NANO LETTERS
Density Functional Theory Study of Optical Transitions in Semiconducting Single-Walled Carbon Nanotubes
2005 Vol. 5, No. 8 1621-1624
Vero´nica Barone, Juan E. Peralta, Michael Wert, Jochen Heyd, and Gustavo E. Scuseria* Department of Chemistry, Rice UniVersity, Houston, Texas 77005-1892 Received April 5, 2005; Revised Manuscript Received May 6, 2005
ABSTRACT We present a density functional theory study of optical transitions in semiconducting single-walled carbon nanotubes. We utilize recently developed exchange-correlation functionals in a set of 21 tubes that includes large and chiral nanotubes. The novel TPSSh meta-generalized gradient approximation hybrid functional accurately reproduces optical excitations with mean absolute errors of 0.024 and 0.065 eV for first and second transitions, respectively. We also report predictions for higher order optical transitions.
The stability and quasi-one-dimensional nature of singlewalled carbon nanotubes (SWNT) make them unique systems from an experimental and theoretical point of view. SWNTs present novel electronic, optical, and mechanical properties that can potentially be exploited in a wide range of applications such as energy storage and nanoelectronics.1 Optical properties of SWNTs are powerful means for determining their structures. Optical transitions in SWNTs, together with their Raman spectra, provide a tool for assigning the structural parameters of a certain nanotube, i.e., indices (n,m), or chiral angle and diameter. The relation between ith-order transitions (Eii) and the diameter of the nanotube (Kataura plot2) is a useful guide for experimentalists. Recently, Bachilo et al.3 assigned the optical spectra of 33 semiconducting SWNTs dispersed in aqueous surfactant suspensions. That pioneering work explicitly showed the limitations of Kataura plots based on conventional tightbinding (TB) calculations. The same authors also presented an empirically fitted Kataura plot that is expected to be more reliable than the model-based plots, particularly for SWNTs with diameters larger than 0.5 nm. Other experiments4,5 have been recently reported with assignments in excellent agreement with those of Bachilo et al.3 Weisman and Bachilo6 have presented preliminary assignments of third (E33) and fourth (E44) order optical transitions in a set of 15 semiconducting nanotubes. However, they did not establish any empirical relation between higher order transitions and the diameter and chirality of the corresponding SWNT. Reliable theoretical approaches are desirable for describing the electronic vertical transitions of SWNTs. A few interesting papers using quasi-particle formalisms have been reported recently, predicting optical transitions of small SWNTs in 10.1021/nl0506352 CCC: $30.25 Published on Web 07/01/2005
© 2005 American Chemical Society
agreement with available experiments.7,8 In addition, calculations performed on small SWNTs using time-dependent density functional theory (TD-DFT) have shown that local field effects and induced exchange and correlation (XC) components beyond the random phase approximation (RPA) are negligible in these systems for light polarized parallel to the nanotube axis.9,10 The bottleneck of these two approaches is the limited size of the systems that can be treated. Typically, the number of carbon atoms in the unit cell of the SWNT studied in those works is smaller than 50.7 However, the unit cell corresponding to (n,m) SWNTs assigned by Bachilo et al.3 may contain from hundreds to thousands of carbon atoms. Some studies describing SWNT electronic structure from a one-particle approach have been presented using density functional theory within the local density approximation (LDA) and the generalized-gradient approximation (GGA), with results in qualitatively good agreement with experiment.11,12 However, the LDA misses some important ingredients that may yield results even closer to experiment. For instance, hybrid density functionals (those including a portion of Hartree-Fock type of exchange) have been shown to perform better than pure functionals in describing the electronic structure of semiconducting bulk materials13,14 and SWNTs.15 In this letter, we present an extensive study of the optical properties of 21 semiconducting SWNTs. We calculate E11 and E22 transitions in excellent agreement with the experimental assignments,3 and, more importantly, we successfully predict E33 and E44 transitions. We have utilized periodic boundary conditions and Gaussian-type orbitals as implemented in the development version of the Gaussian program.16,17 We optimized the geometric
Figure 2. Calculated optical spectrum (a) and band structure (b) for the (10,8) SWNT at the TPSSh/3-21G level. Figure 1. First optical transitions obtained with different functionals as a function of the corresponding experimental values.
structure of all SWNTs at the LDA/STO-3G level.18 Further geometry optimization using other functionals and basis sets produces rather small changes in the energy gap (typically less than 1%). Also, we have performed energy calculations on the small (10,0) and (8,4) SWNTs using the 3-21G and the 6-31G* basis sets,18 which show that the energy band gap depends slightly on the basis set choice (differences are smaller than 0.01 eV). The unit cell sizes of the SWNTs involved in this work (up to 488 carbon atoms) make the computation very demanding. We have thus employed the smaller 3-21G basis set (containing 6s and 3p Gaussian basis functions contracted to 3s and 2p) and the LDA/STO-3G optimized structures for the rest of the calculations. To test the performance of different functionals, we have first chosen a test set of eight chiral semiconducting nanotubes where experimental optical gaps are available. For this set, we employed seven different functionals: LDA, PBE,19 TPSS20 (the novel meta-GGA functional recently developed by Tao, Perdew, Staroverov, and Scuseria), and the hybrid functionals PBEh (PBE hybrid),21 B3LYP,22,23 TPSSh (TPSS hybrid),24 and the screened exchange HSE.25 In Figure 1, we present the calculated optical band gaps (E11) as a function of the corresponding experimental values for the test set. This lowest optical transition corresponds directly to the energy band gap in all cases except the smaller diameter zigzag nanotubes (see below). We observe the wellknown trend that nonhybrid functionals underestimate the experimental band gap due to the discontinuity of the exchange-correlation Kohn-Sham potential.26,27 On the other hand, the hybrid PBEh and B3LYP functionals overestimate band gaps by about 0.3 eV. The performance of HSE is much better; it presents absolute errors of approximately 0.15 eV. TPSSh significantly improves the agreement with experiment: deviations are in all cases less than 0.05 eV. We stress that the portion of the Hartree-Fock exchange in TPSSh (10%) was set to reproduce the thermochemistry of a large set of molecules and remained fixed at that value in all our calculations.24 Empirically, one could scale LSDA, PBE, and TPSS values by a factor of about 1.2 to obtain band gaps in SWNT with significantly smaller errors. 1622
In view of these results, we have evaluated all optical transitions using TPSSh. To obtain the optical spectrum, we have employed the first-order noniterative random phase approximation expression for the imaginary part of the dielectric function ,28 Im() )
1 2
ω
|〈ψκo|p|ψκu〉|2 δ(κo - κu - ω) ∑κ ∑ o,u
(1)
where p is the linear momentum operator and the indices o and u stand for occupied and unoccupied Bloch orbitals, respectively. Although, in a general case, there is little justification for using the single-particle approach adopted in this work for calculating optical transitions as band energy differences, SWNTs seem to represent an exception. Recent papers on the optical properties of SWNTs obtained with TD-DFT have shown that local field effects and induced exchangecorrelation are both negligible in these systems for parallel polarization.9,10 However, for perpendicular polarization, these effects are crucial for a proper description of depolarization effects that suppress the absorption peaks.9 Therefore, in our simple picture, we neglect perpendicular contributions. Figure 2 shows, as a representative example, the optical spectrum and band structure of the chiral (10,8) SWNT obtained using the TPSSh functional. This system contains 488 atoms and 4392 Gaussians in the unit cell, and is the largest DFT calculation on a SWNT reported in the literature to date. First, second, and third optical transitions are indicated with arrows in Figure 2-b. In Table 1, we present the TPSSh results for the optical transitions of 21 zigzag and chiral SWNTs and compare the results with the corresponding experimental values. When experimental values are not available, we compare our results with those obtained from the empirical fitting of Weisman et al.6 Our TPSSh results are in excellent agreement with experiment. We obtain deviations from the experimental values (or those extrapolated from experiment) that result in mean absolute errors (MAE) of 0.024 eV for E11 and 0.065 eV for E22. We observe the largest deviations for first transitions in the zigzag (7,0) and (8,0) SWNT when comparing with the extrapolated values. However, these empirical values could be missing Nano Lett., Vol. 5, No. 8, 2005
Table 1. First, Second, Third, and Fourth Optical Transitions in Semiconducting SWNT Calculated at the TPSSh/3-21G Levela E11 (n,m)
expt.b,c
calc.d
(4,3) (6,2) (6,4) (6,5) (7,0) (8,0) (8,4) (8,6) (10,0) (10,5) (10,6) (10,8) (11,0) (12,4) (12,8) (13,0) (14,0) (14,7) (15,10) (16,8) (20,4) MEg MAEh
(1.771) (1.387) 1.420 1.272 (1.289) (1.598) 1.114 1.058 (1.073) 0.992 0.898 0.841 (1.196) 0.924 (0.748) (0.896) (0.957) (0.701) (0.624) (0.644) (0.584)
1.78 1.35 1.43 1.27 1.22 1.67 1.09 1.05 1.06 0.98 0.88 0.82 1.20 0.90 0.72 0.88 0.94 0.68 0.60 0.62 0.57 -0.014 0.024
E22
E33
E44
expt.b,c calc.d expt. calc.d expt. calc.d (3.118) (2.963) 2.134 2.187 (3.139) (1.878) 2.112 1.732 (2.307) 1.577 1.640 1.428 (1.665) 1.447 (1.353) (1.831) (1.443) (1.299) (1.072) (1.090) (1.119)
3.24 3.20 2.25 2.29 3.594c 3.03 1.99 2.19 1.79 3.179c 2.46 1.61 1.68 2.71f 1.45 1.72 1.48 1.36 1.89 1.47 1.30 1.07 1.09 1.11 0.052 0.065
3.77 3.33 3.96 3.68 4.133c 3.53 3.29 3.27 3.40 3.542c 3.08 3.33 2.82 2.79 3.36 3.09 2.41 2.75 3.05 2.26 2.12 2.22 1.90
4.40 4.60 4.21 4.32 e 5.12 4.23 3.63 3.87 4.27 3.68 3.14 4.02 3.22 3.09 3.85 3.58 3.03 2.41 2.42 2.72
Figure 3. Calculated (open symbols) and experimental (filled symbols) optical transitions as a function of the SWNT diameter dt. Table 2. Comparison between Predicted E33 and E44 from the Empirical Fitting and the Corresponding Experimental Values (in eV) E33
a
When experimental values are not available, we compare our results with those obtained by an empirical fit (values in parentheses). All values in eV. b Experimental values from ref 3. c Values in parentheses correspond to the empirical fitting from ref 6. d This work. e E44 could not be distinguished in the range 0-10 eV. f Experimental value from ref 5. g Mean error. h Mean absolute error.
some important curvature effects. Curvature effects in narrow zigzag SWNTs are so important that the (5,0) and (4,0) tubes are indeed metallic29 (instead of semiconducting as predicted by the conventional TB approach1). Extended TB models are necessary to describe optical transitions in small diameter SWNTs.30,31 If we do not consider the (7,0) and (8,0) tubes, the MAE for TPSSh drops to 0.018 eV for E11. Limited experimental values are available for higher transitions. However, as shown in Figure 3, we obtain a good trend for E33 and E44: all calculated third and fourth transitions fall within the range of the experimental values. These results have encouraged us to fit our calculated values with a formula similar to that proposed by Bachilo et al.,3 m 107 cm-1 Ci cos(3R) νii ) + Ai + Bi dt dt2
(2)
with dt being the diameter of the SWNT, R the chiral angle, i the order of the transition, and m is the mod function of the given tube. In this work, the parameters A, B, and C corresponding to E33 and E44 have been adjusted using a least squares criterion. We have divided the SWNT studied here (except the narrow (7,0) tube) in two sets, depending on the value of mod (n - m, 3) ) 1 or mod (n - m, 3) ) 2. With the optimized parameters, we estimate E33 and E44 for the chiral nanotubes that also have experimental assignments.6 Results are presented in Table 2. We find a good agreement between the estimated values and experiment with MAE of 0.10 and 0.20 eV for E33 and E44, respectively. We notice Nano Lett., Vol. 5, No. 8, 2005
E44
(n,m)
mod
exp.a
pred.b
exp.a
pred.b
(6,5) (7,6) (8,7) (9,5) (9,8) (11,4) (12,5) (7,5) (8,3) (8,6) (9,4) (9,7) (10,2) (11,3) (11,6) MEc MAEd
1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
3.594 3.342 3.069 3.271 2.818 2.818 2.616 3.690 3.512 3.179 3.444 3.046 3.333 3.229 2.980 -
3.63 3.37 3.14 3.06 2.93 2.84 2.71 3.50 3.46 3.25 3.25 3.03 3.23 3.05 2.87 -0.04 0.10
4.133 3.875 3.594 3.961 3.333 3.583 3.757 4.305 4.012 3.542 3.636 3.324 3.924 3.512 3.084 -
4.05 3.77 3.54 3.65 3.33 3.51 3.31 4.00 4.29 3.71 3.92 3.47 4.10 3.78 3.40 0.02 0.20
From ref 6. b From eq 2: A3 ) 167.7, B3 ) 214.7, C 13 ) -3668.8, C 23 ) -1648.8; A4 ) 188.4, B4 ) 157.7, C 14 ) 519.7, C 24 ) 2212.0. c Mean error. d Mean absolute error. a
that the C parameters do not follow the alternating sign rule as in previous work.6 In summary, we have calculated first, second, third, and fourth optical transitions in several SWNTs using recently developed density functionals. The good agreement between the experimental data and our results obtained as vertical excitation energies (band energy differences) seems to indicate that excitonic effects are not important for these nanotubes. However, we cannot rule out that these effects are compensated by approximations intrinsic to our theoretical approach. The novel hybrid meta-GGA functional, TPSSh,24 accurately reproduces optical excitations with errors smaller than 4% for E11 and 8% for E22. For higher order transitions, we have adapted a previously reported relation3 in order to reproduce available experimental values in a set 1623
of 15 semiconducting chiral SWNTs. Furthermore, preliminary results obtained with TPSSh in armchair nanotubes also suggest an excellent agreement with experiment in the case of metallic SWNTs.5,32 Our results may be useful to derive quantum mechanical based parameters for extended TB models and will provide a helpful guide to experimentalists working on the assignment of optical transitions in SWNT. Acknowledgment. This work was supported by NSF Award Number EEC-0118007, NSF-CHE9982156, NSF Grant No. 0139202, and the Welch Foundation. Part of the calculations were performed on the Rice Terascale Cluster funded by NSF under Grant EIA-0216467, Intel, HP. V.B. also thanks R. B. Weisman, P. Nordlander, and S. M. Bachilo for enlightening discussions. References (1) Dresselhaus, M. S.; Dresselhaus, G.; Avouris, Ph. Topics in Applied Physics, Vol. 80; Springer: Heidelberg, 2001. (2) Kataura, H.; Kumazawa, Y.; Maniwa, Y.; Umezu, I.; Suzuki, S.; Ohtsuka, Y.; Achiba, Y. Synth. Met. 1999, 103, 2555. (3) Bachilo, S. M.; Strano, M. S.; Kittrell, C.; Hauge, R. H.; Smalley, R. E.; Weisman, R. B. Science 2002, 298, 2361. (4) Lebedkin, S.; Hennrich, F. H.; Skipa, T.; Kappes, M. M. J. Phys. Chem. B 2003, 107, 1949. (5) Fantini, C.; Jorio, A.; Souza, M.; Strano, M. S.; Dresselhaus, M. S.; Pimenta, M. A. Phys. ReV. Lett. 2004, 93, 147406. (6) Weisman, R. B.; Bachilo, S. M. Nano Lett. 2003, 3, 1235. (7) Spataru, C. D.; Ismael-Beigi, S.; Benedict, L. X.; Louie, S. G. Phys. ReV. Lett. 2004, 92, 077402. (8) Chang, E.; Bussi, G.; Ruini, A.; Molinari, E. Phys. ReV. Lett. 2004, 92, 196401. (9) Marinopoulos, A. G.; Reining, L.; Rubio, A.; Vast, N. Phys. ReV. Lett. 2003, 91, 046402. (10) Marinopoulos, A. G.; Wirtz, L.; Marini, A.; Olevano, V.; Rubio, A.; Reining, L. Appl. Phys. A 2004, 78, 1157.
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(11) Guo, G. Y.; Chu, K. C.; Wang, D. S.; Duan, C. G. Phys. ReV. B 2004, 69, 205416. (12) Liu, H. J.; Chan, C. T. Phys. ReV. B 2002, 66, 115416. (13) Heyd, J.; Scuseria, G. E. J. Chem. Phys. 2004, 121, 1187. (14) Kudin, K. N.; Scuseria, G. E.; Martin, R. L. Phys. ReV. Lett. 2002, 89, 266402. (15) Avramov, P. V.; Kudin, K. N.; Scuseria, G. E. Chem. Phys. Lett. 2003, 370, 597. (16) Frisch, M. J. et al., Gaussian, Inc., Pittsburgh, PA “Gaussian Development Version, Revision B.07”, 2003. (17) Kudin, K. N.; Scuseria, G. E. Phys. ReV. B 2000, 61, 16440. (18) The basis sets used in this work are STO-3G, 3-21G, and 6-31G* and they consist of (2s1p), (3s2p), and (3s2p1d) contracted Gaussian functions, respectively. (19) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. ReV. Lett. 1996, 77, 3865. (20) Tao, J.; Perdew, J. P.; Staroverov, V. N.; Scuseria, G. E. Phys. ReV. Lett. 2003, 91, 146401. (21) Ernzerhof, M.; Scuseria, G. E. J. Chem. Phys. 1999, 110, 5029. (22) Becke, A. D. J. Chem. Phys. 1993, 98, 5648. (23) Stephens, P. J.; Devlin, F. J.; Chabalowski, C. F.; Frisch, M. J. J. Chem. Phys. 1994, 98, 11623. (24) Staroverov, V. N.; Scuseria, G. E.; Tao, J.; Perdew, J. P. J. Chem. Phys. 2003, 119, 12129. (25) Heyd, J.; Scuseria, G. E.; Ernzerhof, M. J. Chem. Phys. 2003, 118, 8207. (26) Perdew, J. P.; Parr, R. G.; Levy, M.; Balduz, J. L. Phys. ReV. Lett. 1982, 49, 1691. (27) Perdew, J. P.; Levy, M. Phys. ReV. Lett. 1983, 51, 1884. (28) Onida, G.; Reining, L.; Rubio, A. ReV. Mod. Phys. 2002, 74, 601. (29) Barone, V.; Scuseria, G. E. J. Chem. Phys. 2004, 121, 10376. (30) Popov, V. N.; Henrard, L. Phys. ReV. B 2004, 70, 115407. (31) Samsonidze, G. G.; Saito, R.; Kobayashi, N.; Gru¨neis, A.; Jiang, J.; Jorio, A.; Chou, S. G.; Dresselhaus, G.; Dresselhaus, M. S. Appl. Phys. Lett. 2004, 85, 5703. (32) Barone, V.; Peralta, J. E.; Scuseria, G. E. to be published.
NL0506352
Nano Lett., Vol. 5, No. 8, 2005
