Raman spectroscopy of graphene and carbon nanotubes

Abstract

This paper reviews progress that has been made in the use of Raman spectroscopy to study graphene and carbon nanotubes. These are two nanostructured forms of sp² carbon materials that are of major current interest. These nanostructured materials have attracted particular attention because of their simplicity, small physical size and the exciting new science they have introduced. This review focuses on each of these materials systems individually and comparatively as prototype examples of nanostructured materials. In particular, this paper discusses the power of Raman spectroscopy as a probe and a characterization tool for sp² carbon materials, with particular emphasis given to the field of photophysics. Some coverage is also given to the close relatives of these sp² carbon materials, namely graphite, a three-dimensional (3D) material based on the AB stacking of individual graphene layers, and carbon nanoribbons, which are one-dimensional (1D) planar structures, where the width of the ribbon is on the nanometer length scale. Carbon nanoribbons differ from carbon nanotubes is that nanoribbons have edges, whereas nanotubes have terminations only at their two ends.

PACS:

• 78.30.-j Raman Spectroscopy
• 61.48.Gh Structure of graphene
• 61.48.De Structure of nanotubes
• 71.38.-k Polarons and electron-phonam interactions

Keywords:

• resonance Raman spectroscopy
• carbon nanotubes
• graphene nanoribbons
• electron–phonon interaction

Acknowledgements

The MIT authors acknowledge the support under NSF Grant DMR 10-04147. A.J. acknowledges the financial support from the Brazilian agencies CNPq, CAPES and FAPEMIG. R.S. acknowledges a Grant-in-Aid (No. 20241023) from the MEXT, Japan. We would like to thank the referee for his/her suggestions which improved the quality of this publication. For all figures reprinted from American Physical Society material readers may view, browse, and/or download material for temporary copying purposes only, provided these uses are for non-commercial personal purposes. Except as provided by law, this material may not be further reproduced, distributed, transmitted, modified, adapted, performed, displayed, published, or sold in whole or part, without prior written permission from the American Physical Society.

Notes

The B atom in Figure 1 gives a brighter STM images than the A atom, since there are electronic energy bands for the B atom near the Fermi energy.

The G′ peak in the Raman spectra of sp² carbons is often called the 2D peak. It should be noted that the mechanisms involved in the 2D double resonance (DR) processes are different from those for the G′ peak which involves only two phonons. The G′ peak involves only two K point phonons, whereas the 2D feature arises from the second-order D that includes a DR D-band process involving a K point phonon and an elastic scattering process. In this review, we distinguish between the G′ and the 2D scattering process and the actual values of their frequencies.

A Stokes process is a terminology used to denote the loss of photon energy in a scattering process. Here the Stokes photoluminescence process is independent of the Stokes Raman process.

Time-dependent perturbation theory tells us that the amplitude of the wavefunctions for excited states oscillates as a function of time if the photon energy does not match the excited-state energy, which is the physical meaning of a virtual transition.

A notch filter is an optical filter that suppresses a specified range of energies of the incident light.

By connecting monochromators in a serial way, the resolution of a monochromator is significantly improved although the actual signal becomes increasingly weak. Furthermore, extra monochromators can be used as an energy-tunable filter to reject the elastically scattered light in contrast to the notch filters, which are wavelength-specific.

The solution of a forced damped harmonic oscillator is not generally a Lorentzian lineshape but approaches the Lorentzian function for . However, if ω _q approaches Γ _q , the lineshape departs from a Lorentzian function.

The designation π of the π band comes from its value of angular momentum which is 1.

Here mod denotes an integer function for evaluating the modulus for an (n, m) SWNT where we use the notation , and as an illustrative example. Some authors use Mod 1 and Mod 2 to denote a semiconducting nanotube depending on or 2. There is thus a one-to-one correspondence between type I (II) and Mod 2 (1) semiconducting nanotubes appearing in the literature, and this is clarified in Figure 11.

It should be noted that there is a logarithmic 2D van Hove singularity in the density of states of graphene at the saddle point of the energy band near the M points (center of the hexagonal edges) of the BZ.

In time-dependent perturbation theory, the mixing (or transition) of the excited states occurs as a function of time with some finite and often measurable probability. The virtual states are defined by such a linear combination of excited states with some probability. The probability for the occupation of an excited state can be large when the energy difference between the excited states and the energy of the external field is relatively small. In such a case, we can say that the transition is resonant with the excited state.

Here “real absorption” means that the photo-excited electron can be in the excited states for a sufficient time, for example, 1 ns, so that the electron can be probed in the excited state. A material can scatter photons in a virtual process.

It is noted that not all even (odd) vibration modes under inversion symmetry are Raman (IR)-active modes.

The two-phonon process involving one-phonon emission and one-phonon absorption does not contribute to the Raman spectra but rather gives a correction to the effective Rayleigh spectral process.

Here q is the real phonon wavevector, measured from the Γ point, while in defining q _DR, the k and k′ vectors are measured from the K point or alternatively, with respect to the K′ point.

It is only when crystalline disorder is present that the first-order q≠0 phonons can be observed, as discussed in Section 4.3.

From the matrix element , we can deduce another matrix element,

However, the integral in Equation (45a) can also be expressed by α terms 268.

The phonon amplitude is proportional to in which the temperature dependence of the amplitude is expressed by given by Equation (47).

It is noted that the minus sign corresponds to a symmetric wavefunction and that the plus sign corresponds to an anti-symmetric wavefunction.

A virtual state is a linear combination of real states. When a virtual state is close to an exciton state, the virtual state contains a large component of the exciton states. This is the reason for the approximation used in obtaining a representation for the virtual state.

A cutting line is defined by the 1D BZ of an SWNT in the 2D BZ of graphene 32 135 136.

Other formula for can be used here, too. The difference is within 1–3 cm⁻¹.

The subscript htt in T _htt denotes heat treatment temperature.

Here core electrons refer to 1s and σ electrons. The screening by π electrons is independently considered by the polarization function within the RPA (random phase approximation) 148 120. See Section 5.2.1 for further details.

The Bethe–Salpeter equation is independently solved for each value of κ. When we obtain E _ii values as a function of κ by solving the Bethe–Salpeter equation many times, then E _ii with different i values are adapted from the different κ value. Since the E _ii eigenvalues come from different cutting lines, there is no problem with the orthogonality of the wavefunctions.