~ ) Solid State Communications, Vol. 87, No. 3, pp. 219-222, 1993. Printed in Great Britain. 0038-1098/93 $6.00+. 00 Pergamon Press Ltd COLLECTIVE RESONANCES OF CARBON ONIONS P. Apell and D. ()stling Department of Physics, Chalmers University of Technology and University of GSteborg, S-412 96 GSteborg, Sweden G. Muk ihyay Department of Physics, Indian Institute of Technology, Bombay-400 076, India (Received 4 March 1993 by B. Lundqvist) (accepted for publication 20 April 1993) Based on a simple equation of motion for the induced density in a spherical carbon particle we demonstrate the existence of a rich spectrum of collective resonances as a function of the number of shells for n = 1 (C6o), 2, 10 and 40. We also present results for the photoabsorption cross section and discuss the nature of the collective resonances by examining the eigenmodes of the systems. 1 Introduction The latest development in fullerene related research has been prompted by the synthesis of spherical concentric graphitic shells (carbon onions) by Ugarte 1 and coa0dal graphite tubes by Ijima2. The carbon onions can have up to 50-70 shells with diameters up to ,-- 5Ohm and the tubes can have lengths up to prn. Earlier there has been a large interest in the opti- cal and collective properties of the by now well known C60 molecule. Both theoretical3'4's and experimental s'v's studies have been presented. Electron Energy Loss Spec- troscopy (EELS) spectra have best displayed the collec- tive resonances of Cs0 samples. In an earlier study9 we discussed the plasmons of Cs0 using a theoretical model developed by Mukhopadhyay and Lundqvist1°, which em- phasizes the classical electrodynamics and geometrical properties of the system studied. Since the new carbon onions consist of concentric graphitic shells, where the in- nermost shell is believed to be a (76o shell 11, it is natural for us to extend our earlier discussion of collective reso- nances of Css to collective resonances of carbon onions. Due to the present lack of experimental data for the car- bon onions as well as for the tubes we here confine our aim to a systematic theoretical study of characteristic collective excitations of carbon onions of various sizes, to be observed in various spectroscopies using photons or electrons as probes. We defer the treatment of the tubes to a later paper 12 since in most respects the results are very similar to those of the carbon onions. 2 Theory Mukhopadhyay and Lundqvist1° showed how to separate out the dominantly collective part of the density response in an electron gas subjected to an external potential. Their work makes it possible to discuss properties, like density oscillations, where the details of single particle excitation energies and wave functions can be neglected. Using this formulation we get the following equation of motion for the induced density Pied when an electron gas is exposed to an external field with the potential V,.~: , (1) where V(x, w) - V~t (x, w)+Vi,a(x, w), w is the frequency of V,.,, w~(x) = 4~re2p0(x)/m defines the local plasmafre- quency through the ground state electron density po(x) and V/.a is the potential from the induced charge density Pi,,. Letting Ve,~ = 0 in Eq. (1) we get an equation for the self oscillations of the electron gas. Due to the spher- ical symmetry of the carbon onions we use the following ansatz for the ground state electron density of an onion with n shells n po(r) = ~ p,O(r - o~)e(b, - r) , (2) i=1 where ai (hi) is the radius of the inner (outer) surface of the i~ concentric shell to the center of the onion and p~ is the constant electron density in the i~ shell. In our further discussion we will take pl -- p (constant) for all shells, which is supported by Kroto's observation11 that the onions have 60n 2 atoms in their n ~ shell. Since n is proportional to r, the number of electrons in a shell will therefore scale as r 2. Assuming an equal shell thickness for all shells in the onion, the volume of a shell with increasing r will also scale as r 2. Therefore, we assume 219