Properties of Carbon Nanotubes: An ab Initio Study Using Large

ARTICLE pubs.acs.org/JPCC

Properties of Carbon Nanotubes: An ab Initio Study Using Large Gaussian Basis Sets and Various DFT Functionals Raffaella Demichelis,*,† Yves No€el,‡ Philippe D’Arco,‡ Michel Rerat,§ Claudio M. Zicovich-Wilson,^ and Roberto Dovesiz †

Department of Chemistry, Curtin University, GPO Box U1987, Perth, WA 6845, Australia Institut des Sciences de la Terre de Paris (UMR 7193 UPMC-CNRS), UPMC, Paris Universitas, France § Equipe de Chimie Physique, IPREM UMR5254, Universite de Pau et des Pays de l'Adour, F-64000 Pau, France ^ Facultad de Ciencias, Universidad Autonoma del Estado de Morelos, Av. Universidad 1001, Col. Chamilpa, 62209 Cuernavaca, Morelos, Mexico z Dipartimento di Chimica IFM, Universita di Torino and NIS -Nanostructured Interfaces and Surfaces - Centre of Excellence, Via P. Giuria 7, 10125 Torino, Italy ‡

bS Supporting Information ABSTRACT: The structural, electronic, dielectric, and elastic properties of zigzag and armchair single-walled carbon nanotubes are investigated at different DFT levels (LDA, GGA, hybrids) with Gaussian type basis sets of increasing size (from 3-21G to 6-1111G(2d,f)). The longitudinal and transverse polarizabilities are evaluated by using the Coupled Perturbed Hartree
Fock and Kohn
Sham computational schemes, which take into account the orbital relaxation through a selfconsistent scheme. It is shown that the difference between the frequently adopted SOS (sum over states, uncoupled) and the fully coupled results is far from being negligible and varies as a function of the tensor component and the adopted functional. Helical symmetry is fully exploited. This allows simulation of tubes larger (up to 140 atoms in the unit cell) than in previous studies by using extended basis sets and severe computational conditions. All the 12 functionals considered here provide similar results for the structural and the elastic properties and for the relative stability among nanotubes and with respect to graphene. On the contrary, the stability with respect to diamond, which has a quite different density than that of nanotubes, sensitively depends on the adopted functional. The band gap and the longitudinal polarizability are strongly dependent on the level of approximation: hybrid functionals provide the least deviation from experimental data. In general, data obtained for (n, n), (3n, 0), (3n þ 1, 0), and (3n þ 2, 0) rolling directions approach the slab limit for large radii following four distinct trends.

1. INTRODUCTION The performance of 12 functionals, corresponding to three different levels of approximation of the exchange-correlation term (EXC)1,2 within the density functional theory (DFT),3
5 i.e., local density approximation (LDA), generalized gradient approximation (GGA, including newly developed EXC functionals parametrized for solids6
8) and hybrid methods, and the effect of basis sets of increasing size, ranging from 3-21G (nine atomic orbitals per C atom, AOs) to 6-1111(2d,f) (34 AOs), in the description of the electronic, structural, dielectric, and elastic properties of a subset of zigzag and armchair singlewalled (SW) carbon nanotubes (CNTs) are explored. The relative stability of SWCNTs with respect to two other carbon allotropes, i.e., diamond and graphene, is investigated, too. r 2011 American Chemical Society

The calculation of the considered properties is then extended to the series of the (n, 0)-zigzag and (n, n)-armchair SWCNTs with n ranging from 8 to 35 and from 5 to 24, respectively, corresponding to tubes with radii up to 16 Å (up to 140 atoms in the unit cell), using the 6-1111G(d) basis set and the B3PW (B3LYP for polarizabilities) functional. a 1 and B a 2 lattice vectors of The n1 and n2 indices multiply the B the monolayer (graphene in the present case) and are sufficient to define the rolling direction of the nanotube, its radius, and its chirality (see Figure 1 in refs 9
14 and the interactive animations Received: November 9, 2010 Revised: March 24, 2011 Published: April 18, 2011 8876

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The Journal of Physical Chemistry C at www.crystal.unito.it/tutorials/nanotube/). Information about the metallic, quasimetallic, or semiconducting character of a carbon nanotube is implicit in the adopted notation. As discussed in refs 15
18, a nanotube is expected to have a null gap when the sum n1 þ n2 is a multiple of 3, the exception being (n, 0)-zigzag nanotubes, characterized by a very narrow, but non null, band gap. There are many reasons for undertaking this study. Apart from the purely scientific interest aroused by this class of materials, which allows investigation of the fundamental physics of 1Dsystems and exploration of nano and molecular electronics,10 CNTs are promising for many potential applications (from nanodevices19 to medicine20 and energy storage21
23), and ab initio methods are likely to represent a powerful tool for projecting and interpreting experiments. A systematic comparison of the various functionals for periodic systems is still limited to simple ionic, covalent, and metallic solids with one or two atoms in the unit cell.6,24
28 The present analysis, carried out only on a specific class of solids (quasi-1D covalent systems with tubular structures), is part of a larger project aimed to test the performance of DFT functionals in predicting structural, energetic, vibrational, electronic, dielectric, and elastic properties of various classes of solids (H-free and H-containing aluminosilicates,29,30 molecular solids31). Systematic comparisons of the performance of LDA, GGA, meta-GGA, and hybrid functionals in describing structural and electronic properties of SWCNTs were presented in refs 32
37. In many cases, however, the calculations were carried out with small basis sets, and geometry optimization of the structure was avoided because of the high computational cost of nanotube ab initio simulation. The exploitation of the high9,38
41 rototranslational nanotube symmetry, recently implemented in the CRYSTAL09 code,9,41,42 enables the geometry optimization of all the considered SWCNTs with large basis sets and a reduced CPU time.9,41,43 In particular, the analytical coupled perturbed Hartree
Fock (CPHF) or Kohn
Sham (CPKS) method implemented in CRYSTAL09 for calculating the response of periodic and nonperiodic systems to an electric field44
46 allows us to obtain very efficiently accurate values of the transverse and longitudinal polarizability components of the nanotube, improving upon the results presented in the literature, obtained at the SOS (sum over states) or finite field level with smaller basis sets and often without geometry optimization.34,35,37,47,48 The paper is structured as follows. Section 2 reports the computational details. In section 3 the results are shown and commented and section 4 contains the main conclusions.

2. COMPUTATIONAL DETAILS Calculations were performed with the CRYSTAL09 periodic ab initio code,42 which exploits the helical symmetry for automatic generation of the nanotube structure starting from a twodimensional layer, for mono- and bielectronic integral calculation and for Fock matrix diagonalization.9,43 All-electron Gaussian type basis sets of increasing size, from 3-21G49 to 6
1111G(2d,f), were used (3-21G, 6-21G, and 6
21G(d) from ref 49, 6-31G(d) and 6-1111G from ref 50, 6-1111G(d), 6-1111G(2d), and 6-1111G(2d,f) obtained by adding one d, Rd1 = 0.8325 bohr
2, two d, Rd1 = 0.6786, and Rd2 = 1.02 bohr
2, and a further f, Rf = 0.8 bohr
2, orbitals to 6-1111G50). Twelve DFT functionals, corresponding to three

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levels of approximation of the exchange-correlation contribution (LDA: SVWN51,52 and SPWLSD;51,53 GGA: PBE,54 PW91,55 PBEsol,6 SOGGA-8 and WC-7PBE; hybrids: B1WC,56 WC1LYP,29 B3PW,57 B3LY,P58,59 and PBE060), were tested. Further details on the basis sets, the list of functionals, their features, and references are reported as Supporting Information. In CRYSTAL, the level of accuracy in evaluating the Coulomb and Hartree
Fock exchange series is controlled by five parameters,42 for which the 8 8 8 8 18 values were used. The SCF convergence on energy was set to 10
8 Eh. In the adopted code, the DFT exchange-correlation contribution is evaluated by numerical integration over the unit cell volume. Radial and angular points of the grid are generated through Gauss
Legendre radial quadrature and Lebedev two-dimensional angular point distributions. A (75 974)p grid was used,42 corresponding to a pruned grid with 75 radial and 974 angular points. The accuracy in the integration can be estimated by the error of the integrated electronic charge density in the unit cell (Δe = 9  10
4|e| on a total of 576 electrons in the (24, 0) SWCNT B3LYP and 6-1111G(d). Structures were optimized by use of analytical energy gradients with respect to atomic coordinates and unit-cell parameters,61
63 within a quasi-Newton scheme combined with the Broyden
Fletcher
Goldfarb
Shanno algorithm for Hessian updating.64
67 Convergence was checked on both gradient components and nuclear displacements, for which the default values42 were chosen. The second derivative of the energy density (E/a, where a is the nanotube lattice parameter) with respect to the axial strain (εx, i.e., the component along the periodic direction in our convention), was evaluated numerically from analytical gradients. The internal coordinates were reoptimized for each applied strain. Point symmetry was exploited to reduce the number of required deformations.68 Longitudinal (Rxx, along the periodic direction) and transverse (Ryy = Rzz) polarizabilities were evaluated with the CPHF/KS44
46 method. The reciprocal space was sampled according to a regular sublattice with shrinking factor42 (IS) 20 for graphene (44 independent kB vectors in the irreducible part of the Brillouin zone) and 8 for diamond (29 independent kB vectors). The choice of the kB point sampling is crucial for a good description of SWCNTs. The properties of small- and largeradius zigzag and armchair nanotubes ((10, 0), (20, 0), (5, 5), (13, 13)) and of a quasimetallic zigzag nanotube, (12, 0), were calculated with 6, 21, and 41 kB vectors (corresponding to IS = 10, 40, 80, respectively) and reported in Tables 1 and 2. The electronic energy difference between IS = 10 and IS = 40 is on the order of 10
4
10
5 Eh per two C atoms, and it reduces to 10
6 or less between IS = 40 and IS = 80. However, as shown in Table 1, geometry and elastic constants of metallic (armchair) and quasiconducting (12, 0) SWCNTs are accurate enough only at IS = 40. The dependence on the number of kB vectors is dramatic for the longitudinal polarizability of semiconducting zigzag nanotubes: Rxx of (10, 0) and (20, 0) SWCNTs decreases by about 30% and 45%, respectively, when the number of kB vectors, nBk , increases from 6 to 21. The value of the polarizability (a secondorder perturbation energy with respect to the field) is very sensitive to the gap particularly when the latter is small such as in carbon nanotubes. The quality of the description of the reciprocal space around the region where the gap is small is, then, very important. The difference between nBk = 6 and nBk = 21 8877

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Table 1. Lattice Parameter (a, Å), Radius (r, Å), Bond Distance (C
C, Å), Elastic Constant (VCxx, eV per C atom), Longitudinal and Transverse Polarizabilities (rxx and ryy, respectively, Å3 per C atom) of Five SWCNTs Calculated with 41, 21, and 6 kB Points in the Irreducible Part of the Brillouin Zone (B3LYP, 6-1111G(d) basis set)a (10, 0)

nkB

6

a 4.2647 r 3.9521 C
C 1.4201 1.4270 ΔE 4.0  10
5 VCxx 57.70 Rxx 20.34 Ryy 1.02

21 4.2643 3.9524 1.4197 1.4271 2.9  10
5 57.78 14.40 1.01

(12, 0) 41 4.2644 3.9524 1.4198 1.4271 0 57.78 14.38 1.01

6 4.2688 4.7265 1.4234 1.4241 9.3  10
5 52.80
1.13

21 4.2626 4.7335 1.4184 1.4266
1.0  10
5 55.72
1.13

(20, 0) 41 4.2621 4.7338 1.4185 1.4265 0 55.82
1.12

6 4.2641 7.8569 1.4204 1.4235 1.6  10
5 58.96 37.40 1.69

(5, 5)

21

41

4.2643 7.8564 1.4205 1.4235
1.4  10
8 58.60 20.72 1.62

4.2643 7.8564 1.4205 1.4235 0
20.61 1.62

6

(13, 13)

21

2.4575 3.4387 1.4225 1.4320
2.0  10
5 59.39
0.93

41

2.4621 3.4320 1.4256 1.4258 6.4  10
6 57.11
0.94

2.4614 3.4331 1.4250 1.4269 0 57.19
0.94

6 2.4612 8.8440 1.4215 1.4240
8.8  10
5 59.20
1.61

21 2.4621 8.8410 1.4221 1.4228 8.2  10
8 58.29
1.74

41 2.4624 8.8399 1.4223 1.4224 0 58.34
1.75

a ΔE is the energy difference (Eh per two C atoms) with respect to the calculation performed with n kB = 41 points. The longitudinal polarizability of quasimetallic and metallic nanotubes, (12, 0), (5, 5) and (13, 13), tends to infinity and cannot be calculated with CPKS(HF).

on transverse polarizability is much smaller: about 1% for (10, 0), (12, 0) and (5, 5) SWCNTs, but increases for larger radii (to 4% and 9% in (20, 0) and (13, 13) SWCNTs, respectively). Table 2 shows that the accuracy of the optimization is not so much dependent on the shrinking factor. Polarizabilities of (20, 0) and (13, 13) SWCNTs were calculated with increasing IS at a fix geometry. It turns out that differences between polarizabilities calculated at IS = 80 for structures optimized at IS = 10 and IS = 80 are smaller than 0.01 Å3 per C. According to these considerations, structural and elastic properties of semiconducting (conducting and quasiconducting) SWCNTs were calculated using 6 (41) kB points, whereas 41 kB points were used to evaluate longitudinal and transverse polarizabilities of all SWCNTs.

Table 2. Polarizabilities of (20, 0) and (13, 13) SWCNTs as a Function of the kB Points in the SCF Cycle during the Optimization (OPT) and in the Subsequent CPHF Calculationa nkB OPT 6

(20, 0)

(13, 13)

CPHF

Rxx

Ryy

Ryy

6

37.40

1.69

1.61

21

20.65

1.62

1.74

41

20.55

1.62

1.75

61

20.55

1.62




21

6

20.72

1.62

1.74

41

21 41

20.61 20.61

1.62 1.62

1.75 1.75

3. RESULTS AND DISCUSSION

a

A. Effect of the Basis Set. The convergence of a few properties of the (14, 0) SWCNT and graphene (geometry, band gap, elastic constant, and polarizabilities of the optimized structures) with the basis set size is documented in Table 3. The hybrid B3LYP Hamiltonian was used; however, similar convergence is obtained with the other DFT functionals. Obviously, a better description of the core electrons has a strong influence on the total electronic energy (E): when passing from 3-21G to 6-21G, E decreases by 350 mEh for graphene. The total energy of graphene decreases by about 24 mEh when passing from 6-21G to 6-21G(d), and 50 mEh when passing from 6-21G(d) to 6-31G(d). Differences between 6-31G(d) and 6-1111G(d) are on the order of a few mEh, as well as those obtained by adding one more d and an f functions to the 6-1111G(d) basis sets. Similar comments are valid also for the nanotubes, so that the energy difference between graphene and the (14, 0) SWCNT is nearly constant. This is due to the similarity of graphene and nanotubes in electronic and geometrical structure, the only difference being related to the rolling-up effect. The a lattice parameter is chosen as a reference for the discussion of the influence of the basis set on the structure. Table 3 shows that reasonable results are obtained also with poor basis sets. Convergence to less than 10
2 Å, however, requires at least the 6-31G(d) basis set. The elastic constants oscillate between 58.7 and 59.7 eV per C atom and seem to be independent of the choice of the basis set. Band gaps are affected by the addition of d and sp polarization shells, whereas they are not influenced by more than 10
3 eV by the addition of a second d and an f shells. As reported in previous

studies,69 the latter are quite crucial for a correct description of dielectric properties. In this case, however, good results for longitudinal and transverse polarizabilities are obtained with the 6-1111G(d) basis set (convergence on the first digit in Å3 per C atom). B. Performance of the Various DFT Functionals in Describing the Nanotube Properties. Structural, electronic, dielectric, and elastic properties of two SWCNTs, (10, 0) and (15, 0), were calculated with 12 DFT functionals by using the 6-1111G(d) basis set. Geometry optimization of graphene and diamond with the same functionals and basis set were also performed, in order to compute the relative stability of the various carbon allotropes. The experimental results for geometries (radius of the tube) and band gaps reported in refs 70 and 16 are used for comparison. To the authors’ knowledge, experimental results for the relative stability, polarizability, and elastic constants of individual nanotubes are not available. Results are summarized in Table 4. HF results (100% exact exchange, no correlation) are shown as an upper limit for hybrid functionals (the most external sp exponent, in HF calculations, was increased from 0.19 (see ref 50) to 0.29 bohr
2). The comparison between calculated and experimental tube radii can only be qualitative, as the reference experiments, performed with spectroscopic70 and microscopic16 techniques, are affected by large uncertainty ((0.3 Å in ref 16 on the order of the differences between calculated and experimental parameters). Differences among radii calculated at various levels of approximations, however,

Symbols and units as in Table 1, B3LYP functional, 6-1111G(d) basis set.

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Table 3. Effect of the Basis Set on the Various Properties of Graphene and (14, 0) SWCNT (B3LYP is used)a graphene

(14, 0)

E

basis set

a

ΔE

E

a

Rxx

Eg

Ryy

VCxx

3-21G


75.76695

2.468


75.76183

13.43

4.271

0.988

14.63

1.22

59.68

6-21G


76.11744

2.470


76.11222

13.71

4.275

0.972

14.76

1.22

59.67

6-21G(d)


76.14172

2.470


76.13656

13.55

4.276

0.933

15.53

1.24

58.89

6-31G(d)


76.19146

2.463


76.18607

14.15

4.264

0.953

15.15

1.24

59.05

6-1111G(d)


76.19416

2.463


76.18882

14.02

4.263

0.945

15.25

1.26

59.11

6-1111G(2d)


76.19510

2.461


76.18982

13.89

4.260

0.943

15.33

1.26

58.76

6-1111G(2d,f)


76.19759

2.460


76.19190

14.94

4.258

0.944

15.29

1.26

58.68

E: electronic energy [Eh per two C atoms]; ΔE: energy difference with respect to graphene [kJ/mol per two C atoms]; a: lattice parameter [Å]; Eg: band gap [eV]; VCxx: elastic constant [eV per C atom]; R: polarizability along (x) and perpendicular to the tube axis (y = z) [Å3 per C atom]. a

Table 4. Effect of the Considered Functionals on Various Properties of the (10, 0) and (15, 0) SWCNTs (CPHF/KS scheme currently inplemented for HF and four DFT functionals only in CRYSTAL)a a graphene

r (10, 0)

Rxx

Eg (15, 0)

(10, 0)

(15, 0)

(10, 0)

Ryy (10, 0)

VCxx (15, 0)

(10, 0)

SVWN

2.4496

3.929

5.902

0.793

0.017

17.59

1.02

1.32

56.95

SPWLSD

2.4500

3.930

5.823

0.793

0.017










56.88

PBE

2.4712

3.965

5.882

0.779

0.018

18.17

1.05

1.36

56.28

PW91 PBEsol

2.4690 2.4640

3.962 3.953

5.922 5.869

0.776 0.789

0.020 0.015













56.44 55.93

SOGGA

2.4628

3.950

5.877

0.796

0.013










55.99

WC-PBE

2.4622

3.950

5.889

0.787

0.017










56.00

B1WC

2.4528

3.935

5.904

1.042

0.031










58.33

WC1LYP

2.4595

3.947

5.917

1.013

0.039










57.89

B3PW

2.4572

3.942

5.901

1.097

0.038










58.71

B3LYP

2.4625

3.952

5.870

1.074

0.043

14.27

1.02

1.30

58.47

PBE0 HF

2.4545 2.4316

3.937 3.899

5.895 5.900

1.188 2.637

0.040 0.243

13.04 8.04

1.00 0.86

1.28 1.08

57.92 67.05

expt

2.458990

3.9770

5.816

1.07370

0.02916

(0.0005




(0.3




(0.004

a

The 6-1111G(d) basis set was used. The tube radius is indicated as r, in angstroms, the other symbols and units as in Table 3. The experimental values, when available, are shown at the bottom line.

are on the order of 10
2 Å only, suggesting that all of them are adequate for the structural description of SWCNTs. This is confirmed by the comparison of calculated and experimental lattice parameters of graphene (a in Table 4). GGA and hybrid functionals provide lattice parameters differing by less than 0.01 Å from experiment. As expected on the basis of previous studies,29,56,71 PBE and PW91 tend to overestimate the geometrical parameters, whereas LDA underestimates them. The recently proposed GGA functionals (PBEsol, WC-PBE, and SOGGA) improve the performance with respect to PBE and PW91, providing structural results comparable to those obtained with hybrid functionals. The five hybrid functionals give, in general, a better description of the structure, WC1LYP, B3PW, and PBE0 providing the least deviation from experiments. Pure DFT methods, due to the discontinuity of the exchangecorrelation Kohn
Sham potential,72,73 are known to systematically underestimate the band gap (see Table 4), and the inclusion of a fraction of exact HF exchange (hybrid methods) is known to redress this lack, as discussed in refs 34 and 36. The five tested hybrid functionals provide almost equally accurate Eg values for the

nanotubes. In the case of the (10, 0) SWCNT, two hybrid functionals slightly underestimate (WC1LYP, B1WC) and three slightly overestimate (B3LYP, B3PW, PBE0) Eg with respect to the experimental data, PBE0 giving the largest deviation (þ0.115 eV). As expected from previous studies on metallic systems,74,75 hybrid functionals are not the best choice to describe the (15, 0) SWCNT band gap (very small16), whereas pure DFT functionals provide more reasonable results. Polarizabilities were calculated with SVWN, PBE, B3LYP, PBE0, and HF only, as the scheme for the other functionals is not yet available in the CRYSTAL code. Being the polarizability related to the band gap, particularly for the more sensitive longitudinal component, functionals providing good band gap description (B3LYP,PBE0) are expected to give more reasonable results for polarizabilities. Elastic constants are nearly independent of the choice of the DFT functional. Hybrids give values larger by about 2 eV per C atom than those of pure DFT methods. The relative stability (ΔE) between graphene and carbon nanotubes is only marginally affected by the choice of the 8879

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Figure 1. Electronic energy difference (ΔE, kJ/mol per two C atoms) of diamond and two SWCNTs with respect to graphene for the various DFT functionals and HF (6-1111G(d) basis set). Data shown in this figure are tabulated in the Supporting Information.

Figure 2. C
C distances in SWCNTs as a function of the tube radius (r, Å). Δ(C
C) [Å] is the difference with respect to the graphene limit (indicated with a line at Δ(C
C) = 0). The three C
C bonds present in the structures are represented with squares, diamonds, and triangles (B3PW, 6-1111G(d)).

functional, due to the similarity of their structure: Figure 1 shows that LDA and GGA functionals provide very similar results, about 1
3 kJ/mol lower than those obtained with hybrids. On the contrary, ΔE between graphene and diamond appears sensitively dependent on the choice of the functional: the denser phase (diamond) is stabilized by those functionals underestimating the volume and vice versa. This confirms a previous observation that DFT functionals tend to stabilize or destabilize polymorphs on the basis of their density difference.30 As 2D and quasi-1D structures are considered, the term “density” is not the proper one; however, we use it as a measure of the empty space among the atoms in the structure. Graphene and the SWCNTs are then considered to have quite the same “density”, whereas diamond is much denser (diamond: 3.513 g cm
3; graphite: 2.2 g cm
3, from ref 76). Results presented in this paragraph are in general agreement with previous ab initio DFT studies.32
37,77,78 Differences are probably due to two main reasons: in the previous works, (1)

Figure 3. Orientation of C
C bonds with respect to the zigzag and armchair rolling directions. Black bullets represent the C atoms.

geometry was often not optimized, or optimized with poor basis sets; (2) when large unit cell nanotubes were studied, also properties were often calculated with poor basis sets. 8880

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Figure 4. Energy difference with respect to graphene (ΔE) and relaxation energy (δE) in kJ/mol per two C atoms for the zigzag and armchair SWCNTs as a function of the tube radius (B3PW, 6-1111G(d)). Data used for this figure are tabulated in the Supporting Information.

Figure 5. Band gap of (n, 0) nanotubes (B3PW, PBE, and experimental16,70]) as a function of the tube radius (B3PW). Calculations are performed with the 6-1111G(d) basis set. Symbols and units as in Tables 4 and 3. Data used for this figure are tabulated in the Supporting Information.

C. Structure, Stability, Band Gap, and Elastic Properties of Zigzag and Armchair SWCNTs. According to the comments in

the previous paragraphs, the 6-1111G(d) basis set and the hybrid B3PW functional were adopted for a systematic study of the (n, 0) zigzag (n ranging from 8 to 35) and (n, n) armchair (n from 5 to 24) SWCNTs. Polarizabilities were calculated with the same basis set and B3LYP functional (the B3PW scheme for polarizabilities is not yet available in CRYSTAL). As hybrids fail in describing small band gaps, results obtained for this parameter with PBE are also presented. Experimental zigzag tube radii are available in refs 70 and 79 and are in good agreement with the present calculated parameters. The same papers provide experimental measurements of band gaps, which will be compared to calculated values later on. 1. Geometry. In Figure 2, C
C bond lengths in the various SWCNTs and in graphene are compared (Δ(C
C): C
C

lengths in nanotubes minus C
C length in graphene, both calculated). The symmetry equivalence of the three first neighbors of graphene is lost in SWCNTs; C
C distances increase or reduce, depending on the nanotube radius and chirality. Regarding the former, the larger the radius, the more the structure of the nanotube becomes close to that of graphene. Concerning the latter, zigzag and armchair nanotubes exhibit a slightly different behavior, due to the different strain affecting C
C bonds in the curved structure (see Figure 3). Figure 2a shows Δ(C
C) for zigzag nanotubes as a function of the tube radius: distances appear as a dispersed set of points converging to the graphene limit. In particular, two out of three distances are larger than in graphene, and one is shorter. Figure 3 helps to understand this behavior as it shows that in zigzag SWCNTs two out of three C
C bonds have a component along 8881

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Figure 6. Longitudinal and transverse polarizabilities normalized to one C atom (Rxx and Ryy, respectively) as a function of the reciprocal of the band gap, a and c, and of the radius, b and d (B3LYP, 6-1111G(d)). Symbols and units as in Tables 4 and 3. Irregularities of the (3n, 0) curve in panel c are due to the extremely small band gap. Data used for this figure are tabulated in the Supporting Information.

Figure 7. Rxx/a [Å2] as a function of r/Eg [Å/eV].

the rolling direction (and then they become larger in the curved structure), and one is perpendicular to it (the strain does not affect this bond; it becomes shorter as a consequence of the enlargement of the other two). Δ(C
C) results represented in Figure 2a are shown, for a given C
C bond, to belong to three different curves describing the (3n, 0), (3n þ 1, 0), and (3n þ 2, 0) cases. In armchair SWCNTs, (Figure 2b) the rolling vector is oriented along one of the three C
C bonds (see Figure 3), which turns out to be the most affected once graphene is rolled up. The other two bonds, having a component also along the tube axis, undergo minor changes. These results are in agreement with previous studies.80 In the model applied here only the rototranslational symmetry operators are considered. However, it is known38,40 that zigzag

and armchair SWCNTs have a higher symmetry, as vertical and horizontal (σv and σh, perpendicular and parallel to the circumference plane, respectively) mirrors are present. In particular, σh mirrors “cut” the C
C bonds along the arrows shown in Figure 3. These mirrors seem to be maintained during the optimization process, even if not applied in our simulation, as the σh-related C
C distances are very similar in the various tubes (the superposed triangles and squares in Figure 2 correspond to symmetry equivalent carbon atoms) 2. Relative Stability. Two parameters are chosen for the present analysis: ΔE, the energy difference between relaxed SWCNTs and graphene, and δE, the relaxation energy after rigid rolling (energy difference between unrelaxed and relaxed SWCNT). They are represented in Figure 4. Roughly speaking, both ΔE and δE decrease regularly as the tube radius increases, as expected and in line with previous findings.78,81
84 Also in this case, as already discussed in ref 9, a finer analysis shows that δE data belong to four different curves, describing the (n, n), (3n, 0), (3n þ 1, 0), and (3n þ 2, 0) cases. However, differences are quite small and tend to disappear with increasing nanotube diameter. 3. Band Gap. Figure 5 reports calculated (with PBE and B3PW functionals) and experimentally measured band gap values as a function of the tube radius. A good agreement is observed for (3n þ 1, 0) and (3n þ 2, 0) SWCNTs when simulated with the B3PW hybrid functional. PBE provides much less satisfactory band gaps (too small). The situation is different for the tubes of the (3n, 0) family, whose band gap is extremely small (less than 0.1 eV). In this case, PBE provides quite accurate band gaps, whereas the B3PW ones tend to be too large. The band gap decreases regularly as the tube radius increases, with (3n, 0) nanotubes belonging to a different curve at lower values. Eg data for (3n þ 1, 0) and (3n þ 2, 0) nanotubes belong to two different curves in the explored range of radii. 8882

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Figure 8. Elastic constants of zigzag and armchair SWCNTs as a function of the radius (B3PW, 6-1111G(d)). The dashed line represents the graphene limit. Symbols and units as in Tables 4 and 3. Data used for this figure are tabulated in the Supporting Information.

The calculated Eg of (8, 0) is smaller than expected also when hybrid functionals are used. This should be due to interactions across the tube that become more important at small radii.85 4. Polarizability Tensor. The two nonequivalent components of the polarizability tensor are discussed separately in the following. The notation Rii/C and Rii/a (ii stands for xx, yy, or zz) will be used to indicate the polarizability per carbon atom (in Å3) and per cell parameter length (a, in Å2), respectively. a. Longitudinal Component. Metallic (armchair) and narrow-gap ((3n, 0), with Eg < 0.1 eV) SWCNTs cannot be treated with the perturbative approach applied here. For the other nanotubes, a linear relation between Rxx/C and the inverse of the gap is observed (see Figure 6a). A similar linear behavior, also found by Brothers et al.34,37 in their study of CNT with r < 10 Å at the SOS level, is observed between Rxx/C and r, the nanotube radius (see Figure 6b). This obviously implies that r and 1/Eg are linearly related, too. A r/Eg2 law was proposed for Rxx/a about 15 years ago by Benedict et al.47 (tight-binding model) and confirmed recently by Kozinsky et al.47 (Density functional perturbation theory, PBE functional) for tubes with r < 10 Å . This dependence is observed neither here, nor by Brothers et al.34 (see their Figure 2). The failure of the r/Eg2 law is not surprising: having shown that Rxx/C is linear in 1/Eg, as Rxx/a is obtained from Rxx/C by multiplying by r, then Rxx/a is expected to obey a r/Eg (instead of r/Eg2) law, as confirmed by the fit in Figure 7. Due to the linear relation between r and 1/Eg discussed above, the r/Eg law can also be formulated as r2 or (r/ Eg2)2/3. As a cross-check, our Rxx/a data have been fitted vs (r/Eg2)w providing w = 0.63 (quite close to 2/3 = 0.66), in agreement with the value obtained in a recent work86 (w = 0.64) where tubes with r < 10 Å were considered. The above discussion applies to large r values, and not to the very small ones, as the tube at this limit collapses with discontinuity. For a general family of nanotubes, when r goes to infinity, Eg becomes constant and equal to that of the corresponding flat structure (monolayer or slab). At this limit, when the curvature becomes negligible, Rxx/a should depend only on the number of atoms on the circumference, and then the polarizability per atom is expected to be constant. However, carbon nanotubes are a special case, because when r increases, the gap tends to zero and Rxx/C increases linearly in the interval explored here. At the r f

¥ limit, Rxx/C will approach the parallel component of the polarizability of graphene, which is infinite. Relaxation (coupling) effects on Rxx are very small. They can be calculated as the ratio of the SOS and the CPHF (KS) polarizabilities, that increases from 0.91 for the (10, 0) nanotube to 0.96 for the (23, 0) one (see Supporting Information). b. Transverse Component. In the explored radius range (r up to 16.3 Å) also Ryy/C is perfectly linear in r (Figure 6d). It is also linear in 1/Eg (in this case the numerical noise is larger, see Figure 6c), with very different slopes according to the more or less metallic character of the nanotube family. This means that Ryy/a increases with r2 in the interval explored here, confirming the results of previous works for tubes with r up to 10 Å.35,47,48,86 Ryy wave function relaxation effects, evaluated as the ratio between SOS and CPKS polarizability values, are very large47 and vary from 3.8 to 4.1, depending on the rolling direction (see Supporting Information). Note that for carbon nanotubes with a large radius the ratio RSOS/RCPKS remains nearly constant and close to 4 (see Supporting Information). By using this information within the formula proposed by Benedict et al.47 relating SOS and CPKS polarizabilities per length: RCPKS ¼ yy

RSOS yy

~2 1 þ 2RSOS yy =R

ð1Þ

~ =r for large r, one obtains that RSOS and imposing R yy /a increases 2 /a as (3/8)r2. A best fit of this latter as (3/2)r , and then RCPKS yy component with respect to r using the data of Table 7 of the Supporting Information leads to a coefficient equal to 0.368 (very close to 3/8 = 0.375). 5. Elastic Properties. Results obtained for VCxx (Figure 8) are in good agreement with those calculated in previous works.77,87
89 No strong correlation is observed between either radius or chirality and the elastic constants, differences being small and within the limit of accuracy of the calculation. Values similar to graphene (around 60 eV, see refs 77 and 88) are obtained for all nanotubes. The exception is the case of narrow-gap (3n, 0) SWCNTs, exhibiting VCxx slightly smaller than for the other tubes. The value of VCxx increases very slowly with the r and seems to approach the graphene limit. However, the relatively large numerical noise does not permit a clear extrapolation to that limit. 8883

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4. CONCLUSIONS This work provides a general overview of the performance of some of the many DFT functionals proposed in the literature in describing zigzag and armchair SWCNTs. The exploitation of the helical symmetry, which strongly reduces the computational cost, allows the simulation of nanotubes containing up to 140 atoms (r up to 16.3 Å) with rich basis sets and severe computational conditions. All the considered properties are well reproduced with a 6-1111G(d) basis set (6-1111G(2d) for the longitudinal polarizability). The results are only marginally affected by the addition of f-type polarization functions to the adopted basis set. In general, all the considered functionals provide a reasonable description of SWCNT properties, with hybrid functionals giving the best performance. The electronic structure of narrow-gap nanotubes, however, is described with higher accuracy by pure DFT methods. The implicit tendency of DFT functionals to stabilize polymorphs on the basis of their density difference is confirmed for diamond
graphene
SWCNTs: functionals overestimating the volumes tend to stabilize the less dense polymorphs and vice versa. The systematic investigation of zigzag and armchair SWCNT series shows the following: • Data obtained for (n, n), (3n, 0), (3n þ 1, 0), and (3n þ 2, 0) SWCNTs approach the slab limit with four distinct trends (particularly evident at r < 10 Å). • The performance of density functionals in the description of electronic properties (in this case band gap and polarizability) depends on the band gap and the radius: metallic and large-gap nanotubes are better described with GGA and hybrid functionals, respectively. • The linear dependence of the polarizability per C atom (both longitudinal and transverse) on both 1/Eg and r is confirmed for tubes with r larger than those investigated in previous studies. The comparison between the polarizability calculated with the CPHF scheme and the SOS confirms that wave function relaxation effects are extremely important for the transverse component, nearly negligible for the parallel component. These results should be considered an improvement of those already present in the literature, in general agreement with ours, as richer basis sets and better performing DFT functionals are applied in a systematic way to the simulation of tubes larger than in previous studies. The effect of accurate geometry optimization and wave function relaxation on polarizabilities is also explored. ’ ASSOCIATED CONTENT

bS

Supporting Information. Additional information as noted in the text. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: raff[email protected].

’ ACKNOWLEDGMENT Roberto Dovesi and Raffaella Demichelis acknowledges Italian MURST for financial support (Cofin07 Project 200755ZKR3_004, coordinated by Professor C. Giacovazzo) and Cineca supercomputing

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center (ISCRA Award no. HP10AC4ZGA.2010) for computing resources.

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