One-phonon resonant Raman scattering in graphene

One-phonon resonant Raman scattering in graphene – effect of strain

Fig. 1. Left: Atomic structure of graphene. Right: part of the Brillouin zone of graphene with K and K’ special points; the circles are cross-sections of the Dirac cones; θ is the angle of the strain direction relative to the zigzag of carbon bonds (Z direction); θ = 30° is the direction of the armchair line of bonds (A direction).

The Raman scattering in graphene is always resonant because for all laser excitations in the visible range there are pairs of valence and conduction states matching the laser photon energy. The relevant states belong to the conduction and valence bands, crossing at the Fermi energy and forming conic surfaces in space, so-called Dirac cones.

The calculation of the Raman spectrum of graphene is performed within a non-orthogonal tight-binding (NTB) model applied to the electronic band structure [1,2], phonon dispersion [3], electron-photon and electron-phonon matrix elements [4], electronic broadening parameter [5], and the quantum-mechanical expression for the intensity [6]. This approach was originally applied to single-walled carbon nanotubes.

Graphene has a single one-phonon Raman band, called the G band. It originates from the Raman-active in-plane optical phonon of symmetry E_2g. The Raman shift does not change with the laser excitation, while the intensity of the G band is an increasing function of the laser excitation. The shift and intensity of the G band do not change with the change of the angles of the incident and scattered light [7].

This is no longer the case in the case of uniaxially strained graphene. In order to study the polarization dependence of the G band, let and are the polarization vectors of the incident and scattered light for backscattering geometry, and is the polarization angle of the Raman-active phonon. Then, the light polarization vectors are,, and the polarization vectors of the two components of the E_2g phonon are and . It is convenient to define the angles with respect to the strain direction, i.e., , , and . In strained graphene, the G band splits into two components: G⁻and G⁺, which are strictly longitudinal and transverse with respect to the strain direction, i.e., . Then, substituting the polarization vectors in , the polarization dependence of the intensity of the two components becomes [7]:

References:

1. V. N. Popov, New J. Phys. 6 (2004) 17.

2. V. N. Popov and L. Henrard, Phys. Rev. B 70 (2004) 115407.

3. V. N. Popov and Ph. Lambin, Phys. Rev. B 73 (2006) 085407.

4. V. N. Popov, L. Henrard, and Ph. Lambin, Phys. Rev. B 72 (2005) 035436.

5. V. N. Popov and Ph. Lambin, Phys. Rev. B 74 (2006) 075415.

6. V. N. Popov and Ph. Lambin, Phys. Rev. B 73 (2006) 165425.

7. V. N. Popov, L. Henrard, and Ph. Lambin, Carbon 54 (2013) 86.

Fig. 2. The calculated Raman intensity of the G⁻and G⁺ components of the G band (symbols) vs light polarization angle ϕ for parallel light polarization, for several strain directions θ. The lines are best fits of the theoretical equations (see text).

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Valentin Popov

September 21, 2014