NANO LETTERS
Dependence of Optical Transition Energies on Structure for Single-Walled Carbon Nanotubes in Aqueous Suspension: An Empirical Kataura Plot
2003 Vol. 3, No. 9 1235-1238
R. Bruce Weisman* and Sergei M. Bachilo Department of Chemistry, Center for Nanoscale Science and Technology, and Center for Biological and EnVironmental Nanotechnology, Rice UniVersity, 6100 Main Street, Houston, Texas 77005 Received June 20, 2003; Revised Manuscript Received July 16, 2003
ABSTRACT Spectrofluorimetric data for identified single-walled carbon nanotubes in aqueous SDS suspension have been accurately fit to empirical expressions. These are used to obtain the first model-independent prediction of first and second van Hove optical transitions as a function of structure for a wide range of semiconducting nanotubes. To allow for convenient use in support of spectral studies, the results are presented in equation, graphical, and tabular forms. These empirical findings differ significantly from Kataura plots computed using a simple tightbinding model. It is suggested that the empirically based results should be used in preference to conventional model-based predictions in spectroscopic nanotube research.
Introduction. One of the most intriguing and potentially useful features of single-walled carbon nanotubes (SWNT) is the sensitivity of their electronic and optical properties to physical structure.1 A nanotube’s diameter (dt) and chiral angle (R) are uniquely related to a pair of integers (n, m) that describe its construction as a rolled-up graphene sheet. Except for very small diameter nanotubes, it is well known that structures for which n - m is evenly divisible by 3 display metallic or semimetallic behavior, whereas other tubes only slightly different in structure have significant band gaps and show semiconducting behavior.2 The quasi-onedimensionality of SWNT gives sharp van Hove peaks in the density of electronic states. Optical properties of SWNT are dominated by transitions between corresponding van Hove peaks on opposite sides of the Fermi level. Knowing the energies of these Eii van Hove transitions for specific (n, m) structures is important not only for absorption spectroscopy but also for the widely used method of resonance Raman spectroscopy in which matches between the incident or scattered light frequencies and van Hove transitions expose vibrational frequencies of selected nanotubes.3-5 Nanotube investigators using optical methods are commonly guided by Kataura plots of Eii transition energies vs tube diameter.6 In the absence of detailed experimental data, these have been generated from parametrized model calculations of SWNT electronic structure. * Corresponding author. E-mail:
[email protected]. 10.1021/nl034428i CCC: $25.00 Published on Web 08/08/2003
© 2003 American Chemical Society
Recent breakthroughs in SWNT spectroscopy now allow the construction of experimentally based Kataura-type plots. In semiconducting SWNT, the lowest-energy van Hove transition (at E11) corresponds to absorption or luminescence directly across the band gap. Since the discovery of bandgap photoluminescence from isolated SWNT in bulk aqueous suspensions,7 it has become possible to use spectrofluorimetry to explore the E11 and E22 transitions of a sizable number of semiconducting SWNT species. As was reported earlier from this laboratory, distinct first and second van Hove transition wavelengths have been observed for 33 semiconducting SWNT in a surfactant-suspended bulk sample, and each of these species has been assigned specific (n, m) structural indices.8 Spectral transitions of six additional SWNT species were reported by Lebedkin et al. in a study of slightly larger diameter nanotubes in a different surfactant.9 Because the identified nanotubes span a rather wide range of structures (ranging from 0.62 to 1.41 nm in diameter and from 3 to 28° in chiral angle), their measured transition frequencies can be used to anchor a robust empirical fitting function that allows extrapolation beyond the set of measured data. This provides the basis for a model-independent, empirically based Kataura plot that gives reliable predictions of optical transition frequencies versus (n, m) for semiconducting SWNT in aqueous surfactant suspensions. To enable easy use of the new predictions, we present them below as empirical functions of diameter and chiral angle, as graphs, and as tabulated values for more than 100 (n, m) structures.
Table 1. Structures and First and Second van Hove Optical Transitionsa for Semiconducting SWNT Structures with Diameters between 0.48 and 2.0 nm (n, m)
dt (nm)
λ11 R νj11 (deg) modb (nm) (cm-1)
λ22 νj22 E11 (eV)c (nm) (cm-1)
(4, 3) (5, 3) (5, 4) (6, 1) (6, 2) (6, 4) (6, 5) (7, 0) (7, 2) (7, 3) (7, 5) (7, 6) (8, 0) (8, 1) (8, 3) (8, 4) (8, 6) (8, 7) (9, 1) (9, 2) (9, 4) (9, 5) (9, 7) (9, 8) (10, 0) (10, 2) (10, 3) (10, 5) (10, 6) (10, 8) (10, 9) (11, 0) (11, 1) (11, 3) (11, 4) (11, 6) (11, 7) (11, 9) (11, 10) (12, 1) (12, 2) (12, 4) (12, 5) (12, 7) (12, 8) (12, 10) (12, 11) (13, 0) (13, 2) (13, 3) (13, 5) (13, 6) (13, 8) (13, 9) (13, 11) (13, 12) (14, 0) (14, 1) (14, 3) (14, 4) (14, 6) (14, 7) (14, 9) (14, 10)
0.483 0.556 0.620 0.521 0.572 0.692 0.757 0.556 0.650 0.706 0.829 0.895 0.635 0.678 0.782 0.840 0.966 1.032 0.757 0.806 0.916 0.976 1.103 1.170 0.794 0.884 0.936 1.050 1.111 1.240 1.307 0.873 0.916 1.014 1.068 1.186 1.248 1.377 1.444 0.995 1.041 1.145 1.201 1.321 1.384 1.515 1.582 1.032 1.120 1.170 1.278 1.336 1.457 1.521 1.652 1.719 1.111 1.153 1.248 1.300 1.411 1.470 1.594 1.658
25.28 21.79 26.33 7.59 13.90 23.41 27.00 0.00 12.22 17.00 24.50 27.46 0.00 5.82 15.30 19.11 25.28 27.80 5.21 9.83 17.48 20.63 25.87 28.05 0.00 8.95 12.73 19.11 21.79 26.33 28.26 0.00 4.31 11.74 14.92 20.36 22.69 26.70 28.43 3.96 7.59 13.90 16.63 21.36 23.41 27.00 28.56 0.00 7.05 10.16 15.61 17.99 22.17 24.01 27.25 28.68 0.00 3.42 9.52 12.22 17.00 19.11 22.85 24.50
1.771 1.721 1.485 1.900 1.387 1.420 1.270 1.289 1.546 1.250 1.211 1.107 1.598 1.191 1.303 1.116 1.057 0.981 1.359 1.090 1.126 0.999 0.938 0.879 1.073 1.177 0.993 0.993 0.900 0.844 0.797 1.196 0.980 1.036 0.904 0.887 0.818 0.767 0.729 1.060 0.900 0.924 0.827 0.803 0.748 0.703 0.671 0.896 0.949 0.828 0.834 0.760 0.733 0.689 0.648 0.622 0.957 0.826 0.857 0.764 0.759 0.701 0.674 0.638
1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1
700 720 835 653 894 873 976 962 802 992 1024 1120 776 1041 952 1111 1173 1265 912 1138 1101 1241 1322 1410 1156 1053 1249 1249 1377 1470 1556 1037 1265 1197 1371 1397 1516 1617 1702 1170 1378 1342 1499 1545 1657 1765 1848 1384 1307 1498 1487 1632 1692 1799 1912 1994 1295 1502 1447 1623 1633 1768 1840 1942
14 283 13 884 11 974 15 323 11 183 11 452 10 244 10 397 12 468 10 083 9768 8930 12 886 9603 10 508 8997 8525 7908 10 964 8790 9086 8055 7567 7093 8652 9493 8006 8006 7262 6805 6428 9644 7906 8353 7295 7157 6597 6183 5877 8549 7255 7452 6670 6473 6036 5666 5412 7228 7651 6676 6723 6127 5909 5558 5230 5015 7721 6660 6910 6162 6123 5655 5435 5148
398 522 483 632 418 578 566 395 626 505 645 648 660 471 665 589 718 728 691 551 722 672 793 809 537 737 632 788 754 869 889 745 610 793 712 858 836 947 969 799 686 855 793 930 917 1024 1049 677 858 764 922 874 1004 997 1102 1128 859 748 920 842 992 955 1080 1078
25 147 19 147 20 697 15 828 23 900 17 312 17 667 25 318 15 977 19 820 15 496 15 441 15 146 21 226 15 029 16 981 13 928 13 727 14 466 18 155 13 843 14 883 12 610 12 362 18 606 13 574 15 834 12 695 13 262 11 502 11 248 13 431 16 388 12 617 14 036 11 661 11 968 10 564 10 320 12 516 14 575 11 693 12 605 10 751 10 910 9763 9535 14 770 11 661 13 095 10 843 11 441 9956 10 027 9071 8862 11 640 13 364 10 867 11 875 10 078 10 476 9261 9279
E22 (eV) 3.118 2.374 2.566 1.962 2.963 2.146 2.190 3.139 1.981 2.457 1.921 1.914 1.878 2.632 1.863 2.105 1.727 1.702 1.794 2.251 1.716 1.845 1.563 1.533 2.307 1.683 1.963 1.574 1.644 1.426 1.395 1.665 2.032 1.564 1.740 1.446 1.484 1.310 1.280 1.552 1.807 1.450 1.563 1.333 1.353 1.210 1.182 1.831 1.446 1.624 1.344 1.419 1.234 1.243 1.125 1.099 1.443 1.657 1.347 1.472 1.250 1.299 1.148 1.150
(n, m)
dt (nm)
λ11 λ22 R νj11 νj22 E11 (deg) modb (nm) (cm-1) (eV)c (nm) (cm-1)
(14, 12) (14, 13) (15, 1) (15, 2) (15, 4) (15, 5) (15, 7) (15, 8) (15, 10) (15, 11) (15, 13) (15, 14) (16, 0) (16, 2) (16, 3) (16, 5) (16, 6) (16, 8) (16, 9) (16, 11) (16, 12) (17, 0) (17, 1) (17, 3) (17, 4) (17, 6) (17, 7) (17, 9) (17, 10) (18, 1) (18, 2) (18, 4) (18, 5) (18, 7) (18, 8) (18, 10) (19, 0) (19, 2) (19, 3) (19, 5) (19, 6) (19, 8) (19, 9) (20, 0) (20, 1) (20, 3) (20, 4) (20, 6) (20, 7) (21, 1) (21, 2) (21, 4) (21, 5) (22, 0) (22, 2) (22, 3) (22, 5) (23, 0) (23, 1) (23, 3) (24, 1) (24, 2) (25, 0)
1.789 1.857 1.232 1.278 1.377 1.431 1.546 1.606 1.730 1.795 1.927 1.994 1.270 1.357 1.405 1.508 1.564 1.680 1.741 1.867 1.932 1.350 1.391 1.483 1.533 1.641 1.697 1.816 1.877 1.470 1.515 1.611 1.663 1.773 1.831 1.951 1.508 1.594 1.641 1.741 1.795 1.907 1.966 1.588 1.629 1.719 1.768 1.872 1.927 1.708 1.752 1.847 1.897 1.747 1.831 1.877 1.975 1.826 1.867 1.956 1.946 1.990 1.985
27.46 28.78 3.20 6.18 11.52 13.90 18.14 20.03 23.41 24.92 27.64 28.86 0.00 5.82 8.44 13.17 15.30 19.11 20.82 23.90 25.28 0.00 2.83 7.99 10.33 14.56 16.47 19.93 21.49 2.68 5.21 9.83 11.93 15.75 17.48 20.63 0.00 4.95 7.22 11.39 13.29 16.76 18.35 0.00 2.42 6.89 8.95 12.73 14.46 2.31 4.50 8.57 10.44 0.00 4.31 6.31 10.02 0.00 2.11 6.05 2.02 3.96 0.00
2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 2 1 2 2 1 1
2059 2141 1426 1622 1589 1752 1779 1907 1987 2086 2207 2287 1623 1561 1746 1732 1884 1925 2046 2134 2231 1552 1744 1699 1873 1875 2019 2072 2188 1682 1868 1838 2003 2020 2156 2218 1867 1816 1994 1979 2135 2164 2295 1808 1990 1952 2122 2121 2269 1938 2116 2090 2253 2114 2070 2243 2229 2064 2238 2205 2193 2365 2362
4856 4672 7011 6165 6294 5708 5621 5245 5032 4793 4532 4373 6163 6405 5728 5775 5307 5194 4887 4685 4483 6443 5733 5886 5340 5332 4952 4827 4571 5944 5353 5439 4993 4952 4638 4509 5357 5507 5015 5052 4684 4621 4358 5531 5024 5124 4712 4715 4406 5161 4727 4786 4439 4731 4830 4459 4487 4845 4467 4535 4560 4229 4233
0.602 0.579 0.869 0.764 0.780 0.708 0.697 0.650 0.624 0.594 0.562 0.542 0.764 0.794 0.710 0.716 0.658 0.644 0.606 0.581 0.556 0.799 0.711 0.730 0.662 0.661 0.614 0.599 0.567 0.737 0.664 0.674 0.619 0.614 0.575 0.559 0.664 0.683 0.622 0.626 0.581 0.573 0.540 0.686 0.623 0.635 0.584 0.585 0.546 0.640 0.586 0.593 0.550 0.587 0.599 0.553 0.556 0.601 0.554 0.562 0.565 0.524 0.525
1181 1208 920 822 986 921 1064 1035 1156 1158 1259 1288 815 984 898 1055 1000 1138 1115 1233 1238 984 886 1050 974 1125 1079 1212 1195 1048 958 1118 1052 1197 1159 1288 953 1114 1033 1187 1130 1271 1238 1113 1023 1181 1108 1258 1208 1178 1095 1250 1184 1090 1245 1168 1320 1244 1160 1314 1311 1231 1228
8470 8278 10 864 12 163 10 140 10 858 9396 9663 8650 8635 7942 7767 12 264 10 162 11 139 9480 9999 8788 8968 8111 8077 10 167 11 289 9525 10 263 8886 9264 8248 8367 9543 10 433 8947 9508 8351 8629 7765 10 492 8979 9683 8422 8853 7869 8076 8989 9775 8466 9025 7947 8280 8487 9134 8000 8445 9171 8030 8560 7575 8039 8620 7612 7631 8121 8146
E22 (eV) 1.050 1.026 1.347 1.508 1.257 1.346 1.165 1.198 1.072 1.071 0.985 0.963 1.521 1.260 1.381 1.175 1.240 1.090 1.112 1.006 1.001 1.261 1.400 1.181 1.272 1.102 1.149 1.023 1.037 1.183 1.294 1.109 1.179 1.035 1.070 0.963 1.301 1.113 1.201 1.044 1.098 0.976 1.001 1.114 1.212 1.050 1.119 0.985 1.027 1.052 1.132 0.992 1.047 1.137 0.996 1.061 0.939 0.997 1.069 0.944 0.946 1.007 1.010
aAs predicted by the empirical formulas in the text. b The value of mod(n - m, 3). c E 11 values represent emission peaks; to obtain absorption peak energies, add ca. 0.004 eV.
We suggest that these empirically based spectral predictions should be used in preference to model-based results because of their significantly higher accuracy. Methods and Results. In our prior report, spectral transitions for 33 semiconducting species were fit semiquan1236
titatively using simple functions of dt and R to illustrate systematic structural trends. By allowing the exponents of cos(3R) and dt to vary from their original values of 1 and 2, respectively, in the second term, we have now deduced expressions that reproduce the experimental data much more Nano Lett., Vol. 3, No. 9, 2003
closely. As before, different parameters apply according to whether mod(n - m, 3) equals 1 or 2 (mod 1 or mod 2 species, respectively). The refined empirical functions for first and second van Hove transition frequencies are (for dt in nm) νj11(mod 1) )
[cos(3R)] 1 × 107 cm-1 - 771 cm-1 157.5 + 1066.9dt d 2.272 t
νj11(mod 2) )
-1
[cos(3R)] 1 × 10 cm + 347 cm-1 157.5 + 1066.9dt d 2.129 7
t
νj22(mod 1) )
-1
[cos(3R)] 1 × 10 cm + 1326 cm-1 145.6 + 575.7dt d 1.809 7
t
νj22(mod 2) )
[cos(3R)] 1 × 107 cm-1 - 1421 cm-1 145.6 + 575.7dt d 2.497 t
1.374
(1a) 0.886
(1b) 0.828
(2a) 1.110
(2b)
These expressions mimic our experimental findings with average errors of 10 cm-1 (1.3 meV) for νj11 and 40 cm-1 (5 meV) for νj22. Compared to the previous fit with integer exponents, these expressions eliminate systematic fitting discrepancies and reduce average errors by approximately 70 and 40%. It seems likely that this accuracy will be retained on extrapolation to larger-diameter tubes, for which trigonal warping and curvature exert smaller spectral effects. Although extrapolation to smaller tube diameters is less secure, we expect the expressions to be useful down to ca. 0.5 nm. Table 1 lists all 127 semiconducting nanotube structures having diameters between 0.48 and 2.00 nm (assuming a C-C distance of 0.144 nm) in order of increasing n and m values. Following the columns that show the diameter, chiral angle, and mod(n - m, 3) values, we give the wavelengths, frequencies, and photon energies for first van Hove emission and second van Hove absorption as predicted by the equations above. Note that first van Hove absorption energies will be ca. 4 meV higher than the listed emission values. In Figure 1, we plot the predicted and measured E11 and E22 values for semiconducting SWNT with diameters between 0.48 and 2.5 nm. A small set of measured values for E33 and E44 are also plotted. Empirical expressions for the energies of E33 and E44 transitions, which can also be important in resonance Raman studies of larger-diameter nanotubes, await further experimental measurements and analysis. Discussion. Equations 1 and 2 are intended only as empirical fitting functions, and we do not suggest that the deduced parameters have any simple meaning within SWNT electronic structure models. Instead, the peculiar exponents deduced from fitting the data may reflect combinations of effects such as trigonal warping, curvature, and exciton binding.10-14 Our fits and predictions apply to samples of individual SWNT in aqueous SDS suspension. Data measured by Lebedkin et al.9 and in our laboratory suggest that small systematic spectral shifts of less than 2% can be Nano Lett., Vol. 3, No. 9, 2003
Figure 1. Optical transition energies vs diameter for semiconducting SWNT. Solid symbols are experimental data; open squares and circles are predictions of E11 and E22, respectively, from the empirical fitting functions.
Figure 2. Comparison between model-based and empirical values of optical transition energies for semiconducting SWNT. Open symbols were computed using a simple tight binding (TB) model with γ0 ) 2.90 eV, and solid symbols were obtained from the empirically based fitting functions.
expected for individual SWNT suspended in other aqueous surfactants. Note, however, that our spectral fits are not applicable to bundled suspended nanotubes, for which strong perturbations between tubes lead to significantly broadened and red-shifted absorption spectra and quenched band-gap luminescence.7 Although Figure 1 qualitatively resembles model-based results, there are significant quantitative differences. To illustrate these, Figure 2 shows plots of E11 and E22 computed from a simple tight-binding model (as commonly used for Kataura plots) overlaid with the empirically based values from Figure 1. Both for E11 and E22 transitions, the tight1237
binding model badly underestimates the apparent scatter arising from chiral variations at a given diameter. Empirically, these chiral variations in transition energy are at least twice as large as predicted by the tight-binding model for tube diameters of 1.5 nm or less. It is also clear that the tight-binding model (with γ0 ) 2.90 eV) underestimates E11 transition energies by up to 25% relative to the empirical values. Tight-binding E22 predictions are ca. 15% below the empirical values for 2.5-nm diameter tubes but as much as 25% aboVe empirical values for some of the smallestdiameter species. Thus, adjusting γ0 cannot bring the simple model predictions into accord with the empirically based transition energies. In our view, these deficiencies in modelbased Kataura plots appear large enough to cause serious interpretation problems in some optical nanotube investigations. We therefore recommend that researchers performing optical experiments with surfactant suspensions of SWNT instead rely on the findings reported here to guide their work. Acknowledgment. This research has been supported by the NSF (grants CHE-9900417 and CHE-0314270), the NSF Center for Biological and Environmental Nanotechnology (grant EEC-1008007), and the Robert A. Welch Foundation (grant C-0807). We thank the Carbon Nanotechnology Laboratory at Rice University for assistance in sample preparation and C. Kane for tight-binding model calculations. Supporting Information Available: We present as Supporting Information a version of Table 1 extended to include nanotube structures through (25, 24) and the Eii
1238
values measured to date in room-temperature aqueous SDS suspensions. This material is available free of charge via the Internet at http://pubs.acs.org. References (1) Saito, R.; Dresselhaus, G.; Dresselhaus, M. S. Physical Properties of Carbon Nanotubes; Imperial College Press: London, 1998. (2) Hamada, N.; Sawada, S.; Oshiyama, A. Phys. ReV. Lett. 1992, 68, 1579-1581. (3) Rao, A. M.; Richter, E.; Bandow, S.; Chase, B.; Eklund, P. C.; Williams, K. A.; Fang, S.; Subbaswamy, K. R.; Menon, M.; Thess, A.; Smalley, R. E.; Dresselhaus, G.; Dresselhaus, M. S. Science 1997, 275, 187-191. (4) Dresselhaus, M. S.; Dresselhaus, G.; Jorio, A.; Souza Filho, A. G.; Pimenta, M. A.; Saito, R. Acc. Chem. Res. 2002, 35, 1070-1078. (5) Kuzmany, H.; Plank, W.; Hulman, M.; Kramberger, Ch.; Gruneis, A.; Pichler, Th.; Peterlik, H.; Kataura, H.; Achiba, Y. Eur. Phys. J. B 2001, 22, 307-320. (6) Kataura, H.; Kumazawa, Y.; Maniwa, Y.; Umezu, I.; Suzuki, S.; Ohtsuka, Y.; Achiba, Y. Synth. Met. 1999, 103, 2555-2558. (7) O’Connell, M.; Bachilo, S. M.; Huffman, C. B.; Moore, V.; Strano, M. S.; Haroz, E.; Rialon, K.; Boul, P. J.; Noon, W. H.; Kittrell, C.; Ma, J.; Hauge, R. H.; Weisman, R. B.; Smalley, R. E. Science 2002, 297, 593-596. (8) Bachilo, S. M.; Strano, M. S.; Kittrell, C.; Hauge, R. H.; Smalley, R. E.; Weisman, R. B. Science 2002, 298, 2361-2366. (9) Lebedkin, S.; Hennrich, F. H.; Skipa, T.; Kappes, M. M. J. Phys. Chem. B 2003, 107, 1949-1956. (10) Saito, R.; Dresselhaus, G.; Dresselhaus, M. S. Phys. ReV. B 2000, 61, 2981-2990. (11) Reich, S.; Thomsen, C. Phys. ReV. B 2000, 62, 4273-4276. (12) Ichida, M.; Mizuno, S.; Tani, Y.; Saito, Y.; Nakamura, A. J. Phys. Soc. Jpn. 1999, 68, 3131-3133. (13) Ando, T. J. Phys. Soc. Jpn. 1997, 66, 1066-1073. (14) Kane, C. L.; Mele, E. J. Phys. ReV. Lett. 2003, 90, 207401/1-4.
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