Electronic structures and optical properties of arsenene and antimonene under strain and an electric field

Abstract

Using density functional and many-body perturbation theories, we explore the strain and electric field effects on the electronic structures and optical properties of hexagonal arsenene (β-As) and antimonene (β-Sb). The calculations show that they can transform from indirect into direct bandgap semiconductors, and even semimetals under biaxial tensile strain and an electric field perpendicular to the layer. In particular, under a stronger electric field, their bandgaps gradually close owing to the field-induced motion of nearly free electron states. More interestingly, increasing the strain can significantly red-shift the optical absorption spectra and even enhance the optical absorption in the energy region of 1.2–2.2 eV (including infrared and partial visible light). Under a stronger electric field, their optical absorptions are enhanced and a large exciton binding energy can be retained. Such dramatic characteristics in the electronic structures and optical properties suggest great potential of β-As and β-Sb for novel electronic and optoelectronic devices.

---

1 Introduction

Atomically thin two-dimensional (2D) materials have attracted continuous attention due to their rather unique properties and potential applications.^1–5 Typically, semimetal graphene can possess an excellent carrier mobility of up to 23 600 cm² V⁻¹ s⁻¹,⁶ which enables its superior performance in electrochemical electrodes.⁷ Several semiconducting monolayers of MX₂ (M = Mo, W; X = S, Se)^8–10 and black phosphorus¹¹ exhibit substantial changes in their electronic structure and optoelectronic properties compared to their bulk. For example, bulk MoS₂ has an indirect bandgap of 1.2 eV.¹² When it is thinned to a monolayer, a direct bandgap of 1.8 eV and a strong photoluminescence emerge.^13,14 Very recently, arsenic monolayers (β-As) and antimony monolayers (β-Sb) with a hexagonal lattice structure have been predicted to be highly stable.¹⁵ Experimentally, β-Sb has also been fabricated by micromechanical exfoliation and shows high stability under ambient conditions.¹⁶ In addition, β-As (β-Sb) has an appropriate bandgap of about 2.64 (2.28) eV,^15,17 which fills the bandgap vacancy among existing 2D semiconductors, such as black phosphorus (1.3 eV),¹¹ MoS₂ (1.8 eV),¹³ and BN (5.5 eV).¹⁸ However, the indirect bandgaps in β-As and β-Sb might impede their potential application in optoelectronic devices.

Strain and an electric field, as an effective means of tuning material properties, have been widely used to attain controllable bandgaps in various 2D materials, like graphene, silicene, MoS₂, phosphorene, etc. For example, graphene can exhibit a bandgap of about 100 meV under a tensile strain.¹⁹ The bandgap of silicene with semimetallic characteristics could be tuned in the order of tens of meV under an external electric field.^20,21 An increasing vertical electric field in a rippled MoS₂ monolayer would decrease the bandgap significantly.²² The application of a vertical electric field to blue phosphorene can even induce a semiconductor–metal transition when the electric field is as large as about 0.7 V Å⁻¹.²³ Moreover, experimental observations demonstrate that strain and an external electric field also have a great influence on the optical properties of 2D materials.^24–26 A tensile strain induces lower excitation energy for the same excitation in GaN epitaxial layers on SiC substrates in comparison to strain-free bulk GaN.²⁷ An increased gate voltage in monolayer MoS₂ can lead to an enhancement in optical absorption near about 660 nm.^24,25 Such electronic band structure and optical absorption engineering in 2D materials are essential to promote their application in various nanoelectronic and nanophotonic devices. Therefore, it is highly desirable to study the electronic and optical properties of β-As (β-Sb) and their response to strain and an electric field.

In this work, by employing density functional theory (DFT) combined with G₀W₀ approximation and Bethe–Salpeter equation (BSE) calculations, we provide a systematic study on the electronic structures and optical properties of β-As (β-Sb) and their response to tensile strain and a perpendicular electric field. Our calculations show that the nature of the indirect bandgap of β-As and β-Sb monolayers can be converted by strain and an electric field. Their band structures and optical absorptions are very sensitive to tensile strain, while they are insensitive to a perpendicular electric field in the range of 0–0.5 V Å⁻¹.

2 Computational details

A three-step procedure was employed to calculate the electronic and optical properties of β-As and β-Sb. Firstly, we employed the general gradient approximation (GGA) together with the Perdew–Burke–Ernzerh (PBE) exchange–correlation functional²⁸ to obtain the energies and wave functions of their ground states, as implemented in Quantum Espresso.²⁹ A plane-wave basis set with a kinetic energy cutoff of 80 Ry, a k-point grid sampling of 27 × 27 × 1, and a norm-conserving Troullier Martins pseudopotential³⁰ for ion–electron interactions were used. A vacuum region of 20 Å along the c direction (perpendicular to the sheet) was constructed to eliminate spurious interactions between periodic images. The total energy was converged within 10⁻⁴ eV and the force acting on each atom was less than 0.01 eV Å⁻¹. An external sawtooth potential was used in the c direction to simulate the effect of an applied electric field perpendicular to the sheet.³¹ The stabilities of the structures were tested at high temperatures using ab initio molecular dynamics (MD) simulations, where the velocities of atoms were rescaled at a given temperature through the Nosé algorithm with a time step of 1 fs and the structures were held for 5 ps at each temperature, as implemented in the SIESTA code.³²

Then, the G₀W₀ approximation³³ was treated to obtain quasi-particle (QP) energies of β-As and β-Sb monolayers. The convergence of the quasi-particle band gap with respect to the number of empty bands, the size of the dielectric matrix and the Monkhosrt–Pack grid were carefully examined and a convergence within 0.1 eV was assured. Finally, the coupled electron–hole excitation energies and exciton wave functions were obtained by solving the Bethe–Salpeter equation (BSE).^34,35 The involved unoccupied band number (960) was used to attain the converged dielectric function within the random phase approximation (RPA).³⁶ A fine k-grid (54 × 54 × 1), 5 valence bands and 15 conduction bands were included to obtain the converged optical spectra. The G₀W₀ and BSE calculations were performed with the YAMBO code.³⁷

3 Results and discussion

Fig. 1(a) and (b) show the optimized geometrical structures of β-As and β-Sb, which share some features with silicene, like a buckled honeycomb lattice. The buckled hexagonal structure increases their stability and they own a high symmetric space group of P3m1. Their optimized structural parameters are presented in Table 1. The lattice constant (a), bond length (l), and buckled height (Δ) significantly increase from β-As to β-Sb, while the bond angle (θ) follows the order of θ (As) > θ (Sb). Stability and experimental feasibility have to be considered for β-As and β-Sb. So, we calculated their adhesive energy, which is defined as E_a = E_tot/n − E_As/Sb, where E_tot is the total energy of β-As or β-Sb while E_As/Sb is the energy of a single As or Sb atom and n is the number of As or Sb atoms. The adhesive energies (E_a) of β-As and β-Sb are estimated to be −3.41 and −2.87 eV per atom, respectively. On the other hand, no soft phonon modes are available in the computed phonon dispersion spectra of the β-As and β-Sb monolayers.^15,38 The negative adhesive energy and phonon dispersion indicate that they are thermodynamically stable and should be achievable in experiments.

Fig. 1 (a and b) Top and side views of the optimized geometric structures of β-As and β-Sb. The blue and orange balls represent the As or Sb atoms of different layers. (c) The first Brillouin zone of β-As or β-Sb. (d) Bond length and buckling height are labeled as l and Δ, respectively.

Table 1 Structural parameters and energy bandgaps of β-As and β-Sb: lattice a (Å), bond length l (Å), buckling height Δ (Å), and bond angle θ (°); E_g-PBE (eV) and E_g-G0W0 (eV) at the DFT-PBE and G₀W₀ levels, respectively. “i” shows the indirect bandgap

Name	a	l	Δ	θ	E g-PBE	E g-G0W0
β-As	3.635	2.5216	1.4053	91.951	1.61 (i)	2.70 (i)
β-Sb	3.982	2.8047	1.6110	90.295	1.10 (i)	2.25 (i)

---

Fig. 2(a) and (b) show the band structures of β-As and β-Sb from the DFT-PBE results. They are predicted to be semiconductors with indirect bandgaps of 1.61 eV (β-As) and 1.10 eV (β-Sb), respectively, which agree well with those reported in the literature.^39–41 However, the Kohn–Sham states of standard DFT seriously underestimate the bandgap of semiconductors. To obtain accurate band structures of β-As and β-Sb, we further utilize the G₀W₀ method to gain their quasi-particle band structures, as plotted in Fig. 2(c and d). Compared with the DFT-PBE results, the corrected conduction (valence) band edge is shifted up (down), resulting in increased bandgaps of 2.70 eV (β-As) and 2.25 eV (β-Sb). The quasi-particle bandgap of β-As agrees well with the reported result of 2.64 eV.¹⁷ Such a significant quasi-particle correction to the DFT-PBE energy gap is the result of reduced electronic screening in the low-dimensional system, which can significantly enhance Coulomb interactions in β-As and β-Sb. Similar significant QP corrections in the atomically thin systems have also been observed in MoS₂ and phosphorene.^42,43

Fig. 2 Band structures of β-As and β-Sb: the DFT-PBE results (a and b), and the G₀W₀ results (c and d). The VBMs of the two structures are shifted to zero.

The moderate band gaps of β-As and β-Sb can be utilized in optoelectronic devices. However, they will suffer from poorly efficient light emission due to the nature of indirect band gaps. Strain and electric field have been proved to be an effective means of tuning the electronic properties in 2D layered materials. Encouragingly, we firstly checked the gap variation of β-As and β-Sb under biaxial tensile strain (δ), as shown in Fig. 3(a). They will undergo an indirect to direct bandgap transition under a relatively large critical strain δ_c (δ_c = 0.08 for β-As and δ_c = 0.09 for β-Sb). The critical strain of 0.08 in β-As is slightly larger than the recently reported result,⁴⁴ and different from the reported critical strains of 0.02³⁹ and 0.04.¹⁵ The difference of critical strain should be attributed to the lattice constant obtained from different methods and computational details. The band structures of β-As under tensile strains of 0.02 and 0.04 are presented in Fig. S1 (ESI†). There is about a 10 meV energy difference between the functional gap and the energy gap at the Γ point under a tensile strain of 0.02, while it will increase to 60 meV under a tensile strain of 0.04. In Fig. 3(b), we present the changes in the two band edges of β-As under a tensile strain of 0–0.12. Under 0–0.02 tensile strain, the conduction band minimum (CBM) of β-As remains in the middle of the Γ–M pathway and the valence band maximum (VBM) occupies the Γ point (for details, see Fig. S1, ESI†). However, upon further increasing the strain up to 0.08, the CBM and the VBM shift to the Γ point and a certain point along the Γ–K pathway, respectively. And the band gap linearly decreases in the tensile strain range from 0.02 to 0.08, which is attributed to the persistent downward shift of the CBM at the Γ point. Under a critical strain of 0.08, β-As becomes a direct band-gap semiconductor at the Γ point and the semiconductor characteristics persisted under a tensile strain of 0.08 to 0.12. Under a tensile strain of 12%, we examine the stability of β-As by calculating its phonon dispersion spectrum and carrying out first-principle molecular dynamics (MD) simulations. The results are shown in Fig. S2(a) and S3 (ESI†), suggesting that β-As with a strain of 12% exhibits dynamical and thermal stability. Moreover, other theoretical studies have also shown that β-As is dynamically stable when the biaxial tensile strain is smaller than 0.184,⁴⁴ suggesting that it can endure a great tensile strain. Experimentally, a great tensile strain in 2D materials can also be realized by epitaxy on a substrate or by mechanical loading.^45,46 The overall trends in the changes of the band gap and band edge of β-Sb in response to biaxial tensile strain (Fig. 3(a) and (c)) are very similar to those of β-As. And the phonon dispersion spectrum and high-temperature MD simulation of β-Sb under a strain of 12% also indicate that it is stable dynamically and thermally, as shown in Fig. S2(b) and S3 (ESI†).

Fig. 3 (a) The bandgap variation under biaxial tensile strain. The dot dash line shows the position of the critical strain of indirect-direct gap transition. (b and c) The positions of conduction and valence band edges under different strains and electric fields for β-As and β-Sb, respectively. The valence band maximum shifts to zero. (d) Bandgaps of β-As and β-Sb obtained from the DFT-PBE method as a function of perpendicular electric field.

As we know, applying a perpendicular electric field to silicene (a buckled structure similar to β-As and β-Sb) can obtain a controllable bandgap and the opened bandgap is 80 meV under an electric field of 0.51 V Å⁻¹.^47,48 The tunable bandgap in silicene is attributed to a significant separation between two non-equivalent Si-sublattices under the perpendicular electric field.²⁰ It is also an interesting problem to investigate how the band structures of β-As and β-Sb are modified under a perpendicular electric field. For this purpose, a perpendicular electric field (see the inset in Fig. 3(d)) is applied to β-As and β-Sb, respectively, ranging from 0 to 0.70 V Å⁻¹ for β-As and 0 to 0.63 V Å⁻¹ for β-Sb. Unlike that of silicene,²⁰ the buckled height (Δ) between two As (Sb) sublattices shown in Fig. 1(d) is insensitive to the perpendicular electric field. For example, for β-As, the buckled height only changes by about 0.35% upon increasing the strength of the electric field from 0 to 0.70 V Å⁻¹. Other structural parameters like the bond length (l) and the bond angle (θ) only have a slight difference, as summarized in Table S1 (ESI†). On the other hand, their bandgaps only have a difference of 10 meV below the electric field of 0.5 V Å⁻¹ (see Fig. 3(d)). However, their electronic band structures undergo a substantial change, particularly under a stronger applied electric field (Fig. 4). When the electric field is larger than 0.5 V Å⁻¹, the CBMs of β-As and β-Sb shift to the Γ point, converting them to semiconductors with a direct bandgap. In fact, an indirect-to-direct gap transition occurs at 0.51 V Å⁻¹ and 0.50 V Å⁻¹ for β-As and β-Sb, respectively. And the bandgap decreases linearly until it closes at a critical electric field of 0.70 V Å⁻¹ for β-As and 0.63 V Å⁻¹ for β-Sb, respectively. The significant decrease of the bandgap originates from the rapid drop of the nearly free electron band, which directly results in a semiconductor–semimetal transition occurring at 0.70 (β-As) and 0.63 (β-Sb) V Å⁻¹. The phenomenon that the bandgap of a system significantly decreases due to the emergence of the nearly free electron band has also been observed in other low-dimensional systems, such as BN nanoribbons⁴⁹ and monolayer blue phosphorene.⁵⁰ In addition, the VBM retained in the Γ point does not change significantly even under a stronger electric field, which shows that the VBM is very insensitive to the electric field compared to that of the biaxial tensile strain.

Fig. 4 Band structures at an external electric field of 0, 0.5, and 0.6 V Å⁻¹: (a–c) β-As, (d–f) β-Sb. The valence band maximum is set to zero and the near free electron band is shown by the red open circles.

Fig. 5 shows the optical absorption spectra of β-As and β-Sb with electron–hole interactions included (G₀W₀ + BSE) for incident light polarized along lattice vector a and c directions. For light polarized along the a direction, their optical absorption spectra are dominated by enhanced excitonic states, which are attributed to the weakened electronic screening in low-dimensional systems. And the excitation energy corresponding to the first absorption peak has a significant difference between them, 2.44 eV for β-As and 2.25 eV for β-Sb. The first absorption peak corresponds to a bound excitonic state, which mainly originates from the inter-band transition between the VBM and CBM at the high symmetry Γ point. The binding energy of the bound exciton is as large as 0.81 eV for β-As and 0.73 eV for β-Sb, respectively. The exciton binding energy is defined as the difference between the excitation energy and the QP energy difference. The large exciton binding energy in β-As and β-Sb can effectively confine both electrons and holes, and thus prevent the rapid recombination of photogenerated electrons and holes. This suggests that β-As and β-Sb have potential to be used in optoelectronics devices. More importantly, there is a wide optical absorption in the energy range of 2–4 eV (including the main part of visible and ultraviolet light) in β-As and β-Sb, which may make them potential candidates for photovoltaic devices. On the other hand, compared with that of β-As, the absorption onset of β-Sb has a significant red shift, and thus a global red shift of the whole spectrum is seen. This phenomenon should be attributed to the smaller bandgap of β-Sb. For light polarization perpendicular to the surface (along the c direction), their optical absorptions are inactive around the bandgap due to the forbidden transition, leading to a negligible absorbance in the observed optical spectral range, indicating that the β-As and β-Sb are directionally optically transparent materials.

Fig. 5 Optical absorption spectra of β-As and β-Sb for incident light polarization parallel (a) and perpendicular (c) to the surface. The green arrow shows the position of the first absorption peak. A Lorentzian broadening of 0.12 eV is adopted.

The optical absorptions of the β-As and β-Sb under biaxial tensile strain are also investigated, as illustrated in Fig. 6(a) and (b). A significant red-shift with increasing strain is observed in the optical spectra of β-As and β-Sb, suggesting that their optical properties are sensitive to the biaxial tensile strain. For example, the first absorption peaks are dependent on the tensile strain. They follow the order of (2.44 eV, δ = 0) > (2.07 eV, δ = 0.04) > (1.73 eV, δ = 0.08) > (1.62 eV, δ = 0.10) for β-As and similar results are also observed in β-Sb, as listed in Table 2. The position of the first absorption (E_e) does not follow the same trend of the bandgap, which possibly originates from the nature of the indirect bandgap under strain. Although the position of the first absorption has an evident difference, the exciton binding energy (E_b) corresponding to the first absorption peak is not very sensitive to the biaxial tensile strain, and only changes 0.09 (0.05) eV from δ = 0 to δ = 0.10 for β-As (β-Sb), as shown in Table 2. Additionally, applying tensile strains on β-As and β-Sb can significantly enhance the optical absorption in the 1.2–2.2 eV region, which indicates that strain on β-As and β-Sb is an effective way to enhance the absorption properties in the infrared and partial visible light regions.

Fig. 6 Optical absorption spectra of β-As and β-Sb for light polarized along the a direction under biaxial tensile strain (a and b) and an external electric field (c and d). A Lorentzian broadening of 0.12 eV is adopted.

Table 2 The excitation energy E_e (eV) and exciton binding energy E_b (eV) corresponding to the first absorption peak of β-As and β-Sb under different strains

	β-As				β-Sb
	δ = 0	δ = 0.04	δ = 0.08	δ = 0.10	δ = 0	δ = 0.04	δ = 0.08	δ = 0.10
E e	2.44	2.06	1.74	1.62	2.25	1.98	1.72	1.60
E b	0.81	0.79	0.75	0.72	0.73	0.71	0.70	0.68

---

Next, we consider the effects of a perpendicular electric field on the optical absorption of β-As and β-Sb. Here, electric field strengths of 0.51 and 0.6 V Å⁻¹ (0.50 and 0.6 V Å⁻¹) are applied on β-As (β-Sb), respectively. Fig. 6(c) and (d) display the optical absorption spectra of β-As and β-Sb under these electric fields. As depicted in Fig. 6(c) and (d), the positions of the characteristic peaks in the absorption spectra are almost independent of the strength of the electric field. Compared to those of zero electric field, their optical absorptions are almost unchanged at 0.51 V Å⁻¹ (β-As) and 0.50 V Å⁻¹ (β-Sb) due to their unchanged bandgaps. The fixed bandgap results in the slight change of the oscillator strength and the number of excited states in β-As and β-Sb, as shown in Fig. S4(b) and (e) (ESI†). This indicates that the optical absorptions of β-As and β-Sb are insensitive to the external electric field. Interestingly, when a large electric field of 0.6 V Å⁻¹ is applied, the absorption for the photon between 1.0 and 2.5 eV is strengthened in β-As and β-Sb, suggesting that the external electric field can tune the optical excitations of β-As and β-Sb. To clearly see the case of the optical excitation, the relative oscillator strengths of excited states in β-As and β-Sb are depicted in Fig. S4(c) and (f) (ESI†). From the figures, it is found that the number of excited states in the energy range of 1.0–2.5 eV is increased by the applied electric field. The increased number of excited states will induce enhanced optical absorption in β-As and β-Sb. Moreover, under an electric field of 0.6 V Å⁻¹, the position of the first absorption peak slightly decreases to 2.35 eV for β-As and 2.18 eV for β-Sb, respectively. Although the position of the first absorption peak has a shift of about 100 meV compared to those of zero electric field, the corresponding exciton binding energy can be as large as 0.86 eV for β-As and 0.74 eV for β-Sb, respectively. These exciton binding energies are almost the same as those of β-As and β-Sb under zero electric field. Under a stronger electric field, the first bound exciton in β-As and β-Sb is of particular interest due to the large binding energy and oscillator strength, which make β-As and β-Sb hold promise for possible important applications in optoelectronics.

4 Conclusions

In conclusion, we have performed DFT combined with G₀W₀ and BSE calculations to systematically investigate the electronic structures and optical properties of β-As and β-Sb. β-As and β-Sb are both indirect bandgap semiconductors with a wide size. Their optical absorption spectra display an excellent absorption within the visible–ultraviolet spectral region. And their exciton binding energies can be as large as 0.81 eV for β-As and 0.73 eV for β-Sb, which can effectively confine both electrons and holes. More importantly, the width of their bandgaps, as well as their indirect/direct characteristics, can be tuned by tensile strain and a perpendicular electric field. Upon increasing the strain, the optical absorption spectra can be red-shifted significantly, the optical absorption can be enhanced in the energy region of 1.2–2.2 eV, and large exciton binding energy can be retained in strained β-As and β-Sb. In addition, the optical absorptions of β-As and β-Sb are very insensitive to perpendicular electric fields below 0.5 V Å⁻¹. Under a stronger electric field of 0.6 V Å⁻¹, their optical absorptions are further enhanced in the energy range of 1.0–2.5 eV (including infrared and the main parts of visible light). Therefore, they are promising materials for applications in electronics and optoelectronics.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work is supported by the University Fund Project of Jiangsu Province (13KJB140004) in China and the Open Research Fund of State Key Laboratory of Bioelectronics (2055031601) in Southeast University. The authors appreciate Professor Jinlan Wang of Southeast University for fruitful discussions.

References

1. A. K. Geim and K. S. Novoselov, Nat. Mater., 2007, 6, 183–191.
2. F. Schwierz, Nat. Nanotechnol., 2010, 5, 487–496.
3. B. Radisavljevic, A. Radenovic, J. Brivio, V. Giacometti and A. Kis, Nat. Nanotechnol., 2011, 6, 147–150.
4. Z. Yin, H. Li, H. Li, L. Jiang, Y. Shi, Y. Sun, G. Lu, Q. Zhang, X. Chen and H. Zhang, ACS Nano, 2012, 6, 74–80.
5. L. Li, Y. Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng, X. H. Chen and Y. Zhang, Nat. Nanotechnol., 2014, 9, 372–377.
6. L. Liao, J. Bai, Y. Qu, Y.-C. Lin, Y. Li, Y. Huang and X. Duan, Proc. Natl. Acad. Sci. U. S. A., 2010, 107, 6711–6715.
7. S. Das Sarma, S. Adam, E. H. Hwang and E. Rossi, Rev. Mod. Phys., 2011, 83, 407–470.
8. Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman and M. S. Strano, Nat. Nanotechnol., 2012, 7, 699–712.
9. P. Tonndorf, R. Schmidt, P. Böttger, X. Zhang, J. Börner, A. Liebig, M. Albrecht, C. Kloc, O. Gordan, D. R. T. Zahn, S. Michaelis de Vasconcellos and R. Bratschitsch, Opt. Express, 2013, 21, 4908–4916.
10. H. Schmidt, S. Wang, L. Chu, M. Toh, R. Kumar, W. Zhao, A. H. Castro Neto, J. Martin, S. Adam, B. Özyilmaz and G. Eda, Nano Lett., 2014, 14, 1909–1913.
11. X. Wang, A. M. Jones, K. L. Seyler, V. Tran, Y. Jia, H. Zhao, H. Wang, L. Yang, X. Xu and F. Xia, Nat. Nanotechnol., 2015, 10, 517–521.
12. K. K. Kam and B. A. Parkinson, J. Phys. Chem., 1982, 86, 463–467.
13. A. Splendiani, L. Sun, Y. Zhang, T. Li, J. Kim, C.-Y. Chim, G. Galli and F. Wang, Nano Lett., 2010, 10, 1271–1275.
14. K. F. Mak, C. Lee, J. Hone, J. Shan and T. F. Heinz, Phys. Rev. Lett., 2010, 105, 136805.
15. S. Zhang, Z. Yan, Y. Li, Z. Chen and H. Zeng, Angew. Chem., 2015, 127, 3155–3158.
16. P. Ares, F. Aguilar-Galindo, D. Rodríguez-San-Miguel, D. A. Aldave, S. Díaz-Tendero, M. Alcamí, F. Martín, J. Gómez-Herrero and F. Zamora, Adv. Mater., 2016, 28, 6332–6336.
17. D. Kecik, E. Durgun and S. Ciraci, Phys. Rev. B, 2016, 94, 205409.
18. L. Song, L. Ci, H. Lu, P. B. Sorokin, C. Jin, J. Ni, A. G. Kvashnin, D. G. Kvashnin, J. Lou, B. I. Yakobson and P. M. Ajayan, Nano Lett., 2010, 10, 3209–3215.
19. G. Gui, J. Li and J. Zhong, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 78, 075435.
20. Z. Ni, Q. Liu, K. Tang, J. Zheng, J. Zhou, R. Qin, Z. Gao, D. Yu and J. Lu, Nano Lett., 2012, 12, 113–118.
21. J.-A. Yan, S.-P. Gao, R. Stein and G. Coard, Phys. Rev. B: Condens. Matter Mater. Phys., 2015, 91, 245403.
22. J. Qi, X. Li, X. Qian and J. Feng, Appl. Phys. Lett., 2013, 102, 173112.
23. B. Ghosh, S. Nahas, S. Bhowmick and A. Agarwal, Phys. Rev. B: Condens. Matter Mater. Phys., 2015, 91, 115433.
24. A. K. M. Newaz, D. Prasai, J. I. Ziegler, D. Caudel, S. Robinson, R. F. Haglund Jr. and K. I. Bolotin, Solid State Commun., 2013, 155, 49–52.
25. K. F. Mak, K. He, C. Lee, G. H. Lee, J. Hone, T. F. Heinz and J. Shan, Nat. Mater., 2012, 12, 207–211.
26. J. S. Ross, P. Klement, A. M. Jones, N. J. Ghimire, J. Yan, D. G. Mandrus, T. Taniguchi, K. Watanabe, K. Kitamura, W. Yao, D. H. Cobden and X. Xu, Nat. Nanotechnol., 2014, 9, 268–272.
27. W. Shan, R. J. Hauenstein, A. J. Fischer, J. J. Song, W. G. Perry, M. D. Bremser, R. F. Davis and B. Goldenberg, Phys. Rev. B: Condens. Matter Mater. Phys., 1996, 54, 13460–13463.
28. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865–3868.
29. P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. Dal Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P. Seitsonen, A. Smogunov, P. Umari and R. M. Wentzcovitch, J. Phys.: Condens. Matter, 2009, 21, 395502.
30. S. Goedecker, M. Teter and J. Hutter, Phys. Rev. B: Condens. Matter Mater. Phys., 1996, 54, 1703–1710.
31. L. Bengtsson, Phys. Rev. B: Condens. Matter Mater. Phys., 1999, 59, 12301–12304.
32. E. Artacho, E. Anglada, O. Diéguez, J. D. Gale, A. García, J. Junquera, R. M. Martin, P. Ordejón, J. M. Pruneda, D. Sánchez-Portal and J. M. Soler, J. Phys.: Condens. Matter, 2008, 20, 064208.
33. L. Hedin, Phys. Rev., 1965, 139, A796–A823.
34. E. E. Salpeter and H. A. Bethe, Phys. Rev., 1951, 84, 1232.
35. M. Rohlfing and S. G. Louie, Phys. Rev. B: Condens. Matter Mater. Phys., 2000, 62, 4927–4944.
36. R. Ahuja, S. Auluck, J. M. Wills, M. Alouani, B. Johansson and O. Eriksson, Phys. Rev. B: Condens. Matter Mater. Phys., 1997, 55, 4999–5005.
37. A. Marini, C. Hogan, M. Grüning and D. Varsano, Comput. Phys. Commun., 2009, 180, 1392–1403.
38. Z. Zhu and D. Tománek, Phys. Rev. Lett., 2014, 112, 176802.
39. L. Kou, Y. Ma, X. Tan, T. Frauenheim, A. Du and S. Smith, J. Phys. Chem. C, 2015, 119, 6918–6922.
40. C. Kamal and M. Ezawa, Phys. Rev. B: Condens. Matter Mater. Phys., 2015, 91, 085423.
41. D. Singh, S. K. Gupta, Y. Sonvane and I. Lukačević, J. Mater. Chem. C, 2016, 4, 6386–6390.
42. A. Ramasubramaniam, Phys. Rev. B: Condens. Matter Mater. Phys., 2012, 86, 115409.
43. V. Tran, R. Soklaski, Y. Liang and L. Yang, Phys. Rev. B: Condens. Matter Mater. Phys., 2014, 89, 235319.
44. H. Zhang, Y. Ma and Z. Chen, Nanoscale, 2015, 7, 19152–19159.
45. H. Shi, H. Pan, Y.-W. Zhang and B. I. Yakobson, Phys. Rev. B: Condens. Matter Mater. Phys., 2013, 87, 155304.
46. G. Plechinger, F.-X. Schrettenbrunner, J. Eroms, D. Weiss, C. Schüller and T. Korn, Phys. Status Solidi RRL, 2012, 6, 126–128.
47. N. D. Drummond, V. Zólyomi and V. I. Fal’ko, Phys. Rev. B: Condens. Matter Mater. Phys., 2012, 85, 075423.
48. M. Ezawa, New J. Phys., 2012, 14, 033003.
49. Z. Zhang and W. Guo, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 77, 075403.
50. B. Ghosh, S. Nahas, S. Bhowmick and A. Agarwal, Phys. Rev. B: Condens. Matter Mater. Phys., 2015, 91, 115433.

---

Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c7tc04072e

---