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Ab Initio Electron Energy-Loss Spectra and Depolarization Effects: Application to Carbon Nanotubes E. AMZALLAG,1 D. EHINON,2 H. MARTINEZ,2 M. RE´RAT,2 I. BARAILLE2 1

Laboratoire d’Etude des Mate´riaux Hors Equilibre, ICMMO UMR8182 CNRS, Universite´ Paris-Sud 11, Baˆt 410, F91405 Orsay Cedex, France 2 Equipe de Chimie Physique, IPREM UMR5254, Universite´ de Pau et des Pays de l’Adour, 2, Avenue du pre´sident Pierre Angot, 64053 Pau Cedex 9, France Received 20 April 2011; accepted 6 June 2011 Published online 21 November 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/qua.23205

ABSTRACT: We perform ab initio calculations of the optical absorption and electron energy-loss spectra of (m, 0) single-walled carbon nanotubes (with m = 3n for m ¼ 725) in the framework of a ‘‘sum over states’’ (SOS) treatment of the Kohn-Sham (KS) single-particle orbitals and energies (CRYSTAL program). This approach tested on hexagonal boron nitride enables to fully assign the interband transitions in the imaginary part of the dielectric constant, in terms of atomic orbitals. As these calculations could not take into account the local field effects (depolarization effects), which take place for perpendicular polarizations in 2D and 1D periodic systems, we apply a simple method based on the Clausius-Mossotti formula, relating the SOS and coupled-perturbed KS polarizability values. This approach reproduces the main features of the spectra of boC 2011 Wiley Periodicals, Inc. Int J ron nitride (001) surface and carbon nanotubes. V Quantum Chem 112: 2171–2184, 2012 Key words: ab initio calculations; dielectric function; carbon nanotube; boron nitride

1. Introduction

E

lectron energy-loss spectroscopy (EELS) that measures the energy lost by electrons after they have passed through a thin section of a mateCorrespondence to: I. Baraille; e-mail: [email protected]

rial is related to electronic, optical, and mechanical properties of the observed material [1]. In particular, excitations corresponding to low energy losses (50 eV) involve collective plasma oscillations (which correspond to a collective excitation of the valence electrons) or single electronic transitions of valence electrons (interband transitions). Spatially resolved EELS is thus well suited to furnish

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AMZALLAG ET AL. information on the electronic structure of materials. For example, in the case of semiconducting and insulating materials, the low energy-loss domain at bandgap energies is of great interest as the electrical and optical properties are directly related to the structure of both valence (VB) and conduction bands (CB). When multiple scattering of the probe electron in the sample can be neglected or removed numerically, the single scattering distribution in the low loss region can be described in terms of the energy-loss function (ELF) [2] that measures the imaginary part of the inverse dielectric function e(x) as ELFðxÞ ¼ Imð1=eðxÞÞ where the energy loss relates to the frequency as E ¼ -hx. The complex dielectric function e(x) ¼ e1(x)þi e2(x) deduced from the low-loss EEL spectrum allows to calculate the optical properties of the material, such as reflectivity and absorption. Theoretical approaches based on band structure calculations within the framework of density functional theory (DFT) are the most widely used to model the dielectric function and deduce the EELS low energy spectra. The full-potential linear augmented plane method (FP-LAPW-DFT) implemented in the WIEN2K code [3] is one of the most used in the literature to simulate the ELF spectra [4]. In most cases, this approach reproduces fairly well the experimental spectra but the nature of the interband transitions is rarely investigated [5, 6]. Another approach to calculate the ELF function is based on the ‘‘sum over states’’ (SOS) calculation of static and dynamic polarizability using the one-electron eigenfunctions and eigenvalues provided by the periodic code CRYSTAL09 [7]. This method [8, 9] allows the calculation of the dielectric constant as a function of real or imaginary frequencies [10–14] and nonlinear susceptibilities of periodic systems [15]. Note that these approaches based on the independent-particle approximation are not sufficient for full interpretation of the experiments, especially for the 2D (surfaces) or 1D (nanotubes) periodic systems. These calculations, although they successfully described the spectra for the polarizations parallel to the periodic directions (surface plane or tube axis), could not take into account the depolarization effects which take place for perpendicular polarizations. This shortcoming is because of the local field effects (LFEs) that can strongly modify the results. For example, LFEs are crucial to explain the suppression of absorption peaks, which occur for perpendicular polarization in the optical spectra (e2(x)) of small-diameter single-walled carbon nanotubes (SWCNs) [16]. Today, in the periodic ab initio framework, two

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main approaches include these effects: Time-Dependent Density Functional Theory (TDDFT) or quasi particle formalisms which, for example, predict optical transitions of small-diameter SWCNs in agreement with available experiment [16–18]. The bottleneck of these two approaches is the limited size of the systems that can be treated. An important check of the reliability of the approximations adopted for the calculation of the response properties, that is, frequency-dependent dielectric function, is based on the comparison of the e(0) value obtained with the experimental value deduced from the refractive index pffiffiffiffiffiffiffiffi n ¼ eð0Þ. The analytical coupled-perturbed Hartree-Fock (HF) or Kohn-Sham (CPHF or CPKS) method for periodic systems [19, 20] has been recently implemented in the ab initio CRYSTAL code [7]. This method is based on the self-consistent evaluation of the second derivative of energy per cell with respect to an applied time-independent electric field, that is, the static polarizability, a(0). The dielectric constant can be approximated as eðxÞ ¼ 1 þ 4pNaðxÞ where N is the number of moieties per volume unit, especially for the static quantities where x ¼ 0. Interestingly, the SOS estimate of the static polarizability turns out to be the zero step of the iterative CPKS process. Studies on either zigzag SWCN (m, 0) (with m = 3n) [21, 22] or boton nitride (BN) slabs [23] have shown that the perpendicular polarizability value (parallel to the nonperiodic direction) calculated when orbital relaxation effects are taken into account as in CPKS is very different from the unrelaxed SOS one. In both cases, the authors have shown that this important relaxation effect can be modeled by the Clausius-Mossotti formula relating the SOS and CPKS polarizability values with a depolarization factor equal to 2p (1D system) or 4p (2D system) when the field is perpendicular to the tube or the slab, respectively. On the contrary, when the field is parallel to the periodic direction(s), the effect is much weaker leading to SOS and CPKS values very similar to each other (the depolarization factor is zero in this case). In this work, we propose to calculate the dielectric functions and ELF spectra of 1D or 2D systems in the framework of the SOS approximation and to take into account the LFE by extending the Clausius-Mossotti formula relating the SOS and CPKS static values to dynamic polarizabilities in the 0–35 eV energy range. Our purpose is to show that this strategy, although based on a simple model, is pertinent and allows the treatment of extended systems

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AB INITIO EELS SPECTRA AND DEPOLARIZATION EFFECTS as semiconductor zigzag SWCN (m, 0) (with m = 3n for m ¼ 7–25). The response to electric field is important for understanding the gas adsorption on carbon nanotube as shown by Girardet and coworkers [24, 25]. Some of us developed a new ab initio method to evaluate the long-range dispersion coefficient for a molecule adsorbing on a biperiodic system modeling a crystalline surface [14, 23]. The imaginary frequency dependence of the polarizabilities was calculated for each system, separately and the dispersion contribution to the interaction potential was evaluated via the Casimir-Polder formula [26]. Our initial objective is to extend this method to the physisorption of small molecules (H2, N2, O2) on the zigzag (m, 0) SWCN modeled as periodic systems (with nonvanishing gaps for m = 3n). For this purpose, we need to calculate the dynamic polarizabilities of these nanotubes, within the SOS method and the resulting values must be corrected to include the depolarization effect. First, to check the validity of the SOS method, we compare the results of both Linear Combination of Atomic Orbitals (LCAO)-DFT and FP-LAPW-DFT theoretical approaches to the experimental spectra available in the literature for the hexagonal BN (h-BN). We also investigate the influence of the DFT Hamiltonian on the calculation of the optical properties. Moreover, we show how the electronic transitions occurring in the ELF spectra can be fully interpreted in terms of crystalline orbitals within the LCAO-DFT approach. Then the BN slabs and semiconductor zigzag SWCN (m, 0) (with m = 3n for m ¼ 7–25) will be considered here. As already stated in previous work, the reason for choosing BN resides in the semimetal character of the graphite and graphene which leads to infinite parallel static polarizability in the SOS technique. Therefore h-BN is isoelectronic to graphite and has the same layered structure but with different stacking. The article is organized as follows. Method and Computational details are presented in Section 2. Results for h-BN, BN slabs, and semiconductor zigzag SWCN (m, 0) (with m = 3n for m ¼ 7–25) are analyzed in Section 3. Finally, conclusions are summarized in Section 4.

2. Method and Computational Details In the CRYSTAL09 sofware [7], the crystalline orbitals are expanded in terms of localized atomic Gaussian basis set, in a way similar to the LCAO

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method currently adopted for molecules. The number of k points in the first irreducible Brillouin zone (BZ) in which the DFT Hamiltonian matrix is diagonalized, is 133 for h-BN, 61 for the BN slabs, and 27 for the carbon nanotubes. All electron basis sets already optimized in early studies (7-321G(d) for N [27], 6-211G(d) for B [28] and 6-21G(d) for C [22]) were used for orbital expansion solving the KS equation iteratively. We set the total energy tolerance to 107 hartree and eigenvalue tolerance to 106 hartree in the iterative solution of the KS equation. The level of accuracy in evaluating the Coulomb series is controlled by five parameters, for which standard values (i.e., 6 6 6 6 12) have been used. The SOS method derived from the linear response theory calculates the mean dynamic polarizability [a(x)] as a function of the electric field frequency (x) [8, 9] and the dielectric constant e(x) is deduced from eðxÞ ¼ 1 þ 4pNaðxÞ where N is the number of moieties per volume unit. The unit cell volume is not well defined for slabs or nanotubes. This will be discussed in the following section. The choice of the effective unit cell volume would only change the magnitude of the dielectric function. The shape and energy position of the features in the optical dielectric function would not be affected by the chosen value although the energy position of the plasmon peaks would be shifted to lower or higher energy according to the unit cell volume used. In the SOS method, the polarizability [a(x)] (in a.u.) is computed from the knowledge of the single-particle orbitals and energies approximated by the solutions of the KS equations. Assuming the one-electron, rigid band approximations, neglecting electron polarization effects (Koopmans’ approximation) and in the limit of linear optics in the visible-ultraviolet region, the polarizability can be expressed as follows: Z aðxÞ ¼

dk BZ

X i;j

2 fij~k 4

De2ij~k x2 þ igx ðDe2~ x2 Þ2 þ x2 g2

3 5

(1)

ijk

Only vertical transitions between an occupied initial crystalline orbital |i[ and unoccupied |j[ crystalline orbitals for a k point in the first BZ are considered and Deij~k ¼ ej~k ei~k are the corresponding vertical transition energies. g is the damping factor which accounts for finite life time in the excited states. It avoids resonances in the absorption spectrum

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AMZALLAG ET AL. (imaginary part of dielectric constant). In the dipole approximation (valid when the wavelength of the electric field is much larger than the Bohr length), the oscillator strength fij~k is then given by: ~k j ji~2 fij~k ¼ ð2=3Þðej~k ei~k ÞhijX k

(2)

~k ¼ iei~k:~r r ~k is the Hermitian ~k ei~k:~r ¼ ~ where X r þ ir operator corresponding to the field perturbation for periodic systems [29–31]. As in our previous works ~r þ i~ k used by Gajdos [11, 12], the velocity operator r et al. [32] is preferred, and the transition moments between orthogonal crystalline orbitals are equal to ~k j ji~ as if the ~r j ji~=ðe ~ e ~Þ, instead of hij~ r þ ir hi j r jk ik k k hypervirial theorem was checked [33]. If the integrals are easier to calculate in such a gauge, the transition moment values are very sensitive to the transition energy values between occupied and unoccupied crystalline orbitals. The values of the oscillator strengths fij~k permit to identify the nature of electronic transitions detected on the ELF spectra (active electronic transitions). When 2D or 1D systems are studied, the relaxation effects have been found to be very large when the field direction is perpendicular to the slab or the nanotube, while the depolarization field component parallel to the periodic direction(s) should be tiny, as shown in previous works [23, 34, 35]. Then, the use of SOS method for calculating parallel components of the dynamic polarizability (axx ðxÞ) is well justified provided that the gap is well described. This is not true for the perpendicular component (aSOS zz ðxÞ) and corrections to the SOS method are needed in this case. Following the model of Benedict et al. [21], we have used the Clausius-Mossotti relation to retrieve the ‘‘corrected’’ value of the perpendicular polarizability from the SOS estimate: azz ðxÞ ¼

aSOS zz ðxÞ SOS 1 þ pazzV ðxÞ

(3)

where the depolarization factor due to the medium is p ¼ 4p (2D systems) or p ¼ 2p (1D systems). The volume V is determined by fitting the static value (azz ð0Þ) in the previous equation to the value (aCPKS ð0Þ) computed with the CPKS. zz V¼

p aSOS1ð0Þ aCPKS ð0Þ 1

zz

(4)

Z e2 ðxÞ ¼ 4p

dk BZ

X h jjra jiihijrb j ji r

i;j

Deij~k

r

dðDeij~k xÞ (5)

As for the LCAO approach, transition moments are calculated for each k-point of the first BZ. Finally, a Kramers-Kronig analysis is performed to obtain the real part e1(x).

3. Results and Discussion

zz

The CPKS method has recently been implemented in CRYSTAL to compute analytically the response to static fields with periodic boundary

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conditions [19, 20]. The CPKS static polarizability can be considered as reference value including both major contributions: (i) the noninteracting single-particle excitations which give rise to the unscreened static value as computed in the SOS method and (ii) the interactions between singleparticle excitations which make the total field felt by the electrons lower than the externally applied field by producing a depolarization local field. As discussed in Introduction section, we compare the dielectric functions obtained by both LCAO-GGA and FP-LAPW-GGA methods for hBN crystals. To make the comparison quite coherent, the Generalized Gradient Approximation (GGA) exchange and correlation potentials correspond to the formulation proposed by Perdew et al. [36] implemented in the FP-LAPW approach (PW-GGA). The crystalline orbitals are developed on k-points grid obtained using the improved tetrahedron method given by Blo¨chl et al. [37] (we used between 170 and 200 k-points). They are expanded in radial function time spherical harmonics up to lmax¼10 inside the muffin-tin spheres with radius Rmt. The plane-wave cut-off (RMTKmax) was set to 7. The convergence criteria are 105 eV on the total energy of the system and 104 eV on the atomic charge. The WIEN2K code computes the optical properties using the independent-particle random phase approximation [38], which corresponds to a perturbation method based on the same approximations as those previously described. LFEs are not included. The momentum matrix elements are calculated from the single electron states and the integration over the irreducible BZ is performed to calculate the imaginary part of the dielectric function e2(x) (expressed in a.u.):

3.1. H-BN To access the accuracy of the present independent-particle approach to the optical properties, we


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AB INITIO EELS SPECTRA AND DEPOLARIZATION EFFECTS TABLE I Properties of h-BN computed within different DFT functionals. LDA Band-gap (eV) exx ezz e n

4.31 4.81 2.47 4.03 2.01

(4.94) (2.78) (4.22) (2.05)

GGA 4.41 (4.23) (2.25) (3.57) (1.89)

BLYP 4.49 4.16 2.04 3.45 1.86

(4.22) (2.28) (3.57) (1.89)

PBE 4.42 4.75 2.43 3.98 1.99

(4.84) (2.73) (4.14) (2.03)

Experiment 5.971a 3.55b 2.20b 3.10b 1.76b

Comparison between the SOS and CPKS dielectric constants and refractive indexes. exx and ezz are the components of the dielectric constant parallel and perpendicular to the BN sheet, respectively. The SOS values are given in parentheses. a Ref. [41]. b Ref. [51].

first calculated the self-consistent band structure and also the dielectric function for h-BN. All the calculations were performed with the experi˚ and c ¼ mental geometric structure (a ¼ 2.50 A ˚ 6.65 A) [39]. The electronic properties of h-BN have been extensively studied both experimentally and theoretically but complete agreement has not yet been achieved on basic electronic properties such as the values of the lowest excitation energies. The large spread of experimental bandgap energies reported in literature, ranging from 3.1 to 7.1 eV, is currently explained by the difficulty of synthesizing high quality h-BN crystals or by the difficulty of extracting the bandgaps from experiment [40]. The most accurate results have been obtained by Watanabe et al. [41] who studied h-BN at very low temperature (8 K) by luminescence and supposed a direct bandgap energy measured at 5.971 eV. On the other hand, the most recent calculations using the all electron GW approximation [42] which include electron–hole interaction effects predict an indirect bandgap of 5.95 eV between the bottom of the CB at the M point and the top of the VB near the K point. The lowest direct interband transition of 6.47 eV is located at the H point. In view of these discrepancies, further confrontation between experimental and theoretical results will continue. As the DFT bandgaps cannot reproduce the experimental ones—the so-called bandgap problem—our aim is not to continue this debate. Nevertheless, a brief analysis of the gap between VB and CB is necessary to evaluate the errors induced by the DFT methods. As the SOS calculation of the dynamic polarizability needs the vertical transition energies, we consider only the direct bandgap values at the H point which are underevaluated in agreement with the well known bandgap problem of

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DFT [43–45]. The LCAO-DFT approaches predict a direct bandgap at the H point ranging between 4.31 eV (LDA) and 4.49 eV (BLYP) as shown in Table I. The FP-LAPW-GGA bandgap at 4.30 eV is found to be lower than the LCAO-GGA value (4.41 eV). To evaluate the LFEs, we report the SOS and CPKS polarizabilities calculated at the LDA [46, 47], PBE [48], and BLYP [49, 50] levels in Table I. The CPKS method based on the PW-GGA functional is not yet implemented in the CRYSTAL code. The comparison between the SOS and CPKS values shows that the LFE affect only weakly the components of the static dielectric function whatever the considered KS Hamiltonian. The relative difference between the SOS values and the CPKS values are less than 5% confirming that the dielectric function could reasonably be computed within the SOS approximation for these systems. The CPKS refractive indexes are slightly higher than the experimental value (1.76) [51]. The imaginary e2(x) and real e1(x) parts of dielectric functions have been calculated in the FP-LAPW-GGA and various LCAO-DFT (LDA, PBE, BLYP, GGA) approaches. The ELF function has thus been deduced for the photon energy ranging up to 35 eV as in the experimental spectra reported in Ref. 52. All these functions reported in Figure 2 are plotted with the same small Lorentzian function—the broadening g being equal to 0.3 eV—and are all very similar, showing that the choice of the DFT Hamiltonian is not a crucial point to compute these properties. The interpretation of interband transitions on e2(x) has been carried out within the framework of LCAO-GGA method. This analysis is supported by the band structures computed along ALMCAHKC path in the hexagonal BZ (see Fig. 1), the transition energies and the oscillator strengths


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FIGURE 1. LCAO-GGA band structure of BN along the path ALMCAHKC of the first hexagonal Brillouin zone. Energies are given in atomic unit (a.u.). The numbers on the lines referring to valence bands and first conduction bands are used to assign the electronic transitions in Table II. calculated at specific k-points of the BZ (see Table II). The LCAO-GGA band structure displayed in Figure 1 is very similar to those reported previously [53, 54]. The r and p states in the VBs (see Fig. 1) are mainly developed on the 2p(N) with a significant contribution of the 2p(B) atomic orbitals. The r* and p* states in the CB) exhibit a major 2p(B) participation as previously reported in the literature [55]. The imaginary part e2(x) [see Fig. 2(a)] mainly exhibits three bands around 7.3, 12.5, and 15.7 eV, common to the FP-LAPW-GGA and all the LCAO-DFT plots and in agreement with the experimental spectra [52]. The analysis of the electronic transitions (position and intensity) at LCAO-GGA level (see Table II and Fig. 1) leads to the following allocation. &

The band at 7.30 eV is associated to p–p* transitions between the last two VB states (7, 8)

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&

&

and the first two CB states (9, 10), along the LM direction with oscillator strengths equal to 2.56 a.u. at L and 2.21 or 2.97 a.u. at M. The band at 12.5 eV associated to r–p* transitions around C between bands number 4 and 9, the oscillator strength at C being equal to 1.74 a.u. which can be attributed to the peak around 12 eV. The band at 15.7 eV can be associated to r–r* transitions along the LM direction with oscillator strengths equal to 3.26 a.u. at L and 3.42 a.u. at M.

The study performed on h-BN at the FPLAPW-LDA level by Moreau and Cheynet [5] leads to similar spectra. The discrepancies between the two approaches are linked to oscillator strengths which are very sensitive to the


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AB INITIO EELS SPECTRA AND DEPOLARIZATION EFFECTS TABLE II Optical absorption spectrum of h-BN: electronic transitions (in eV) at the special points of first Brillouin zone. Aa

L

Cb

H

M

K

(7,8) ! (9,10)

10.7

7.0*

9.5–12.8

5.6

6.0

(7,8) ! (11,12)

14.3

12.2*

14.2

15.2

(7,8) ! (13,14)

15.5

17.4

14.3

16.8

(5,6) ! (9,10) (5,6) ! (11,12) (4,5) ! (9,10)

– – 14.6

10.4 15.7* 15.3

– – 12.3*–15.6

13.1 22.6 13.7

6.1–7.4* 7.2*–8.5 11.2–12.1* 12.3–13.2 15.6–16.4 16.7–17.5 10.6–11.2 15.0–15.9* 14.8–16.1

14.4 15.8–19.0 13.3 21.7 14.1

The numbers used to identify the transitions refer to the band structure in Figure 1. The red, blue, and green values refer to the three main peaks identified on the experimental spectrum: 7.3, 12.5, and 15.7 eV, respectively. The stars identify active electronic transitions corresponding to the highest oscillator strengths (greater than 1). a At A point, the bands 5, 6, 7, and 8 are degenerated. b At C point, the bands 5, 6, 7, and 8 are degenerated, but the bands 4, 5 and 9, 10 are not degenerated.

nature of the occupied and virtual crystalline orbitals involved in the corresponding transition. Moreover one may recall that, in the Self Consistent Field (SCF) method, the status of virtual orbitals remains questionable. In the FP-LAPW approach, the nature of the transition should be more difficult to analyze in terms of atomic orbitals because of the intersphere contribution outside the muffin-tin atomic spheres. The LCAO-GGA e1(x) function [Fig. 2(b)] is in a good agreement with the experimental plot while oscillations of larger amplitudes are observed at the FP-LAPW level of calculation. However, both theoretical curves have negative values (between 15.1 and 17.0 eV for FP-LAPW and between 16.6 and 17.1 eV for LCAO). The calculated ELF spectra [Fig. 2(c)] reproduces fairly well the characteristics of the experimental plot which exhibits three main peaks with maxima at 8.4, 11.9, and 24.9 eV; the latter peak is much larger with three maxima at 19.6, 23.3, and around 31 eV. The peaks in the ELF spectra correspond either to maxima in e2 or to zeros or small values of e1. The plasmon energy (the maximum of the loss function) is identified as occurring at the point where e1(x) ¼ 0 with a positive slope and the interband transitions induce transition bands in e2(x). For BN, the two first bands correspond to first minima of the real part e1(x) (7.85 and 12.9 eV at LCAO level). They can be qualified as plasmon-like in nature and can be assigned to the p electrons. In fact, these behaviors occur when there is a major exhaustion of interband transitions in the imaginary part e2(x). As far as

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FIGURE 2. Imaginary (a) and real (b) parts of the dielectric function and ELF spectrum (c) for h-BN with different LCAO-DFT approaches: LDA, PBE, GGA, and BLYP. Comparison with FP-LAPW-GGA (Wien) results. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]


AMZALLAG ET AL. TABLE III Properties of the BN slabs Sn (n 5 1, 2, 5, and 13) and bulk calculated at the LDA level: comparison between the CPKS and SOS components of the polarizability tensor (in a.u.).

Band gap (eV) aSOS (0) xx aCPKS (0) xx azzSOS (0) azzCPKS (0)

S1

S2

S5

S13

Bulk

4.78 32.2 31.2 8.6 3.7

4.47 32.4 31.3 10.6 4.2

4.39 32.4 31.3 11.9 4.5

4.40 32.4 31.4 12.5 4.6

4.31 32.4 31.4 12.9 10.5

the band structure is concerned, these plasmons peaks rely on a major intensity decrease in the imaginary part. The LCAO approach leads to a broad structured band with two main maxima at 22.3 and 24.7 eV involving the p and r electrons, in very good agreement with the experimental spectrum. At this point, we have checked the ability of our SOS method to reproduce and assign the ELF spectrum of h-BN. 3.2. 2D STRUCTURES: BN SLABS The investigated 2D structures are obtained by simulating the semi-infinite crystal by the slab model. The influence of the thickness of the slab is evaluated by considering four systems with different numbers of atomic layers (Sn with n ¼ 1, 2, 5, and 13 layers) parallel to the (001) face of the h-BN. The computed electric properties are compared to those of the bulk. The four Sn systems correspond to unrelaxed slabs in which the experimental parameters of the bulk have been maintained. To our knowledge, no experimental value of the gap is available for the BN slabs. The LDA method gives direct bandgaps at the K point (see Table III), slightly lower than the vertical energies calculated at the K point for the bulk (4.78 eV). The SOS and CPKS values of the static polarizabilities of the BN slabs and bulk are reported in Table III. Only the values performed at the LDA level are reported. They are quasi the same as those previously calculated using the BLYP method [23]. These results provide indication on the adequacy of the LDA/SOS method. The comparison between the unscreened (SOS) and screened (CPKS) values shows that the LFEs affect only weakly the parallel component (axx ð0Þ). As in Ref. 23, the perpendicular polarizabilities (azz ð0Þ) present very large discrepancy between the SOS and CPKS values. This effect has been previously discussed. The nonconvergence of ð0Þ values to the bulk value with increasthe aCPKS zz

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ing slab thickness is because of the difference between the displacement external field for the slabs and the macroscopic internal field for the bulk in the direction perpendicular to the slab. Their ratio should be equal to the perpendicular component of the dielectric constant of the bulk (ezz ¼2.2) and the ratio between the aCPKS ð0Þ valzz ues for the bulk and S13 is precisely of 2.3. The calculated absorption spectra of S1, S13, and bulk are shown in Figure 3. For more clarity, the spectra of S2 and S5 are not reported as they are similar to S13. The spectra of the BN slabs for the in-plane polarization [see Fig. 3(a)] resemble closely the in-plane polarization spectrum of the h-BN bulk. They are dominated by intense peak structures around 5 and 15 eV corresponding to p–p* and r–r* transitions, respectively. When LFE are neglected on the out-of-plane polarization spectra [see Fig. 3(b)], the peak structures of the Sn (n ¼ 2, 5, 13) in the low-energy range look very similar to the bulk: absorption peaks are observed around 5, 12, and 14 eV. The Clausius-Mossoti correction suppresses almost completely the absorption peaks at 10–15 eV and the BN slabs become transparent below 10 eV for the out-of-plane polarization. These results are in agreement with the dielectric functions calculated at the TD-LDA level by Marinopoulos et al. [17], showing that the Clausius-Mossotti (CM) correction should be able to account for the LFE. They corroborate the fact that: (i) LFE is negligible in the in-plane direction and independent-particle approach should be sufficient to describe the optical properties of BN sheets in the periodic directions and (ii) the LDA calculations are unable to reproduce the shift of oscillator strength induced by LFE below 15 eV for the out-of plane polarization. The ELF functions of S1, S13, and bulk are shown in Figure 4 for both in-plane (ELFxx) and out-of-plane (ELFzz) polarizations. The ELFxx function exhibits two prominent bands for all the


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AB INITIO EELS SPECTRA AND DEPOLARIZATION EFFECTS 3.3. 1D STRUCTURES: SINGLE-WALLED CARBON NANOTUBES (M,0) In a previous work [22], we have used the CPKS method to compute both in-plane and outof-plane static polarizabilities of the (m, 0) SWCN series with m = 3n for m ¼ 7–25. As stated in previous studies, the comparison with SOS values

FIGURE 3. Imaginary part of the dielectric function for BN slabs (S1 and S13) and h-BN computed at the LDA level. (a) In-plane polarization spectrum (b) out-of plane polarization spectrum. The green, blue, and red curves are associated to S1, S13, and bulk, respectively. The solid lines denote the functions including the Clausius-Mossotti correction while the dashed lines refer to noncorrected data. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

systems (slabs and bulk). The small one at 7.5 eV has been attributed to the collective excitation of the p electrons. The large broad band around 26 eV corresponds to plasma oscillations involving both p and r electrons [56]. The main effect of the CM correction on the ELFzz function is to decrease the intensities of the bands beyond 10 eV.

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FIGURE 4. ELF function for BN slabs (S1 and S13) and h-BN computed at the LDA level. (a) In-plane polarization (ELFxx) (b) out-of plane polarization (ELFzz). The green, blue, and red curves are associated to S1, S13, and bulk, respectively. The solid lines denote the functions including the Clausius-Mossotti correction while the dashed lines refer to non corrected data. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]


AMZALLAG ET AL.

FIGURE 5. Imaginary part of the dielectric function for the (10,0) carbon nanotube computed at the LDA level. The black, red, and blue curves are associated to the in-plane, out-of plane, and average spectra, respectively. The solid lines denote the functions including the Clausius-Mossotti correction while the dashed lines refer to noncorrected data. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.] obtained in the same computational conditions clearly confirms that the in-plane static polarizabilities are slightly sensitive to the polarization effect. On the other hand, the unscreened SOS out-of-plane polarizabilities which are found to be four to five times larger than the screened CPKS ones have been corrected to account for the LFE. This correction based on the classical ClausiusMossotti relation applied to a cylindrical volume ~ 2 L) has been fitted on the CPKS val(V ¼ p R ues taken as reference, assuming a linear variation ~ with the square of the effective tube radius R which takes into account the delocalization of the electrons—mainly the p electrons—that participate to the screening. The adjustable parameter ~ R (R being the radius of the nanotube) dR ¼ R represents the radial extension of the p orbitals. In ˚ ) is this work, the same correction (dR ¼ 0.79 A applied to the dynamic polarizability a(x) to calculate the dielectric function e(x). All the calculations were performed at the LDA level and all nanotube geometries were generated by TubeGen [57]. The unit cell volume X needed to deduce e(x) from a(x) is not well defined for isolated nanotubes and one must consider nanotube bundles to compare the theoretical results with the experimental findings. We propose to construct

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the nanotube bundles from one individual (m,0) tube placed in a hexagonal unit cell. The smallest ˚ distance between the walls is fixed to 3.35 A which corresponds to the interlayer distance in graphite. For a deeper analysis of the LFE, the calculated optical absorption spectra for the (10,0) nanotube are reported in Figure 5. The in-plane spectrum exhibits a very intense peak structure at low energy up to 5 eV and also another broader peak structure beyond 10 eV with a pronounced peak at 13.5 eV. As the 1D dimensionality of SWNT gives sharp van Hove peaks in the density of electronic states, the peak structure up to 5 eV is explained from the occurrence of direct interband transitions between these van Hove singularities. For the perpendicular polarization, when LFE are not accounted for, the peak positions at low energy are 1.5 and 4.1 eV. The CM correction applied to the out-of-plane spectrum suppresses these absorption peaks and renders the tube almost transparent below 4 eV. The main effect of the CM is to shift the transitions to higher energies and to decrease the intensity of the peaks, in agreement with TDLDA or Random Phase Approximation (RPA) with LFE studies [16, 17]. As a consequence, the resulting absorption spectrum including CM correction is appreciably modified at low energy up to 10 eV. It is similar to the in-plane spectrum from 10 to 17 eV and to the out-of-plane one beyond 17 eV in agreement with the shift of oscillator strength to higher energies. The absorption spectra of the whole (m,0) series reported in Figure 6 present the same features as those underlined for the (10,0) tube. The calculated ELF functions are reported in Figure 7 for the whole series. All of them share a common feature with prominent plasmon structures: p plasmon around 5.5–6 eV and (p þ r) plasmon around 20–25 eV. The main effects of the CM correction are to decrease the intensity of the p plasmon peaks and to shift the (p þ r) plasmon peaks toward higher energy in connection with the shift of the out-of-plane transitions to higher energies on the absorption spectra. Existing experimental data of the EELS spectra for SWCN have given p and (p þ r) bands at 6 and 21–24 eV, respectively which correspond to our predicted energies [58, 59]. These results show the effect of the nanotube radius on the (p þ r) plasmon peak. The results reported here are representative of individual nanotubes and not of a bundle of interacting nanotubes. This is an important point because the (p þ r) plasmon position is extremely


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FIGURE 6. Imaginary part of the average dielectric function for the (m,0) SWCN series (with m = 3n for m ¼ 7–23) computed at the LDA level. The solid lines denote the functions including the Clausius-Mossotti correction while the dashed lines refer to non corrected data. sensitive to the intertube interactions. This dependence of the plasmon position on the intertube distance induces a strong shift toward higher energies from 22 to 28 eV. As we assume a common spacing between the nanotube, the larger radii give rise to a displacement of the (p þ r) plasmon toward lower energies in agreement with the results of Marinopoulos et al.

4. Conclusions Despite the drastic approximations arising from the DFT ground state and the SOS method, the LCAO-DFT band-structure calculations are able to simulate the ELF spectra quite well, for

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band wide gap materials such as h-BN. Moreover, as both FP-LAPW-GGA and LCAO-GGA methods give very similar dielectric functions, these results show that the optical properties should not be dependent on the nature of the basis set. The interpretation of interband transitions which appear to be easier in the LCAO approach is based on the jointed study of transition energy and oscillator strengths. It enables us to identify and assign the interband transitions in terms of atomic orbitals. This ability to interpret features in the low-energy region combined with good resolution of experimental spectra should be a very powerful tool for the measurement of the electronic structure, in particular to identify states in the bandgap occurring from defects in materials, for example.


AMZALLAG ET AL.

FIGURE 7. ELF function for the (m,0) SWCN series (with m = 3n for m ¼ 7–23) computed at the LDA level. The solid lines denote the functions including the Clausius-Mossotti correction while the dashed lines refer to non corrected data. The other important point concerns the inclusion of the LFEs in the calculation of the dielectric functions and ELF spectra of 2D or 1D periodic systems. We have developed a simple four-pronged strategy. i. Calculate the perpendicular static polarizabilities using CPKS and SOS methods. ii. Determine the effective volume V in the Clausius-Mossotti formula. iii. Apply the Clausius-Mossotti formula to the perpendicular dynamic polarizability iv. Calculate the dielectric function using a wellsuited unit cell volume. This method should be applied to the treatment of extended systems as more sophisticated approaches such as TDDFT are limited by the size of

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the systems that can be treated. The results obtained for the semiconductor zigzag SWCN (m, 0) (with m = 3n for m ¼ 7–25) reproduces the features of the absorption and ELF spectra. In a next step, this strategy will be used to calculate the variations of the dynamic polarizabilities versus imaginary frequencies of SWCN to evaluate the long range dispersion coefficient for a molecule adsorbing on these systems, via the Casimir-Polder formula.

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