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Tuning Electronic Properties of Hydro-Boron-Carbon Compounds by Hydrogen and Boron Contents: A First Principles Study Yi Ding and Jun Ni* Department of Physics and Key Laboratory of Atomic and Molecular Nanoscience (Ministry of Education), Tsinghua UniVersity, Beijing 100084, People’s Republic of China ReceiVed: April 13, 2009; ReVised Manuscript ReceiVed: July 26, 2009
We have investigated the electronic structures of hydro-boron-carbon compounds by first principles calculations. We predict that hydro-boron-carbon compounds have stable semiconducting and metallic conformations. During step by step hydrogenation, semiconductor-semiconductor-metal transitions appear in the hydro-BC3 compounds and metal-semiconductor-metal transitions appear in the hydro-BC5 and hydroBC7 compounds. We find that the bonding status of the boron pz orbitals play an important role in tuning the electronic properties of hydro-boron-carbon compounds. Introduction Graphene, a two-dimensional hexagonal carbon sheet, is a semimetal with unique electronic property for applications in nanoelectronics.1-3 The graphene nanoribbons can be high-yield produced and have an efficient edge-reconstruction process.4 Chemically modified graphene has attracted lots of research interests recently.5-11 For the hydrogenation of graphene, theoretical studies predict that the most stable conformations are the ones with 100% coverage for two-side functionalization and 25% coverage for one-side functionalization.7-9 In experiments, Ryu et al. demonstrated that graphene can chemically react with hydrogen and the hydrogenation of graphene is reversible.10 More recently, Elias et al. exposed graphene under hydrogen plasma surroundings and synthesize a two-dimensional hydrocarbon, which is called graphane.11 Graphane is a hydrocarbon with 1:1 C/H ratio. Carbon atoms in graphane are all in sp3 hybridization.8 The most stable conformation of graphane is chairlike in which the carbon sheet is buckled and hydrogen atoms alternate on both sides of the carbon sheet.8,12,13 Transforming from the semimetal of graphene, graphane is a semiconductor with a generalized gradient approximation (GGA) band gap of 3.5 eV and a GW band gap of 5.4 eV.8,9,12,13 Boron atoms are widely used as dopants in carbon nanostructures for building functional materials.14,15 Pontes et al. find that boron atoms can substitute carbon atoms in graphene sheet without activation barrier.16 The zigzag graphene nanoribbons with boron edges exhibit half-metallic behavior irrespective of the ribbon width or electric field.17 Using carbon nanotubes as frames, B-doped nanotubes with different concentrations are produced by the substitution reaction in experiments.18 For the highly B-doped concentrations, the BC3 ordering structure forms instead of a homogeneous random boron distribution.19 The hexagonal BC3 sheet has been grown in an epitaxial way on the NbB2 (0001) surface.20 The BC3 nanoribbons with the armchair shaped edges are all semiconductors, while BC3 nanoribbons with the zigzag shaped edges are semiconductors or metals depending on the edge atoms.21 By coevaporation of boron and carbon atoms, boron carbides are formed and hexagonal-like structures are kept when the boron content is * To whom correspondence
[email protected].
should
be
addressed.
E-mail:
less than 50%.22 For the B-doped graphene, the binding with hydrogen is strengthened.23 Therefore, hydro-boron-carbon compounds are quite likely realized by the substitution reaction on graphane or the hydrogenation of boron-carbon sheets in a similar way as that of graphane. For graphane, the reversible hydrogenation can change semimetallic or semiconducting behavior. While for the hydro-boron-carbon compounds, the electronic properties can be tuned by the variation of both the boron and hydrogen contents. Thus, it is important to explore the promising properties of hydro-boron-carbon compounds for application of nanodevices. In this paper, we investigate the electronic properties of hydroboron-carbon compounds. Since the BCx ordering structures exist with boron content ranging about 15-25%,24 the hexagonal BC3,21,25 BC5,26 and BC727 sheets are chosen as precursors for hydro-boron-carbon compounds. We find that the hydroboron-carbon compounds with only carbon atoms bonding with hydrogen atoms are semiconductors. Different from graphane, when all the atoms bond with hydrogen atoms, the hydroboron-carbon compounds become metals. During step by step hydrogenation, semiconductor-semiconductor-metal transitions appear in the hydro-BC3 compounds and metalsemiconductor-metal transitions appear in the hydro-BC5 and hydro-BC7 compounds. Methods First principles calculations are performed by VASP (the Vienna ab initio simulation package).28 The approach is based on an iterative solution of the Kohn-Sham equation of the density-function theory in a plane-wave set with the Vanderbilt ultrasoft pseudopotentials. We use the regular pseudopotentials with a core radius of 1.81 au for carbon, 1.90 au for boron, and 1.25 au for hydrogen. In our calculations, the Perdew-Wang 91 exchange-correlation functional of the GGA is adopted. We set the plane-wave cutoff energy to be 350 eV and the convergence of the force on each atom to be less than 0.01 eV/ Å. The optimizations of the lattice constants and the atom coordinates are made by the minimization of the total energy. The supercells are used to simulate the isolated sheet and the sheets are separated larger than 10 Å in order to avoid interactions. The Monkhorst-Pack scheme is used to sample the Brillouin zone. In the calculations, the structures are fully
10.1021/jp903384m CCC: $40.75 2009 American Chemical Society Published on Web 10/05/2009
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Figure 1. The structures, energy bands and densities of states for (a) the BC3 sheet, (b) the C-H type of hydro-BC3 compound, and (c) the all-H type of hydro-BC3 compound. The Fermi level is indicated as the line at E ) 0.0 eV.
relaxed with a mesh of 12 × 12 × 1, and the mesh of k space is increased to 15 × 15 × 1 in the static calculations. The cutoff energy and k-point sampling have been tested to be of sufficient precision for our calculated systems (see Figure S1-S4 in Supporting Information). The atomic structures and partial charge density are depicted by VESTA.29 We have performed spin-polarized calculations for hydro-boron-carbon complexes. However, there is no magnetic state found in the calculated systems. Results For boron-carbon sheets, there are three types of hydroboron-carbon compounds, C-H type, B-H type, and all-H type. For the C-H type (B-H type) of hydro-boron-carbon compounds, only carbon (boron) atoms bond with hydrogen atoms, while for the all-H type, both boron and carbon atoms bond with hydrogen atoms. Similar to graphane,8,12,13 we find that the chairlike conformations are more stable than the boatlike ones in the hydro-boron-carbon compounds. Thus, we focus on the properties of the chairlike hydro-boron-carbon compounds. We have calculated the formation energy εH2 to determine the stability of hydrogenation. The formation energy is defined as εH2 ) Ehydro - Esheet - (nH/2)EH2. Ehydro (Esheet) is the total energy of the hydro-boron-carbon compound (boron-carbon sheet) and EH2 is the energy of an isolated hydrogen molecule. For the hydro-BC3 compounds, the C-H type and all-H type have lower formation energies, which are -0.081 and 0.008
eV/atom, respectively. While for the B-H type, εH2 is 0.161 eV/atom. It indicates that hydrogen atoms prefer to binding with carbon rather than boron atoms. The C-H type and all-H type of hydro-BC3 compounds are more likely to be synthesized in experiment. The B-H type always has higher free energy under different hydrogen conditions as discussed below. Thus, we mainly focus on the C-H type and all-H type of hydro-BC3 compounds. In Figure 1a-c, we show the structures of the pristine BC3 sheet, the C-H type, and all-H type of hydro-BC3 compounds. The unit cell of the BC3 system is indicated by the lines as shown in Figure 1a. The lattice constant is 5.15 Å for the pristine BC3 sheet. It increases to 5.28 and 5.17 Å for the C-H type and all-H type of hydro-BC3 compounds, respectively. The C-C bond length is 1.42 Å in the pristine BC3 sheet and is elongated after hydrogenation. In the hydro-BC3 compounds, the BC3 sheet becomes buckled and atoms are in sp3 hybridization. In the C-H type, the out-of-plane displacements are 0.22 and 0.11 Å for carbon and boron atoms. These values increase to 0.27 and 0.34 Å in the all-H type of hydro-BC3 compound, respectively. The C-C bond lengthes are close in both compounds, which is 1.55 Å in the C-H type and 1.53 Å in the all-H type. For the B-C bond length, it is 1.56 Å in the pristine BC3 sheet and only changes a little to 1.57 Å in the C-H type of hydro-BC3 compound. While in the all-H type of hydro-BC3 compound, the B-C bond length elongates to 1.67 Å due to the formation of the σ B-H bonds.
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Figure 2. The Gibbs free energies G versus chemical potential µ for the hydro-BC3 compounds.
The pristine BC3 sheet is an indirect-gap semiconductor with a GGA gap of 0.76 eV. The valence band maximum (VBM) is located at the Γ point and the conduction band minimum (CBM) is located at the M point as shown in Figure 1 (a). The top valence bands have the σ-character and the bottom conduction bands have the π- and π*-characters, respectively.25 In the C-H type of hydro-BC3 compound, those π- and π*-character bands vanish and the CBM is located at the Γ point as shown in Figure 1b. The C-H type of hydro-BC3 compound is a direct-gap semiconductor with a GGA gap of 2.11 eV. While for the all-H type of hydro-BC3 compound, different from the C-H type and graphane, it becomes a metal as shown in Figure 1c. Although a gap exists between the σ- and σ*-character bands, the Fermi level falls into the σ-character bands, which leads to the metallic behavior. We have also preformed calculations within the local density approximation (LDA). The calculated band structures are consisted with the GGA ones, except that the LDA gaps of the BC3 sheet and C-H type decrease to 0.54 and 1.94 eV, respectively. In the experiment, graphane is synthesized by graphene reacted with hydrogen plasma.11 For hydro-boron-carbon compounds, the possible synthetic paths are similar to that of graphane. Now, we consider the stabilities of hydro-boron-carbon compounds related with the environmental hydrogen effects. The Gibbs free energies, which are defined as GH2 ) εH2 -(nH)/ (2)µH2, are compared to determine the relative stability referring to hydrogen molecule. The chemical potential µH2 stands for the environmental influences, which is a function of the absolute temperature T and the partial hydrogen pressure P as µH2 ) H°(T) - H°(0) - TS°(T) + kBT ln(P/P°).30 H° and S° are the enthalpy and entropy at the pressure of P° ) 1 bar, values of which can be obtained from the CRC handbook.31 In presence of atomic hydrogen gas, it should use GH ) εH - nHµH, where εH ) εH2 - nH × 2.28 eV by our first principles calculations. The Gibbs free energies G of hydro-BC3 compounds as functions of chemical potential µ are shown in Figure 2. For a given value of µ, the stable structure has the lowest value of G. We can see that the B-H type always has larger Gibbs free energy and is hard to be synthesized. At small chemical potential of µH2, the pristine BC3 sheet is stable. As the chemical potential µH2 increases, the C-H type of hydroBC3 compound become stable. At µH2 ) -0.38 eV, the pristine BC3 sheet transforms to the C-H type of hydroBC3 compound. A semiconductor-semiconductor transition occurs, which changes the hydro-BC3 compound from a
Ding and Ni semiconductor with a small indirect band gap to a semiconductor with a large direct band gap. When µH2 increases, the Gibbs free energy of the all-H type decreases faster than that of the C-H type. At µH2 ) 2.50 eV, the C-H type transforms to the all-H type of hydro-BC3 compound and a semiconductormetal transition occurs. At large chemical potential of µH2, the metallic all-H type becomes stable. From the formula of µH2, it can be seen that under the hydrogen molecule circumstances, µH2 is normally less than zero. When µH2 > 0 eV, it needs the hydrogen atoms instead of the hydrogen molecules to hydrogenate the compounds. Thus, step by step hydrogenation of hydro-boron-carbon compounds corresponds to two stages. In the first stage, the compounds is under hydrogen molecule gas circumstances and a semiconductor-semiconductor transition occurs. In the second stage, the compounds is under hydrogen atom gas circumstances and a semiconductor-metal transition occurs. The semiconductor-semiconductor-metal transitions appear in the hydro-BC3 compounds during this processing of step by step hydrogenation. If the hydrogenation is reversible in a similar way as that of graphane, the hydro-BC3 compounds can offer the switch of indirect band gap semiconductor, direct band gap semiconductor and metal for the applications of nanodevices. In graphane, the sp3 hybridization causes semiconducting behavior with a large band gap between the σ-and σ*-character bands.8,13 In hydro-BC3 compounds, we find that the bonding status of the boron pz orbitals play an important role in the electronic properties, which attributes to two differences with graphane. (i) For the C-H type, the bottom conduction bands mainly originate from the boron pz orbitals, which leads to a small band gap. As shown in Figure 3a, the bottom conduction bands of the C-H type are prevailed by the boron pz states. The partial charge density at the CBM, plotted in the Figure 3c, indicates these pz states are composed of the boron pz orbitals. Let us consider one boron atom. Its three valence p electrons participate in the σ-bonds with nearby carbon atoms and the boron pz orbital is left to be unoccupied. Since there are enough electrons for the σ-B-C bonds, all σ-character bands are fully filled in the C-H type of hydro-BC3 compound, which leads to a semiconducting hydro-BC3 compound. However, the energy level of the unoccupied boron pz orbital is located between those of the σ- and σ*-B-C bonds. The band gap depends on conduction bands originating from boron pz orbitals. Thus, the C-H type has a band gap smaller than the gap value between usual σ- and σ*-character bands. (ii) The all-H type has metallic behavior due to the bonding character of the boron pz orbitals. In the all-H type of hydro-BC3 compound, each boron atom bonds with one hydrogen atom. A σ-B-H bond is formed by a boron pz orbital and a hydrogen s orbital. Comparing the densities of states (DOSs) in Figure 3a,b, the boron pz states do not appear in the conduction bands but appear about 2 eV deep in the valence bands. The schematic energy diagram for the hydro-BC3 compounds is plotted in Figure 3 (d). The energy level of the σ B-H bond in the all-H type is lower than that of the σ B-C bond and falls below the Fermi level. One boron atom forms three σ-B-C bonds and one σ-B-H bond now. The pz orbitals take electrons from the σ-B-C bonds to fill the σ-B-H bond. Thus, the formation of B-H bonds weakens the plane bonding strongly. This induces shorter lattice constant and longer B-C bond for the all-H type of hydro-BC3 compound compared to those of the C-H type. Because of the electron deficiency at boron, the σ-character bands are not fully filled and the all-H type of hydro-BC3 compound becomes a metal.
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Figure 3. The DOSs of (a) the C-H type of hydro-BC3 compound and (b) the all-H type of hydro-BC3 compound. The Fermi level is indicated as the line at E ) 0.0 eV. (c) The partial charge density at the CBM of the C-H type of hydro-BC3 compound. The isosurface is 50% of the maximum of the density. (d) The schematic energy diagram for hydro-BC3 compounds.
Figure 4. The band structures of (a) the C-H type of hydro-BC5 compound, (b) the all-H type of hydro-BC5 compound, (c) the C-H type of hydro-BC7 compound, (d) the all-H type of hydro-BC7 compound. Bands with predominantly boron pz characters are highlighted in circle. The Fermi level is indicated as the line at E ) 0.0 eV.
Similar phenomena are also found in the hydro-BC5 and hydro-BC7 compounds. As shown in Figure 4a,d, the C-H types of hydro-BC5 and hydro-BC7 compounds are direct band gap
semiconductors. The lowest conduction bands of the C-H types have significant boron pz characters. The value of band gaps varies with the boron contents. The GGA gaps are 2.62 and
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Ding and Ni
TABLE 1: The Electronic Properties of the Stable Hydro-Boron-Carbon Compoundsa compounds
pristine
C-H type
all-H type
hydro-BC3 hydro-BC5 hydro-BC7
SC 0.76 eV metal metal
SC 2.11 eV SC 2.62 eV SC 2.37 eV
metal metal metal
a SC represents semiconductor and the corresponding value of band gap is listed following it.
2.37 eV in the C-H type of hydro-BC5 and hydro-BC7 compounds, respectively. While in the all-H types of hydroBC5 and hydro-BC7 compounds, these boron pz character bands fall below the Fermi level due to the formation of the σ-B-H bonds. Similar to that of hydro-BC3 compounds, the all-H types of hydro-BC5 and hydro-BC7 compounds become metals. By comparing the formation energies, we find that both the C-H type and all-H type are stable in the hydro-BC5 and hydro-BC7 compounds. εH2 are -0.131 and -0.043 eV/atom for the C-H type and all-H type of hydro-BC5 compounds, respectively. For the hydro-BC7 compounds, εH2 are -0.121 and -0.070 eV/atom, respectively. The electronic properties of stable hydro-boron-carbon compounds are summarized in Table 1. Different from the BC3, the pristine BC5 sheet is a metal, which agrees with the previous calcutions.26 For the BC5 sheet, a metal-semiconductor transition occurs at µH2 ) -0.57 eV and the C-H type of hydro-BC5 compounds is stable in the chemical potential range of [-0.57, 3.87] eV. When µH2 increases, a semiconductor-metal transition occurs and the all-H type becomes stable in the H-rich condition of µH2 > 3.87 eV. Thus, the metal-semiconductor-metal transitions occur in the hydrogenation of the BC5 sheet. Similar to the hydroBC5 compounds, the hydro-BC7 compounds have three stable conformations, which are metallic pristine sheet (when µH2 < -0.52 eV), semiconducting C-H type (when µH2 in [-0.52, 3.06] eV), and metallic all-H type (when µH2 > 3.06 eV). The metalsemiconductor-metal transitions also occur in the hydrogenation of the BC7 sheet. similar to the BC3 systems, step by step hydrogenations will also control these metal-semiconductormetal transitions. In the first stage, the compounds are also under hydrogen molecule gas circumstances. The pristine ones can transform to the C-H types and the metal-semiconductor transition occurs in this stage. In the second stage, the compounds are under atomic hydrogen gas circumstances. The C-H types can transform to the all-H ones and the semiconductormetal transition occurs in this stage. Conclusions In summary, we have investigated the electronic structures of hydro-boron-carbon compounds. The hydro-boron-carbon compounds with only carbon atoms bonding with hydrogen atoms are all semiconductors. When both boron and carbon atoms bond with hydrogen atoms, all the hydro-boron-carbon compounds become metals. The electronic properties of hydro-boron-carbon compounds depend on the bonding status of boron pz orbitals. During step by step hydrogenation, semiconductor-semiconductor-metal transitions appear in the hydro-BC3 compounds and metal-semiconductor-metal transitions appear in the hydro-BC5 and hydro-BC7 compounds. Because of the rich electronic properties, the hydroboron-carbon compounds are the potential low-dimensional materials for nanoelectronic and device applications. Acknowledgment. This research was supported by the National Science Foundation of China (Grant Nos. 10974107 and 10721404) and MOST (Grant No. 2006CB605105).
Supporting Information Available: Figures S1-S4 show the convergence tests for plane-wave cutoff energies and k-point samplings. Figure S5 shows the energy bands of BC3 sheet and hydro-BC3 compounds obtained by PAW pseudopotentials. This material is available free of charge via the Internet at http:// pubs.acs.org. References and Notes (1) Avouris, P.; Chen, Z.; Perebeinos, V. Nat. Nanotechnol. 2007, 2, 65. (2) Geim, A. K.; Novoselov, K. S. Nat. Mater. 2007, 6, 183. (3) Neto, A. H. C.; Guinea, F.; Peres, N. M. R.; Novoselov, K. S.; Geim, A. K. ReV. Mod. Phys. 2009, 91, 109. (4) (a) Campos-Delgado, J.; et al. Nano Lett. 2008, 8, 2773. (b) Tung, V. C.; Allen, M. J.; Yang, Y.; Kaner, R. B. Nat. Nanotechnol. 2009, 4, 25. (c) Hernandez, Y.; et al. Nat. Nanotechnol. 2008, 3, 563. (d) Li, X.; Zhang, G.; Bai, X.; Sun, X.; Wang, X.; Wang, E.; Dai, H. Nat. Nanotechnol. 2008, 3, 538. (e) Jia, X.; Hofmann, M.; Meunier, V.; Sumpter, B. G.; CamposDelgado, J.; Romo-Herrera, J. M.; Son, H.; Hsieh, Y. P.; Reina, A.; Kong, J.; Terrones, M.; Dresselhaus, M. S. Science 2009, 323, 1701. (5) Park, S.; Ruoff, R. S. Nat. Nanotechnol. 2009, 4, 217. (6) Boukhvalov, D. W.; Katsnelson, M. I. J. Am. Chem. Soc. 2008, 130, 10697. (7) Boukhvalov, D. W.; Katsnelson, M. I. J. Phys.: Condens. Matter 2009, 21, 344205. (8) Sofo, J. O.; Chaudhari, A. S.; Barber, G. D. Phys. ReV. B 2007, 75, 153401. (9) Boukhvalov, D. W.; Katsnelson, M. I.; Lichtenstein, A. I. Phys. ReV. B 2008, 77, 035427. (10) Ryu, S.; Han, M. Y.; Maultzsch, J.; Heinz, T. F.; Kim, P.; Steigerwald, M. L.; Brus, L. E. Nano Lett. 2008, 8, 4597. (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.; Novoselov, K. S. Science 2009, 323, 610. (12) Legoas, S. B.; Autreto, P. A. S.; Flores, M. Z. S.; Galva˜o, D. S. arXiv:0903.0278v1 2009. (13) Lebe`gue, S.; Klintenberg, M.; Eriksson, O.; Katsnelson, M. I. Phys. ReV. B 2009, 79, 245117. (14) Panchakarla, L. S.; Govindaraj, A.; Rao, C. N. R. ACS Nano 2007, 1, 494. (15) Subrahmanyam, K. S.; Panchakarla, L. S.; Govindaraj, A.; Rao, C. N. R. J. Phys. Chem. C 2009, 113, 4257. (16) Pontes, R. B.; Fazzio, A.; Dalpian, G. M. Phys. ReV. B 2009, 79, 033412. (17) Dutta, S.; Pati, S. K. J. Phys. Chem. B 2008, 112, 1333. (18) Borowiak-Palen, E.; Rummeli, M.; Gemming, T.; Knupfer, M.; Kalenczuk, R. J.; Pichler, T. Synth. Met. 2005, 153, 345. (19) Fuentes, G. G.; Borowiak-Palen, E.; Knupfer, M.; Pichler, T.; Fink, J.; Wirtz, L.; Rubio, A. Phys. ReV. B 2004, 69, 245403. (20) (a) Yanagisawa, H.; Tanaka, T.; Ishida, Y.; Matsue, M.; Rokuta, E.; Otani, S.; Oshima, C. Phys. ReV. Lett. 2004, 93, 177003. (b) Tanaka, H.; Kawamataa, Y.; Simizua, H.; Fujitaa, T.; Yanagisawaa, H.; Otanic, S.; Oshima, C. Solid State Commun. 2005, 136, 22. (21) Ding, Y.; Wang, Y.; Ni, J. Appl. Phys. Lett. 2009, 94, 073111. (22) Caretti, I.; Gago, R.; Albella, J. M.; Jime´nez, I. Phys. ReV. B 2008, 77, 174109. (23) (a) Miwa, R. H.; Martins, T. B.; Fazzio, A. Nanotechnology 2008, 19, 155708. (b) Zhou, Y. G.; Zu, X. T.; Gao, F.; Nie, J. L.; Xiao, H. Y. J. Appl. Phys. 2009, 105, 014309. (24) Magri, R. Phys. ReV. B 1994, 49, 2805. (25) (a) Miyamoto, Y.; Rubio, A.; Louie, S. G.; Cohen, M. L. Phys. ReV. B 1994, 50, 18360. (b) Wang, Q.; Chen, L. Q.; Annett, J. F. Phys. ReV. B 1996, 54, R2271. (26) (a) Way, B. M.; Dahn, J. R.; Tiedje, T.; Myrtle, K.; Kasrai, M. Phys. ReV. B 1992, 46, 1697. (b) Hu, Q.; Wu, Q.; Ma, Y.; Zhang, L.; Liu, Z.; He, J.; Sun, H.; Wang, H. T.; Tian, Y. Phys. ReV. B 2006, 73, 214116. (27) Lowther, J. E. J. Phys.: Condens. Matter 2005, 17, 3221. (28) (b) Lowther, J. E.; Zinin, P. V.; Ming, L. C. Phys. ReV. B 2009, 79, 033401. (29) (a) Kresse, G.; Furthmuller, J. Comput. Mater. Sci. 1996, 6, 15. (b) Kresse, G.; Furthmuller, J. Phys. ReV. B 1996, 54, 11169. (30) Momma, K.; Izumi, F. J. Appl. Crystallogr. 2008, 41, 653. (31) Wassmann, T.; Seitsonen, A. P.; Saitta, A. M.; Lazzeri, M.; Mauri, F. Phys. ReV. Lett. 2008, 101, 096402.
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