This Demonstration plots the observables relevant for understanding the optical properties of graphite and related materials (carbon nanotubes) over the hexagon of the 2D Brillouin zone (BZ) of a single graphitic sheet (graphene):
• The constant energy contour plot of the 2D electronic band structure, which shows a trigonal structure around the points, where the bands cross the Fermi level (energy eV)
• The dipole vector field for vertical transitions (a real vector); it rotates clockwise around the points, counterclockwise around the points, and is zero at the point , where an optical transition is forbidden
• The oscillator strength contour plot, which is the square modulus of the dipole vector and is higher at the edges of the hexagonal 2D BZ—at the edge midpoints or points and at the corners and —and zero at the point
• The optical absorption intensity, whose maxima and minima rotate by the azimuthal angle of the polarization vector of incident light; this shows that there are nodes in the optical absorption intensity around the and points, because of the 2D profile of the dipole vector field, which is a peculiarity of graphitic materials
The calculated functions are based on a tight binding model for the electronic structure of graphene, as explained in the references in the Details section.
The calculations and plots given in this Demonstration were implemented and reproduced from the equations in the following references:
A. Grüneis, Resonance Raman Spectroscopy of Single Wall Carbon Nanotubes, Ph.D. thesis, Tohoku University, 2004.
A. Grüneis, R. Saito, Ge. G. Samsonidze, T. Kimura, M. A. Pimenta, A. Jorio, A. G. Souza Filho, G. Dresselhaus, and M. S. Dresselhaus, "Inhomogeneous Optical Absorption around the K Point in Graphite and Carbon Nanotubes," Phys. Rev. B67(16), 2003 165402.
R. Saito, A. Grüneis, Ge. G. Samsonidze, G. Dresselhaus, M. S. Dresselhaus, A. Jorio, L. G. Cançado, M. A. Pimenta, and A. G. Souza, "Optical Absorption of Graphite and Single-Wall Carbon Nanotubes," Appl. Phys. A78(8), 2004 pp. 1099–1105.
R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Physical Properties of Carbon Nanotubes, London: Imperial College Press, 1998.