Characterization of nanographite and carbon nanotubes by polarization dependent optical spectroscopy
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Characterization of nanographite and carbon nanotubes by polarization dependent optical spectroscopy Alexander Grüneis,1 Riichiro Saito,1,2 Georgii G. Samsonidze,3 Marcos A. Pimenta,6 Ado Jorio,4,6 Antonio G. Souza Filho,4,7 Gene Dresselhaus,5 and Mildred S. Dresselhaus3,4 1
Department of Electronic Engineering, University of Electro-Communications, Tokyo, 182-8585, Japan 2 CREST, JST, Japan 3 Department of Electrical Engineering and Computer Science, 4Department of Physics, 5 Francis Bitter Magnet Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, U.S.A. 6 Departamento de Física, Universidade Federal de Minas Gerais, Belo Horizonte, MG 30123-970, Brazil 7 Departamento de Física, Universidade Federal do Ceará, Fortaleza, CE 60455-760, Brazil ABSTRACT The optical absorption for π electrons as a function of the electron wavevector k is investigated by first order perturbation theory in graphite and single wall carbon nanotubes (SWNTs). The matrix element connecting two states in the valence and conduction bands is found to be significantly anisotropic in k-space and polarization dependent. In the case of graphite, the absorption shows a node around the equi-energy contour, and in the case of SWNTs we obtain selection rules that allow only transitions between certain pairs of subbands. The strength of the optical absorption is not only diameter dependent but also chirality dependent. The implications of the optical absorption matrix element on the resonant conditions are discussed. INTRODUCTION For application as a possible nanometer sized semiconducting device, the rapid and non-destructive characterization of single wall carbon nanotubes (SWNTs) is essential to determine their electronic structure or chirality.[1] Resonance Raman spectroscopy was found to fulfill these requirements. [2] In the Raman process, two kinds of interactions take place: one is the electron-photon interaction, which is the topic of this paper, and the other one is the electron-phonon interaction. The electron-photon interaction creates a hole and an electron at a k state in the valence and in the conduction band, respectively. Because the wavevector of light is almost zero compared to the dimensions of the first Brillouin zone (BZ), optical transitions occur vertically, conserving k. In the case of graphite, the transition between the π and the π* band is relevant for visible photon energies. In SWNTs the electronic bands can be obtained by a zone-folding procedure which essentially means cutting through the graphite bands along N lines in k space which are exclusively determined by the chiral indices (n,m) of a SWNT.[3] As a result, we get N valence and N conduction bands where 2N is the number of carbon atoms in the unit cell of the nanotube. Because all these bands are one dimensional (1D), a large number of optical transitions can occur, in principle, i.e. from one given valence band state to N conduction band states, if we do not consider selection rules. In the following section we show which of these
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