Quasiparticle Energies and Optical Excitations in Chevron-Type

Article pubs.acs.org/JPCC

Quasiparticle Energies and Optical Excitations in Chevron-Type Graphene Nanoribbon Shudong Wang and Jinlan Wang* Department of Physics & Key Laboratory of MEMS of the Ministry of Education, Southeast University, Nanjing 211189, China ABSTRACT: The electronic structure and optical properties of very recently synthesized chevron-type graphene nanoribbon (CGNR) are investigated within many-body Green’s function and Bethe−Salpeter equation formalism. The CGNR can effectively confine both electrons and holes, leading to its exciton binding energy larger than that of regular GNRs. The excitonic peaks owing to electron−hole interactions dominate the optical spectra with a significant blue-shift and a different line shape in CGNR compared with those in regular GNRs. Moreover, the singlet−triplet exciton splitting of CGNR is also larger than that of the regular GNRs, which is expected to show high fluorescence luminescence efficiency. The enhanced excitonic effects in CGNR should be of great importance in optoelectronic applications.

I. INTRODUCTION Because of its unique electronic properties,1 graphene, the twodimensional (2D) single layer, holds great potential applications in nanoelectronics; 2−8 e.g., graphene is broadly considered as the most promising candidate to replace Si in future electronic devices. However, the gapless feature of graphene normally prevents the dream of graphene microelectronics because of the low on/off ratio of graphene-based field effect transistors. Great efforts such as chemical functionalization9−11 or nanopatterning12,13 are exploited to open the band gap of graphene. Among the methods of nanopatterning, tailoring the 2D graphene as stripsgraphene nanoribbons (GNRs)is a widely used method. Because of quantum confinement effects, quasi-one-dimensional (1D) GNRs are of finite band gaps,12 and their electronic and optical properties exhibit rich size and edge dependence as well.14−17 Earlier studies18,19 reveal that to have a comparable band gap of conventional semiconductors like silicon, the width of GNRs should be less than 3 nm or even more narrow. Currently, many attempts including chemical,20−22 sonochemical,13 and lithographic12,23 methods as well as unzipping of carbon nanotubes24−29 have been extensively exploited to synthesize narrow GNRs. However, it remains great challenge to fabricate reliable production of GNRs smaller than 10 nm. Recently, Cai et al.30 have successfully synthsized atomically precise GNRs with different topologies and widths via a bottom-up approach. One of resultant structures is chevron-type GNRs (CGNRs) which are comprised of two regular GNRs with alternating widths, e.g., CGNR (6A, 9A) is built by two armchair GNRs (AGNRs) with the widths of N = 6 and N = 9 (Figure 1a). Very recently, ab initio calculations reveal that some CGNRs are semiconducting, and their band gaps and magnetic properties can be engineered via different edge topologies, offering a broader space for practical application.31 © 2012 American Chemical Society

Figure 1. Geometrical structures of (a) chevron-type graphene nanoribbon (6A, 9A) and regular armchair graphene nanoribbon (N = 15). The unit cells are presented as dashed rectangles.

Electronic and optical properties of materials are two important physical quantities for technological applications, and thus an accurate description of them is very essential. In low-dimensional nanostructures, it is well-known that manybody effects like electron−electron and electron−hole interactions play a crucial role in electronic structures and optical absorption.32−35 Earlier theoretical and experimental explorations demonstrate that excitonic effects (electron−hole interactions) can dramatically influence the position and intensity of the optical absorption spectra of low-dimensional structures, such as quasi-1D silicon nanowires,36 carbon nanotubes,34,35,37 graphene nanoribbons,16,17,38 BN nanotubes,39 and nanoribbons,40 and 2D graphene,41 hexagonal BN single layer.42 Therefore, it is highly desirable to include many-body effects for the investigation of the electronic and optical properties of CGNRs. Accordingly, we perform firstprinciples calculations via a GW (Green function G and Received: December 30, 2011 Revised: April 15, 2012 Published: April 19, 2012 10193

dx.doi.org/10.1021/jp2125872 | J. Phys. Chem. C 2012, 116, 10193−10197

The Journal of Physical Chemistry C

Article

Kx) separately.46 Then the singlet−triplet exciton splitting ΔS−T can be obtained. The spin-triplet states tend to have lower energies than the spin-singlet counterparts due to the lack of repulsive term in the former case. The optical absorption spectra are calculated including 15 valence bands and 20 conduction bands with the light polarization being along x direction. To calculate the static screening in W, we exploit 45 000 (110 000) and 3600 (4000) RL vectors for exchange and attraction matrices in AGNRs (CGNR), respectively. A box-shape truncated Coulomb interaction is used to simulate truly isolated nanoribbons.47 A k-point sampling of 9 × 1 × 1 [45 × 1 × 1] is used for the CGNR (6A, 9A) [AGNR (N = 6, 9, 15)] during the GW and BSE calculations implemented in the YAMBO code.48

screened Coulomb interaction W) + Bethe−Salpeter equation to explore the electronic and optical properties of CGNR (6A, 9A), which we take as an example of semiconducting CGNRs derived from ref 30, and compare with the regular AGNR (N = 6, 9, 15). Interestingly, the exciton binding energy in CGNR (6A, 9A) is found to be 19−98% larger than that of AGNR (N = 6, 9, 15), indicating the enhanced electron−hole interactions in CGNR (6A, 9A). Moreover, the singlet−triplet exciton splitting ΔS−T of CGNR (6A, 9A) is at least 40% larger than that of AGNR (N = 6, 9, 15), which is expected to be of high fluorescence luminescence efficiency.

II. THEORIES AND CALCULATION DETAILS We consider the CGNR (6A, 9A) assembled alternately by armchair GNRs (denote by subscript A) with widths of N = 6 (oblique) and N = 9 (parallel) (Figure 1a). The regular AGNR with the width N = 15 is also included (Figure 1b). In these structures, dangling bonds at the edges are saturated with atomic hydrogen. We carry out a three-step procedure to study the electronic and optical properties by using many-body perturbation theory approach.43 As a starting point, the groundstate electronic properties of the relaxed CGNR (6A, 9A) and AGNR (N = 6, 9, 15) are obtained by performing DFT-LDA (local density approximation) calculations as implemented in the QUANTUM-ESPRESSO package44 using plane waves and norm-conserving pseudopotentials with a 45 Ry kinetic energy cutoff. The atomic structures are fully relaxed with no symmetry restriction until forces acting on each atom being less than 0.01 eV/Å. Periodic boundary condition is applied, and a vacuum thickness of about 15 Å is taken along the nonperiodic direction to avoid interaction between neighboring cells. The Brillouin zone integration is done by 9 × 1 × 1 [25 × 1 × 1] k-points for the CGNR (6A, 9A) [AGNR (N = 6, 9, 15)] within the Monkhorst−Pack scheme. In the second step, quasiparticle energies EQP are calculated i within the G0W0 approximation by EiQP = EiKS + ⟨ψi|Σ(EiQP) − Vxc|ψi ⟩

III. RESULTS AND DISCUSSION Figure 2 depicts the electronic band structures, valence band maximum (VBM), and conduction band minimum (CBM) of

(1)

where Vxc is the DFT exchange-correlation potential and selfenergy Σ is the product of the noninteracting one-electron Green’s function G and the screened Coulomb potential W. The screening in W is treated within the plasmon-pole approximation.45 400 (800) bands, 2500 (3600) RL vectors for correlation part, and 65 000 (100 000) RL vectors for exchange part are used for calculation dielectric matrices and self-energies in AGNRs (CGNR). Finally, the electron−hole interaction is included by solving the Bethe−Salpeter equation (BSE)43,46 S (Eck − Evk )A vck +

∑

Figure 2. Electronic band structures, charge distribution of valence band maximum (VBM), and conduction band minimum (CBM) of (a−c) AGNR (N = 6, 9, 15) and (d) CGNR (6A, 9A) [the yellow (light) areas denote the high electrons density areas, while the pink (dark) areas denote the low ones]. The red (solid) and the blue (dot solid) lines in the band structures depict at LDA and GW levels, respectively. The Fermi level is set to zero.

⟨vck|K d + 2K x|v′c′k′⟩A vS′ c ′ k ′

AGNR (N = 6, 9, 15) and CGNR (6A, 9A) derived from LDA calculations. It can be clearly seen, from right panel of Figure 2a−c, that the electrons of AGNR (N = 6, 9, 15) at VBM and CBM are delocalized, while the electrons in CGNR (6A, 9A) are more localized and they mostly reside around inner carbon atoms (along the axial direction) (right panel of Figure 2d). These localized electrons make the bands less dispersed than that in regular AGNR (N = 6, 9, 15). Based on the GW formalism, the band gaps of AGNR (N = 6, 9, 15) increase to 2.74, 2.13, and 1.62 eV, respectively, and for CGNR (6A, 9A), the gap increases from 1.61 to 3.74 eV, which is significantly larger than that of its components, regular AGNR (N = 6, 9). In such reduced dimensionality systems, these large quasiparticle corrections are derived from the enhanced Coulomb

k′v′c′ S = ΩSA vck

(2)

where Eck and Evk are the quasiparticle energies for the conduction and valence band states, ΩS and ASvck are the excited eigenvalues and eigenfunctions, respectively. Kd, related to the screened Coulomb interaction W, is the direct interaction term which is responsible for the attractive nature of electron−hole interaction. Kx is the repulsive exchange term of the electron− hole interaction which contains the bare Coulomb interaction v. If the spin−orbit interaction is negligible with respect to the electron−hole interaction, the BSE Hamiltonian can be solved for spin-singlet and spin-triplet (where omit the exchange term 10194



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This might provide a new way to tune the optical properties through the new structure of CGNRs. The binding energy of the first exciton in CGNR (6A, 9A) is 1.76 eV, larger than that of its constituents AGNR (N = 6, 9) (1.48/1.10 eV with N = 6/9). The larger exciton binding energy stems from the stronger electron−hole interaction in CGNR (6A, 9A) as compared with that in AGNR (N = 6, 9). This suggests that CGNR can effectively confine both electrons and holes. The position of the first absorption peaks E11 and their corresponding binding energies of AGNR (N = 6, 9, 15) and CGNR (6A, 9A) are listed in Table 1.

interaction. In addition, the band structure stretching of AGNR (N = 6, 9, 15) and CGNR (6A, 9A) are about 20% and 30%, respectively, similar to early reported values.16,49 Optical absorption spectra of AGNR (N = 6, 9, 15) and CGNR (6A, 9A) are shown in Figure 3. GW + RPA (without

Table 1. Band Gap (Second and Third Column), the First Peak Position (Fourth Column), Binding Energy of the First Exciton, and Singlet−Triplet Exciton Splitting for AGNR (N = 6, 9, 15) and CGNR (6A, 9A) (Units in eV) type

LDA

GW

BSE

Eb

ΔS−T

AGNR (N = 6) AGNR (N = 9) AGNR (N = 15) CGNR (6A, 9A)

1.11 0.78 0.58 1.61

2.74 2.13 1.62 3.74

1.26 1.03 0.73 1.98

1.48 1.10 0.89 1.76

0.25 0.15 0.07 0.35

To further investigate the spatial correlation of the electrons and holes, the real-space charge distribution of the excitons at fixed hole position (black dot) for AGNR (N = 6, 9, 15) and CGNR (6A, 9A) is plotted in Figure 4. For AGNR (N = 6, 9, 15), the electron distribution of the exciton E11 spreads over the

Figure 3. Optical absorption spectra of (a−c) AGNR (N = 6, 9, 15) and (d) CGNR (6A, 9A) with (GW + BSE, black solid area) and without (GW + RPA, red slash area) the electron−hole interaction.

electron−hole interaction) and GW + BSE (with electron−hole interaction) are both considered. With the electron−hole interaction absence, the first absorption peaks of AGNR (N = 6, 9, 15) and CGNR (6A, 9A) locate at 1.62−2.74 and 3.74 eV, respectively. When the electron−hole interaction is included, the entire absorption spectra of AGNR (N = 6, 9, 15) and CGNR (6A, 9A) both dramatically change, accompanying the appearance of some new peaks below the onset of the continuum. These dominating new peaks in the absorption spectra are associated with excitonic effects in these reduced dimensional systems. Compared to the GW + RPA results, the spectra obtained within the GW + BSE formalism have a redshift with a change in the relative position of the peaks. For the AGNR (N = 6), the first absorption peak E11 locates at 1.26 eV, which is originated from optical transition between the first valence band and the first conduction band around the Γ point. The binding energy of the corresponding exciton is 1.48 eV, larger than that of AGNR (1.10 eV for N = 9 and 0.89 eV for N = 15) due to the narrower ribbon width. For the CGNR (6A, 9A), the first absorption peak E11 appears at 1.98 eV, which is in visible light region. While the second strongest peak E22, resulting from optical transition between the second valence band and the second conduction band, is at 2.34 eV. These two strongest peaks E11 and E22 separate by about 0.36 eV, remarkably smaller than the values of AGNR (N = 6, 9, 15) (0.78−2.06 eV). Moreover, as it can be clearly seen from Figure 3, when the AGNRs with N = 6 and N = 9 assembled as CGNR (6A, 9A), the absorption spectra move to higher energy region.

Figure 4. Charge distribution of the exciton E11 at fixed hole position (black dot): (a−c) AGNR (N = 6, 9, 15); (d) CGNR (6A, 9A); (e, f) two lowest-energy dark excitons of CGNR (6A, 9A) [the yellow (light) areas denote the high electrons density areas, while the pink (dark) areas denote the low ones]. 10195



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than that in AGNR (N = 6, 9, 15).50 As a consequence, the fluorescence luminescent yield of CGNR (6A, 9A) is expected to be higher than that of AGNR (N = 6, 9, 15).

whole ribbon width (Figure 4a−c), while for the exciton E11 of CGNR (6A, 9A), the most probable site of finding the electron is located on the nearest neighbors of the hole (Figure 4d). These two distinguishable features between AGNR (N = 6, 9, 15) and CGNR (6A, 9A) suggest that excitons in CGNR have less extending area than those in regular GNRs because of the larger band gaps and heavier electron effective mass of CGNR. We also detect two lowest-energy dark excitons between the first E11 and the second peak E22 (Figure 4 e,f). The electron and the hole in these two dark excitons spread separately, and their wave functions have a small overlap, which may result in the optically inactive feature. The charge distribution of the first exciton E11 in these two type ribbons is also depicted to provide a quantitative picture on the extending range Figure 5. The projected electron density

IV. CONCLUSIONS In summary, we have carried out first-principles calculations via a GW + Bethe−Salpeter equation to investigate the electronic and optical properties of chevron-type graphene nanoribbon (6A, 9A) and compared it with regular armchair graphene nanoribbon (N = 6, 9, 15). Our calculations reveal that the many-body effects have significant impacts on the electronic and optical absorption properties of CGNR (6A, 9A). The excitonic peaks stemming from the electron−hole interactions dominate the optical spectra. The line shape of absorption spectra of CGNR (6A, 9A) remarkably differs from that of AGNR (N = 6, 9, 15). Most importantly, the exciton binding energy in CGNR (6A, 9A) is found to be 19−98% larger than that of its constituent regular AGNR (N = 6, 9) showing the stronger electron−hole interactions in CGNR (6A, 9A). Besides, the singlet−triplet exciton splitting ΔS−T of CGNR (6A, 9A) is at least 40% larger than that of AGNR (N = 6, 9, 15), suggesting that CGNR (6A, 9A) might have high fluorescence luminescence efficiency. These intriguing results demonstrate that this experimentally available CGNR (6A, 9A) can be used to tune the optical properties and is of promising potential in nano-optoelectronics.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by NBRP (2011CB302004, 2010CB923401, and 2009CB623200), the NSF (21173040 and 11074035), SRFDP (20090092110025), and Peiyu Foundation of SEU. The authors thank the computational resource at Department of Physics, SEU, and National Supercomputing Center in Tianjin.

Figure 5. Electron distribution of the first exciton for (a−c) AGNR (N = 6, 9, 15) and (d) CGNR (6A, 9A) for a fixed hole position (rh = 0) through integrating out coordinates perpendicular to the ribbon axis (x).

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REFERENCES

(1) Geim, A. K.; Novoselov, K. S. Nat. Mater. 2007, 6, 183. (2) Schedin, F.; Geim, A. K.; Morozov, S. V.; Hill, E. W.; Blake, P.; Katsnelson, M. I.; Novoselov, K. S. Nat. Mater. 2007, 6, 652. (3) Ö zyilmaz, B.; Jarillo-Herrero, P.; Efetov, D.; Abanin, D. A.; Levitov, L. S.; Kim, P. Phys. Rev. Lett. 2007, 99, 166804. (4) Huard, B.; Sulpizio, J. A.; Stander, N.; Todd, K.; Yang, B.; Goldhaber-Gordon, D. Phys. Rev. Lett. 2007, 98, 236803. (5) Gorbachev, R. V.; Mayorov, A. S.; Savchenko, A. K.; Horsell, D. W.; Guinea, F. Nano Lett. 2008, 8, 1995. (6) Trauzettel, B.; Bulaev, D. V.; Loss, D.; Burkard, G. Nat. Phys. 2007, 3, 192. (7) Lin, Y. M.; Jenkins, K. A.; Valdes-Garcia, A.; Small, J. P.; Farmer, D. B.; Avouris, P. Nano Lett. 2009, 9, 422. (8) Zhu, L. Y.; Wang, J. L.; Zhang, T. T.; Ma, L.; Lim, C. W.; Ding, F.; Zeng, X. C. Nano Lett. 2010, 10, 494. (9) Wu, X. S.; Sprinkle, M.; Li, X. B.; Ming, F.; Berger, C.; A.de Heer, W. Phys. Rev. Lett. 2008, 101, 026801. (10) Jung, I.; Dikin, D. A.; Piner, R. D.; Ruoff, R. S. Nano Lett. 2008, 8, 4283. (11) Elias, D. C.; Nair, R. R.; Mohiuddin, T. M. G.; Morozov, S. V.; Blake, P.; Halsall, M. P.; Ferrari, A. C.; Boukhvalov, D. W.; Katsnelson, M. I.; Geim, A. K.; et al. Science 2009, 323, 610.

is plotted along the ribbon axis (x) through integrating out the other two directions and fixing the hole in the black dot as shown in Figure 4 (we set the hole position rh = 0 along the ribbon axis). It can be seen clearly that the extent of the first exciton of CGNR (6A, 9A) is ∼25 Å, markedly smaller than that of AGNR (N = 6, 9, 15) which is in the range of ∼48−55 Å. The smaller spatial extent of the first exciton of CGNR (6A, 9A) leads to the larger binding energy. In the emission process, the low-lying triplet states can mix with a small amount of singlet through spin−orbit coupling and allows the luminescence to have a long lifetime. The calculated singlet−triplet exciton splitting ΔS−T is 0.07−0.25 eV for AGNR (N = 6, 9, 15) while it is as large as 0.35 eV for CGNR (6A, 9A). The large singlet−triplet exciton splitting in CGNR (6A, 9A) stems from the strong electron−hole overlapping, which it in turn enhances the repulsive exchange interaction Kx. The large singlet−triplet splitting also brings about less probability of the nonradiatively transitions from singlet state to triplet state (e.g., intersystem crossing) in CGNR (6A, 9A) 10196



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(12) Han, M. Y.; Ozyilmaz, B.; Zhang, Y. B.; Kim, P. Phys. Rev. Lett. 2007, 98, 206805. (13) Li, X. L.; Wang, X. R.; Zhang, L.; Lee, S. W.; Dai, H. J. Science 2008, 319, 1229. (14) Nakada, K.; Fujita, M.; Dresselhaus, G.; Dresselhaus, M. S. Phys. Rev. B 1996, 54, 17954. (15) Son, Y.-W.; Cohen, M. L.; Louie, S. G. Phys. Rev. Lett. 2006, 97, 216803. (16) Prezzi, D.; Varsano, D.; Ruini, A.; Marini, A.; Molinari, E. Phys. Rev. B 2008, 77, 041404(R). (17) Yang, L.; Cohen, M. L.; Louie, S. G. Nano Lett. 2007, 7, 3112. (18) Barone, V.; Hod, O.; Scuseria, G. E. Nano Lett. 2006, 6, 2748. (19) Yang, L.; Park, C. H.; Son, Y. W.; Cohen, M. L.; Louie, S. G. Phys. Rev. Lett. 2007, 99, 186801. (20) Datta, S. S.; Strachan, D. R.; Khamis, S. M.; Johnson, A. T. C. Nano Lett. 2008, 8, 1912. (21) Campos-Delgado, J.; et al. Nano Lett. 2008, 8, 2773. (22) Yang, X. Y.; et al. J. Am. Chem. Soc. 2008, 130, 4216. (23) Chen, Z. H.; Lin, Y. M.; Rooks, M. J.; Avouris, P. Physica E 2007, 40, 228. (24) Jiao, L. Y.; Zhang, L.; Wang, X. R.; Diankov, G.; Dai, H. J. Nature 2009, 458, 877. (25) Kosynkin, D. V.; et al. Nature 2009, 458, 872. (26) Elías, A. L.; et al. Nano Lett. 2009, 10, 366. (27) Jiao, L.; Wang, X.; Diankov, G.; Wang, H.; Dai, H. Nat. Nanotechnol. 2010, 5, 321. (28) (a) Wang, J.; Ma, L.; Yuan, Q.; Zhu, L.; Ding, F. Angew. Chem., Int. Ed. 2011, 50, 8041. (b) Wang, J.; Ma, L.; Yuan, Q.; Zhu, L.; Ding, F. Angew. Chem. 2011, 123, 8191. (29) (a) Ma, L.; Wang, J.; Ding, F. Angew. Chem., Int. Ed. 2012, 51, 1161. (b) Ma, L.; Wang, J.; Ding, F. Angew. Chem. 2012, 124, 1187. (30) Cai, J.; et al. Nature 2010, 466, 470. (31) Girão, E. C.; Liang, L.; Cruz-Silva, E.; Filho, A. G. S.; Meunier, V. Phys. Rev. Lett. 2011, 107, 135501. (32) Rohlfing, M.; Louie, S. G. Phys. Rev. Lett. 1999, 82, 1959. (33) Ruini, A.; Caldas, M. J.; Bussi, G.; Molinari, E. Phys. Rev. Lett. 2002, 88, 206403. (34) Spataru, C. D.; Ismail-Beigi, S.; Benedict, L. X.; Louie, S. G. Phys. Rev. Lett. 2004, 92, 077402. (35) Chang, E.; Bussi, G.; Ruini, A.; Molinari, E. Phys. Rev. Lett. 2004, 92, 196401. (36) Yang, L.; Spataru, C. D.; Louie, S. G.; Chou, M. Y. Phys. Rev. B 2007, 75, 201304(R). (37) Spataru, C. D.; Ismail-Beigi, S.; Benedict, L. X.; Louie, S. G. Appl. Phys. A: Mater. Sci. Process. 2004, 78, 1129. Spataru, C. D.; Ismail-Beigi, S.; Capaz, R. B.; Louie, S. G. Phys. Rev. Lett. 2005, 95, 247402. Deslippe, J.; Spataru, C. D.; Prendergast, D.; Louie, S. G. Nano Lett. 2007, 7, 1626. (38) Yang, L.; Cohen, M. L.; Louie, S. G. Phys. Rev. Lett. 2008, 101, 186401. Zhu, X.; Su, H. J. Phys. Chem. C 2010, 41, 17257. (39) Wirtz, L.; Marini, A.; Rubio, A. Phys. Rev. Lett. 2006, 96, 126104. Park, C.-H.; Spataru, C. D.; Louie, S. G. Phys. Rev. Lett. 2006, 96, 126105. (40) Wang, S. D.; Chen, Q.; Wang, J. L. Appl. Phys. Lett. 2011, 99, 063114. (41) Yang, L.; Deslippe, J.; Park, C.-H.; Cohen, M. L.; Louie, S. G. Phys. Rev. Lett. 2009, 103, 186802. Trevisanutto, P. E.; Holzmann, M.; Côté, M.; Olevano, V. Phys. Rev. B 2010, 81, 121405(R). Park, C.-H.; Louie, S. G. Nano Lett. 2010, 10, 426. Yang, L. Phys. Rev. B 2011, 83, 085405. (42) Arnaud, B.; Lebègue, S.; Rabiller, P.; Alouani, M. Phys. Rev. Lett. 2006, 96, 026402. (43) Onida, G.; Reining, L.; Rubio, A. Rev. Mod. Phys. 2002, 74, 601. (44) Giannozzi, P.; et al. J. Phys.: Condens. Matter 2009, 21, 395502. (45) Godby, R. W.; Needs, R. J. Phys. Rev. Lett. 1989, 62, 1169. (46) Rohlfing, M.; Louie, S. G. Phys. Rev. B 2000, 62, 4927 and references therein. (47) Rozzi, C. A.; Varsano, D.; Marini, A.; Gross, E. K. U.; Rubio, A. Phys. Rev. B 2006, 73, 205119.

(48) Marini, A.; Hogan, C.; Grüning, M.; Varsano, D. Comput. Phys. Commun. 2009, 180, 1392. (49) Prezzi, D.; Varsano, D.; Ruini, A.; Molinari, E. Phys. Rev. B 2011, 84, 041401(R). (50) Zener, C. Pros. R. Soc. London 1932, 137, 696.

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Quasiparticle Energies and Optical Excitations in Chevron-Type

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