J. Phys. Chem. C 2007, 111, 3285-3289
3285
Static and Optical Transverse and Longitudinal Screened Polarizabilities of Boron Nitride Nanotubes Lu Wang,† Jing Lu,*,†,‡ Lin Lai,† Wei Song,† Ming Ni,† Zhengxiang Gao,*,† and Wai Ning Mei‡ Mesoscopic Physics Laboratory, Department of Physics, Peking UniVersity, Beijing 100871, P. R. China, and Department of Physics, UniVersity of Nebraska at Omaha, Omaha, Nebraska 68182-0266 ReceiVed: August 30, 2006; In Final Form: NoVember 17, 2006
We have investigated static and optical transverse and longitudinal screened polarizabilities of single-walled (SWBNNT) and multiwalled (MWBNNT) boron nitride nanotubes by using the density functional perturbation theory. The static and optical transverse and longitudinal screened polarizabilities of SWBNNTs all increase linearly with the radius and are smaller than those of similar-sized single-walled carbon nanotubes. The screened optical transverse polarizabilities of SWBNNTs are found to be about half that of the unscreened ones. In addition to the screened polarizabilities themselves, the ratio between the static and optical screened polarizabilities of SWBNNTs also exhibits anisotropy. In contrast to multiwalled carbon nanotubes, where the inner tube is almost completely shielded by the outer tube, in MWBNNTs, the inner tube is only partially shielded by the outer tube when a transverse electric field is applied.
Introduction Since their discovery1 in 1991, carbon nanotubes (CNTs) have been extensively investigated due to their special mechanical and electronic properties. Single-walled carbon nanotubes (SWCNTs) can be metallic or semiconducting depending on their helicity and diameter.2 Controlled synthesis of SWCNTs with similar electronic properties or postsynthesis separation of metallic and semiconducting SWCNTs is needed for their wide application. This causes extra work, although significant progress has been made in the separation of metallic and semiconducting SWCNTs.3 Boron nitride nanotubes (BNNTs) have structures similar to those of CNTs. They were predicted in 19944,5 to be wide band gap semiconductors, with weak dependence on their chirality, diameters, and the number of walls of the tube, and were subsequently synthesized in 1995.6 In addition to their stable electronic properties, BNNTs are also more resistant to oxidation than CNTs and can survive in up to 800 °C in air.7 Many experimental8,9 and theoretical10-15 works have been reported for BNNTs in recent years. The dielectric properties of nanotubes are of great importance for their electronic and optical applications. In recent years, the response of infinitely long SWCNTs and multiwalled carbon nanotubes (MWCNTs) have been studied by the tight-binding16 and first principles approaches.17-20 These studies show that the screened transverse static polarizability of SWCNTs is smaller than the unscreened one by a factor of 4-5.16 Both the unscreened16 and screened17,19 transverse static polarizabilities of SWCNTs per unit length are insensitive to the band gap and proportional to the square of the effective radius, R. The longitudinal static polarizability16,19 per unit length is found to scale as R|(0) ∼ R/Eg2, where Eg is the band gap, whereas the longitudinal polarizability per atom scales as R|(0) ∼ 1/Eg.20 In MWCNTs, the outer tube dominates the transverse static * Corresponding author. E-mail:
[email protected] (J.L.); zxgao@ pku.edu.cn (Z.G.). † Peking University. ‡ University of Nebraska at Omaha.
polarizability and almost completely shields the inner tube.18,19 Frequency-dependent screened polarizabilities of finitely long single-walled boron nitride nanotubes (SWBNNTs) have been calculated with a semiempirical dipole-dipole interaction model and ab initio calculations at the SCF level.10 They found that SWBNNTs have smaller polarizabilities than those of SWCNTs with the same geometry and number of atoms, and the geometry of the tubes has a large influence on the anisotropy of the polarizability components, whereas the mean polarizability remains almost unaffected when the geometrical configuration is modified. Guo et al.13 calculated the unscreened optical polarizabilities (they called them static polarizabilities, but they neglected the ionic contribution, and what they obtained were actually optical polarizabilities) of infinitely long perfect SWBNNTs through density functional theory (DFT) with the independent-particle approximation. They found that both the transverse and longitudinal unscreened optical polarizabilities are almost independent of the chirality and roughly proportional to the tube radius. Three open issues are left for the dielectric properties of infinitely long BNNTs: (1) What is the screened transverse optical polarizability of SWBNNTs? The local field effects in a transverse electric field are not negligible from the results of SWCNTs.16 (2) What is the static screened polarizability of SWBNNTs? (3) Is the inner tube shielded by the outer tube in multiwalled boron nitride nanotubes (MWBNNTs) as in MWCNTs? In order to address the above open issues, in this article, we study the static and optical screened polarizabilities of infinitely long SWBNNTs and MWBNNTs by using the density functional perturbation theory (DFPT) calculations21,22 within the local density approximation (LDA). Computational Details We use norm-conserving pseudopotential23 plane-wave basis sets, implemented in CASTEP code.24 The energy cutoff of the plane-wave is 550 eV. We construct a hexagonal supercell with periodic boundary conditions for BNNTs. The tube is placed in the supercell with its axis along the z-direction. Both the atom
10.1021/jp065644t CCC: $37.00 © 2007 American Chemical Society Published on Web 02/02/2007
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Wang et al.
positions and the lattice parameter along the tube axis are fully optimized by a conjugate gradient technique.25 In all the geometry optimizations, the closest distance between the nearest tubes L is greater than 7 Å, and a 1 × 1 × 7 MonkhorstPack26 k-point grid is used. After optimization, the max force on each atom is less than 0.03 eV/Å, and the max stress on the supercell is less than 0.05 GPa. Band structure and dielectric property calculations are performed for the optimized geometries with a uniform 1 × 1 × 19 k-point. In the DFTP scheme, a weak electric field is applied to the BNNT system as a perturbation, and the first-order change in wavefunction is computed by minimizing the second-order perturbation in the total energy.22 The electronic polarizability is obtained from the second derivative of the total energy with respect to the electric field perturbation while all the ions are clamped.21 Since BNNTs are anisotropic due to their quasi-one-dimensional structures, the electric field perturbation is applied along and perpendicular to the axis. Therefore we have the following equations:20,21
) Rb,ele ⊥ Rb,ele |
∂E2 ∂2x
∂E2 ) 2 ∂ z
Ω ( - 1) 4πc ⊥
polarizability of a BNNT in a periodic array Rb⊥. The conversion relation is
(1) Ω ⊥ - 1 ) R⊥ ) 2πc ⊥ + 1 (2)
where E is the total energy, and x and z are the induced electric and Rb,ele fields in the x- and z-directions, respectively. Rb,ele ⊥ | are the transverse and longitudinal electronic polarizabilities of a periodically repeated array of BNNTs (due to the periodic boundary conditions), respectively. The response of the whole electronic system is achieved self-consistently. Hence, local field effects have been taken into account, and we get screened polarizabilities. The ionic contributions are negligible at high frequencies because of the inertia of the ions.27 As a result, the dielectric response at optical frequencies arises almost entirely from the electronic polarizability. In other words, the optical polarizability is equivalent to the electronic polarizability, and Rb⊥(∞) ) Rb,ele namely, Rb| (∞) ) Rb,ele | ⊥ . Static polarizability consists of the electronic and ionic polarizabilities, namely, Rb| (0) ) Rb,ele + Rb,ion and Rb⊥(0) ) Rb,ele + Rb,ion | | ⊥ ⊥ . Therefore, we perform a gamma-point phonon calculation in the DFPT scheme and obtain the longitudinal and transverse ionic polar21 The polarizabilities we obtain in izabilities, Rb,ion and Rb,ion | ⊥ . the calculations are the values per supercell. For comparison purposes, we convert the polarizabilities per supercell to the values per unit length along the axis. Since calculations provide us the transverse response of a periodically repeated array of BNNTs rather than an isolated BNNT, it is necessary to remove the depolarization fields originating from the periodic images. The relation between the polarizability per unit length and the dielectric constant is
R⊥ )
Figure 1. Convergence of the screened transverse polarizabilities per angstrom of Rb⊥(0), R⊥(0), Rb⊥(∞), and R⊥(∞) with respect to the lattice parameter a of the supercell for the (4,4) SWBNNT.
(3)
where Ω is the volume of the supercell, c is the lattice parameter along the tube axis, and ⊥ is the dielectric constant, which depends on the lattice parameter a. We can get the polarizability of an isolated BNNT by taking the limit a f ∞, but it is extremely time-consuming when the lattice parameter a becomes very large. We solve this problem by using a classical twodimensional Clausius-Mossotti correction19,28 relating the single-tube transverse polarizability R⊥ and the transverse
Rb⊥ 2πc b R 1+ Ω ⊥
(4)
Figure 1 illustrates the dependence of the two polarizabilities on the lattice parameter a for the (4,4) SWBNNT. The convergence of Rb⊥(0) and Rb⊥(∞) with respect to a is slow due to the long-range electrostatic interactions between image tubes, whereas both R⊥(0) and R⊥(∞) converge within 0.01 Å2 when the lattice parameter a is larger than 16 Å (the corresponding closest distance between the nearest tubes L is 10.4 Å). The longitudinal screened polarizability of an isolated BNNT can be obtained directly from the value of the periodically arranged BNNTs, namely, R| ) Rb| , because the response of nanotubes to longitudinal electric fields is additive.19 Our test confirmed that the longitudinal polarizability remains unchanged with increasing a. According to the above convergence tests, we set L to 11 Å in all the polarizability calculations. Results and Discussion In the optimized SWBNNTs, the B and N atoms form two cylinders with different radii (RN and RB). The radius of a SWBNNT is taken as R ) (RN + RB)/2. The separation between the two cylinders, defined as ∆R ) RN - RB, decreases from 0.12 to 0.02 Å as the radius of SWBNNTs increases from 2.06 to 10.12 Å (Table 1). Such a tendency is also reported in a previous work.29 The calculated band gaps of SWBNNTs are listed in Table 1 and are also shown in Figure 2 as a function of the radius. All the calculated band gaps agree well with the previous LDA calculations.13,29 In general, the calculated band gaps of the SWBNNTs increase with their radii and approach the calculated band gap of an isolated hexagonal boron nitride sheet at R > 7 Å. The four types of screened polarizabilities per unit length of the isolated BNNTs are listed in Table 1. The longitudinal screened polarizabilities are much larger than the transverse ones, exhibiting apparent anisotropy. The ratio between the static and optical screened polarizabilities of SWBNNTs also exhibits anisotropy. The ratio between the static and optical transverse screened polarizabilities R⊥(0)/R⊥(∞) ranges from 1.13 to 1.20, while that between the static and optical longitudinal screened polarizabilities, R|(0)/R|(∞), ranges from 1.47 to 1.58, close to
DFPT Analysis of Screened Polarizabilities of BNNTs
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TABLE 1: Radii, Radius Separations ∆R (Defined as the Difference Between the Radius of N Atoms and the Radius of B Atoms), Band Gaps, and Optical and Static Transverse and Longitudinal Screened Polarizabilities (per unit length) of BNNTs as a Function of the Chiral Vector (n,m)a (n,m)
R (Å)
∆R (Å)
Eg (eV)
R⊥(∞) (Å2)
Run ⊥ (∞) (Å2)
R⊥(0) (Å2)
R|(∞) (Å2)
Run | (∞) (Å2)
R|(0) (Å2)
(5,0) (6,0) (7,0) (8,0) (9,0) (12,0) (15,0) (18,0) (21,0) (3,3) (4,4) (5,5) (6,6) (7,7) (8,8) (9,9) (10,10) (12,12) (15,15) (4,2) (6,0)@(15,0) (4,4)@(9,9) (7,7)@(12,12) (10,10)@(15,15)
2.06 2.43 2.82 3.21 3.60 4.77 5.96 7.13 8.32 2.10 2.78 3.45 4.13 4.82 5.50 6.19 6.87 8.25 10.12 2.15 5.95b 6.20b 8.23b 10.29b
0.12 0.09 0.08 0.07 0.06 0.04 0.04 0.03 0.02 0.12 0.08 0.06 0.05 0.04 0.04 0.03 0.03 0.03 0.02 0.11 0.04b 0.04b 0.03b 0.02b
2.15 2.73 3.32 3.51 3.73 4.15 4.35 4.48 4.55 4.34 4.28 4.23 4.31 4.38 4.48 4.51 4.61 4.67 4.70 3.29 2.32 3.84 4.16 4.26
3.6 4.4 5.3 6.1 7.1 10.1 13.3 16.7 20.3 3.8 5.2 6.8 8.5 10.3 12.1 14.1 16.1 20.2 26.6 3.9 15.7 17.0 26.1 36.4
8.5 10.8
4.1 5.0 6.0 7.0 8.1 11.7
13.0 15.3 17.7 20.1 22.5 29.7 37.1 44.4 51.8 11.9 16.4 20.8 25.1 29.4 33.8 38.1 42.4 51.0 63.9 13.0 52.7 55.0 80.8 106.8
13.2 16.2
20.6 23.7 27.0 30.3 33.7 44.2
15.3 20.4 25.6
18.1
35.6 44.0 9.3
4.3 5.9 7.8 9.7 11.9 14.2 16.6 19.1 24.2
22.1 29.4 36.5
24.6
49.7 61.8 12.6
17.9 24.3 30.6 36.9 43.3 49.6 55.9 62.3 74.9
un The unscreened optical transverse (Run ⊥ (∞)) and longitudinal (R| (∞)) polarizabilities obtained in a previous work (ref 13) are also given for b comparison. For the outer tube. a
Figure 2. Calculated band gap versus radius of SWBNNTs. The calculated band gap of an isolated hexagonal boron nitride sheet (4.7 eV) is shown as a dashed horizontal line.
the calculated ratio (1.48) of the in-plane static and optical screened polarizabilities of an isolated hexagonal boron nitride sheet. The unscreened transverse and longitudinal polarizabilities obtained by using DFT within the independent-particle approximation13 are also given in Table 1 for comparison. The difference between the unscreened polarizability and screened polarizability is due to the difference between the total and applied electrical field. This arises because the virtual singleparticle excitations have electrical charge and produce a local field. In other words, the unscreened polarizability only accounts for the polarization of the individual single-particle wavefunctions, while the screened one includes their mutual interaction as well.16 What the experiment measures is the screened polarizability. It has been pointed out that the local field effects along the axis of nanotubes are negligible because an external electric field along the axis will not induce any bound charge for infinitely long nanotubes.16 We find a good agreement between the screened and unscreened13 longitudinal optical
polarizabilities. Local field effects perpendicular to the nanotube axis are significant due to the buildup of bound surface charge when a transverse electric field is applied.16 There is approximately a factor of 1.7-2.4 difference between the unscreened13 and screened transverse optical polarizabilities of SWBNNTs, nearly half the factor of 4-5 for SWCNTs.16 Therefore, local field effects are less significant in SWBNNTs than in SWCNTs. This difference is attributed to the different electronic properties between SWCNTs and SWBNNTs. In SWCNTs, the polarizability is mostly contributed by the fully delocalized π-electrons.19 On the other hand, SWBNNTs have a wide energy gap, and their valence electrons are tightly bounded to the B and N atoms. The delocalized π-electrons in SWCNTs can generate a stronger local field than the localized valence electrons in SWBNNTs. Figure 3a,b shows the longitudinal and transverse screened polarizabilities per unit length versus the radius of an SWBNNT, respectively. All the examined screened polarizabilities vary approximately linearly with the radii of SWBNNTs and can be fitted with these relations: R⊥(0) ) 3.19(R - 0.95), R⊥(∞) ) 2.77(R - 0.94), R|(0) ) 8.92R, and R|(∞) ) 6.31R, with R in Å2 and R in Å. The screened polarizabilities per pair of B and N atoms are R⊥(∞)/Npair ) 1.58-2.21 Å3, R⊥(0)/Npair ) 1.752.51 Å3, R|(∞)/Npair ) 4.97-5.56 Å3, and R|(0)/Npair ) 7.448.80 Å3. Alternatively, the transverse static screened polarizability per unit length of SWCNTs is proportional to the square of the radius,16,17,19 whereas the longitudinal screened static polarizability scales as R|(0) ∼ R/Eg2.16,19 The behavior of the transverse polarizability of SWCNTs can be understood in terms of an empty lattice model of electrons moving freely on a cylinder of infinitesimal thickness.16 The behavior of the longitudinal polarizability of SWCNTs can be derived from linear-response theory30 in combination with the fact that SWCNTs have a small band gap.16,19 We plot the optical and static longitudinal polarizabilities per unit length of the widegap SWBNNTs as a function of R/Eg2 in Figure 4. R|(∞) and
3288 J. Phys. Chem. C, Vol. 111, No. 8, 2007
Figure 3. (a) Transverse screened polarizability per angstrom versus the radius of SWBNNTs. (b) Longitudinal screened polarizability per angstrom versus the radius of SWBNNTs. The solid lines are linear least-squares fits.
Figure 4. (a) Optical and (b) static longitudinal screened polarizabilities per angstrom of SWBNNTs, R|(∞) and R|(0), as a function of R/Eg2, where R is the radius, and Eg is the band gap. The solid lines are linear least-squares fits for the armchair SWBNNTs.
R|(0) of the small-radius zigzag and chiral SWBNNTs do not have linear relations with R/Eg2, while those of the armchair and large-radius zigzag SWBNNTs do. The band gaps of the
Wang et al.
Figure 5. (a) Comparison of the screened static transverse polarizabilities per angstrom versus the radius between SWCNTs (blue triangles) and SWBNNTs (red circles). (b) Comparison of the screened static longitudinal polarizabilities per angstrom versus the radius between semiconducting SWCNTs (blue diamonds) and SWBNNTs (red stars).
armchair and large-radius zigzag SWBNNTs are approximately constant, and, in this case, the two relations R|(0) ∼ R/Eg2 and R|(∞) ∼ R are actually equivalent. Therefore, the applicability of R|(0) ∼ R/Eg2 for the armchair and large-radius zigzag SWBNNTs appears to be a coincidence. The comparison of the static screened polarizabilities between SWBNNTs and SWCNTs19 is made in Figure 5. Both the transverse and longitudinal screened polarizabilities of SWCNTs are greater than those of similar-sized SWBNNTs, and the differences increase with increasing radius. The difference in the longitudinal screened polarizability between similar-sized SWCNTs and SWBNNTs is significantly larger than that in the transverse screened polarizability. The smaller screened polarizability of SWBNNTs relative to that of SWCNTs is also ascribed to the localized valence electrons in SWBNNTs versus the delocalized π-electrons in SWCNTs. Depolarization effects along the tube axis are negligible, and the dielectric interaction along the tube axis between constituent tubes of MWBNNTs is quite weak. As shown in Table 1, the longitudinal screened polarizabilities of all the MWBNNTs are approximately the sum of the longitudinal screened polarizabilities of the constituent tubes. The additive property of the longitudinal response is also calculated for MWCNTs.19 The transverse screened polarizabilities of the four investigated MWBNNTs are smaller than the sums of the values of their components but larger than the values of the outer tubes; thus the inner tube is partially shielded by the outer tube in MWBNNTs when a transverse electric field is applied. Alternatively, the transverse screened polarizability of MWCNTs is nearly equal to the value of the outer tube,18,19 suggestive of a nearly complete shielding of the inner tube by the outer tube
DFPT Analysis of Screened Polarizabilities of BNNTs
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when a transverse electric field is applied. Following a previous work,18 we define shielding S as
S)
Rinner + Router - Rtotal ⊥ ⊥ ⊥ Rinner ⊥
× 100%
(5)
where Rinner and Router are the transverse polarizabilities of the ⊥ ⊥ is the total inner and outer tubes, respectively, and Rtotal ⊥ transverse polarizability of MWBNNTs. We obtained S ) 46, 44, 43, and 39% for the (6,0)@(15,0), (4,4)@(9,9), (7,7)@(12,12), and (10,10)@(15,15) MWBNNTs, respectively. The shielding efficiency of the outer tube decreases with increasing radius. On the other hand, MWCNTs have a much larger shielding S of 90-95%.18 The nearly complete shielding effects in MWCNTs are attributed to their delocalized π-electrons.19 A macroscopic insulating cylinder hardly produces shielding effects. The 39-46% shielding in insulating MWBNNTs probably originates from the extremely large surface volume ratio at nanoscale. Finally, we calculate the screened optical polarizability of the (4,4) SWBNNT under a finite electric field of 0.0051 V/Å by using the Berry phase scheme31,32 implemented in ABINIT package.33 We find an agreement within 1% between the Berry phase and the DFPT values, confirming the reliability of the DFPT approach. Conclusions In summary, we have calculated the linear response of isolated infinite SWBNNTs and MWBNNTs to electric field perturbation within DFPT. The static and optical transverse and longitudinal screened polarizabilities of insulating SWBNNTs all have linear relationship with the radius. The response of insulating SWBNNTs to an electric field is weaker than small-gap or metallic SWCNTs with delocalized π-electrons, and this is reflected from three aspects: (1) Local field effects are weaker in SWBNNTs than in SWCNTs. (2) The screened static polarizabilities of SWBNNTs are smaller than those of similar-sized SWCNTs. (3) The shielding efficiency in MWBNNTs is merely half that in MWCNTs in a transverse electric field. Acknowledgment. This work was supported by the NSFC (Grant Nos. 10474123, 10434010, and 90606023), National 973 Projects (No. 2002CB613505, MOST of China), 211, 985 and Creative Team Projects of MOE of China, and the Nebraska Research Initiative (No. 4132050400) of the USA. Our calculations were partially carried out in the HP Cluster of the Calculation Center of Science and Engineering and the Cluster of the Institute of Condensed Matter and Material Physics, Peking University.
References and Notes (1) Iijima, S. Nature 1991, 354, 56. (2) Baughman, R. H.; Zakhidov, A. A.; de Heer, W. A. Science 2002, 297, 787. (3) Maeda, Y.; Kimura, S.; Kanda, M.; Hirashima, Y.; Hasegawa, T.; Wakahara, T.; Lian, Y.; Nakahodo, T.; Tsuchiya, T.; Akasaka, T.; Lu, J.; Zhang, X.; Gao, Z.; Yu, Y.; Nagase, S.; Kazaoui, S.; Minami, N.; Shimizu, T.; Tokumoto, H.; Saito, R. J. Am. Chem. Soc. 2005, 127, 10287. (4) Blase, X.; Rubio, A.; Louie, S. G.; Cohen, M. L. Europhys. Lett. 1994, 28, 335. (5) Rubio, A.; Corkill, J. L.; Cohen, M. L. Phys. ReV. B 1994, 49, 5081. (6) Chopra, N. G.; Luyken, R. J.; Cherrey, K.; Crespi, V. H.; Cohen, M. L.; Louie, S. G.; Zettl, A. Science 1995, 269, 966. (7) Han, W. Q.; Mickelson, W.; Cumings, J.; Zettl, A. Appl. Phys. Lett. 2002, 81, 1110. (8) Ishigami, M.; Sau, J. D.; Aloni, S.; Cohen, M. L.; Zettl, A. Phys. ReV. Lett. 2005, 94, 056804. (9) Zhi, C.; Bando, Y.; Tang, C.; Golberg, D. J. Phys. Chem. B 2006, 110, 8548. (10) Kongsted, J.; Osted, A.; Jensen, L.; Astrand, P. O.; Mikkelsen, K. V. J. Phys. Chem. B 2001, 105, 10243. (11) Wirtz, L.; Rubio, A.; de la Concha, R. A.; Loiseau, A. Phys. ReV. B 2003, 68, 045425. (12) Marinopoulos, A. G.; Wirtz, L.; Marini, A.; Olevano, V.; Rubio, A.; Reining, L. Appl. Phys. A 2004, 78, 1157. (13) Guo, G. Y.; Lin, J. C. Phys. ReV. B 2005, 71, 165402. (14) Park, C. H.; Spataru, C. D.; Louie, S. G. Phys. ReV. Lett. 2006, 96, 126105. (15) Wirtz, L.; Marini, A.; Rubio, A. Phys. ReV. Lett. 2006, 96, 126104. (16) Benedict, L. X.; Louie, S. G.; Cohen, M. L. Phys. ReV. B 1995, 52, 8541. (17) Brothers, E. N.; Kudin, K. N.; Scuseria, G. E.; Bauschlicher, C. W. Phys. ReV. B 2005, 72, 033402. (18) Brothers, E. N.; Scuseria, G. E.; Kudin, K. N. J. Chem. Phys. 2006, 124, 041101. (19) Kozinsky, B.; Marzari, N. Phys. ReV. Lett. 2006, 96, 166801. (20) Brothers, E. N.; Scuseria, G. E.; Kudin, K. N. J. Phys. Chem. B 2006, 110, 12860. (21) Gonze, X.; Lee, C. Phys. ReV. B 1997, 55, 10355. (22) Gonze, X. Phys. ReV. B 1997, 55, 10337. (23) Lin, J. S.; Qteish, A.; Payne, M. C.; Heine, V. Phys. ReV. B 1993, 47, 4174. (24) Segall, M. D.; Lindan, P. J. D.; Probert, M. J.; Pickard, C. J.; Hasnip, P. J.; Clark, S. J.; Payne, M. C. J. Phys.: Condens. Matter 2002, 14, 2717. (25) Pfrommer, B. G.; Cote, M.; Louie, S. G.; Cohen, M. L. J. Comput. Phys. 1997, 131, 233. (26) Monkhorst, H. J.; Pack, J. D. Phys. ReV. B 1976, 13, 5188. (27) Kittel, C. Introduction to Solid State Physics, 7th ed.; John Wiley & Sons, Inc.: New York, 1996. (28) Garcia-Vidal, F. J.; Pitarke, J. M.; Pendry, J. B. Phys. ReV. Lett. 1997, 78, 4289. (29) Xiang, H. J.; Yang, J. L.; Hou, J. G.; Zhu, Q. S. Phys. ReV. B 2003, 68, 035427. (30) Penn, D. R. Phys. ReV. 1962, 128, 2093. (31) Nunes, R. W.; Gonze, X. Phys. ReV. B 2001, 63, 155107. (32) Souza, I.; IÄn˜iguez, J.; Vanderbilt, D. Phys. ReV. Lett. 2002, 89, 117602. (33) Gonze, X.; Beuken, J. M.; Caracas, R.; Detraux, F.; Fuchs, M.; Rignanese, G. M.; Sindic, L.; Verstraete, M.; Zerah, G.; Jollet, F. Comput. Mater. Sci. 2002, 25, 478.
