Electronic and optical properties of surface-functionalized armchair graphene nanoribbons

Abstract

First-principle calculations with quasiparticle corrections are performed on the optical and electronic properties of functionalized armchair graphene nanoribbons (AGNRs). W8, W9 and W10 AGNRs are chosen based on the width (W) index, n. The functional groups (X = CH₃, NH₂, NO₂ and OH) are selected for the functionalized AGNRs (W(n)-X). Most of the functional groups enlarge the GW band gaps of W8 and W9, and reduce the GW band gap of W10. The variation of band gaps is analyzed in terms of the bonding characteristics. In all of the W10-X structures, most of the exciton wavefunctions are located on the bigger segments divided by the functional groups. Additionally, W10-NO₂ is a potential candidate for luminescence and photovoltaic devices due to its strong optical absorption and small exciton binding energy.

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Introduction

Graphene, the atomic-thick graphite sheet, has been intensively investigated^1–9 since its discovery,^1,2 but its optical applications are limited due to the zero band gap in pristine graphene. In order to extend the optical realm, various approaches have been explored. Graphene nanoribbons (GNRs), strips of graphene with nanoscale width, can be turned into semiconducting quasi one-dimensional structures. GNRs can be produced by unzipping carbon nanotubes through plasma etching or solution-based oxidation,^10,11 or by cracking graphene through electron writing¹² or plasma etching¹³ along one specific direction.

Armchair GNRs (AGNRs), as one of the most investigated GNR structures, possess a unique “family effect”¹⁴ in their electronic properties, which can be classified into three families identified by the ribbon width: n = 3p − 1, 3p and 3p + 1, where n is the width index and p is a positive integer. Due to their special structure, AGNRs have direct band gaps and can also be used to generate and detect light, providing potential for future optoelectronic devices.^3,15–18

Chemical functionalization, such as atomic^19,20 or molecular^21–23 doping, helps the metal-to-insulator transition^21,23 or band re-alignment with respect to the Fermi energy.^19,20,22 Therefore, the chemical functionalization approach is considered to be a promising strategy to tune the electronic and optical properties. Graphene functionalization with particular groups including hydroxyl,^24–29 amine,^30,31 methyl³² and nitro groups³³ has been experimentally performed. Previous work has also pointed out that chemical functionalization can induce significant doping on graphene and can display reversible control of current intensity.³⁴ In this work, we perform a first principle study on the electronic and optical properties of surface-functionalized AGNRs, which possess common functional groups.

Theoretical methods and models

To obtain the optical properties, first-principle calculations were performed using a many-body perturbation approach, which possesses a three-step procedure:³⁵ (i) firstly, the ground-state electronic properties of the fully relaxed system are studied within a local density approximation (LDA) based on Quantum ESPRESSO code.³⁶ Each atomic structure is fully relaxed with the forces per atoms converged to 0.01 eV Å⁻¹. Separable ultrasoft pseudopotentials and a plane-wave basis set are carried out for the calculations. The kinetic energy cutoff is set as 30 Ry and 121 × 1 × 1 is used for the k-point parameter. The convergence studies for the particular choices of energy cut-off and k-point mesh are shown in Fig. S1 and S2 (ESI†). (ii) Secondly, LDA, as a good approximation, is chosen for the quasiparticle wavefunctions. The quasiparticle corrections to the LDA eigenvalues are performed based on the G0W0 approximation for the self-energy operator with consideration of the plasmon-pole approximation for the screening.³⁷ (iii) Thirdly, to calculate the electron–hole interaction, the Bethe–Salpeter equation (BSE) is solved in the basis set of quasi-electron and quasi-hole states, where the static screening in the direct term is calculated within the random-phase approximation.³⁸ All the GW and BSE calculations are performed with the Yambo code.³⁹ Twenty valence bands and twenty conduction bands are considered for the GW calculations. Five valence bands and five conduction bands are included for the BSE calculation, which are enough for our calculations since only the low range of exciton and optical transitions are discussed in the work. A rectangular-shaped truncated Coulomb interaction is applied to eliminate the image effect between adjacent supercells to mimic isolated AGNRs,⁴⁰ due to the usage of the supercell method in the calculations.

Several functional groups including CH₃, NH₂, NO₂ and OH are considered to adsorb on the surface of AGNRs, as shown in Fig. 1(a)–(d). The reason for the chosen molecules/groups is that they are common in chemistry and are always used for discussion due to their different electronic effects.^41–43 The black rectangles schematically present the unit cells in Fig. 1. The period direction is along the x axis. The unit length along the y axis and the height of a vacuum layer along the z axis are set as 30 Å. The convergence studies about the separation along the y and z-axis are shown in Fig. S3 (ESI†). The functionalized AGNRs are defined as W(n)-X, where n is the width index of the AGNRs¹⁴ (marked in Fig. 1). In this study, we consider three series of AGNRs (n = 8, 9 and 10), which cover the distinct three families (n = 3p − 1, 3p and 3p + 1) of pristine AGNRs according to previous works.^40,44 The dangling σ bonds at the AGNR edges are passivated by hydrogen atoms. It is noted that there are various adsorption positions on the surface of AGNRs for the chemical groups and the band gaps can be tuned by the adsorption positions. Herein we only consider the systems which have the smallest band gap, which can be roughly decided through the effective width for the sp² network in W(n)-X.^45,46 Our chosen systems, which are similar to the hydrogenated AGNRs in the previous works,^45,46 can further help understand the effect of functionalization on AGNRs and also provide a better strategy to tune the electronic and optical properties.

Fig. 1 Schematic structures of (a) W9-CH₃, (b) W9-NH₂, (c) W9-NO₂ and (d) W9-OH from the top- and side-views. Cyan, yellow, red and blue balls represent the hydrogen, carbon, oxygen and nitrogen atoms, respectively. The black rectangles schematically present the unit cell.

Results and discussion

The analysis of the variation of formation energies and band gaps

The related formation energies (E_formation) for the W(n)-X AGNRs listed in Table 1, are defined as E_W(n)-X − E_W(n) − 2E_X. The negative formation energies demonstrate that the reactions are exothermic and that the selected W(n)-X AGNRs are stable and can possibly be synthesized.

Table 1 LDA band gaps, GW band gaps, BS energies (peak position), exciton binding energies (E_b) and the formation energies (E_formation) for the W(n)-X AGNRs, where n is 8, 9 or 10, and X is CH₃, NH₂, NO₂ or OH. The data in parentheses have been reported in previous work.⁴⁰ The formation energies (E_formation) for the W(n)-X AGNRs are defined as E_W(n)-X − E_W(n)

Case	LDA (eV)	GW (eV)	BSE (eV)	Eb (eV)	Eformation (eV per atom)
W8	0.26, (0.28)	1.10, (1.00)	0.48, (0.42)	0.62, (0.58)	—
W8-CH3	0.20	1.21	0.47	0.74	−0.1600
W8-NH2	0.17	1.19	0.47	0.72	−0.1875
W8-NO2	0.17	1.18	0.43	0.75	−0.0313
W8-OH	0.08	1.08	0.39	0.69	−0.2180
W9	0.81, (0.78)	2.25, (2.10)	1.10, (0.99)	1.15, (1.11)	—
W9-CH3	0.93	2.91	1.56	1.35	−0.1507
W9-NH2	0.61	2.26	1.26	1.00	−0.1766
W9-NO2	1.12	2.91	1.75	1.16	−0.0307
W9-OH	0.94	2.74	1.53	1.21	−0.2087
W10	1.10, (1.16)	2.80, (2.82)	1.50, (1.51)	1.30, (1.31)	—
W10-CH3	0.22	1.10	0.35	0.75	−0.1823
W10-NH2	0.22	1.07	0.33	0.74	−0.2042
W10-NO2	0.16	0.91	0.19	0.72	−0.0770
W10-OH	0.24	1.11	0.36	0.75	−0.2378

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Table 1 summarizes the LDA band gaps and GW band gaps for the W(n)-X structures. Both the LDA and GW band structures of the W(n) and W(n)-X AGNRs are plotted in Fig. 2. Each band line goes from Γ (0, 0, 0) to X (π/a, 0, 0), where a is the unit length along the ribbon axis. It is obvious that band gap corrections in the GW band structures appear, and that the GW band structures possess the main characteristics of the LDA band structures. All the structures have direct band gaps. For the pristine AGNRs, the GW band gaps of W8, W9 and W10 are 1.10, 2.25, and 2.80 eV, respectively, which are in agreement with previous results.^40,44,45

Fig. 2 LDA (black) and GW (red) band structures for the W(n)-X AGNRs, where n is 8, 9 or 10, and X is CH₃, NH₂, NO₂ or OH. The top of the valence bands in the LDA and GW band structures are set to zero. Γ (0, 0, 0) and X (π/a, 0, 0), where a is the unit length along the ribbon axis.

In comparison to the pristine AGNRs, most of the GW band gaps of W8-X and W9-X are enlarged, while those of W10-X are reduced. The functionalization causes the band gap differences. The binding of the carbon atoms with functional groups results in an sp-hybridization transformation from sp² to sp³. Due to the different groups, there are complex carbon hybridizations in W(n)-X. Here we only analyze the hybridization roughly from the bond length and the bond angle, which can also be evaluated accurately from Wannier functions. The bond lengths of C_α–X (X = CH₃, NH₂, NO₂ or OH) are 1.53 Å, 1.46 Å, 1.64 Å and 1.43 Å, respectively, where C_α is the carbon atom connecting with X. The bond lengths between two carbon atoms C_α–C′_α, both of which connect to X groups, increase from 1.42 Å to 1.54 Å, 1.53 Å, 1.52 Å and 1.53 Å, respectively. For all the absorption cases, the sp³ carbons tilt out of the plane (see the middle of Fig. 1) due to the hybridization, and form a ∼30° torsion configuration. Additionally, it is noted that the functional groups are not standing straight in the z-direction, and the bond angles deviate from 90 degrees (see the bottom of Fig. 1) due to the partial sp² and sp³ hybridizations. The complex chemical bonding wipes out the family effect of pristine AGNRs and gives some clues for the regular sorting about the electronic structures.

The quasi-particle GW energy corrections are around 1 eV, as shown in Table 1 and Fig. 2. The GW corrections of the W9 series are larger than those of the W8 and W10 series. Additionally, the GW corrections with bigger LDA band gaps tend to be larger.

The analysis of the optical transitions

By solving the Bethe–Salpeter (BS) equation based on a set of quasi-electron and quasi-hole states,³⁵ optical transitions can be calculated with the light polarization along the ribbon, as shown in Fig. 3. The BS energies, defined as the first peak positions of the transition energies, are listed in Table 1. The absolute values of the BS energies are around half of the GW band gaps, revealing a strong binding of the electrons and holes in the W(n)-X AGNRs.

Fig. 3 Optical absorption spectra of the W(n)-X AGNRs, where n is 8, 9 or 10, and X is CH₃, NH₂, NO₂ or OH.

The respective exciton binding energy, which is computed by E_GW − E_BS, is used to evaluate the exciton binding. It is well known that the screening in one or quasi-one dimensional materials is an order of magnitude weaker than that in bulk materials, thus the exciton binding strength is expected to be higher in AGNRs.⁴⁷ As listed in Table 1, the exciton binding energies of the W9-X AGNRs are larger than 1 eV, while those of the W8-X and W10-X AGNRs are a little smaller than 1 eV.

The GW band gaps of W9-CH₃ and W9-OH are about 3 eV, resulting in a fairly weak screening to reduce the Coulomb interaction between the electrons and holes.⁴⁸ Therefore, their exciton binding energies are the strongest in the W9 series. Additionally, the exciton binding energy of W8-OH is the weakest in the series due to the stronger screening.

As discussed above, the adsorbed chemical groups can tune the electronic and optical properties. In the W9-X structures, the chemical adsorption procedures increase the band gap, the transition energies and the corresponding exciton binding energies. On the other hand, in the W10-X structures, the chemical functionalization procedure has the opposite effect. The origination of the different trends of W9-X and W10-X are complicated and still under our investigation. Additionally, the strong optical absorption intensity and small exciton binding energy of W10-NO₂ reveal the potential optical applications including luminescence and photovoltaics.

The analysis of CBMs and VBMs

In order to well understand the electronic properties, the valence band maximum (VBM) and conduction band minimum (CBM) at Γ points are plotted in Fig. 4. The chemical groups modify the hybridization of the connected carbon atoms from sp² into sp³ type and increase the related carbon–carbon bond length, and further affect the electronic properties. In the W8 cases, the VBMs are located in both segments, while most of the CBMs are located in one segment. In most of the W9 cases, the CBMs and VBMs are confined to two different segments, providing promising usage for organic photovoltaic devices. Additionally, in the W10 series, the CBMs and VBMs are confined to the bigger segments. Moreover, this confinement is very similar to that of hydrogenation on pristine AGNRs.^45,49,50

Fig. 4 VBMs and CBMs for the W(n)-X AGNRs, where n is 8, 9 or 10, and X is CH₃, NH₂, NO₂ or OH. The pink arrows point out the positions of the functional groups on the AGNRs.

Discussion of exciton wavefunctions

To better understand the effects of the electron–hole interactions, the exciton wavefunctions with the lowest exciton peaks are plotted in Fig. 5, each with one fixed hole position in the middle of a π bond (marked as a black spot). The exciton wavefunction, which depends on both the valence bands and conduction bands, is written as follows:³⁸
where and are wavefunctions of the valence band and conduction band for holes (h) and electrons (e), respectively. Consequently, the spatial distributions of the CBMs and VBMs can help to roughly understand the exciton wavefunctions.

Fig. 5 The lowest exciton wavefunctions for the W(n)-X AGNRs, where n is 8, 9 or 10, and X is CH₃, NH₂, NO₂ or OH. The pink arrows point out the positions of the functional groups on the AGNRs.

The distribution of exciton wavefunctions along the x-axis (ribbon length) is determined by the Coulomb interaction with quasi-one-dimensional character, which is similar to carbon nanotubes⁵¹ and pure AGNRs.⁴⁰ The general patterns of wavefunction distribution are quite different in the W(n)-X systems. For the W10 series, most of the exciton wavefunctions are located on the bigger segment divided by the functional groups. Additionally, these features of the spatial distribution can be understood by the VBMs and CBMs (Fig. 4). Both the VBMs and CBMs prefer to localize at the same edge. Consequently, the exciton wavefunctions are expected to be located near the same edge. Additionally, the W8 series possesses quantum confinement effects on the excitons, except for W8-CH₃. The exciton wavefunctions of the W8 series can be easily understood by the overlap of the CBMs and VBMs, since both the CBMs and VBMs have the same spatial distribution parts and the exciton wavefunctions depend on both the CBMs and VBMs. A similar explanation can be proposed for W9-CH₃. The other three W9 AGNRs have separated CBMs and VBMs, which induce different originations of the exciton wavefunctions. Though it is interesting to note that indeed the wavefunctions of the CBM closely mimic those of the excitons for almost all the W9 structures, except W9-NO₂, the exciton wavefunctions depend on both the valence bands and conduction bands.

It is noted that the different groups have various electronic effects though they adsorb on the same positions, as a result, the exciton distributions are quite significant in the W8 and W9 series. However, the exciton distributions in the W10 series exhibit similar appearances, demonstrating that the functional groups play a weaker role in W10 than those in W8 and W9. Additionally, the effect of the width cannot be ignored.

To figure out the exciton type of the W(n)-X AGNRs, we first compare their binding energy (Table 1) with that of a Frenkel exciton,⁵² which is estimated as E^Frenkel_b ∼ (2μ²)/(4πε₀ε_rr³), where μ is the dipole moment, ε₀ is the vacuum permittivity, ε_r ∼ 2.5 is the static dielectric constant of the AGNR, and r ∼ 1.4 Å is the interatomic distance.⁴⁵ The calculated dipole moment for μ is around 1 × 10⁻²⁹ C m, which is an order larger than the typical one in a Frenkel exciton.⁵³ Thus, we know that the excitons do not belong to Frenkel ones. Due to the radius (∼10 Å) and the binding energy (∼1 eV) of the exciton, it may be a charge-transfer exciton.⁵⁰

The calculation results help us understand how to tune the electronic and optical properties and how to explore the future applications. For example, the functionalized W9 structures are not suitable for luminescence since the calculated optical absorption intensities are quite weak (Fig. 3), compared to the W8 and W10 series. The differences in the optical absorption positions and intensities in the spectra (Fig. 3), and the distributions of wavefunctions can provide possible ways to identify the AGNRs. Moreover, the exciton binding energy is an important parameter to define the separation of electrons and holes, which can help to understand the optoelectronic performance and to find a direction to improve the devices. Herein, the calculations show that the W8 and W10 series have potential for optical applications.

Conclusions

In summary, the optical and electronic properties of functionalized AGNRs have been investigated. Most functional groups enlarge the GW band gaps of the W8 and W9 AGNRs, and reduce the GW band gaps of the W10 AGNRs. The variation of band gaps is analyzed in terms of the bonding characteristics. In all the W10-X structures, most of the exciton wavefunctions are located on the bigger segments divided by the functional groups. Additionally, W10-NO₂ can be a potential candidate for luminescence and photovoltaic devices due to its strong optical absorption and small exciton binding energy.

Acknowledgements

This work was financially supported by the National Natural Science Foundation of China (Grant No. 21203154 and 21375108) and Fundamental Research Funds for the Central Universities (Grant No. XDJK2016B001). Computation resource is supported by Faculty of Materials and Energy (Southwest University).

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Footnote

† Electronic supplementary information (ESI) available: The convergence studies of energy cut-off, k-point mesh and the separation layers along y and z directions. See DOI: 10.1039/c5ra22701a

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