12860
J. Phys. Chem. B 2006, 110, 12860-12864
Longitudinal Polarizability of Carbon Nanotubes Edward N. Brothers,*,† Gustavo E. Scuseria,† and Konstantin N. Kudin‡ Department of Chemistry, Mail Stop 60, Rice UniVersity, Houston, Texas 77005-1892, and Department of Chemistry and Princeton Institute for the Science and Technology of Materials, Princeton UniVersity, Princeton, New Jersey 08544 ReceiVed: January 19, 2006; In Final Form: April 24, 2006
The longitudinal polarizabilities of carbon nanotubes are determined using first principles density functional theory. These results demonstrate that the polarizability per atom of a nanotube in the axial direction is primarily determined by the band gap. In fact, polarizability per atom versus inverse band gap yields a linear trend for all nanotubes and methods utilized in this study, creating a universal relationship for longitudinal polarizability. This can be explained by examining the terms in the sum over states equation used to determine polarizability and noting that the vast majority of the polarizability arises from a few elements near the band gap. This universal trend is then used with experimentally determined band gaps to predict the experimental polarizability of carbon nanotubes.
Introduction There have been a number of studies of the polarizability of carbon nanotubes.1-7 While many of these papers primarily consider band gap modification in the presence of electric fields or polarizability perpendicular to the direction of periodicity, two of these papers discuss the polarizability in the direction of periodicity, i.e., the longitudinal polarizability. In the 1995 paper of Benedict, Louie, and Cohen, longitudinal polarizability per unit length is reported as being proportional to radius over band gap squared, and was determined using tight binding methods with four basis functions per atom.1 In a 2004 paper Guo et al. reported, using local spin density approximation (LSDA) augmented plane wave calculations, that longitudinal polarizability was proportional to nanotube radius squared.2 This difference of opinion may be resolved through higher-level calculations, and such are reported herein. This study is a search for trends in the data, to create empirical rules for polarizability. This has been done in two of our recent publications, where transverse polarizability was found to be proportional to nanotube radius squared regardless of band gap,3 (confirming a previous prediction1) and a study of double-wall nanotubes showing shielding of the inner nanotubes from electric fields regardless of the band gap of the outer nanotube.4 The emphasis of this paper, like the previous two, is thus on the breadth of nanotubes and functionals considered in a search for trends. However, this paper also utilizes experimental band gaps presented by Weisman and Bachilo8 with the discovered trends to allow easy comparison with experimental values for longitudinal polarizability, when such become available. There are three important things that must be considered before discussing the results of this study. First, Hartree-Fock (HF)9 and hybrid density functional theory (DFT) methods10-12 must be included in any study of extended system polarizability as regular DFT13 has been shown to significantly overestimate polarizability of extended systems.14 Second, semiconducting * Author to whom correspondence should be addressed. † Rice University. ‡ Princeton University.
chiral nanotubes, which have larger and more complicated unit cells than those of zigzag nanotubes, must be included to avoid forming a biased test set. (Armchair nanotubes are not included in this work as all are metallic, and the metallic zigzag nanotubes are also avoided, since the longitudinal polarizability of a metallic system would be infinite.15) Third, any trend for polarizability must be normalized either per atom or per unit length, so that all nanotubes can be considered on an equal footing. Method All calculations reported in this paper were carried out using the Gaussian16 suite of programs modified locally to allow calculations with periodic boundary conditions (PBCs)17 to respond to electric fields.18 While it would be possible to perform nonperiodic calculations and an extrapolation to include cap effects on polarizabilities, extremely long fragments would be necessary due to the slow convergence of polarizability.19 To ensure accurate polarizabilities, program defaults were set tighter for high accuracy.20 All nanotube geometries were generated by TubeGen21 and used without geometry optimization, as previous experience has shown polarizability is not sensitive to small geometric changes when compared with sensitivity to functional choice.3 Finally, a Gaussian-type basis set consisting of 3s2p atomic orbitals (3-21G)22 was chosen as a compromise between speed and accuracy. While a larger basis set would provide better converged polarizabilities, this basis set is sufficient to demonstrate trends and allows a larger test set of bigger nanotubes to be studied. A large set of nanotubes were examined (see Table 1 for a complete list) using the density functionals LSDA,23 PBE,24 TPSS,25 and VSXC,26 the hybrid density functionals PBEh10,11 (also referred to as “PBE0” or “PBE1PBE” in the literature) and TPSSh,12 the screened hybrid density functional HSE,27 and HF.9 The use of several functionals was undertaken as functional choice has been shown to greatly affect the results of nanotube studies. In our previous papers on nanotube polarizability,3,4 polarizability has been calculated numerically using the standard
10.1021/jp0603839 CCC: $33.50 © 2006 American Chemical Society Published on Web 06/14/2006
Longitudinal Polarizability of Carbon Nanotubes
J. Phys. Chem. B, Vol. 110, No. 26, 2006 12861
TABLE 1: Longitudinal SOS Polarizability (rzz, Å3) and Minimum Direct Bandgaps (Eg, eV) for the Nanotubes Considered in This Study LSDA (10,0) (11,0) (13,0) (14,0) (16,0) (17,0) (19,0) (20,0) (22,0) (23,0) (6,2) (6,4) (8,4) (10,5) (12,4)
PBE
TPSS
VSXC
PBEh
TPSSh
HSE
HF
Rzz
Eg
Rzz
Eg
Rzz
Eg
Rzz
Eg
Rzz
Eg
Rzz
Eg
Rzz
Eg
Rzz
Eg
631 729 964 1114 1386 1583 1893 2135 2486 2771 1340 1954 1724 2545 4127
0.87 0.84 0.70 0.67 0.58 0.55 0.50 0.47 0.43 0.41 1.12 1.06 0.85 0.71 0.65
624 726 956 1109 1375 1577 1880 2127 2470 2761 1323 1941 1709 2532 4109
0.88 0.85 0.71 0.67 0.59 0.55 0.50 0.47 0.43 0.41 1.13 1.06 0.86 0.72 0.66
608 708 931 1082 1339 1537 1831 2074 2406 2693 1289 1893 1665 2469 4007
0.90 0.87 0.73 0.69 0.60 0.57 0.51 0.48 0.45 0.42 1.16 1.09 0.88 0.74 0.67
623 677 945 1039 1352 1481 1840 2003 2410 2605 1313 1842 1673 2398 3875
0.89 0.91 0.72 0.71 0.60 0.59 0.51 0.50 0.45 0.44 1.14 1.12 0.88 0.76 0.70
366 437 557 655 792 916 1070 1218 1389 1560
1.55 1.48 1.27 1.19 1.06 1.00 0.92 0.87 0.81 0.77
478 564 729 853 1042 1201 1416 1608 1848 2073
1.17 1.12 0.95 0.89 0.79 0.74 0.68 0.64 0.59 0.56
3.64 3.44 3.00 2.82 2.54
1.51 1.27
1314 1952
1.14 0.95
1.10 1.05 0.88 0.83 0.72 0.68 0.61 0.58 0.53 0.50 1.45 1.34 1.07 0.89 0.81
162 199 248 295 351
1011 1504
492 578 765 891 1109 1273 1523 1723 2006 2242 1011 1528 1355 2033 3306
expression for response polarizability (also called “exact” or “screened” polarizability) using methods previously described.18,30,31
Rii )
∂ET2
(1)
∂2Fi
In this expression, ET is the total self-consistent field (SCF) energy and Fi is an electric field in Cartesian direction i. However, for the case of longitudinal polarizabilities of carbon nanotubes, there are a number of reasons why using this method is suboptimal. Previous studies of transverse polarizabilities have utilized electric fields of 2.5 × 10-3 au. These fields are too large for use in calculating longitudinal polarizabilities, as they perturb the band structure too much to allow accurate calculation of polarizability. In fact, it was difficult to find a field strength such that the numerical accuracy was acceptable while remaining inside the linear response range. This in and of itself does not preclude the use of numerical derivatives, as several smaller values for electric field could be used to generate a curve and the derivative found in eq 1 can be obtained from that. However, given the large number of nanotubes and functionals to be used in this study a more time-efficient method was sought. Because of this, the use of sum over states (SOS, also known as “unscreened” or “approximate”) polarizabilities was investigated. Formally, SOS polarizabilities are
Tests of the SOS polarizability versus response polarizability were undertaken for the zigzag nanotubes used in this study with the functionals LSDA and TPSSh, to examine the least and most sophisticated functionals. As can be seen in Figure 1, response polarizability varies linearly with SOS polarizability, and the slopes of the lines are close to unity, implying that for this property SOS polarizabilities are acceptable.
Figure 1. Sum over states polarizabilities versus response polarizabilities (Å3) for a series of carbon nanotubes calculated with the LSDA (red) and TPSSh (blue) functionals. Best fit lines are also plotted.
Results
RAzz )
2a
|Ωar(kz)|2
occ unocc
∑ ∑ ∫BZ (k ) - (k ) dkz π a r r
z
a
(2)
z
where Ωar(kz) are the occupied-virtual intraband dipole matrix elements, which in periodic systems can be defined according to Blount as32
〈
Ωar(kz) ) ua(kz)
|
d u (k ) dk r z
〉
(3)
and ua and ur are the cell periodic parts of the band coefficients. Sum over states polarizabilities do not include the response of the density matrix to an electric field and thus are unacceptable for the study of properties such as shielding, as approximate excited states generated from the ground state by swapping a pair of occupied and virtual orbitals are used rather than the true excited states. However, as shown below, the SOS method seems adequate for examining longitudinal polarizabilities.
Table 1 lists the longitudinal polarizabilities and minimum direct band gaps for all nanotubes examined in this study. The blanks in the table are present as the largest nanotubes in the test set were not used with the hybrid functionals, as hybrid methods make the calculations much more expensive, and their inclusion was judged to add no additional information to the study. A cursory examination of the band gaps in Table 1 reveals two trends. If a specific nanotube is selected and Table 1 is examined across a specific row, then the inclusion of exact exchange increases the band gap. If a specific method is selected and Table 1 is examined going down a column, for the zigzag nanotubes the band gap decreases, the radius increases, and the polarizability increases. This trend for polarizability is lost as the column enters the chiral nanotubes, as these have much larger unit cells than than the zigzag columns, which illustrates the necessity of some method of normalization for unit cell size so that the zigzag and chiral nanotubes can be compared.
12862 J. Phys. Chem. B, Vol. 110, No. 26, 2006
Brothers et al. To generate a simple trend for longitudinal SOS polarizability, several possibilities were considered. An example of a rejected trend, in this case generated with PBE, is presented in Figure 2. While numerically the fit is acceptable, the points themselves can be seen to be somewhat nonlinear, and thus this fit was rejected. The results presented in the paper are of a higher theoretical level than previous calculations, and thus the differences between the results presented here and those of previous papers can be attributed to the more advanced methods used. Ultimately, the best fit (numerically and aesthetically) was found to be inverse band gap versus approximate polarizability per atom for all semiconducting tubes. The fit of this trend is quite good for all tested nanotubes. Attempts were also made to separate the nanotubes into Mod(n - m, 3) ) 1 and Mod(n - m, 3) ) 2 nanotubes to see if it improved the fit, but this was found to make little difference numerically and was thus rejected.
Figure 2. Radius over band gap squared (Å/eV2) versus SOS polarizability per unit length (Å2) for a series of carbon nanotubes calculated with the PBE functional.
Figure 3. Inverse direct band gap (1/eV) versus SOS polarizability per atom (Å3) for all the nanotubes considered in this study with all the various functionals.
TABLE 2: Number Elements above a Certain Size and Proportion of Total Approximate Polarizability Contained in Those Elements >1
>5
>10
>25
>50
>100
>200
nanotube
total elements
functional
no.
fraction
no.
fraction
no.
fraction
no.
fraction
no.
fraction
no.
fraction
no.
fraction
(10,0)
2 275 200
(6,2)
4 283 136
(6,4)
7 485 696
LSDA PBE TPSS VSXC PBEh TPSSh HSE HF LSDA PBE TPSS VSXC HSE LSDA PBE TPSS VSXC HSE
291 289 284 284 242 254 241 186 414 420 407 405 325 585 581 578 554 478
0.892 0.890 0.888 0.894 0.833 0.861 0.871 0.725 0.900 0.899 0.897 0.902 0.874 0.912 0.912 0.911 0.909 0.894
129 125 122 130 86 107 96 32 176 175 176 173 155 245 244 240 236 222
0.805 0.796 0.793 0.809 0.667 0.745 0.758 0.410 0.846 0.843 0.842 0.849 0.813 0.866 0.865 0.863 0.863 0.844
58 58 58 54 30 38 38 20 121 124 120 119 107 170 169 167 167 140
0.688 0.684 0.684 0.681 0.503 0.595 0.632 0.342 0.803 0.803 0.796 0.807 0.763 0.824 0.824 0.821 0.822 0.786
18 18 18 18 16 16 18 6 52 50 47 47 27 66 64 62 63 57
0.569 0.564 0.564 0.577 0.423 0.508 0.563 0.173 0.672 0.661 0.653 0.671 0.568 0.694 0.690 0.685 0.687 0.667
14 14 14 14 10 12 14 0 22 22 22 22 21 46 46 45 45 41
0.535 0.530 0.530 0.544 0.337 0.457 0.526 0.000 0.564 0.559 0.560 0.582 0.538 0.642 0.642 0.637 0.635 0.609
10 10 10 10 2 8 8 0 17 17 17 17 15 30 28 26 27 19
0.466 0.461 0.461 0.475 0.085 0.363 0.397 0.000 0.523 0.520 0.519 0.541 0.474 0.552 0.537 0.521 0.524 0.449
6 4 4 6 0 0 0 0 12 12 11 12 9 14 14 14 14 12
0.336 0.237 0.237 0.344 0.000 0.000 0.000 0.000 0.444 0.441 0.417 0.462 0.346 0.399 0.399 0.398 0.392 0.351
Longitudinal Polarizability of Carbon Nanotubes
J. Phys. Chem. B, Vol. 110, No. 26, 2006 12863
TABLE 3: Prediction of Nanotube Longitudinal Polarizability Based on Band Gaps Given in Ref 8 nanotube
band gap (eV)
tube radius (Å)
Rzz/atom (Å3)
Rzz/length (Å2)
(10,0) (11,0) (13,0) (14,0) (16,0) (17,0) (19,0) (20,0) (22,0) (23,0) (6,2) (6,4) (8,4) (10,5) (12,4)
1.07 1.20 0.90 0.96 0.76 0.80 0.61 0.69 0.59 0.60 1.39 1.42 1.12 0.99 0.92
3.93 4.32 5.10 5.49 6.23 6.67 7.45 7.84 8.62 9.01 2.84 3.43 4.16 5.19 5.66
12.64 11.53 14.78 13.95 17.02 16.35 20.73 18.75 21.60 21.14 10.19 10.00 12.23 13.51 14.39
119 119 180 183 255 261 369 352 446 456 69 82 121 168 195
More striking is the result in Figure 3, which contains all of the data in Table 1 in graphic form. All the of the SOS polarizabilities follow a single universal trend, such that once a band gap is known, the polarizability per atom is immediately available. To find the cause of this trend, we examined at each k-point the individual contributions to the polarizability from occupied-virtual band pairs that appear in eq 2. It turns out that most of the approximate polarizability arises from a very few large terms. For all of the largest terms, the occupied level is either the highest occupied crystalline orbital (HOCO) or HOCO - 1, and the virtual level is either the lowest unoccupied crystalline orbital (LUCO) or LUCO + 1. The k-point where total polarizability is the largest depends on where the denominator is the smallest, which is the k-point defining the band gap. Thus for different chirality nanotubes with similar band gaps the denominators for the largest terms are roughly the same and are equal to the band gap, implying that this is not simply a denominator effect. The dipole transition moment squared in the numerator of eq 2, however, comes out to be proportional to the number of carbon atoms and is almost independent of the functional. This is because for all functionals the shapes of the orbitals near the HOCO and the LUCO are similar. Therefore the trend shown in Figure 3 is entirely due to the fact that the sum of the squares of the transition dipole moments per carbon atom for contributing band pairs near the gap is very similar in all tubes with all functionals. The fact that most polarizability arises from a few terms is demonstrated in Table 2, where it is shown that most of the polarizability arises from a few terms in the SOS polarizability expression, and as stated above, these terms always arise from the HOCO-LUCO region. Reliable experimental band gaps have been presented for all the nanotubes utilized in this study.8 As a result, the trend used found in Figure 3 was applied to these experimental band gaps, and they are presented in Table 3. Specifically, the polarizabilities are presented in three ways: per atom, per unit cell, and per unit length. It is hoped that these values can be compared to experimental polarizability values as they become available. Conclusion For longitudinal polarizabilities of carbon nanotubes, SOS polarizabilities are adequate and far less computationally intensive than response polarizabilities calculated through numerical second derivatives. Longitudinal polarizabilities follow a simple trend and are determined entirely by band gap when the polarizability is normalized for the number of atoms
in the unit cell. This can be explained both by physical expectations and by noting that most of the polarizability arises from a very few HOCO-LUCO terms in the SOS expression. A set of values using this trend and experimental band gaps are provided for comparison purposes when experimental values become available. Acknowledgment. This work was supported by the National Science Foundation (NSF-CHE-0457030), the Nanoscale Science and Engineering Initiative of the National Science Foundation under NSF Award Number EEC-0118007, and the Welch Foundation. References and Notes (1) Benedict, L. X.; Louie, S. G.; Cohen, M. L. Phys. ReV. B 1995, 52, 8541. (2) Guo, G. Y.; Chu, K. C.; Wang, D.-s.; Duan, C.-g. Comput. Mater. Sci. 2004, 30, 269. (3) Brothers, E. N.; Kudin, K. N.; Scuseria, G. E.; Bauschlicher, C. W., Jr. Phys. ReV. B 2005, 72, 33402. (4) Brothers, E. N.; Kudin, K. N.; Scuseria, G. E. J. Chem. Phys. 2006, 124, 041101. (5) Zhou, X.; Chen, H.; Zhong-Can, O.-Y. J. Phys.: Condens. Matter 2001, 13, L635. (6) O’Keeffe, J.; Wei, C.; Cho, K. Appl. Phys. Lett. 2002, 80, 676. (7) Li, Y.; Rotkin, S. V.; Ravaioli, U. Nano Lett. 2003, 3, 183. (8) Weisman, R. B.; Bachilo, S. M. Nano Lett. 2003, 3, 1235. (9) Roothan, C. C. J. ReV. Mod. Phys. 1951, 23, 69. (10) Ernzerhof, M.; Scuseria, G. E. J. Chem. Phys. 1999, 110, 5029. (11) Adamo, C.; Barone, V. J. Chem. Phys. 1999, 110, 6158. (12) Staroverov, V. N.; Scuseria, G. E.; Tao, J.; Perdew, J. P. J. Chem. Phys. 2003, 119, 12129. (13) (a) Hohenberg, P.; Kohn, W. Phys. ReV. B 1964, 136, 864. (b) Jon, W.; Sham, L. J. Phys. ReV. A 1965, 140, 1133. (14) Mori-Sanchez, P.; Wu, Q.; Yang, W. J. Chem. Phys. 2003, 119, 11001. (15) Nanotubes are named based on a chiral vector (n, m) that denotes which two carbons are superimposed when a sheet of graphene is rolled into a nanotube. Zigzag nanotubes are all of the form (n, 0), while armchair nanotubes are all of the form (n, n). Chiral nanotubes are all the nanotubes that are neither zigzag or armchair. Note that nanotubes with Mod(n - m, 3) ) 0 are metallic, while Mod(n - m, 3) ) 1 and Mod(n - m, 3) ) 2 nanotubes are semiconducting and in some instances have been considered as three separate categories of nanotubes instead of two (metallic and semiconducting). (16) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A., Jr.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J. A. Gaussian 03, revision C.02; Gaussian, Inc.: Wallingford, CT, 2004. (17) Kudin, K. N.; Scuseria, G. E. Phys. ReV. B 2000, 61, 5141. (18) Kudin, K. N.; Scuseria, G. E. J. Chem. Phys. 2000, 113, 7779. (19) Kudin, K. N.; Car, R.; Resta, R. J. Chem. Phys. 2005, 122, 134907. (20) The integration grid was set to “ultrafine,” which requests a grid of 99 radial shells and 590 angular points per shell pruned for computational efficiency, and the SCF convergence criterion was set for the root-meansquare density change between cycles to be less than 1.0 × 10-9. The Gaussian default number of k-points was used at all times, which is 79 for zigzag nanotubes. (21) Frey, J. T.; Doren, D. J.; TubeGen, version 3.1; University of Delaware: Newark, DE, 2003. (22) Binkley, J. S.; Pople, J. A.; Hehre, W. J. J. Am. Chem. Soc. 1980, 102, 939. (23) Vosko, S. H.; Wilk, L.; Nusair, M. Can. J. Phys. 1980, 58, 1200. (24) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. ReV. Lett. 1996, 77, 3865.
12864 J. Phys. Chem. B, Vol. 110, No. 26, 2006 (25) Tao, J.; Perdew, J. P.; Staroverov, V. N.; Scuseria, G. E. Phys. ReV. Lett. 2003, 91, 146401. (26) Van Voorhis, T.; Scuseria, G. E. J. Chem. Phys. 1998, 109, 400. (27) Heyd, J.; Scuseria, G. E. J. Chem. Phys. 2004, 120, 7274. (28) Barone, V.; Peralta, J. E.; Wert, M.; Heyd, J.; Scuseria, G. E. Nano Lett. 2005, 5, 1621.
Brothers et al. (29) Barone, V.; Peralta, J. E.; Scuseria, G. E. Nano Lett. 2005, 5, 1830. (30) King-Smith, R. D.; Vanderbilt, D. Phys. ReV. B 1993, 47, 1651. (31) Resta, R. ReV. Mod. Phys. 1994, 66, 899. (32) Blount, E. I. In Solid State Physics; Ehrenreich, H., Seitz, F., Turnbull, D., Eds.; Academic: New York, 1962; Vol. 13, p 305.
