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J. Phys. Chem. C 2010, 114, 12446–12450
Terahertz Spectroscopy of Single-Walled Carbon Nanotubes in a Polymer Film: Observation of Low-Frequency Phonons Sunil Kumar,† N. Kamaraju,† B. Karthikeyan,† M. Tondusson,‡ E. Freysz,‡ and A. K. Sood*,† Department of Physics and Center for Ultrafast Laser Applications, Indian Institute of Science, Bangalore 560 012, India, and UniVersity of Bordeaux 1, CPMOH, UMR CNRS 5798, 351, Cours de la liberation, 33405 Talence cedex, France ReceiVed: April 6, 2010; ReVised Manuscript ReceiVed: June 12, 2010
We investigate the dielectric response of single-walled carbon nanotubes dispersed in poly(vinyl alcohol) matrix by using terahertz time domain spectroscopy. Frequency-dependent real and imaginary parts of the complex dielectric function are measured experimentally in the terahertz regime. The low-frequency phonons of carbon nanotubes, though predicted theoretically, are directly observed for the first time at frequencies 0.26, 0.60, and 0.85 THz. Further, a broad resonance is observed at 1.15 THz associated with the longitudinal acoustic mode of vibration of straight-chain segments of the long polymeric molecules in the film. The latter is observed at 1.24 THz for a pristine polymer film and has been used to derive the size of crystalline lamellae in the film. I. Introduction In recent years, terahertz spectroscopy using picosecond and sub-picosecond terahertz pulses has found widespread applicability, particularly in spectroscopy of solid-state materials,1–8 gases,9 liquids,10–12 and biological systems,13–16 in the spectral range of 0.1-10 THz. Time-domain terahertz spectroscopy (THz-TDS) has been successfully applied to probe the infraredactive low-energy phonons in crystalline CdTe and ZnTe,4 nanostructures of ZnO and ZnS,5,6 vibrational modes of polymers,7 saccharides,8 and biomolecules13–15 by measuring the real and imaginary parts of the dielectric function. Thin films of single-walled carbon nanotubes (SWNTs) have also been characterized in the THz range.17–20 A broad spectral feature observed between 2 and 6 THz in the measured dielectric function has been attributed to low-frequency electronic excitations across curvature-induced band gap in the nanotubes.18–20 The above studies have motivated the present experiments on SWNTs embedded in a polymer matrix to explore if it is possible to observe the low-frequency vibrational modes of carbon nanotubes predicted theoretically21–27 but not observed directly in absorption or scattering experiments so far. The low-frequency phonons have major contributions to the carrier scattering and electrical conductivity, and hence they are important to be studied for potential application of carbon nanotubes in optoelectronics. An isolated SWNT has four acoustic phonons with frequency ω f 0 as wave-vector q f 0. There has been a controversy on the nature of these long wavelength phonons of carbon nanotubes. In force model calculations, it was shown that the four modes, classified as one longitudinal, two transverse, and one torsional (i.e., rigid rotation around the tube axis), have ω ∝ q.21–24 Another set of calculations25–27 predicted one longitudinal (ω ∝ q), one torsional (ω ∝ q), and two flexural modes (the entire tube oscillates sinusoidally) (ω ∝ q2). This discrepancy of not having flexural * To whom correspondence should be addressed. Phone: (91) 80-22932964. Fax: (91) 80-2360-2602. E-mail:
[email protected]. † Indian Institute of Science. ‡ University of Bordeaux 1.
modes in the first set of calculations21–24 has been traced to the potential functions that violate symmetry rules.27 Mahan and Jeon27 have pointed out that phonons in nanotubes have two quantum numbers: the wave vector q along the tube axis and another quantum number, R, related to the angular dependence around the tube axis. The longitudinal and torsional acoustic modes have R ) 0, and the flexural modes have |R| g 1. Two flexural modes with R ) (1 have ω f 0 as q f 0, but higher order modes (R ) (2, (3, (4, etc.) have finite frequency at q ) 0. The latter are relevant to understand THz absorption experiments. For a (10,10) nanotube, the frequencies of the first few flexural modes at q ) 0 are 0.65 THz (R ) (2), 1.86 THz (R ) (3), and 3.58 THz (R ) (4).25,27 These low-frequency phonons (
