGQDs). Manipulation of the optoelectronic properties of PG nanoribbons has recently been done using various computer-aided material design techniques, such as doping,²⁸ edge termination,²⁹ physisorption and chemisorption,³⁰ and selective vacancies.³¹ Research on PGQDs, the strongest quantum confinement of PG materials, needs more attention. However, to our knowledge, research on the structural stability, electronic, and optical properties of PGQDs is almost rare. These calculations provide guidance for a better understanding of the structural stability, electronic, and optical properties of PGQDs.

In this study, the electronic and optical properties of edge-functionalized PGQDs using density functional theory will be investigated. Based on these calculations, we uncovered the changes in geometrical structures, band structures, (partial) density of states, and optical absorption spectra of PGQDs induced by various edge-functionalized groups of atoms including hydrogen, halogen (fluorine, chlorine, and bromine), and hydroxyl functional groups. We show that these quantum dots, especially those passivated by hydrogen atoms, are thermally stable in vacuum. All investigated PGQDs exhibit semiconductor properties, and the bandgaps decrease with an increase in the size of the quantum dots. Under different incident light polarization, the highly anisotropic morphology of PGQDs induced by the edge-functionalized groups results in strong optical absorption anisotropy. These results highlight the use of edge-functionalized groups to design the next generation of optoelectronic devices.

This paper is organized as follows: The methodology is presented in Sec. II. Section III covers the results and discussion of the optimized geometric structures, and the electronic and optical properties of these quantum dots. Finally, Sec. IV presents the conclusions.

To explore the effect of edge-functionalized groups on the optoelectronic changes of penta-graphene quantum dots, from a common 2D penta-graphene pattern,^19,32 a penta-graphene sheet was cut into smaller PGQDs for three different dot sizes including 12, 21, and 36 carbon atoms, respectively. These dots were then passivated by different functionalized groups including one of the following types of atoms: hydrogen, fluorine, chlorine, bromine, and hydroxyl functional groups. The structures were optimized by the density functional theory (DFT) method implemented in the CASTEP package.³³ The generalized gradient approximation (GGA) of the Perdew–Burker–Ernzerhof (PBE) exchange–correlation functional was used.³⁴ Since the PGQDs are confined in three dimensions, we used a k-point grid of 1 × 1 × 1 with an energy cut-off between 500 and 850 eV depending on the types of PGQDs. To avoid the interactions between periodic images of PGQDs, vacuum of 15 Å thickness was applied in the x-, y-, and z-directions. The convergence precision of energy for the maximum energy change, the maximum residual force on each atom, the maximum stress, and the maximum displacement of 10⁻⁶ eV/atom, 0.01 eV/Å, 0.1 GPa, and 0.001 Å, respectively.

To evaluate the stability of the optimized structures, the formation energy is defined as follows:³⁵ 

where E_total is the total Gibbs free energy of the optimized PGQDs. E_C, E_H, and E_X are the chemical potential of C, H, and X atoms (O, F, Cl, and Br), respectively. The n_C, n_H, and n_X indices are the number of C, H, and X atoms in the PGQDs.

The electronic and optical properties of the optimized structures are performed in the Atomistix ToolKit (ATK) software package by DFT computing with the GGA-PBE function and double-zeta polarized basis sets.^36,37 The Brillouin zone is sampled at the G point. The grid integration of cut-off energy is at 1000 eV. The Fermi level (E_F) is set to zero in all calculations.

The optical properties of the investigated structures will be determined by the complex dielectric functions,³⁸ 

where ɛ₁ and ɛ₂ are the real and imaginary parts of the dielectric function, respectively. ω indicates the angular frequency of the incident photon. The dielectric function is related to susceptibility as ɛ(ω) = 1 + χ(ω). The susceptibility χ(ω) is calculated by the Kubo–Greenwood formula,³⁹ 

where $πnmi,j$ means i, j-components of the dipole matrix element between the n and m states, Γ is the broadening, V is the volume, f(E_m/E_n) is the Fermi function, and E_m(E_n) corresponds to eigenvalues of the m(n) state. The extinction coefficient κ is calculated as follows:

From the extinction coefficient κ, we can calculate the optical absorption coefficient α of the materials using the following equation:

Here, c is the velocity of light in a vacuum.

Figure 1 presents the most energetically favorable geometries of PGQDs with various edge-functionalized groups. It is obvious that the edge-functionalized groups, as well as the odd–even number of carbon atoms, play a significant role in controlling the morphology of these stabilized quantum dots. As presented in Table I, the formation energy of all investigated PGQDs varies between −3.88 and −7.22 eV/atom, indicating that these optimized PGQDs are thermally stable, in agreement with Abdelsalam et al.’s report⁴⁰ for edged-functionalized silicene quantum dots. Furthermore, the larger the quantum dots, the more negative the formation energy on stability could reach, thus forming thermodynamically more stable quantum dots. As shown in Fig. S1 (see the supplementary material), the formation energy, in general, is scaled linearly with $1/n$⁠, where n is the number of carbon atoms in the PGQDs, which falls into the common behavior of formation energy for quantum dots.⁴¹ Among the selected edge-functionalized groups, hydrogen atoms stand out as an excellent candidate to stabilize these PGQDs, resulting from the largest absolute formation energy as presented in Table I.

Taking into account the contribution of the number of carbon atoms in the PGQDs, it indicates that these dots with an even number of carbon atoms form more compact symmetric structures than those formed by the odd number of carbon atoms (Fig. 1). It should be noted that PGQDs with 12 and 36 carbon atoms are composed of an even number of sp² and sp³ carbon atoms, while PGQDs with 21 carbon atoms are formed by an even number of sp³ and an odd number of sp² ones. To further explore this interesting observation, we measure the bond lengths from each carbon atom to its surrounding atoms and average these values over identical functional groups of atoms. To be precise, $dsp2−sp2$⁠, $dsp2−sp3$⁠, and $dsp3−sp3$ are the average bond lengths between two sp², a sp², and a sp³, and two sp³ hybridized carbon atoms, respectively. The $dsp2−edge$ and $dsp3−edge$ distances are the average bond lengths between sp² or sp³ carbon atoms and their nearest edge-functionalized atoms. As presented in Table I, while most of the average bond lengths remain almost unchanged for each type of quantum dot, the $dsp2−sp3$ bond lengths of PGQDs passivated by H, Br, and OH functional groups with an odd number of carbon atoms are noticeably larger than the other ones, illustrating that the PGQDs with an even number of carbon atoms are apparently more compact than the odd systems. Such compactness induced by the odd and even numbers of electrons was also observed for gold clusters.^42,43

The band structures and the frontier orbital energies of penta-graphene quantum dots with various edge-functionalized groups are presented in Fig. 2 and Table II. It is obvious that all electronic states are localized as illustrated by non-dispersive band curves in Fig. 2. This non-dispersive behavior is typically caused by the quantum confinement of quantum dots as presented in recent reports for graphene quantum dots⁴⁴ and CdS/ZnSe quantum dots.⁴⁵ Furthermore, all investigated edge-functionalized PGQDs exhibit semiconductor properties with bandgaps varying between 2.35 and 4.42 eV, which is in excellent agreement with the bandgaps obtained from penta-graphene sheets functionalized by H, F, and OH groups.⁴⁶ The largest bandgap opening is obtained for H termination and followed by halogen atoms (F, Cl, and Br). In contrast, OH-edge PGQDs have the smallest bandgaps, resulting from the vertical downshift of the lowest unoccupied molecular orbital (LUMO) states of oxygen toward the Fermi level, whereas the highest occupied molecular orbital (HOMO) states do not alter considerably. To understand the bandgap modification upon interactions, we calculated the (partial-) density of states (DOS and PDOS) for these PGQDs. Figures 2, S2, and S3 (see the supplementary material) demonstrate that the bandgap reduction is primarily caused by orbital hybridization between p-states of edge-functionalized groups and carbon atoms. These results are in line with previous investigations for various edge-functionalized penta-graphene nanoribbons^8,9 as well as for carbon phosphide nanoribbons with H, halogen (F and Cl), and hydroxyl edge functionalization groups.^47,48

Furthermore, investigations are carried out regarding the system size effect on the electronic properties of PGQDs. As shown in Fig. S1, the bandgaps of small PGQDs increase linearly with $1n$ as in conventional quantum dots,^49,50 where n is the number of carbon atoms in the quantum dots. This indicates that the HOMO and LUMO energy states become closer to the Fermi level in larger systems. In addition, as discussed in Sec. III A, the morphologies of edge-functionalized PGQDs differ significantly with the odd and even numbers of carbon atoms. Such odd–even behaviors result from the inherent electron pairing in molecular orbitals. Even-sized systems have an even number of valence electrons, while the odd-sized systems possess singly occupied states (H-21 and OH-21 panels in Fig. S3), resulting in the willingness to attach or detach additional electrons to form larger even-sized systems. Such cluster formation was recently reported for small silver⁵¹ and gold clusters.⁴³ 

To gain insights into the charge distributions on each atom, the electron density (ED) and electron difference density (EDD) are calculated,^52–54 which are a means to estimate the amount of electron transfer between the edge atoms and the principal components upon interactions. As discussed in Sec. III A, the orbital hybridization is more pronounced for PGQDs with the odd number of carbon atoms, accordingly merely the charge analyses for PGQDs with 21 carbon atoms in Figs. 3 and 4. In Fig. 3, the electron density values of H-21, Cl-21, and Br-21 are almost identical while the quantities for OH-21 and F-21 are about 3 times and 4.5 times higher.

In Fig. 4, it is apparent that charge accumulations at all carbon bonds do not alter significantly while noticeable changes are observed at the edges. For H-21, OH-21, F-21, and Cl-21, charge clouds are localized on top of the edged atoms. Particularly, charge accumulation prevails at oxygen atoms while charge depletion increases on top of hydrogen atoms when comparing the EDD of OH-21 and H-21 PGQDs. In contrast, no clear signature of either charge depletion or accumulation at the edges of F-21 PGQDs is observed. These results reveal the underlined mechanism of a noticeable bandgap difference between PGQDs passivated by hydrogen and hydroxyl groups of atoms.

Here, the attention is focused on the optical properties of PGQDs induced by various edge-functionalized groups. It is well-accepted that the optical properties of materials obtained from the complex dielectric functions³⁸ are closely related to the materials’ band structures. Figure 5 presents the real and imaginary parts of these complex dielectric functions. The static dielectric constant of these quantum dots, ɛ₁(0), varies between 1.005 and 1.022, which is smaller than that of the monolayer penta-graphene⁵⁵ and the penta-graphene nanoribbon³¹ as well. Taking account of the effect of edge-functionalized groups on PGQDs, we conceive that the first peaks of ɛ₁(ω) of hydrogen passivated PGQDs are blue-shifted toward higher energy values. More interestingly, the first peaks of ɛ₁(ω) of PGQDs passivated by halogen atoms or the hydroxyl functional groups are in the visible light region, pointing out the possibility of using these quantum dots for the application in solar cell devices. As the size of PGQDs increases, these first peaks are redshifted toward the lower energy regimes.

In terms of the imaginary parts of the dielectric functions, the first peaks of ɛ₂(ω) reflect the electronic bandgaps of these investigated PGQDs. The first peaks of H-12 hydrogen passivated PGQDs ɛ₂(ω) occur at the highest energies of 4.5 eV while the equivalent peaks obtained from the PGQDs passivated by hydroxyl functional groups are at the lowest energies of ∼2.5 eV. These values for the halogen edge-functionalized PGQDs vary between 2.8 and 3.2 eV with the following order of Br-PGQDs < Cl-PGQDs < F-PGQDs. In addition, as the sample size increases, the first peaks of ɛ₂(ω), following a similar trend to the first peaks of ɛ₁(ω), shift toward the lower energy values. While the real and imaginary parts of the dielectric functions exhibit isotropic polarizations for PGQDs with an even number of carbon atoms, a noticeable spatial anisotropy of these optical components for PGQDs-21 is observed. This optical anisotropy results from the effect of directional bond orientations on the electron difference density induced by the odd number of hybridized carbon atoms.

The absorption coefficients of the edge-functionalized PGQDs derived from the real and imaginary parts of the dielectric functions are shown in Fig. 6. In general, with an increase in the system sizes, the major peaks of the absorption spectra are red-shifted toward lower energy regimes. While hydrogen passivated PGQDs are sensitive in the ultraviolet region, halogen passivated PGQDs stand out as excellent optical traps in the visible light range (380–420 nm). Furthermore, because of the strong p-orbital hybridization between carbon and oxygen atoms, the hydroxyl functional groups induce additional optical absorption peaks in the visible light range of 400 and 500 nm. These results highlight the crucial impact of edge-functionalized groups on the electronic and optical properties of PGQDs.

In summary, the DFT calculations have been successfully applied to systematically investigate the effect of various edge-functionalized groups, such as hydrogen and halogen (fluorine, chlorine, and bromine) atoms and hydroxyl groups, on the structural stability and optoelectronic properties of penta-graphene quantum dots. It has been proven that larger PGQDs are thermodynamically more stable than smaller ones. At the same system sizes, in order to increase the absolute values of formation energies, the PGQDs can be passivated by the following Br-, Cl-, F-, OH-, and H-atoms, which illustrates that hydrogen edge-functionalized quantum dots are the most stabilized structures compared to all the investigated PGQDs. Furthermore, it has been indicated that we can actively tune the electronic bandgaps as well as the absorption spectra of PGQDs by passivating these quantum dots with various edge-functionalized groups. PGQDs were passivated by the hydroxyl groups and have the smallest bandgaps, resulting from the downshift of LUMO states toward the Fermi level while the HOMO energies remain almost unchanged, which is caused by the strong hybridization effect of carbon and oxygen p-orbitals of these quantum dots. Furthermore, the orbital interactions together with spatial directional bond orientations induce optical anisotropy of PGQDs with an odd number of carbon atoms. The results emphasize the role of edge-functionalized groups in the development of the next-generation optoelectronic devices using penta-graphene quantum dots.

See the supplementary material for the additional information about the formation energy, bandgap, and projected density of states of the edge-functionalized PGQDs.

This work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 103.01-2018.308. This work was carried out on the Can Tho University high-computing system with the support of the Information and Network Management Center at Can Tho University.

The authors declare that they have no known competing financial interest or personal relationships that could have appeared to influence the work reported in this paper.

The data that support the findings of this study is available from the corresponding author upon reasonable request.