Electronic and optical properties of doped graphene

Electronic and optical properties of doped graphene

Physica E 118 (2020) 113894 Contents lists available at ScienceDirect Physica E: Low-dimensional Systems and Nanostructures journal homepage: www.el...

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Physica E 118 (2020) 113894

Contents lists available at ScienceDirect

Physica E: Low-dimensional Systems and Nanostructures journal homepage: www.elsevier.com/locate/physe

Electronic and optical properties of doped graphene Po-Hsin Shih a , Thi-Nga Do b,c ,∗, Godfrey Gumbs d , Ming-Fa Lin a a

Department of Physics, National Cheng Kung University, Tainan, 701, Taiwan Laboratory of Magnetism and Magnetic Materials, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, Viet Nam c Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Viet Nam d Department of Physics and Astronomy, Hunter College of the City University of New York, 695 Park Avenue, New York, NY 10065, USA b

ARTICLE Keywords: Doped graphene Electronic properties Optical spectra Tight-binding model


ABSTRACT The diverse electronic and optical properties of doped graphene with chosen densities of Si, B, and N guest atoms are investigated with the use of the tight-binding model. The main characteristics of the energy dispersion are substantially modified by doping. This includes the opening of a band gap, a change in the number of energy subbands, and a shift in Fermi levels. The presence of different dopants in graphene greatly diversifies the optical-absorption spectra through changes in the spectral frequency, intensity, and absorption structures. Doping gives rise to more excitation channels, leading to unique spectral structures as well as fluctuation in the absorption frequency and intensity. Our theoretical predictions provide a clear understanding of the electronic and optical properties of graphene-related systems, which could be helpful in the designing of novel electronic, photonic and optoelectronic devices based on these materials.

1. Introduction Graphene, a monolayer of carbon atoms, has been receiving considerable attention from both experimentalists and theoreticians due to its interesting characteristics and important device applications. Its unusual band structure with zero band gap and linear energy–momentum relation near the Dirac point gives rise to several novel physical effects, especially electronic and optical properties [1–6]. So far, fundamental nanoelectronic applications of graphene have found their way into sensors [7], energy storage [8], and transparent electrodes [9]. However, opening a band gap in this material is paramount for additional applications in graphene-based devices, such as p–n junction diodes and transistors as well as solar cells. Various types of defect-producing mechanisms have been proposed to generate the tunability of the electronic properties of graphene. These include lateral confinement, edge disorder, dislocations and grain boundaries, as well as doping [10–21]. Among these procedures, substitutional doping with heteroatoms is the most effective way to manipulate the energy dispersion. Additionally, doped graphene provides physics [1–6] due to the robust relationship between the material properties and crystal structure. This leads to new possible applications in nanoelectronics and energy-related devices [7–9]. So far, Si, B, and N have been demonstrated to be the most suitable dopants for graphene since the resulting doped graphene has been successfully synthesized [16–19]. Although in practice the distribution of guest atoms within graphene is random [22,23], theoretical studies have made some

interesting predictions based on periodic arrangements of dopants [24– 27]. With the current advances in experimental technique, ones might be able to control the exact possible positions of the substitutional guest atoms on graphene sheets in the near future [28,29]. So far, the energy bands of graphene doped with various dopants have been investigated with the use of first-principle calculations [20, 21]. However, as far as we are aware, there is still lacking a comprehensive comparison of the dependence of the band structures and optical excitations on both the type and concentration of dopants. In this work, we present interesting results for the electronic and optical properties of graphene in the presence of Si, B, and N doping atoms, based on the tight-binding model. This framework has been proven to be an efficient calculational tool which yields results that are comparable with experiments and other theoretical methods for many condense matter systems [30,31]. Here, we build up the sets of hopping interactions for doped graphene in order to reproduce the band structures from the first-principles results. However, beyond the first-principles method, these tight-binding parameters could be applied for arbitrary doping concentration and extended to diverse configurations, such as layered systems [31], nanotubes [32], nanoribbons [33] and others. Evidently, this is groundwork for further studies of various fundamental physical properties, including the optical absorption spectra and thermal conductivitym [34] Coulomb excitations [35], lifetimes [36], external field effect and quantum Hall effect [37]. For the investigation

∗ Corresponding author. E-mail address: [email protected] (T.-N. Do).

https://doi.org/10.1016/j.physe.2019.113894 Received 3 September 2019; Received in revised form 28 October 2019; Accepted 13 December 2019 Available online 20 December 2019 1386-9477/© 2019 Elsevier B.V. All rights reserved.

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2. Method We now consider several doped graphene systems generated by periodically substituting C atoms by Si, B, or N atoms. The doping concentration is defined as the extension of the primitive unit cell along the lattice vector (𝑎⃗1 and 𝑎⃗2 ) as 𝑁0 × 𝑁0 in size. Here, the cell multiplicity 𝑁0 takes various values of 2, 3, 4, and 5, as illustrated in Fig. 1(a) through 1(d) for the extended unit cells. A primitive cell consists of 2𝑁02 sites, covering both host and guest atoms with the ratio of dopant/cell = 1/2𝑁02 . We employ a tight-binding model for the parameter fitting in order to reproduce the band structures from the first-principles results. The parameter set comprises the non-uniform nearest-neighbor hopping interactions and site energies, implying non-equivalence in distribution of the host and guest atoms.[44] When an electromagnetic wave impinges on a doped graphene sheet, it results in vertical optical excitations from occupied to unoccupied states. Following Fermi’s golden rule, such excitations have finite intensity which could be expressed, at zero temperature, as ⟨ ⟩2 ∑ 𝑑𝐤 | 𝑐 | 𝐄̂ ⋅ 𝐏 | 𝑣 | 𝐴(𝜔) ∝ 𝛹 (𝐤, 𝑚′ )| 𝛹 (𝐤, 𝑚) | | | 2| | 𝑚𝑒 | | ∫ ′ 1𝑠𝑡𝐵𝑍 (2𝜋) 𝑐,𝑣,𝑚,𝑚

[ 𝑓 (𝐸 𝑐 (𝐤, 𝑚′ )) − 𝑓 (𝐸 𝑣 (𝐤, 𝑚)) ] . (1) 𝐸 𝑐 (𝐤, 𝑚′ ) − 𝐸 𝑣 (𝐤, 𝑚) − 𝜔 − 𝑖𝛤 ⟩ ⟨ |̂ | 𝑣 Here, the velocity matrix element, 𝛹 𝑐 (𝐤, 𝑚′ )| 𝐄⋅𝐏 |𝛹 (𝐤, 𝑚) , plays | 𝑚𝑒 | a decisive role in the existence of possible excitations. It is calculated using the gradient approximation [45]. On ] the other hand, the joint [ 𝑐 (𝐤,𝑚′ ))−𝑓 (𝐸 𝑣 (𝐤,𝑚)) , is closely related to the density of states, Im 𝐸𝑓𝑐(𝐸 ′ 𝑣 (𝐤,𝑚 )−𝐸 (𝐤,𝑚)−𝜔−𝑖𝛤 energy dispersion of the system through the energies of initial (𝐸 𝑣 ) and final (𝐸 𝑐 ) electronic states of transition. 𝜔 is the excitation frequency and 𝑓 (𝐸 𝑐,𝑣 (𝐤, 𝑚)) are the Fermi–Dirac distribution functions. It is noticed that in our current work, the spectral broadening effect is not included self-consistently since we neglect the interaction between the optical excitations and the atomic configuration. Consequently, the broadening parameter 𝛤 is chosen to be sufficiently small (10 meV) so that it does not affect the resonant peak position in the spectral structure [46]. ×𝐼𝑚

Fig. 1. (Color online) Schematic illustrations of doped graphene crystals with chosen concentrations. The green and red spheres represent host and guest atoms, respectively. The original unit cell is extended by (a) 2 × 2, (b) 3 × 3, (c) 4 × 4, or (d) 5 × 5 along the lattice vectors. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

3. Results and discussion Pristine graphene was demonstrated to be semi-metallic with a special gapless cone-like energy band where the Fermi level is located right at the Dirac point [1–3]. By calculating the band structures of graphene doped with different atoms, we now show that the electronic properties of graphene can be efficiently manipulated by doping. We employ Si, B, and N guest atoms with various dopant densities so that the primitive cell is extended by 2 ×2, 3×3, 4× 4, or 5 × 5 of the original one. The critical characteristics of the energy dispersion are clearly modified by changing the dopant or doping concentration, as illustrated in Fig. 2. The substituted guest atoms could change the crucial physics of graphene by opening a band gap near zero energy. Such behavior presents a robust connection with the dopant density as well as the size of the primitive unit cell. We observe that regardless of the dopant type, a band gap is opened at the Dirac point with the size being reduced by increasing doping the concentration. Furthermore, our calculations show that the size of the band gap is hardly affected by the doped atoms. This is in agreement with previous reports for graphene in which carbon atoms are substituted with dopants, whereby the band gap depends only on the doping concentration and atomic configurations [20,21]. As an exception for the 3 × 3 enlarged unit cell, the valence and conduction bands cross at the 𝛤 point. This is consistent with the 3𝑁0 rule proposed by previous theoretical first-principles calculations [21]. The changes in the number of energy subbands in the low-energy regime and in the Fermi level deserve a detailed investigation since they help to determine the optical spectra of graphene. Increasing the

of electronic and optical properties, the advantage of the tight-binding method is to uncover the connection between the energy dispersion and the absorption channels. This is critical for a clearer exploration of the optical spectral structures. We will show that modification of doping concentration for various dopants leads to band gap opening, change in the number of energy bands, and the shift of the Fermi levels. These are responsible for the feature-rich spectral structures of the absorption spectra. Our theoretical predictions provide a clear understanding of the electronic and optical properties of graphene-related systems. This could be helpful in the design of future experimental measurements as well as novel electronics, photonics and optoelectronics applications involving the materials [7–9,38–41]. The essential electronic and optical functions of many condensedmatter systems have been deduced with the use of various experimental techniques, especially optical experiments [5,42,43]. Optical investigations provide verification not only of the absorption spectra but also the electronic band structures of materials. Optical spectroscopy has been demonstrated to be a powerful method to probe optical absorption [5,42,43]. For graphene, the absorption arising from the intraband and interband optical excitations has been investigated over a wide spectral range, from the far infrared to the ultraviolet [42,43]. Our numerical prediction of the electronic and optical properties of doped graphene are expected to be verified by optical measurements. 2

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Fig. 3. (Color online) The absorption Si, (b) B, and (c) N guest atoms for 2 optical excitation channels are shown the occupied bands to the unoccupied

Fig. 2. (Color online) The band structures of graphene doped with (I) Si, (II) B, and (III) N guest atoms for various concentrations of (a) 2 × 2, (b) 3 × 3, (c) 4 × 4, and (d) 5 × 5 extended primitive cells. The original unit cell is extended by (a) 2 × 2, (b) 3 × 3, (c) 4 × 4, or (d) 5 × 5 along the lattice vectors.

spectra at𝑇 = 0 K of graphene doped with (a) × 2 extended primitive cell. The corresponding in (d) through (f) by the vertical arrows from bands.

Fig. 3(d) through 3(f). We observe diverse spectral structures, including absorption peaks and shoulders with various shapes. As for Si-graphene, the threshold structure, a shoulder marked by a yellow triangle in Fig. 3(a), is due to excitation from the valence maximum to the conduction minimum extremes, indicated by the yellow arrow head in Fig. 3(d). We note that several extreme-related transitions with equivalent frequency (green arrow heads in Fig. 3(d) as an example) could give a spectral peak (the corresponding green triangle in Fig. 3(a)) instead of a shoulder. The sharp peaks (pink and brown arrow heads) and the broader peaks (blue and purple arrow heads) correspond to optical excitations between the flat bands, for which their inequivalent intensities are associated with the relevant DOS. The differences in electronic properties of graphene doped with three types of dopants lead to critical differences in the optical excitation channels among the doped systems. As a matter of fact, the absorption spectra of B- and N-doped graphene (referring to Figs. 3(b) and 3(c) for 2 × 2 systems) are quite different from that of Si-graphene in terms of spectral structure and amplitude. This is attributed to not only the distinct energy dispersion but also the shift in Fermi levels of nand p-doped graphene. It is noticed that, the similarity of the absorption structures in the presence of B and N dopants could be simply explained by the special relation between the band structures, as mentioned above. The threshold excitations of the 2 × 2 B- and N-graphene are revealed as low and broad asymmetric peaks. They are generated from the vertical transitions between the Fermi surfaces and the nearest energy bands, indicated by the yellow arrow heads in Figs. 3(e) and 3(f). Similar optical transitions related to the Fermi surfaces and the next-nearest energy bands (blue arrow heads) yield the absorption shoulders (blue triangles) at higher frequencies. The absorption peaks with different intensities are associated with the transitions from the flat bands, similar to those of Si-graphene.

dopant density remarkably intensifies the number of energy bands near zero energy. This is because the electronic properties at low energy are no longer controlled by only the C-2𝑝𝑧 orbital but also by the guest orbitals of 𝐵 − 2𝑝𝑧 , 𝑆𝑖 − (2𝑝𝑧 , 3𝑝𝑧 ), and 𝑁 − 2𝑝𝑧 . This phenomenon moreover gives rise to significant fluctuation of the Fermi energy through the electron number difference between the host and guest atoms. It is helpful to note that, the Fermi level remains the same as in pristine graphene when it is doped with Si. However, it is shifted downward to the valence bands for p-doped B-graphene or shifted upward to the conduction bands for n-doped N-graphene. The values for the Fermi energy are further varied by changing the density of B and N guest atoms in doped graphene, as clearly shown in Table I of the Supplemental Material. Interestingly, the energy dispersion of B-graphene resembles that of N-graphene with the valence and conduction bands interchanged. This could be explained by the fact that a B atom has one less electron than a C atom while the opposite is true for a N atom. Particularly, when graphene is doped with a B (N) atom, an electron is removed from (added to) the system, leading to the change in the occupied and unoccupied bands as well as the Fermi level. The influence of doping on the optical-absorption spectra could be interpreted by examining the main features of related energy dispersion. The vertical transitions from occupied electronic to unoccupied states lead to the finite amplitude of the absorption function. The appearance of certain special optical-spectral structures is associated with transitions from band-edge states, the saddle points, and the flat bands. Fig. 3(a) through 3(c) show plots of the absorption spectra of 2 × 2 doped graphene for Si, B, and N dopants, respectively. The corresponding excitation channels are marked with arrow heads in 3

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Fig. 5. (Color online) Comparison of threshold absorption structures of doped graphene for various (a) guest atoms and (b) concentrations. Fig. 4. (Color online) The absorption spectra at 𝑇 = 0 K of Si-doped graphene with the unit cell extended by (a) 2 × 2, (b) 3 × 3, (c) 4 × 4, and (d) 5 × 5. The corresponding band structures and excitation channels are shown in (e) through (h)

decreased for B- and N-doped graphene. This comes from the forbidden optical transitions between the highest valence and lowest conduction subbands for n-doped and p-doped systems due to the shift of 𝐸𝐹 . Doping concentration is also one of the critical factors for modifying the spectral features, especially the threshold structure, as demonstrated in Fig. 5(b) for Si-graphene with various dopant densities. Regarding the doped systems with band gap opening, the threshold frequency is closely related to the magnitude of the energy gap, which is substantially reduced when the dopant density is increased. The difference in threshold intensity could be explained by the smaller DOS for higher doping concentration. On the other hand, for gapless doped graphene (green curve for 𝑁0 = 3), the threshold structure corresponds to the transition between the Fermi surface and the nearest subband. Therefore a simple relationship between the threshold frequency and doping concentration is absent.

Optical properties of doped graphene could also be efficiently manipulated by changing the doping concentration. The panel on the left-hand side in Fig. 4 presents the optical-absorption spectra of Sidoped graphene for 2 ×2, 3 × 3, 4× 4, and 5 × 5 extended primitive cells. The panel on the right-hand side presents plots of the corresponding excitation channels. For increased dopant density, the substantial enhancement in subband energy in the low-energy regime and significant modification of energy dispersion features give rise to remarkable changes in the spectral structures. There exist more optical excitation channels between the occupied valence and unoccupied conduction electronic states. As a matter of fact, many of them possess equivalent frequencies. The amalgamation of these spectral structures is responsible for several exotic absorption structures, e.g., those marked by the pink triangle in Fig. 4(c) and brown triangle in Fig. 4(d). Furthermore, the overall spectral intensity is evidently reduced for higher doping concentration. Apparently, the lack of flat bands in the vicinity of 𝐸𝐹 results in the disappearance of high-intensity absorption structures. Similar behaviors are also observed in the absorption spectra of B- and N-doped graphene, as illustrated in Figs. S5 and S6 of the Supplemental Material. The dependence of the threshold absorption structures on the type and density of dopants deserves a thorough discussion. This should provide helpful information for examination by the infrared spectroscopy technique. The threshold excitation channels of B- and N-graphene are dissimilar from that of Si-graphene, leading to differences in absorption structures and frequencies, referring to Fig. 5(a). The former belongs to vertical transitions between the Fermi surfaces and the nearest energy bands, differing from the extreme-related excitation of the latter. With the increase of frequency from where the threshold excitation appears, the amplitude of 𝐴(𝜔) rises for Si-doped but is

4. Concluding remarks In conclusion, we have presented in this paper the results from a comprehensive investigation of the feature-rich electronic and optical properties of doped graphene with the aid of the tight-binding model. By choosing different guest atoms, i.e., Si, B, and N, with various concentrations, we have demonstrated the robust influence played by the doping effect on the feature-rich energy bands and optical-absorption spectra. Doping greatly alters the main characteristics of the energy dispersion, such as the opening of a band gap, the number of energy subbands, and the shift of the Fermi levels. The optical-absorption spectra present diverse spectral structures, including absorption peaks and shoulders with various shapes. The spectral frequency, intensity, and special structures are significantly modified by both the type and density of dopants. The predicted results in this work are expected to be verified by optical measurements and they should provide useful information for future experiments. 4

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Declaration of competing interest

[19] L.S. Panchakarla, K.S. Subrahmanyam, S.K. Saha, A. Govindaraj, H.R. Krishnamurthy, U.V. Waghmare, C.N.R. Rao, Synthesis, structure, and properties of boron- and nitrogen-doped graphene, Adv. Mater. 21 (2009) 4726–4730, http://dx.doi.org/10.1002/adma.200901285. [20] S.S. Varghese, S. Swaminathan, K.K. Singh, V. Mittal, Energetic stabilities, structural and electronic properties of monolayer graphene doped with boron and nitrogen atoms, Electronics 5 (2016) 91, http://dx.doi.org/10.3390/ electronics5040091. [21] Y.-C. Zhou, H.-L. Zhang, W.-Q. Deng, A 3N rule for the electronic properties of doped graphene, Nanotechnology 24 (2013) 225705, http://dx.doi.org/10.1088/ 0957-4484/24/22/225705. [22] X. Wang, X. Li, L. Zhang, Y. Yoon, P.K. Weber, H. Wang, J. Guo, H. Dai, N-doping of graphene through electrothermal reactions with ammonia, Science 324 (2009) 768–771, http://dx.doi.org/10.1126/science.1170335. [23] X. Li, X. Wang, L. Zhang, S. Lee, H. Dai, Chemically derived, ultrasmooth graphene nanoribbon semiconductors, Science 319 (2008) 1229–1232, http: //dx.doi.org/10.1126/science.1150878. [24] P.A. Denis, Band gap opening of monolayer and bilayer graphene doped with aluminium, silicon, phosphorus, and sulfur, Chem. Phys. Lett. 492 (2010) 251–257, http://dx.doi.org/10.1016/j.cplett.2010.04.038. [25] R.B. dos Santos, R. Rivelino, F. de B. Mota, G.K. Gueorguiev, Effects of N doping on the electronic properties of a small carbon atomic chain with distinct 𝑠𝑝2 terminations: a first-principles study, Phys. Rev. B. 84 (2011) 075417, http://dx.doi.org/10.1103/PhysRevB.84.075417. [26] R.B. dos Santos, R. Rivelino, F. de B. Mota, G.K. Gueorguiev, Exploring hydrogenation and fluorination in curved 2D carbon systems: a density functional theory study on corannulene, J. Phys. Chem. A. 116 (2012) 9080–9087, http: //dx.doi.org/10.1021/jp3049636. [27] S. Casolo, R. Martinazzo, G.F. Tantardini, Band engineering in graphene with superlattices of substitutional defects, J. Phys. Chem. C. 115 (2011) 3250–3256, http://dx.doi.org/10.1021/jp109741s. [28] L. Dössel, L. Gherghel, X. Feng, K. Müllen, Graphene nanoribbons by chemists: nanometer-sized, soluble, and defect-free, Angew. Chem. International Edition 50 (2011) 2540–2543, http://dx.doi.org/10.1002/anie.201006593. [29] J. Cai, P. Ruffieux, R. Jaafar, M. Bieri, T. Braun, S. Blankenburg, M. Muoth, A.P. Seitsonen, M. Saleh, X. Feng, K. Müllen, R. Fasel, Atomically precise bottom-up fabrication of graphene nanoribbons, Nature 466 (2010) 470–473, http://dx.doi.org/10.1038/nature09211. [30] T. Taychatanapat, K. Watanabe, T. Taniguchi, P. Jarillo-Herrero, Quantum hall effect and landau-level crossing of dirac fermions in trilayer graphene, Nat. Phys. 7 (2011) 621–625, http://dx.doi.org/10.1038/nphys2008. [31] F. Zhang, B. Sahu, H. Min, A.H. MacDonald, Band structure of 𝐴𝐵𝐶-stacked graphene trilayers, Phys. Rev. B. 82 (2010) 035409, http://dx.doi.org/10.1103/ PhysRevB.82.035409. [32] L. Yang, J. Han, Electronic structure of deformed carbon nanotubes, Phys. Rev. Lett. 85 (2000) 154–157, http://dx.doi.org/10.1103/PhysRevLett.85.154. [33] M. Ezawa, Peculiar width dependence of the electronic properties of carbon nanoribbons, Phys. Rev. B. 73 (2006) 045432, http://dx.doi.org/10.1103/ PhysRevB.73.045432. [34] P.-H. Shih, T.-N. Do, G. Gumbs, H.D. Pham, M.-F. Lin, Electric-field-diversified optical properties of bilayer silicene, Opt. Lett. OL. 44 (2019) 4721–4724, http://dx.doi.org/10.1364/OL.44.004721. [35] P.-H. Shih, Y.-H. Chiu, J.-Y. Wu, F.-L. Shyu, M.-F. Lin, Coulomb excitations of monolayer germanene, Sci. Rep. 7 (2017) 40600, http://dx.doi.org/10.1038/ srep40600. [36] P.-H. Shih, C.-W. Chiu, J.-Y. Wu, T.-N. Do, M.-F. Lin, Coulomb scattering rates of excited states in monolayer electron-doped germanene, Phys. Rev. B. 97 (2018) 195302, http://dx.doi.org/10.1103/PhysRevB.97.195302. [37] T.-N. Do, G. Gumbs, P.-H. Shih, D. Huang, M.-F. Lin, Valley- and spin-dependent quantum hall states in bilayer silicene, Phys. Rev. B. 100 (2019) 155403, http://dx.doi.org/10.1103/PhysRevB.100.155403. [38] F. Bonaccorso, Z. Sun, T. Hasan, A.C. Ferrari, Graphene photonics and optoelectronics, Nat. Photon. 4 (2010) 611–622, http://dx.doi.org/10.1038/nphoton. 2010.186. [39] N.M. Gabor, J.C.W. Song, Q. Ma, N.L. Nair, T. Taychatanapat, K. Watanabe, T. Taniguchi, L.S. Levitov, P. Jarillo-Herrero, Hot carrier–assisted intrinsic photoresponse in graphene, Science 334 (2011) 648–652, http://dx.doi.org/10. 1126/science.1211384. [40] J. Park, Y.H. Ahn, C. Ruiz-Vargas, Imaging of photocurrent generation and collection in single-layer graphene, Nano Lett. 9 (2009) 1742–1746, http://dx. doi.org/10.1021/nl8029493. [41] X. Xu, N.M. Gabor, J.S. Alden, A.M. van der Zande, P.L. McEuen, Photothermoelectric effect at a graphene interface junction, Nano Lett. 10 (2010) 562–566, http://dx.doi.org/10.1021/nl903451y. [42] K.S. Kim, Y. Zhao, H. Jang, S.Y. Lee, J.M. Kim, K.S. Kim, J.-H. Ahn, P. Kim, J.Y. Choi, B.H. Hong, Large-scale pattern growth of graphene films for stretchable transparent electrodes, Nature 457 (2009) 706–710, http://dx.doi.org/10.1038/ nature07719.

None Acknowledgments The authors thank the Ministry of Science and Technology of Taiwan (R.O.C.) under Grant No. MOST 105-2112-M-017-002-MY2. T. N. Do would like to thank the Ton Duc Thang University. Appendix A. Supplementary data Supplementary material related to this article can be found online at https://doi.org/10.1016/j.physe.2019.113894. References [1] P.R. Wallace, The band theory of graphite, Phys. Rev. 71 (1947) 622–634, http://dx.doi.org/10.1103/PhysRev.71.622. [2] J.-C. Charlier, J.-P. Michenaud, X. Gonze, J.-P. Vigneron, Tight-binding model for the electronic properties of simple hexagonal graphite, Phys. Rev. B. 44 (1991) 13237–13249, http://dx.doi.org/10.1103/PhysRevB.44.13237. [3] M. Sprinkle, D. Siegel, Y. Hu, J. Hicks, A. Tejeda, A. Taleb-Ibrahimi, P. Le Fèvre, F. Bertran, S. Vizzini, H. Enriquez, S. Chiang, P. Soukiassian, C. Berger, W.A. de Heer, A. Lanzara, E.H. Conrad, First direct observation of a nearly ideal graphene band structure, Phys. Rev. Lett. 103 (2009) 226803, http://dx.doi.org/ 10.1103/PhysRevLett.103.226803. [4] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A.A. Firsov, Electric field effect in atomically thin carbon films, Science 306 (2004) 666–669, http://dx.doi.org/10.1126/science.1102896. [5] L.A. Falkovsky, Optical properties of graphene, J. Phys.: Conf. Ser. 129 (2008) 012004, http://dx.doi.org/10.1088/1742-6596/129/1/012004. [6] L.A. Falkovsky, S.S. Pershoguba, Optical far-infrared properties of a graphene monolayer and multilayer, Phys. Rev. B. 76 (2007) 153410, http://dx.doi.org/ 10.1103/PhysRevB.76.153410. [7] B. Zhan, C. Li, J. Yang, G. Jenkins, W. Huang, X. Dong, Graphene field-effect transistor and its application for electronic sensing, Small 10 (2014) 4042–4065, http://dx.doi.org/10.1002/smll.201400463. [8] M. Liang, B. Luo, L. Zhi, Application of graphene and graphene-based materials in clean energy-related devices, Int. J. Energ. Res. 33 (2009) 1161–1170, http: //dx.doi.org/10.1002/er.1598. [9] S. Bae, H. Kim, Y. Lee, X. Xu, J.-S. Park, Y. Zheng, J. Balakrishnan, T. Lei, H. Ri Kim, Y.I. Song, Y.-J. Kim, K.S. Kim, B. Özyilmaz, J.-H. Ahn, B.H. Hong, S. Iijima, Roll-to-roll production of 30-inch graphene films for transparent electrodes, Nat. Nanotechnol. 5 (2010) 574–578, http://dx.doi.org/10.1038/ nnano.2010.132. [10] O.V. Yazyev, S.G. Louie, Topological defects in graphene: dislocations and grain boundaries, Phys. Rev. B. 81 (2010) 195420, http://dx.doi.org/10.1103/ PhysRevB.81.195420. [11] A. Cresti, N. Nemec, B. Biel, G. Niebler, F. Triozon, G. Cuniberti, S. Roche, Charge transport in disordered graphene-based low dimensional materials, Nano Res. 1 (2008) 361–394, http://dx.doi.org/10.1007/s12274-008-8043-2. [12] M.Y. Han, B. Özyilmaz, Y. Zhang, P. Kim, Energy band-gap engineering of graphene nanoribbons, Phys. Rev. Lett. 98 (2007) 206805, http://dx.doi.org/ 10.1103/PhysRevLett.98.206805. [13] Y.-H. Zhang, Y.-B. Chen, K.-G. Zhou, C.-H. Liu, J. Zeng, H.-L. Zhang, Y. Peng, Improving gas sensing properties of graphene by introducing dopants and defects: a first-principles study, Nanotechnology 20 (2009) 185504, http://dx.doi.org/10. 1088/0957-4484/20/18/185504. [14] A. Lherbier, X. Blase, Y.-M. Niquet, F. Triozon, S. Roche, Charge transport in chemically doped 2D graphene, Phys. Rev. Lett. 101 (2008) 036808, http: //dx.doi.org/10.1103/PhysRevLett.101.036808. [15] J. Dai, J. Yuan, P. Giannozzi, Gas adsorption on graphene doped with b, n, al, and s: a theoretical study, Appl. Phys. Lett. 95 (2009) 232105, http: //dx.doi.org/10.1063/1.3272008. [16] W. Zhou, M.D. Kapetanakis, M.P. Prange, S.T. Pantelides, S.J. Pennycook, J.-C. Idrobo, Direct determination of the chemical bonding of individual impurities in graphene, Phys. Rev. Lett. 109 (2012) 206803, http://dx.doi.org/10.1103/ PhysRevLett.109.206803. [17] Z.-H. Sheng, H.-L. Gao, W.-J. Bao, F.-B. Wang, X.-H. Xia, Synthesis of boron doped graphene for oxygen reduction reaction in fuel cells, J. Mater. Chem. 22 (2012) 390–395, http://dx.doi.org/10.1039/C1JM14694G. [18] A. Yanilmaz, A. Tomak, B. Akbali, C. Bacaksiz, E. Ozceri, O. Ari, R. T.Senger, Y. Selamet, H.M. Zareie, Nitrogen doping for facile and effective modification of graphene surfaces, RSC Adv. 7 (2017) 28383–28392, http://dx.doi.org/10.1039/ C7RA03046K. 5

Physica E: Low-dimensional Systems and Nanostructures 118 (2020) 113894

P.-H. Shih et al.

[45] L.C. Lew Yan Voon, L.R. Ram-Mohan, Tight-binding representation of the optical matrix elements: theory and applications, Phys. Rev. B. 47 (1993) 15500–15508, http://dx.doi.org/10.1103/PhysRevB.47.15500. [46] M. Rohlfing, S.G. Louie, Electron–hole excitations and optical spectra from first principles, Phys. Rev. B. 62 (2000) 4927–4944, http://dx.doi.org/10.1103/ PhysRevB.62.4927.

[43] X. Li, W. Cai, J. An, S. Kim, J. Nah, D. Yang, R. Piner, A. Velamakanni, I. Jung, E. Tutuc, S.K. Banerjee, L. Colombo, R.S. Ruoff, Large-area synthesis of high-quality and uniform graphene films on copper foils, Science 324 (2009) 1312–1314, http://dx.doi.org/10.1126/science.1171245. [44] P.-H. Shih, T.-N. Do, B.-L. Huang, G. Gumbs, D. Huang, M.-F. Lin, Magnetoelectronic and optical properties of Si-doped graphene, Carbon 144 (2019) 608–614, http://dx.doi.org/10.1016/j.carbon.2018.12.040.