NANO LETTERS
Optical Transitions in Metallic Single-Walled Carbon Nanotubes
2005 Vol. 5, No. 9 1830-1833
Vero´nica Barone, Juan E. Peralta, and Gustavo E. Scuseria* Department of Chemistry, Rice UniVersity, Houston, Texas 77005-1892 Received May 24, 2005; Revised Manuscript Received July 31, 2005
ABSTRACT We report vertical electronic transitions of 20 metallic single-walled carbon nanotubes calculated as band energy differences from Kohn− Sham density functional theory. Our first-order transitions (E11) calculated with hybrid functionals (containing a portion of exact exchange) are in very good agreement with available experimental data. Recently, we have reported similar agreement between experiment and theory for semiconducting tubes. We find that the trigonal warping splitting in the band structure of metallic tubes is about 1.5 to 2 times larger than that reported previously.
Characterization methods for single-walled carbon nanotubes (SWNTs) based on optical absorption have attracted much attention during the past few years. Experiments based on photoluminescence,1,2 resonant Raman spectroscopy,3,4,5 and Rayleigh scattering6 have been reported in which optical transitions are obtained as fingerprints of a certain (n, m) nanotube. In pioneering work, Bachilo et al.1 presented the (n, m) assignment and transition energies Eii (i < 3) of 33 semiconducting SWNTs suspended in aqueous solution. These authors also presented an empirical fit to Eii as a function of the diameter and chiral angle of the tube. Some of the extrapolated values corresponding to the smallest tubes along with the spectrum of narrow SWNTs grown in zeolite channels7 were utilized in several other papers to test the accuracy of theoretical approximations.8-15 In a recent paper, we have presented a systematic study of the performance of density functional theory (DFT) for predicting optical vertical transitions in semiconducting SWNTs.16 We have shown that excellent agreement between theory and experiment can be achieved using hybrid functionals, that is, those containing a portion of Hartree-Fock (HF) type exchange. In particular, the hybrid metageneralized gradient approximation, TPSSh,17,18 yields mean absolute errors of 0.024 eV for E11 and 0.065 eV for E22 in a set of 21 semiconducting SWNTs when compared to experimental data. Functionals based on the local spin density approximation (LSDA) and generalized gradient approximation (GGA) systematically underestimate both sets of transitions by approximately 0.3-0.4 eV. It is natural, therefore, to wonder how the single-particle approach based on hybrid functionals such as TPSSh18 or HSE19,20 performs for optical transitions of metallic SWNTs. The answer to this question is by no means obvious because * Corresponding author. E-mail:
[email protected]. 10.1021/nl0509733 CCC: $30.25 Published on Web 08/17/2005
© 2005 American Chemical Society
the HF approximation has serious shortcomings for describing the metallic behavior of bulk materials.21 Experimentally, optical transitions in metallic SWNTs have been obtained recently by resonant Raman spectroscopy,3,4,5 and they provide a reference point to compare with theoretical results. In this letter, we present the first study on the performance of hybrid functionals for predicting optical transitions in metallic SWNTs. Within the quasi-particle formalism, self-energy corrections and excitonic effects were shown to be important although they tend to partially cancel each other.8 These two corrections together are belived to be one order of magnitude smaller in metallic tubes than in semiconductors. Recently, Wang et al.22 demonstrated the excitonic character of optical transitions in semiconducting SWNTs experimentally. Within the time-dependent DFT framework, local field effects as well as exchange-correlation inductions were shown to be negligible for light polarized along the nanotube axis.10 However, these effects become important for perpendicular polarization and describe the peak supression found in experiment correctly.10 This fact allows us to utilize the simple Kubo formula23 for the imaginary part of the macroscopic dielectric function, 2, to obtain the optical spectrum of a given SWNT considering parallel polarization. All of the calculations presented in this paper were carried using periodic boundary conditions and Gaussian-type orbitals as implemented in the Gaussian suite of programs.24 The structures of the tubes studied in this work were initially generated using the TUBEGEN program25 and then optimized with DFT (see below). We have utilized the small unit cell (7, 7) SWNT to determine the appropriate level of geometry optimization to carry out the optical excitations study. For the (7, 7) metallic tube, we compare in Table 1 the effect of the optimization
Table 1. Geometrical Parameters (in Å and Degrees) Optimized at Different Levels of Theory and First Optical Transitions, E M 11 (in eV) Calculated Using Different Basis Sets for the (7, 7) SWNT geometry optimization level
Baa Bb bond angle translational vector EM 11 TPSSh/3-21G EM 11 TPSSh/6-31G EM 11 TPSSh/6-31G(d) a
LSDA STO-3G
LSDA 3-21G
PBE 6-31G(d)
1.435 1.437 119.4 2.476 2.61 2.60 2.54
1.421 1.420 119.3 2.453 2.67 2.65 2.59
1.432 1.431 119.3 2.482 2.62 2.61 2.55
Bond perpendicular to the tube axis.
level on the geometrical parameters and E M 11. Changes in the absorption peaks due to the geometry are generally smaller than 0.05 eV. However, the LSDA/STO-3G26 level of theory yields very similar structural parameters than the better quality level PBE/6-31G(d),26 and therefore, the changes on E M 11 at the LSDA/STO-3G and PBE/6-31G(d) geometies are less or equal than 0.01 eV. For this reason and because of the large size of the unit cell of many of the nanotubes that we are considering, we adopted the LSDA/ STO-3G optimized geometries for the rest of this work. We have found previously that for semiconducting tubes the use of polarization functions [d-type orbitals on C atoms denoted by (d)] is not critical for determining E11 and E22.16 This is not the case for metallic tubes, where the inclusion of polarization functions noticeably changes E11, as seen in Table 1 for the (7, 7) tube. Moreover, E22 calculated with TPSSh changes from 4.19 to 4.02 eV when using the 3-21G and 6-31G(d) basis set, respectively. Consequently, we evaluate optical transitions using the largest basis set tested, that is, the 6-31G(d) [(10s4p1d) contracted to (3s2p1d)]. In this way, the largest SWNT with available experimental E11 studied in this work is the (11, 8), with 364 carbon atoms and a total of 5460 contracted Gaussian basis functions per unit cell. It should be noted that these are among the longest DFT calculations ever carried out with Gaussian orbitals, periodic boundary conditions, and hybrid functionals. To study the effect of different functionals on E11, we have chosen a set of five metallic tubes with small unit cells (three armchairs: (7, 7), (8, 8), and (9, 9), and two zigzags: (12, 0) and (15, 0)). On the basis of our previous experience in semiconducting16 and narrow metallic SWNTs,15 we have employed five functionals: LSDA, PBE,27 TPSS,17 the TPSSh functional,18 and the screened exchange hybrid HSE.19,20 These functionals are representative of different families of approximate density functionals with characteristic features, each of them adding additional ingredients to the hierarchy (electron density, density gradient, kinetic energy density, and exact exchange, respectively). In Figure 1, we present the results obtained for first-order transitions in metallic SWNTs, E M 11. For comparison purposes, we also include in Figure 1 the first-order transitions Nano Lett., Vol. 5, No. 9, 2005
Figure 1. Calculated versus experimental first optical excitation S energies for metallic (E M 11) and semiconducting (E 11) SWNTs.
for five semiconducting tubes, E S11.16 All nonhybrid functionals (LSDA, PBE, and TPSS) underestimate E M 11 by approximately 0.3 eV. This error is comparable to the error found previously for E S11.16 Interestingly, trends in metallic and semiconducting tubes are similar, even when in principle, excitonic effects22 (not accounted for in our calculations) should be more important in semiconducting tubes.8 The best overall performance is achieved by TPSSh and HSE, which yield similar first-order transitions for metallic SWNTs. The amount of HF exchange contained in TPSSh (10%) is empirical and was fitted to reproduce molecular heats of formation. This functional has been employed successfully in a wide variety of systems18 and was not tuned up or reparametrized in this work to reproduce excitation energies of SWNTs. However, HSE contains 25% of HF exchange but only in the short-range because the long-range exact exchange is neglected in this functional. Metallic tubes other than armchairs present a splitting of the van Hove singularities because of the trigonal warping effect.28,29 This splitting separates the electronic transitions in zigzag and chiral SWNTs in an upper (E + 11) and a lower (E 11) branch with respect to the armchair curve. Recently, Bussi et al.30 found that quantum interference effects should affect the experimental determination of the band splitting by Raman spectroscopy. Fantini et al.4 and Telg et al.5 presented experiments in which only the lower branch of the first transition was observed. However, Strano et al.3 presented experimental transitions with some extrapolated values based on a zone-folding model for the E + 11 and E 11 branches in a set of metallic nanotubes. In Table 2, we summarize our results for E + 11 and E 11 transitions with PBE, TPSSh, and HSE in a set of 20 metallic SWNTs. For comparison, available experimental values are also included in Table 2. We find excellent agreement between our calculated TPSSh and HSE values with the experimental 4 5 E11 of Fantini et al. and Telg et al. 1831
Table 2. Comparisson between E M 11 Calculated Using Different Functionals and the 6-31G(d) Basis Set and Experimental Valuea TPSShb
experimental (n, m) (7, 7) (8, 5) (9, 3) (10, 1) (8, 8) (9, 6) (10, 4) (11, 2) (12, 0) (9, 9) (10, 7) (11, 5) (12, 3) (13, 1) (10, 10) (11, 8) (12, 6) (13, 4) (14, 2) (15, 0)
-c +c -d -e E 11 E 11 E 11 E 11
2.34 2.37 2.47 2.36 2.61 2.33 2.71 2.11 2.15 2.22 2.17 2.33 2.17 2.43 2.16 2.47 1.91 1.96 2.01 1.99 2.10 2.00 2.19 2.01 2.24 1.75 1.80 1.83 1.83 1.91 1.85 1.98 1.86 2.04 1.86 2.06
2.43
2.22 2.22 2.19 2.16 2.03 2.07 2.06 2.04 2.02 1.89 1.90 1.92 1.93 1.92 1.88
E 11
+ E 11
HSEb E 11
+ E 11
2.54 2.54 2.47 2.53 2.82 2.54 2.84 2.43 2.48 3.11 2.49 3.15 2.38 2.43 3.40 2.44 3.45 2.28 2.27 2.24 2.31 2.49 2.30 2.49 2.29 2.71 2.29 2.73 2.27 2.91 2.26 2.94 2.25 3.03 2.24 3.05 2.02 2.07 2.05 2.10 2.23 2.09 2.22 2.07 2.11 2.40 2.09 2.40 2.08 2.10 2.56 2.09 2.56 2.06 2.10 2.67 2.08 2.68 1.89 1.89 1.86 1.93 1.92 2.02 1.91 1.99 1.94 1.95 2.15 1.92 2.13 1.93 1.95 2.28 1.92 2.27 1.95 2.38 1.92 2.37 1.86 1.94 2.42 1.92 2.41
PBEb E 11
+ E 11
2.06 2.06 2.30 2.00 2.57 1.96 2.83 1.85 1.86 2.02 1.85 2.22 1.83 2.40 1.95 2.53 1.67 1.70 1.81 1.70 1.95 1.69 2.10 1.68 2.20 1.52 1.55 1.63 1.56 1.74 1.56 1.85 1.56 1.95 1.56 1.99
Figure 2. Calculated TPSSh/6-31G(d) E M ii as a function of the inverse of the diameter for several (n, m) SWNTs. Different families A ) 2n + m are shown using different symbols.
a Lower and upper branches of E M are indicated as E - and E + 11 11 11 respectively. The upper branch was not observed in the experiments of refs 4 and 5. All values in eV. b This work. c Experimental and extrapolated (italicized) values from ref 3. d Experimental values from ref 4. e Experimental values from ref 5.
According to the selection rules given by Kubo’s formula, the upper branch E + 11 should be obserVed for metallic SWNTs other than armchairs, as can be seen in Table 2. However, our density functional calculations predict a larger splitting than the zone-folding scheme.29,4 This could be attributed to hybridization effects induced by the SWNTs curvature, which are missing in the zone-folding scheme. These effects have been shown to be larger for π* and σ* (unoccupied) than for π and σ (occupied) orbitals,31 and therefore they are expected to have a larger influence on the conduction rather than on the valence bands. The trigonal warping splitting can be understood as a band distortion caused by the folding of a two-dimensional graphene sheet into a one-dimensional structure. The large splitting predicted in this work could provide an explanation of why the upper branches were not observed in some experiments in which they were searched using a narrow laser window while expecting a smaller splitting. In Figure 2, we plot the transitions for metallic SWNTs as a function of the inverse tube diameter (1/d). This plot shows the behavior of different branches for families of nanotubes with a constant family number A ) 2n + m. These “V” shaped patterns can be compared with those predicted by the zone-folding model and with experiments.4 Our density functional patterns exhibit a more pronounced curvature than the zone-folding ones and they resemble those determined experimentally more closely (figure 3 in ref 4). We observe that the agreement with experiment seems to worsen slightly as the SWNTs diameter decreases. Because hybridization effects are taken into account fully in our calculations, we speculate that other effects, for instance, local field effects, might be more important for narrower tubes. 1832
Figure 3. Calculated TPSSh/6-31G(d) absortion specta for SWNTs belonging to the family A ) 27. The diameter of the tubes decreases from botton to top.
The calculated absorption spectra for the family A ) 27 are shown in Figure 3. The first peak of the A ) 27 family remains almost constant for the whole series, whereas the single E11 peak corresponding to the armchair (9, 9) SWNT progressively splits up to 0.60 eV for the (13, 1) tube. The extrapolated splitting from an empirical fitting is ∼0.23 eV.3 As observed in Table 3, the trigonal warping splitting predicted by all of the functionals employed in this study is larger than the zone-folding splitting29 or that presented by Strano et al.3 Notably, the splitting obtained using the PBE functional (based on the GGA), E + 11 - E 11, is close to the one predicted by hybrid functionals. Therefore, our density functional results confirm that the trigonal warping splitting is about 1.5 to 2 times larger than that reported previously,29,3 regardless of the functional utilized to calculate the electronic Nano Lett., Vol. 5, No. 9, 2005
Table 3. Extrapolated, Zone-Folding (ZF), and Calculated -a Trigonal Warping Splittings, E + 11 - E 11 (n, m)
ext.b
ZFc
TPSShd
HSEd
PBEd
(8, 5) (9, 3) (10, 1) (9, 6) (10, 4) (11, 2) (12, 0) (10, 7) (11, 5) (12, 3) (13, 1) (11, 8) (12, 6) (13, 4) (14, 2) (15, 0)
0.10 0.26 0.38 0.07 0.16 0.26 0.31 0.04 0.11 0.18 0.23 0.03 0.07 0.13 0.18 0.20
0.15 0.34 0.49 0.10 0.23 0.35 0.40 0.07 0.16 0.25 0.31 0.05 0.11 0.18 0.23 0.25
0.29 0.63 0.97 0.19 0.42 0.64 0.78 0.13 0.29 0.45 0.57 0.09 0.20 0.33 0.43 0.48
0.30 0.66 1.02 0.19 0.44 0.67 0.81 0.14 0.31 0.48 0.60 0.09 0.21 0.35 0.45 0.49
0.24 0.56 0.87 0.16 0.37 0.57 0.59 0.11 0.26 0.41 0.52 0.08 0.19 0.30 0.39 0.43
a
All Values in eV
b
Taken from ref 3. c Taken from ref 29. d This work.
transitions. However, the specific value of E 11 is reproduced significantly better by hybrids TPSSh and HSE than the other functionals employed in this work. As the family number increases, the difference between the zone-folding model and our density functional splittings decreases. Therefore, a similar prediction for the splitting is expected for very large diameter tubes. Also, our TPSSh results are in better agreement with experiment than those that Popov32 obtained using a symmetry-adapted nonorthogonal tight binding model, which yields values close to our PBE results. In summary, we have calculated E M 11 using hybrid density functionals in 20 metallic SWNTs. In previous work, we have shown that the TPSSh functional performs very well for optical transitions of semiconducting SWNTs. Here, we show that TPSSh also does an excellent job in describing optical transitions in metallic SWNTs. The good agreement between experimental data and our results obtained as band energy differences for both semiconducting and metallic SWNTs is puzzling given that excitonic and other effects beyond our level of theory are expected to be different for the two classes of nanotubes. The method employed here can potentially be used to perform simulations in doped or chemically functionalized SWNTs. Because the TPSSh functional does not contain empirical parameters depending on the particular system, we also expect it to achieve reliable results for optical transitions in chemically modified SWNTs. Acknowledgment. This work is supported by the Nanoscale Science and Engineering Initiative of the National Science Foundation under NSF award no. EEC-0118007, NSF-CHE9982156, and the Welch Foundation. Part of the calculations were carried out at the Rice Terascale Cluster funded by NSF under grant EIA-0216467, Intel, and HP. References (1) Bachilo, S. M.; Strano, M. S.; Kittrell, C.; Hauge, R. H.; Smalley, R. E.; Weisman, R. B. Science 2002, 298, 2361.
Nano Lett., Vol. 5, No. 9, 2005
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