Radial-breathing Mode Frequencies for Single-Walled Carbon Nanotubes of Arbitrary Chirality: First-Principles Calculatio
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0899-N07-25.1
Radial-breathing Mode Frequencies for Single-Walled Carbon Nanotubes of Arbitrary Chirality: First-Principles Calculations H.M. Lawler1, D. Areshkin2, J.W. Mintmire3 and C.T. White1 1
Chemistry Division, Naval Research Laboratory, Washington, DC 20375
2
Department of Chemistry, George Washington University, Washington, DC 20052
3
Department of Physics, Oklahoma State University, Stillwater, OK 74078 ABSTRACT
First-principles calculations are performed for the radial-breathing mode of all 105 single-walled carbon nanotubes within the rolling-geometry diameter range of 0.4 to 1.4 nm. The diameter dependence of the frequencies is analyzed in some detail, and compared with measurable parameters of bulk graphite. The frequencies are compared with those available from other first-principles work, and experimental studies. BODY The radial-breathing mode frequency of single-walled carbon nanotubes, ωRB, is critical to optical characterization, and its dependence on structure has been wellresearched. It is expected to diminish inversely with tube radius,1,2,3 and this behavior is approximately mapped to an in-plane elastic strain of a graphene sheet or bulk graphite. By implementing a density functional method exploiting the helical symmetry of the electronic system,4 we have calculated ωRB for the complete set of nanotube structures with diameters from 0.4 to 1.4 nm. This work may prove useful in correlating spectroscopic signatures with unique structure assignments. The calculations are all-electron and assume ideal, isolated, single nanotubes. They use a 7s3p Gaussian basis set and assign an orbital Bloch phase with application of the screw operation.5 The frozen-phonon displacements implemented are purely radial, and hence do not incorporate symmetry-allowed non-radial polarizations. The structures considered and the radial-mode frequencies are listed in Table I. The uncertainties in frequency are taken to be ± 2%. When the frequencies are fit to an inverse diameter law,
A
ω RB = d ,
(1)
we obtain A = 231 cm-1 nm. The bond distortions accompanying the formation of nanotubes require a strain energy, and this strain energy per atom decreases with the inverse square of the nanotube’s equilibrium diameter.6,7 Assuming that this behavior extends to the case of a continuous strain amplitude, a modification of Eq. 1 is expected:
A
ω RB = d +
B . d3
(2)
0899-N07-25.2
(4,1)
0.3588
661
(7,3)
0.6958
330
(12,0)
0.9395
241
(12,4)
1.129
203
1.137
199
(16,1)
1.293
173
(12,7)
1.303
177
(11,8)
1.293
174
(5,0)
0.3914
602
(9,0)
0.7046
318
(11,2)
0.9492
238
(14,1)
(3,3)
0.4068
584
(8,2)
0.7175
315
(7,7)
0.9492
241
(9,8)
1.153
199
(4,2)
0.4143
576
(6,5)
0.7468
310
(8,6)
0.9524
239
(13,3)
1.153
198
(15,3)
1.308
173
(5,1)
0.4359
548
(9,1)
0.7468
307
(9,5)
0.9620
236
(10,7)
1.159
196
(13,6)
1.317
174
(6,0)
0.4697
492
(7,4)
0.7550
303
(10,4)
0.9778
230
(11,6)
1.169
196
(17,0)
1.331
171
(4,3)
0.4762
491
(8,3)
0.7710
301
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