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Nonlinear optical response of fullerenes
Prog. Crystal Growth and Charact. Vol. 34, pp. 81-93, 1997 © 1997 Published by Elsevier Science Ltd Printed in Great Britain. All rights reserved
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PII: S 0 9 6 0 - 8 9 7 4 ( 9 7 ) 0 0 0 0 6 - 5
NONLINEAR OPTICAL RESPONSE OF FULLERENES Kailash C. Rustagi*, S e l v a k u m a r V. Nair* and L a v a n y a M. R a m a n i a h t * Laser Programme, Theoretical Physics Division, Indore452 013, India I~Centre for Advanced Technology, BhabhaAtomic Research Centre, Bombay 400 085, India
ABSTRACT A critical review of nonlinear optical properties of fullerenes is presented. We explore structure property relationships for optical nonlinearities of these unusual ~r electron systems from two complementary angles: (i) to explore the possibility of obtaining structural information from nonfinear optical experiments and (ii) to assess the potential of fullerenes, fullerites and their derivatives for the practical nonlinear optical devices. In particular we show that substituted fullerenes form an interesting class of molecules for nonlinear optics. We will also show that third harmonic generation of circularly polarised light in the solid fullerites should yield some information on the intermolecular interactions in these solids. We critically examine the various calculations indicating that the screening of the optical fields due to electron-electron interaction may reduce the optical susceptibilities of these molecules by a large factor and suggest experiments that may clarify the issue. KEYWORDS
Nonlinear Optics, r electron systems, Third Harmonic Generation. INTRODUCTION Soon after the discovery (Kratschmer et al., 1990) of a method of making macroscopic quantities of Buckminsterfullerene, the attention of the nonlinear optics community was drawn to their potential as nonlinear materials (Hoshi et al., 1991; Blau et al., 1991; Nair et al., 1991; Wang and Cheng, 1992; Rustagi, 1993). This interest arose from the fact that conjugated organic molecules and polymers are known to owe their large non-resonant optical nonlinearities to their electrons delocalized in one (Rustagi and Ducuing, 1974; Hermann, 1974) or two dimensions (Chemla and Zyss, 1987). Thus, fullerenes with their three-dimensional 7r-electron systems offer an additional avenue in the search for more efficient nonlinear optical materials. Vigorous experimental (Hoshi et al., 1991; Blau et al., 1991; Wang and Cheng, 1992; Kafafi et al., 1992a; Kafafi et al., 1992b; Gong et al., 1992; Yang et al., 1992; Flora et al, 1992; Rosker et al., 1992; Zhang et al., 1992; Vijaya et al., 1992; Tang et al., 1993; Kajzer et al., 1992; Meth et al., 1992; Neher et al., 1992; Wang et al., 1992; Tuft and Kost, 1992; Joshi et al., 1993a,b; Mishra et al.) and theoretical (Rustagi et al., 1993; Wang et al., 1991; Talapatra et al., 1992; Rosen and Westin, 1993; Harigaya and Abe, 1992; Matsuzawa and Dixon, 1992; Ramaniah et al., 1993; Shuai and Bredas, 1992; Ramaniah et al. 1994) research over the last two years or so has produced several surprises. In this article we summarise our present understanding of this effort in the hope that this will sharpen some of the questions that need to be 81
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addressed. We begin with a brief description of the main experimental findings. This is followed by a review of the theoretical attempts to understand the nonlinear optical response of these molecules and the corresponding solids. So far almost all experiments have been performed on C ~ and a few on C~0. Theoretical models should, however, also try to predict the nonlinear optical response of other fullerenes and fullerene derivatives. As more molecules of this family become available, the comparison of experimental and theoretical results is expected to lead to a better understanding of the relation of the nonlinear response of fullerenes to the geometrical structure of their 7r-electron clouds. EXPERIMENTAL OBSERVATIONS
The main nonlinear optical processes investigated in C60 or Cz0 are degenerate four wave mixing (DFWM) (Blau et al., 1991),Kafafi, (Kafafi et al, 1992b; Gong et al., 1992; Yang et al., 1992; Flom et al, 1992; Rosker et al., 1992; Zhang et al., 1992; Vijaya et al., 1992; Tang et al., 1993), third harmonic generation (THG) (Hoshi et al., 1991; Kajzer et al., 1992; Meth et al., 1992; Neher et al., 1992), and optical limiting (Tutt and Kost, 1992; doshi et al., 1993a,b; Mishra et al.). The first two can occur without any excitation of the medium and can be described in terms of a perturbative expansion of the polarization, i.e., the induced electric dipole density, P. Following the standard notation, we write p = p(U + pO} + p(3) + " ' ' ,
(1)
where p(~/ is of order n in the electric field of the incident wave. The Fourier component p(3)(fl) at frequency f~ is related to E(co0, that of the electric field through
P(,Zl(a) = N~ ,,,(3) , ~ , / _ oo,, co,, ~ , ~'3)E~(col)E~(co~)F,(~a),
(2)
where ~ = ~ol + co2 + w3 and Nd is the number of distinct permutations of cot, w2 and ~z. The susceptibility tensor X(z) describes the material response in the third order. The two processes THG and DFWM are described by the tensors X(3)(-3co, co, co, co) and X(3)(-~o,~', -co, 0J) respectively. We recall that the Fourier component at -co is the complex conjugate of that at ~o, i.e., E(-co) = E*(co) and similarly for P(co) etc. For homogeneous molecular solids the susceptibilities are related to the molecular polarizabilities or, t3 and 7 through
~(')(co) = :~L(~),~(co),
(3)
~(2)(-(col + w2),~1,~02) = NL(col + w2) L(coi ) L(co2)fl(-(col + co2), col, 0-12)
(4)
)~(3}(__(COl + ~.U2~t_ ~3), COl, CO2,cj3)
(5) (6)
and =
,~rL(co I _}_co2 + co3) L(gMI) L(o.)2) L(~3) x 7(-(col + ~'2 + co,~),~01,co2,cos)
where L(co), the local field correction factor is the ratio of the total electric field Eloc acting on the molecule to the macroscopic field E and N is the inolecular density. In addition to E, the local field includes the field generated by the induced dipoles at all the other molecules. For molecules occupying atleast a tetrahedral site L(co) = (e(a;) + 2)/3 (Armstrong et al., 1962). For fluids, additional contributions to X"(3) can arise due to the orientation of anisotropic molecules by the incident electric field. The response time for such nonlinearities is typically the inverse of the rotational frequency and is > 10 - I 2 s. For the most challenging applications of nonlinear optics namely ultrafast signal processing, it is desirable to have a large hyperpolarizability 7 which is due to virtual excitation of the electrons in the molecule. Third harmonic generation (THG) measures this directly and is the preferred method for the measurement of byperpolarizabilities of electronic origin, For Ceo such measurements have been performed by Hoshi et al. (1991), Meth et al. (1992)
Nonlinear Optimal Response of Fullerenes
83
and Kajzar et al. (1992) on thin films, and by Neher et al. (1992) in solution. The values of X(3) obtained by THG are shown in Table 1. They are reasonably consistent with each other when we allow for the fact that the measurements are relative to two different standards. The measurements for solutions are intrinsically less sensitive because a larger part of the signal comes from the solvent. For Cr0, the only values of 3'(-3w, w, w, w) avalable are those from solution experiments of Neher et al. (1992) giving 3' ~ 4 × 10 -33 esu in the nonresonant region, while for a fundamental wavelength of 1.064 #m they report {"/) = ( - 7 . 6 + i 5 7 ) × 10 -33 esu. Meth et al., Kajzar et al. (1992) and Neher et al. (1992) all report measurements of X(3) at several wavelengths. Resonance enhancement of THG can increase ] X(3) ] by as much as a factor of 20. Table 1. xt3)(-3w, w, w, w) measured by THG experiments.
(~m) 2.37 1.907 1.3 1.064 1.064 0.85
(X( 3 ) ) 4.1×10 -12 3.2×10 -H 5.1×10 -11 8.2×10 -la 2.0×10 -1° 1.5×10 -~1
Reference (Meth et al., 1992) (Kajzer et al., 1992) (Kajzer et al., 1992) (Kajzer et al., 1992) (Hoshi et al., 1991) (Kajzer et al., 1992)
The value of X(3) = 7.0 x 10 -1~ esu obtained by Kafafi et al. (1992a,b) using DFWM in thin films is consistent with the THG values in the non-resonant regime. It is interesting to observe that Kafafi et al. (1992a, b) estimated an absorption coefficient of 6 cm -1 at A = 1.064#m for their film while the typical value for organics is < 10~cm -1 (Stegman et al., 1989). Many other observations of the third order nonlinearity have been made by DFWM. They often report much larger values of 7While some of them can be associated with thermal or population transfer effects in the absorbing region, for several others this is not a plausible cause. Clearly more work is needed to clarify the situation. Indeed, a very recent paper by Tang eta[. (1993) reports that a very careful analysis of their DFWM results using 30 ps pulses at 1.06 # m yield values that are consistent with those of Kafafi et al. (1992a,b) and are much smaller than those reported by several other groups. One of the most interesting prospects is the use of fullerenes as optical limiters. Tutt and Kost (Tutt and Kost, 1992) observed optical limiting in C60-toluene solution at a relatively low threshold and attributed this to the reverse saturable absorption mechanism [RSA]. Subsequent work by our group (Joshi et al., 1993a,b) and others (Mclean et al, 1993) has shown that RSA is inadequate to explain the observed results in solutions. While a complete picture is yet to emerge it is strongly indicated that nonlinear scattering and self induced defocussing plays an impotant role (Mishra et al.). THEORETICAL
CONSIDERATIONS
S y m m e t r y of t h e P o l a r i z a b i l i t y T e n s o r s For any molecule, the various components of the polarizability tensors c~,/3, and 7, are related to each other by symmetry. The symmetry group of C60 is ]h and that of Cr0 is Dsh. Both these point groups are not among the crystallographic point groups and are thus not included in the standard tables of symmetry relations for nonlinear polarizabilities (Butcher, 1965). We have shown (Ramaniah et al., 1993) that c~ and 7 for C60 (Ih group) have structures identical to those of an isotropic system while for CT0 (Dsh group) their structure is the same as that for an inversion symmetric hexagonal system of D6h symmetry. Surprisingly, fl vanishes for Dsh syimnetry although this group does not contain the inversion operation. In contrast, for all the 21 non-centrosymmetric crystallographic point groups at least one component of fl is nonvanishing by symmetry. For C70, 3' has 21 non-zero components with l0 of them independent. However, when dispersion can be neglected, only three components of 3' are independent which may be taken to be 3'. . . . . 3'.... and 7 . . . . For C60, we find (Ramaniah et al., 1993) a ~ = c~yv = c ~ , i.e., same as for an isotropic or a cubic
84
K.C. Rustagi et aL
material, and 7 has 21 non-zero components, of which only 3 are independent, again as for an isotropic system. Under Kleinman symmetry only one component is independent with %~= = 3 % ~ and all the components related by an interchange or permutation of x, y, z are equal. We note that this is exactly valid for the static (zero-frequency) polarizabilities and provides a check on the theoretical calculations (See e.g, Ref. Shuai and Bredas, 1993). The fact that 7 of C6o has the same symmetry structure as that for an isotropic material offers an interesting apphcation of nonlinear optics viz., to probe the solid state effects on the electronic density of C60 (Ramaniah et al., 1993). This possibility arises because no crystalline material has a strictly isotropic nonlinear susceptibility )/(3). However, C60 forms a molecular solid so that one would expect that its X(a) can be obtained as a sum of the hyperpolarizabilities (7) of the individual molecules to a good approximation. Thus for solid C6o, although only those symmetry restrictions on X(3) demanded by its cubic crystallographic point group are strictly valid, X(3) will be isotropic to a good degree of accuracy. And, the deviation from isotropy of X(3) of solid C60 will give a measure of the inter-molecular interactions. In particular, we (Ramaniah et al., 1993) have suggested that since third harmonic generation from an isotropic medium vanishes for circularly polarized light (Bey and Rabin, 1967), the ratio of the third harmonic generated by circularly and linearly polarized light in Ceo solid will give a measure of the cubic perturbation of the molecular electron density due to inter-molecular interactions. Electronic Structure and Polarizabilities Starting from the early classic work of Platt (1953), it has been realized that the dipole polarizabilities c~,fl and ")' of conjugated r-electron systems are sensitive to the ~r-electron density p(r) and it is generally believed that an accurate representation of p(r) is necessary to obtain reliable estimates of the polarizabilities. Consequently, several sophisticated computational methods have been used to determine the electronic structure and polarizabilities of conjugated r-electron molecules (Chemla and Zyss, 1987). Such calculations, however, often involve not-so-transparent approximations. Only a few can be said to have reached such a level of reliability that their disagreement with experimental observations are taken with sufficient seriousness. On the other hand, simple free-electron or Hiickel molecular orbital theories have predicted significant structure-property connections which have been verified experimentally as well as by more elaborate theories. Similar simple models of the electronic structure of fullerenes have been used for obtaining valuable information oll the optical response of fullerenes (Nair et al., 1991),Wang, (Gallup, 1991; Saito et al., 1992; Savina et al., 1993). For C60, due to its high symmetry and the delocalized nature of the r-electrons, a natural extension of the free electron theory is possible and has been attempted by several groups. We present here the model of Nair and Rustagi (NR) (Nair et al., 1991; Rustagi et al., 1993) which appears most reasonable on physical grounds. There is a close parallel between our approach for conjugated molecules and the empirical pseudopotential method (EPM) (Cohen and Chelikowsky, 1988) used for the electronic structure of semiconductors. The essential point in the EPM is to describe the effective potential seen by valence electrons in terms of a few Fourier components of the lattice periodic potential. Ill C60, instead of linear translational invariance we have spherical symmetry in the empty-lattice limit. And in place of the lattice periodic potential in the solid we have a perturbation of icosahedra} symmetry in C~o. This pseudopotential can be described in terms of only 3 parameters as for the group IV semiconductors. We believe such an approach is complementary to the sophisticated numerical calculations of electronic structure and polarizabilities since it provides an understanding of the whole range of optical properties including absorption spectra and thus facilitates greater interaction between theory and experiment. Secondly, it suggests chemical means of increasing the nonlinearity in some cases. As in the empirical pseudopotential method (Cohen and Chelikowsky, 1988) for semiconductors, NR first make an "empty lattice" description for the C6o molecule and then treat the icosahedral pseudopotential perturbatively. The "empty lattice" in this case has spherical symmetry, and they take
NonlinearOptimal Responseof Fullerenes
85
this zeroth order model to be 60 r-electrons confined in a spherical shell of mean radius R = 3.55/~ and thickness t = 3A, consistent with other estimates of the electron density. The eigenfunctions for this system are (Nair, 1993) ¢=tm(r) = f.t(r)Ytm(O, ~), (7) with
j,(k.,n_) 'k ,"~
f~t(r) = N jt(k~tr) - yl(k.lR_)y~( .t ) ) ,
(8)
where jt and Yl are the spherical bessel and n e u m a n n functions and Ytm(0,~p) denote the spherical harmonics. N is the normalization constant, k.t's are determined by f(R+) = 0, and t ~ = R 4- t/2 are the outer and the inner radii of the shell. The energy levels are given by
E.e = h2k~l
(9)
2m0 The energy levels are well approximated by (Nalr, 1993)
E~ = (hV2m0)
(n27r 2 l(l 4- 1)'~ --h-- + R~ ]
(10)
n and l being the radial and angular quantum numbers. As shown in Fig.l, the n = 1 radial functions are nodeless whereas the n = 2 radial functions have a node near r = R. Since ideally the r-electrons have vanishingly small overlap with the a-electron skeleton, NR choose them to occupy the n = 2, l = 0 to 5 energy levels. The n = 1 levels which lie below the n = 2, l = 0 level have maximum charge density at r = R and are assumed to be filled by the a-electrons. As the (r-electrons are expected to have a much smaller radial spread than the r-electrons, the relative position of the a and r levels is not expected to be accurately depicted by the model potential having a uniform radial width. On the other hand, the total band width occupied by the a-electrons or the 7r-electrons does not depend on this width and is well depicted by the model potential. Later calculations (Yabana and Bertsch) using the spherical jellium model have shown that the core and the a-electrons indeed occupy n = 1, l < 12 levels with l = l l and 12 partially occupied. The 22-fold degenerate (including the spin degeneracy factor), l = 5 level is only partially occupied by l0 electrons so that the r-electron structure is sensitive to all perturbations which split this level. The perturbation due to the presence of the 60-vertex r-electron skeleton has icosahedral symmetry (group lh) which splits the l -- 5 level into 3 levels of tl~, t3~ and h. symmetry (Harter and Weeks, 1989). Since the radial wavefunctions are found to be nearly independent of l, the unperturbed states may be approximated by f~=2(r)Ylm(0, ~). The icosahedral perturbing potential can also be expanded into a series of spherical harmonics. Only those l values will occur in the expansion which contain the identity representation of the icosahedral group Ih. Since the levels close to the Fermi level have either I = 5 or l = 6, it suffices to retain only the I = 6, l0 and 12 components of the potential. The l = 0 term may be ignored because it is a constant. The next non-zero contribution would come from l = 16. Thus to see the effect of the icosahedral perturbation V in the n = 2 subspace one can write,
(l,~lVIl'm') =
A6( lmlV~ll',,~') + A,o(1,nlV, oll'm') + d,:(ImlV,2ll'm')
ill)
where V6, Vm and Vl2 are those linear combinations of I = 6, l0 and 12 spherical harmonics, respectively, which remain invariant under the operations of lh (Nair, 1993). The r-electron energy levels are then obtained by diagonalising the Hamiltonian numerically. A truncated basis set with l ~ 9 is used. The use of a rigid confining shell in the zeroth-order approximation makes the separation between energy levels of different I values somewhat larger than that expected from a more realistic sphericallyaveraged jellium potential (Y~bana and Bertsch). The effect of the icosahedral pseudopotential is
86
K.C. Rustagi et al.
mainly to split the multiplets of a given l. So NR fix the pseudopotential parameters such that the splitting of the l = 5 empty lattice level gives a HOMO-LUMO gap of 1.8 eV and the lowest electric-dipole allowed transition occurs at ~ 3.7 eV. This leaves one parameter free which is chosen to minimize the discrepancy with the next optical absorption peak. Inspire of the simplicity of the starting spherically symmetric potential, the calculated energy levels show reasonable agreement with the valence photo-ionization spectrum (Nair, 1993). The values of the parameters used are A6 = -0.035 Ry, A10 = 0.4 Ry and AI~ = 0.15 Ry. The calculated r-electron density (Fig.2) shows a concentration over the a-bond skeleton. The electron density per solid angle at the centre of a hexagon is less than 2 compared to about 5.5 at the centre of a pentagonal edge and about 6.5 at the centre of a bond shared by two hexagons. These estimates are in very good agreement with a recent LDA calculation (Yabana and Bertsch). It is interesting to note that in the spherical harmonic expansion for the electron density, although the leading correction to the 1 = 0, spherically symmetric term is the l = 6 term, its coefficient is substantially smaller than that of the next term with l = 10. A good approximation to the r-electron density can be obatained by retaining only the 1 = 0 and l = I0 terms which in the present model is 6O p.(r) = 47 - 3"881f2(r)l~v'°(O'~)' (12) where f2(r) is the normalized radial eigenfunction with n = 2 for tile empty lattice problem described above. The weak/-dependence of f2(r) is neglected, as mentioned earlier. There are a number of independent calculations of the electronic structure of Cs0 using the model described above. However, most of them make some unphysical assumptions. Saito et al. (1992) and Savina et al. (1993) take the r-electrons to occupy the n = 1 states of the spherical shell whereby their r-electron density lacks the required antisymmetry about tile nuclear position. The radial relectron density obtained by them is thus unphysical. Saito et al. (1992) also take the mean radius of the shell to be 4.05~ instead of the known value of 3.55~. Further, they apply this model to higher fullerenes of icosahedral symmetry. This is unreasonable as higher fullerenes are expected to have non-spherical structure with the carbon atoms lying at different distances from the centre. Gallup (1991) uses zero-range potentials at the nuclear positions to simulate the potential due to the ioncores and the a-electrons, and estimates its effect on the r-electro,Is using first order perturbation theory. The icosahedral potential is not weak enough to justify such a treatment and as a consequence their r-electron density deviates only marginally from sphericity, in contradiction to the results of more sophisticated calculations. Another simple model for describing the electronic structure of fullerenes is the tight-binding model. Unlike the nearly-free-electron model descibed above, the tight-binding method is more versatile since it can be easily applied to any geometry. A nmnber of tight-binding parametrization schemes for pure carbon structures have been formulated by various authors (Tomanek and Schluter, 1991; Menon and Subbaswamy, 1991; Xu et al., 1992). All these schemes obtained by detailed comparison with ab initio calculations, provide very satisfying results for the structure of many carbon clusters. Even relatively simple molecular dynamics calculations for fullerenes using a tight-binding model have been shown to provide reliable results (Wang et al., 1992). This versatility of the tight-binding method makes it very appropriate for calculating the polarizablity of fullerenes, especially for the higher and substituted fullerenes. In the tight-binding method for Cso the occupied valence electron states and the lowest few excited states are described by the 240 x 240 Hamiltonian matrix in the basis set provided by 240 orthogonal atomic like orbitals - - four (s,p~,p~,pz) at each atonfic site. Since only the nearest-neighbour interag.tions are retained the only parameters involved in the Hamiltonian are the atomic site energies E v and E, which are characteristic of the chemical species occupying the site and the hopping integrals V,,, V,v, Vpv, and Vp~ which are usually scaled to the corresponding interatomic distance (Harrison, 1980). Various tight-binding models used for fullerenes differ in the way in which the parameters are obtained. We describe calculations based on the scheme developed by Tomgnek and Schluter (TS)
Nonlinear Optimal Response of Fullerenes
87
(Tomanek and Schluter, 1991) by detailed comparison with LDA calculations for various crystalline and molecular structures of carbon. Using this model for determining the structure of carbon clusters Cn for n < 60, TS found that the cage geometry of fullerenes becomes more stable than one- and twodimensional structures for n > 20, in very good agreement with the experimental observations. The TS model was also used by Bertsch et al. (1991) to calculate the electronic structure and linear optical response of C6o within the random phase approximation (RPA) and they predicted an unusually large screening correction to the linear response. We defer a detailed description of screening in C6o to the next section. Using the TS model Wang et al. (1991) calculated the nonlinear polarizability of C6o to be X(3} 2 × 10-34esu, in good agreement with the result of NR. Later Ramaniah et al. (1994) used this model to calculate the electronic structure and nonlinear optical properties of boron and nitrogen substituted fullerenes (Cs9N and CsgB) (Rustagi et al., 1993). Since CsgB has 239 valence electrons and C59N has 241, both have partially-filled levels. The lowest gap in both the molecules is the impurity level splitting which is very small but the oscillator strength of this transition is negligible. In C59B the most important new electric-dipole transition is between the two impurity levels split-off from the 2 highest-filled C60 levels of hg and h~ symmetry, which we may refer to as the split-off hg and split-off h~ levels. In C59N the new strong transition is between the impurity levels split-off from the 2 lowest unfilled levels of C6o of tl~ and tlg symmetry. The corresponding transition energies are 0.9 eV and 0.6 eV in C59B and Cs9N respectively. The polarizabilities calculated for C6o using the pseudopotential model (Nair et al., 1991; Rustagi et al., 1993) and for substituted C60 using the tight-binding model (Ramaniah et al. 1994) reveal several interesting points. First we compare the numbers for C6o with those for a one-dimensional r-electron system : fl-carotene with 22 r-electrons delocalized over a zig-zag chain of about 24 bond lengths. For fl-earotene, the calculated values for the polarizabilities, in the free-electron model (Rustagi and Ducuing, 1974) are : c~z~ = 1.8 × l0 -~2 cm 3 and %z~ = 4.8 × l0 -33 esu, where z is the molecular axis. The contributions of c ~ , c~uy, "~. . . . and "/u~yu are expected to be much smaller because the r-electron extension perpendicular to the molecular axis is much smaller. Thus we may approximate (c~) ~ c~:/3 = 60 ~t3 so that (c~)/N~, the linear polarizability per r-electron is nearly the same for the two molecules. In contrast ("/)/N, is much smaller for C60. This is partly because 7 increases steeply with increasing delocalization length and one-dimensional r-electron systems provide a larger conjugation length per r-electron. From this view point tubules would be the preferred r-electron systems. However, the electronic structure of such molecules is still not well understood. Secondly, the hyperpolarizability of C60 is also reduced by the high symmetry of the molecule and consequent selection rules as well as by the fact that the HOMO-LUMO transition is parity-forbidden. This indicates that adding electrons to the tl~ (LUMO) state as in Cs9N or C~0 state or removing electrons from the h~ (HOMO) state as in C59B or C+0 will increase % The tight-binding calculations for CsgB and CsgN indeed confirm this. Tiros as for symmetric cyanines (Mehendale and Rustagi, 1979; Stevenson et al., 1988) perturbation of a r-electron system by chemical substitution appears to be an effective means of increasing the hyperpolarizability of a system if it is much below that of other equally polarizable systems. Further, the lack of inversion symmetry allows the doped molecules to have an electric dipole moment which is negative in CsgB and positive in C59N. The dipole moment is not sensitive to the difference between the long and short C-C bond lengths but does reduce noticeably when boron is moved out in CsgB. However, the values for the dipole moment obtained by Ramaniah et al. (Ramaniah et al. 1994) are about five times larger than those reported in Ref. (Andreoni et al., 1992). Secondly the tight-binding results for c~ and -~ are sensitive to the difference in the long and short bond lengths. In particular, 7 of C60 increases by 46% when the more realistic, experimentally determined, bond lengths are used. The boron displaced structure of C59B is found to have a relatively high nonresonant 7 ~ S x 10-33 esu. This is found to be 30 times larger than that estimated for tim same molecule if the atomic positions are assumed to be the same as in C60.
88
K.C. Rustagi et al.
Finally we note that in doped fullerenes, the hyperpolarizability 13 is large and relatively insensitive to the atomic positions. These values are comparable to the largest reported for organic molecules. Using the density and the local field correction corresponding to the solid C6o, this value of/3 corresponds to a nonlinear susceptibility X(~) of 10-s esu for both CssB and CssN. This will reduce by a factor of ~ 10 by screening but will increase somewhat ~lue to dispersion. Interestingly, while in the one-dimensional case the HOMO-LUMO gap is strongly affected by the bond alternation, for Cs0 the gap is not sensitive to bond alternation.
Screening In the above discussion, screening due to electron-electron interaction has not been considered and all the polarizability values presented are the unscreened or bare values i.e., it has been assumed that the field seen by electrons in a fullerene molecule is the same as the one applied to it. In reality this is not so, because when the electron cloud is polarized, it also modifies the charge density and hence the self-consistent potential seen by each electron. Bertsch et al. (1991) used the RPA neglecting the radial spread in electron density to estimate that for C60 the effect of including this screening correction is to reduce a by a factor of f = (1 + ot/R 3) ~ 6. Nair (Nair, 1993) has also reported a similar calculation using the EPM wavefunctions and found that the screening is very sensitive to the radial spread in the electron density, the screening decreasing with increasing shell thickness. For the linear response he found that f = (1 + aa/R z) to a good accuracy where a is a constant which depends on the shell thickness. For t = 3, he found a = 0.73. It can be much smaller for a more realistic radial density. For 3' the screening correction was found to be slightly smaller than f4. From the measured refractive index (2.25) (Saeta et al., 1992) of C60 thin films, we estimate c~. . . . ca ~ 100 flk 3 using the Claussius-Mossotti relation. With able ~ 200A 3, we estimate the screening factor f to be ~ 2 which implies a reduction of ~t by a factor of 16 from the unscreened values. It may be argued that unless an accurate understanding of the screening correction can be made, no useful estimate of hyperpolarizability can be made. This is only partially true. The screening corrections in Cso and CssB are expected to be similar. Thus, if the unscreened calculations give/3 and 3' for C59B a conservative estimate can be made for the ratio of the hyperpolarizabilities of the two molecules. The effect of screening on the frequency dependent (dynamic) response is more dramatic. Bertsch et al. (1991) have analysed this problem in the RPA and found that the absorption spectra are strongly modified by the screening of the applied electric field by the electrons themselves. All the absorption peaks occur shifted towards higher energies from the energy gaps in the single particle spectra and the oscillator strengths of the low energy transitions are suppressed in favour of the higher energy ones. A strong resonance very similar to the Mie type plasmon resonance in a metal particle characterises the response at higher energies (Bertsch et al., 1991; Steger et al., 1992). These observations point to the similarity of fullerenes to semiconductor quantum dots for which unusual effects in the optical response due to the resonances in the electric field penetration have been predicted (Chemla et al., 1986; Nair and Rustagi, 1987; Ramaniah et al., 1989). Similar classical electrodyanamic treatment of the screening in fullerenes has been attempted by Ramaniah et al. (1992) by modelling fullerenes as spherical or ellipsoidal shells and using the bare polarizabilities as given by tight-binding calculations. These calculations show that screening is responsible for the fact that C70 and higher fullerenes all have only weak absorption bands in the visible and near infrared. This feature of the experimental spectra of higher fullerenes is otherwise surprising as all of them have a low symmetry and their HOMO-LUMO gaps are comparable to that of C60. Extension of these analyses to the nonlinear response would be very interesting. Naively, using the standard analysis of the local field correction to nonlinear susceptibilitites of condensed media (Armstrong et al., 1962), one would expect the screening correction to the n th order polarizability to be given by a product of n linear screening factors corresponding to the (n + 1) frequencies involved.
Nonlinear Optimal Response of FuUerenes
89
This would imply many unusual features in the near-resonant nonlinear optical response of fullerenes. For example, the one-photon and three-photon resonances corresponding to the same pair of singleparticle states would be expected to occur at slightly different frequencies. A detailed analysis of these features is still awaited. Future O u t l o o k
After some initial controversy, reliable estimates of the hyperpolarizability 3' of C6o are now available. However, theoretical estimates are generally smaller than the experimental values and also several experimental results are not yet understood. Still C~0 is just one member of a whole family of new molecules and much more remains to be done. Symmetry properties of/3 and 7 turn out to be rather special. C6o and other molecules of lh symmetry are the only systems for which c~, fl and 7 are isotropic like atoms. Unlike non-inversion symmetric crystallographic point groups, the symmetry group of C70 (Dsh) has fl = 0. We have suggested that harmonic generation in solid C60 and C70 could provide very interesting insights into their intermolecular interactions. Studies on these molecules have also resulted in a better appreciation of screening effects due to the electron-electron interaction. However, much needs to be done on this aspect. From practical considerations, the rather large ~ values anticipated for CsgB, CsgN and CTs appears to hold much promise specially for C7~ as it does not have a dipole moment and thus may easily crystallize in a structure lacking inversion. Finally, it seems fair to conclude that the study of nonlinear optical properties of these molecules with a variety of geometrical shapes and sizes is at the initial stages. With more molecules becoming available, much more exciting developments are anticipated. REFERENCES
Andreoni, W., F.Gygi and M.Parrinel]o. (1992). hnpurity states in doped fullerenes: C~gB and CsgN. Chem. Phys. Lett., 190, 159. Armstrong, J.A., N.Bloembergen, J.Ducuing and P.S.Pershan. (1962). Phys. Rev., 127, 1918. Bertsch, F., A.Bulgac, D.Tomdnek and Y.Wang. (1991). clusters. Phys. Rev. Lett., 67, 2690.
Collective Plasma Excitations in C6o
Bey, P.P., and H.Rabin. (1967). Phys. Rev., 162, 794. Blau, W.J., H.J.Byrne, D.J.Cardin, T.J.Dennis, J.P.Hare, H.W.Kroto, R.Taylor and D.R.M.Walton. (1991). Large Infrared Nonlinear Optical Response of C60. Phys. Rev. Lett., 67, 1423. From fig.2 of this paper we estimate 7 (C60)/7 (Benzene) ~ 1.7× 104 which implies 7 (C60) ~ 6 x 1 0 -32 esu. This is much smaller than the value claimed in this paper. Butcher, P.N., Nonlinear Optical Phenomena (Ohio Stae University, Ohio, 1965). For corrections of some errors in Butcher's tables for the third order susceptibility see Y.Zhao. (1986). IEEE J. Quant. Elect., QE-22, 1012. Chemla, D.S., and D.A.B.MilIer. (1986). Linear and nonlinear optical response of spherical anisotropic semiconductor microcrystallites. Opt. Lett., bf 11,522 Chemla, D.S. and J.Zyss eds. (1987). Crystals. Academic, Orlando.
Nonlinear Optical Properties of Organic Molecules and
Cohen, M.L., and J.R.Chelikowsky. (1988). Electronic Structure and Optical Properties of Semiconductors, Springer-Verlag, Berlin. Flora, S.R.,R.G.S.Pong, F.J.Bartoli and Z.H.Kafafi. (1992). Resonent Nonlinear Optical Response of the fullerenes C60 and C70. Phys. Rev., B46, 15598. Gallup, G.A. (1991). The application of zero-range potentials to the electronic properties of footballene, C60. Chem. Phys. Lett., 157, 187.
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Gong, Q., Y.Sun, Z.Xia, Y.H.Zou, Z.Gu, X.Zhou and D.Qiang. (1992).3. Appl. Phys., 71, 3025. Harigaya, K., and S.Abe. (1992). Dispersion of the third-order nonlinear optical suceptibility in C6o calculated with a tight-binding model. Jap. J. Appl. Phys., 31, L887. Harrison, W.A., Electronic Structure and the Properties of Solids: The Physics of the Chemical Bond (Freeman, San Francisco, 1980). Hatter, W.G., and D.E.Weeks. (1989). Rotation-vibration spectra of icosahedral molecules. I. Icosahedral symmetry analysis and fine structure. J. Chem. Phys., 90, 4727. Hoshi,H., N.Nakamura, Y.Maruyama, T.Nakagawa, S.Suzuki, H.Shiromaru, Y.Achiba. (1991). Optical second and third harmonic generations in C60 film. Jap. J. Appl. Phys. Lett. 30, L 1397. Joshi, M.P., S.R.Mishra, H.S.Rawat, S.C.Mehendale and K.C.Rustagi. (1993a). Investigation of optical limiting in C6o solution. Appl. Phys. Lett., 62, 1763. Joshi, M.P.,S.R.Mishra, H.S.Rawat, S.C.Mehendale and K.C.Rustagi. (1993b). Appl. Phys. Lett., 63, 2578. Hermann, J.P. (1974). Ph.D. Thesis, Univ. Paris-Sud. Kafafi,Z.H., J.R.Lindle, R.G.S.Pong, F.J.Bartoli, L.J.Lingg and J.Milliken. (1992). Off-resonant nonlinear optical properties of C60 studied by degenerate four wave mixing. Chem. Phys. Lett., 188, 492. Kafafi, Z.H.,F.J.Bartoli, J.R.Lindle and R.G.S.Pong. (1992). Comment on "Large Infrared Nonlinear Optical Response of C6o'. Phys. Rev. Lett., 68, 2705. Kajzar, F.,C.Taliani, R.Zamboni, S.Rossini and R.Danieli. (1992). in Fullerenes, ACS symposium series 481, G.S.Hammond and V.J.Kuck, eds., (Amer. Chem. Soc., Washington DC, 1992). Kr£tschmer, W., L.D.Lamb, K.Fostiropoulos and D.R.Huffman. (1990). Solid C60 : A New Form of Carbon. Nature, 347, 354 Matsuzawa, N., and D.A.Dixon. (1992). Local density funtional calulations of the polarizability and second order hyperpolarizability of C6o. J. Phys.Chem., 96, 6872. McLean, D.G., R.L.Sutherland, M.C.Brant, D.M.Brandelik, P.A.Fleitz and T.Pottenger. (1993). Nonlinear absorbtion study of a C60-toluene solution. Opt. Lett., 18, 858. Mehendale, S.C., and K.C.Rustagi. (1979). Optical nonlinearities in substituted symmetric cyamine dyes. Opt. Commun., 28, 359. Menon,M., and K.R.Subbaswamy. (1991). Universal Parameter Tight-Binding Molecular Dynamics: Application to C60. Phys. Rev. Lett., 67, 3487. Meth, J.S., H.Vanherzeele and Y.Wang. (1992). Dispersion of third-order optical nonlinearity of C60. A third harmonic generation study. Chem. Phys. Lett., bf 197, 26. Mishra, S.R., H.S.Rawat, M.P.Joshil and S.C.Mehendale, to be published. Nair. S.V. and K.C.Rustagi. (1987). Superlatt. Mierostruct., 6, 337 Nair,S.V., and K.C.Rustagi. (1991). Proc. Solid State Phys. Symp., Banaras, (Dept. Atomic Energy, Bombay 1991), vol. 34C, p.98. Nair, S.V. (1993). eh.D.Thesis, D.A.University, Indore (Submitted). Neher,D., G.I.Stegeman, F.A.Tinker and N.Peyghambarian. (1992). Nonlinear optical response of C6o and C~0. Opt. Lett., 17, 1491. Platt. J.R. (1953)..J. Chem. Phys., 21, 1597. Ramaniah, L.M., S.V.Nair and K.C.Rustagi, Opt. Commun. (submitted). Ramaniah, L.M., S.V.Nair and K.C.Rustagi. (1989). Phys. Rev., B 40, 12423.
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Ramaniah, L.M., S.V,Nair and K.C.Rustagi, to be published. Ramaniah, L.M., S.V.Nair and K.C.Rustagi. (1993). Symmetry of the polarizability tensors for molecules with DCsh and ICh symmetry. Opt. Commun., 96, 289. Rosen A., and E.Westin. (1993). in Clusters andfullerenes, V.Kumar, T.P.Martin and E.Tosatti, eds., World Scientific, Singapore. Rosker, M.J.,H.O.Marcy, T.Y.Chang, J.T.Khoury, K.Hanen and R.L.Whetten. (1992). Timeresolved degenerate four wave mixing in thin films of Cso and C70 using femtosecond optical pulses. Chem. Phys. Lett., 196,427. Rustagi, K.C., L.M.Ramaniah and S.V.Nair. (1993). in Clusters andfullerenes, V.Kumar, T.P.Martin and E.Tosatti, eds., World Scientific, Singapore. Rustagi, K.C. (1993). (Indian J. Phys. 67A, 493. Rustagi, K.C. and Ducuing, J. (1974). Third order optical polizability of conjugated organic molecules. Opt. Commun., 10, 258. Saeta, P.N., B.I.Greene, A.R.Kortan, N.Kopylov and F.A.Thiel. (1992). Optical studies of singal crystal C60. Chem. Phys. Lett., 190, 184. Saito, R., G.Dresselhans and M.S.Dresselhaus. (1992). Ground states of large icosahedral fullerenes. Phys.Rev., B46, 9906. Savina, M.R., L.L.Lohr and A.H.Francis. (1993). A particle on a sphere model for C60. Chem. Phys. Lett., 205, 200. Shuai, Z., and J.L.BrSdas. (1992). Electronic structure nonlinear optical properties of the fullerenes Cs0 and C~0: A valence effective Hamiltonian study. Phys. Rev., B 46, 16135. Shuai,Z., and J.L.BrSdas. (1993). Electronic structure nonlinear optical properties of the fullerenes Cs0 and C7o: A valence effective Hamiltonian study. Phys. Rev., B 48, 11520 (E). Stegeman, G.I., R.Zanoni, E.M.Wright, N.Finlayson, K.Rochford, J.Ehrlich and C.T.Seaton in Organic Materials for Nonlinear Optics, R.A.Hann and D.Bloor, eds., (Royal Society of Chemistry, London, 1989). Steger, H., J. de Vries, B.Weisser, C.Menze], B.Kamke, W.Kamke and I.V.Hertel. (1992). Gaint Plasmon Excitation in Free C60 and CT0 molecules studied by photoionization. Phys. Rev. Lett., 68, 784. Stevenson, S.H., D.S.Donald and G.R.Meredith in em Nonlinear Optical Properties of Polymers (Mat. Res. Soc., Pittsburgh, 1988) proc. vol. 109, p.103. Talapatra, G.B., N.Manickam, M.Samoc, M.E.Orczyk, S.P.Karna and P.N.Prasad. (1992). Nonlinear optical properties of the C60 molecule: theoretical and experimental studies..l. Phys. Chem., 96, 5206. Tang, N., J.P.Partanen, R.W.Hellwarth and R.J.Knize. (1993). Third-order optical nonlinearity of Cs0, C70 and CS2 in Benzene at 10.6 m. Phys. Rev., B 48, 8404. Tomfinek,D., and M.A.Schluter. (1991). Growth regimes of carbon clusters. Phys. Rev. Lett., 67, 2331. Tutt L.W.,and A.Kost. (1992). Optical limiting performance of C6o and C~0 solutions. Nature 356, 225. Vijaya, R., Y.V.G.S.Murti, G.Sundararajan, C.K.Mathews and P.R.Vasudeva Rao. (1992). Degenerate four wave mixing in the carbon cluster C60. Opt. Commun., 94, 353. Wang, Y., G.Bertsch and D.Tomnek. (1991). Hyperpolarizability of the Cso fullerene cluster. Z. Phys., D 25, 181.
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Wang, C.Z., C.T.Chan and K.M.Ho. (1992). Structure and dynamics of C6o and C70 from tight-
binding molecular dynamics. Phys. Rev., B46, 9761. Wang, X.K.,T.G.Zhang, W.P.Lin, S.Z.Liu, G.K.Wong, M.Kappes, R.P.H.Chang and J.B.Ketterson. (1992). AppI. Phys. Lett., 60,810. Wang, Y., and L.-T.Cheng. (1992). Nonlineax optical properties of fullerenes and charge transfer complexes of fullerenes. J. Phys. Chem. 96, 1530. Xu, C.H., C.Z.Wang, C.T.Chan and K.M.Ho. (1992). A transferable tight-binding potential of carbon. J. Phys. Condens. Matter, 4, 6047. Yabana, K., and G. Bertsch, Michigan University preprint. Yang, S.C.,Q.Gong, Z.Xia, Y.n.Zhou, Y.q.Wu, D.Qiang, Y.L.Sun and Z.N.Gu. (1992). Large third order nonlinear optical properties of CT0 fullerene in the infrared regime. Appl. Phys., B 55, 51. Zhang, Z., D.Wang, P.Ye, Y.Li, P.Wu and D.Zhu. (1992). Studies of third-order optical nonlinearities in Cso-toulene and Cs0-toulene solutions. Opt. Lett., 17, 973.
rl=l
0.2 m
c
0
q.-
-.2
Fig. 1. The radial dependence of the eigenfunctions of the spherical shell model.
Nonlinear Optimal Response of Fullerenes
93
t~0.
30.
-w-
I,I
•
20.
t--D t.tJ t'--I --
t.ld t"3
10.
I---
I--< --.J
0.
-10.
-20. - 3 0' .
-2'0.
- 1 '0 .
8.
T10.
10
.
" 30.
LONGITUDE (DEGREE) Fig. 2. The angular dependence of the rr-electron density i]1 Cso. The values indicated denote the n u m b e r of electrons per unit solid angle• (After Nair, 1993)
Pergamon
0960-8974/97 $32.00
PII: S 0 9 6 0 - 8 9 7 4 ( 9 7 ) 0 0 0 0 6 - 5
NONLINEAR OPTICAL RESPONSE OF FULLERENES Kailash C. Rustagi*, S e l v a k u m a r V. Nair* and L a v a n y a M. R a m a n i a h t * Laser Programme, Theoretical Physics Division, Indore452 013, India I~Centre for Advanced Technology, BhabhaAtomic Research Centre, Bombay 400 085, India
ABSTRACT A critical review of nonlinear optical properties of fullerenes is presented. We explore structure property relationships for optical nonlinearities of these unusual ~r electron systems from two complementary angles: (i) to explore the possibility of obtaining structural information from nonfinear optical experiments and (ii) to assess the potential of fullerenes, fullerites and their derivatives for the practical nonlinear optical devices. In particular we show that substituted fullerenes form an interesting class of molecules for nonlinear optics. We will also show that third harmonic generation of circularly polarised light in the solid fullerites should yield some information on the intermolecular interactions in these solids. We critically examine the various calculations indicating that the screening of the optical fields due to electron-electron interaction may reduce the optical susceptibilities of these molecules by a large factor and suggest experiments that may clarify the issue. KEYWORDS
Nonlinear Optics, r electron systems, Third Harmonic Generation. INTRODUCTION Soon after the discovery (Kratschmer et al., 1990) of a method of making macroscopic quantities of Buckminsterfullerene, the attention of the nonlinear optics community was drawn to their potential as nonlinear materials (Hoshi et al., 1991; Blau et al., 1991; Nair et al., 1991; Wang and Cheng, 1992; Rustagi, 1993). This interest arose from the fact that conjugated organic molecules and polymers are known to owe their large non-resonant optical nonlinearities to their electrons delocalized in one (Rustagi and Ducuing, 1974; Hermann, 1974) or two dimensions (Chemla and Zyss, 1987). Thus, fullerenes with their three-dimensional 7r-electron systems offer an additional avenue in the search for more efficient nonlinear optical materials. Vigorous experimental (Hoshi et al., 1991; Blau et al., 1991; Wang and Cheng, 1992; Kafafi et al., 1992a; Kafafi et al., 1992b; Gong et al., 1992; Yang et al., 1992; Flora et al, 1992; Rosker et al., 1992; Zhang et al., 1992; Vijaya et al., 1992; Tang et al., 1993; Kajzer et al., 1992; Meth et al., 1992; Neher et al., 1992; Wang et al., 1992; Tuft and Kost, 1992; Joshi et al., 1993a,b; Mishra et al.) and theoretical (Rustagi et al., 1993; Wang et al., 1991; Talapatra et al., 1992; Rosen and Westin, 1993; Harigaya and Abe, 1992; Matsuzawa and Dixon, 1992; Ramaniah et al., 1993; Shuai and Bredas, 1992; Ramaniah et al. 1994) research over the last two years or so has produced several surprises. In this article we summarise our present understanding of this effort in the hope that this will sharpen some of the questions that need to be 81
82
K.C. Rustagi et al.
addressed. We begin with a brief description of the main experimental findings. This is followed by a review of the theoretical attempts to understand the nonlinear optical response of these molecules and the corresponding solids. So far almost all experiments have been performed on C ~ and a few on C~0. Theoretical models should, however, also try to predict the nonlinear optical response of other fullerenes and fullerene derivatives. As more molecules of this family become available, the comparison of experimental and theoretical results is expected to lead to a better understanding of the relation of the nonlinear response of fullerenes to the geometrical structure of their 7r-electron clouds. EXPERIMENTAL OBSERVATIONS
The main nonlinear optical processes investigated in C60 or Cz0 are degenerate four wave mixing (DFWM) (Blau et al., 1991),Kafafi, (Kafafi et al, 1992b; Gong et al., 1992; Yang et al., 1992; Flom et al, 1992; Rosker et al., 1992; Zhang et al., 1992; Vijaya et al., 1992; Tang et al., 1993), third harmonic generation (THG) (Hoshi et al., 1991; Kajzer et al., 1992; Meth et al., 1992; Neher et al., 1992), and optical limiting (Tutt and Kost, 1992; doshi et al., 1993a,b; Mishra et al.). The first two can occur without any excitation of the medium and can be described in terms of a perturbative expansion of the polarization, i.e., the induced electric dipole density, P. Following the standard notation, we write p = p(U + pO} + p(3) + " ' ' ,
(1)
where p(~/ is of order n in the electric field of the incident wave. The Fourier component p(3)(fl) at frequency f~ is related to E(co0, that of the electric field through
P(,Zl(a) = N~ ,,,(3) , ~ , / _ oo,, co,, ~ , ~'3)E~(col)E~(co~)F,(~a),
(2)
where ~ = ~ol + co2 + w3 and Nd is the number of distinct permutations of cot, w2 and ~z. The susceptibility tensor X(z) describes the material response in the third order. The two processes THG and DFWM are described by the tensors X(3)(-3co, co, co, co) and X(3)(-~o,~', -co, 0J) respectively. We recall that the Fourier component at -co is the complex conjugate of that at ~o, i.e., E(-co) = E*(co) and similarly for P(co) etc. For homogeneous molecular solids the susceptibilities are related to the molecular polarizabilities or, t3 and 7 through
~(')(co) = :~L(~),~(co),
(3)
~(2)(-(col + w2),~1,~02) = NL(col + w2) L(coi ) L(co2)fl(-(col + co2), col, 0-12)
(4)
)~(3}(__(COl + ~.U2~t_ ~3), COl, CO2,cj3)
(5) (6)
and =
,~rL(co I _}_co2 + co3) L(gMI) L(o.)2) L(~3) x 7(-(col + ~'2 + co,~),~01,co2,cos)
where L(co), the local field correction factor is the ratio of the total electric field Eloc acting on the molecule to the macroscopic field E and N is the inolecular density. In addition to E, the local field includes the field generated by the induced dipoles at all the other molecules. For molecules occupying atleast a tetrahedral site L(co) = (e(a;) + 2)/3 (Armstrong et al., 1962). For fluids, additional contributions to X"(3) can arise due to the orientation of anisotropic molecules by the incident electric field. The response time for such nonlinearities is typically the inverse of the rotational frequency and is > 10 - I 2 s. For the most challenging applications of nonlinear optics namely ultrafast signal processing, it is desirable to have a large hyperpolarizability 7 which is due to virtual excitation of the electrons in the molecule. Third harmonic generation (THG) measures this directly and is the preferred method for the measurement of byperpolarizabilities of electronic origin, For Ceo such measurements have been performed by Hoshi et al. (1991), Meth et al. (1992)
Nonlinear Optimal Response of Fullerenes
83
and Kajzar et al. (1992) on thin films, and by Neher et al. (1992) in solution. The values of X(3) obtained by THG are shown in Table 1. They are reasonably consistent with each other when we allow for the fact that the measurements are relative to two different standards. The measurements for solutions are intrinsically less sensitive because a larger part of the signal comes from the solvent. For Cr0, the only values of 3'(-3w, w, w, w) avalable are those from solution experiments of Neher et al. (1992) giving 3' ~ 4 × 10 -33 esu in the nonresonant region, while for a fundamental wavelength of 1.064 #m they report {"/) = ( - 7 . 6 + i 5 7 ) × 10 -33 esu. Meth et al., Kajzar et al. (1992) and Neher et al. (1992) all report measurements of X(3) at several wavelengths. Resonance enhancement of THG can increase ] X(3) ] by as much as a factor of 20. Table 1. xt3)(-3w, w, w, w) measured by THG experiments.
(~m) 2.37 1.907 1.3 1.064 1.064 0.85
(X( 3 ) ) 4.1×10 -12 3.2×10 -H 5.1×10 -11 8.2×10 -la 2.0×10 -1° 1.5×10 -~1
Reference (Meth et al., 1992) (Kajzer et al., 1992) (Kajzer et al., 1992) (Kajzer et al., 1992) (Hoshi et al., 1991) (Kajzer et al., 1992)
The value of X(3) = 7.0 x 10 -1~ esu obtained by Kafafi et al. (1992a,b) using DFWM in thin films is consistent with the THG values in the non-resonant regime. It is interesting to observe that Kafafi et al. (1992a, b) estimated an absorption coefficient of 6 cm -1 at A = 1.064#m for their film while the typical value for organics is < 10~cm -1 (Stegman et al., 1989). Many other observations of the third order nonlinearity have been made by DFWM. They often report much larger values of 7While some of them can be associated with thermal or population transfer effects in the absorbing region, for several others this is not a plausible cause. Clearly more work is needed to clarify the situation. Indeed, a very recent paper by Tang eta[. (1993) reports that a very careful analysis of their DFWM results using 30 ps pulses at 1.06 # m yield values that are consistent with those of Kafafi et al. (1992a,b) and are much smaller than those reported by several other groups. One of the most interesting prospects is the use of fullerenes as optical limiters. Tutt and Kost (Tutt and Kost, 1992) observed optical limiting in C60-toluene solution at a relatively low threshold and attributed this to the reverse saturable absorption mechanism [RSA]. Subsequent work by our group (Joshi et al., 1993a,b) and others (Mclean et al, 1993) has shown that RSA is inadequate to explain the observed results in solutions. While a complete picture is yet to emerge it is strongly indicated that nonlinear scattering and self induced defocussing plays an impotant role (Mishra et al.). THEORETICAL
CONSIDERATIONS
S y m m e t r y of t h e P o l a r i z a b i l i t y T e n s o r s For any molecule, the various components of the polarizability tensors c~,/3, and 7, are related to each other by symmetry. The symmetry group of C60 is ]h and that of Cr0 is Dsh. Both these point groups are not among the crystallographic point groups and are thus not included in the standard tables of symmetry relations for nonlinear polarizabilities (Butcher, 1965). We have shown (Ramaniah et al., 1993) that c~ and 7 for C60 (Ih group) have structures identical to those of an isotropic system while for CT0 (Dsh group) their structure is the same as that for an inversion symmetric hexagonal system of D6h symmetry. Surprisingly, fl vanishes for Dsh syimnetry although this group does not contain the inversion operation. In contrast, for all the 21 non-centrosymmetric crystallographic point groups at least one component of fl is nonvanishing by symmetry. For C70, 3' has 21 non-zero components with l0 of them independent. However, when dispersion can be neglected, only three components of 3' are independent which may be taken to be 3'. . . . . 3'.... and 7 . . . . For C60, we find (Ramaniah et al., 1993) a ~ = c~yv = c ~ , i.e., same as for an isotropic or a cubic
84
K.C. Rustagi et aL
material, and 7 has 21 non-zero components, of which only 3 are independent, again as for an isotropic system. Under Kleinman symmetry only one component is independent with %~= = 3 % ~ and all the components related by an interchange or permutation of x, y, z are equal. We note that this is exactly valid for the static (zero-frequency) polarizabilities and provides a check on the theoretical calculations (See e.g, Ref. Shuai and Bredas, 1993). The fact that 7 of C6o has the same symmetry structure as that for an isotropic material offers an interesting apphcation of nonlinear optics viz., to probe the solid state effects on the electronic density of C60 (Ramaniah et al., 1993). This possibility arises because no crystalline material has a strictly isotropic nonlinear susceptibility )/(3). However, C60 forms a molecular solid so that one would expect that its X(a) can be obtained as a sum of the hyperpolarizabilities (7) of the individual molecules to a good approximation. Thus for solid C6o, although only those symmetry restrictions on X(3) demanded by its cubic crystallographic point group are strictly valid, X(3) will be isotropic to a good degree of accuracy. And, the deviation from isotropy of X(3) of solid C60 will give a measure of the inter-molecular interactions. In particular, we (Ramaniah et al., 1993) have suggested that since third harmonic generation from an isotropic medium vanishes for circularly polarized light (Bey and Rabin, 1967), the ratio of the third harmonic generated by circularly and linearly polarized light in Ceo solid will give a measure of the cubic perturbation of the molecular electron density due to inter-molecular interactions. Electronic Structure and Polarizabilities Starting from the early classic work of Platt (1953), it has been realized that the dipole polarizabilities c~,fl and ")' of conjugated r-electron systems are sensitive to the ~r-electron density p(r) and it is generally believed that an accurate representation of p(r) is necessary to obtain reliable estimates of the polarizabilities. Consequently, several sophisticated computational methods have been used to determine the electronic structure and polarizabilities of conjugated r-electron molecules (Chemla and Zyss, 1987). Such calculations, however, often involve not-so-transparent approximations. Only a few can be said to have reached such a level of reliability that their disagreement with experimental observations are taken with sufficient seriousness. On the other hand, simple free-electron or Hiickel molecular orbital theories have predicted significant structure-property connections which have been verified experimentally as well as by more elaborate theories. Similar simple models of the electronic structure of fullerenes have been used for obtaining valuable information oll the optical response of fullerenes (Nair et al., 1991),Wang, (Gallup, 1991; Saito et al., 1992; Savina et al., 1993). For C60, due to its high symmetry and the delocalized nature of the r-electrons, a natural extension of the free electron theory is possible and has been attempted by several groups. We present here the model of Nair and Rustagi (NR) (Nair et al., 1991; Rustagi et al., 1993) which appears most reasonable on physical grounds. There is a close parallel between our approach for conjugated molecules and the empirical pseudopotential method (EPM) (Cohen and Chelikowsky, 1988) used for the electronic structure of semiconductors. The essential point in the EPM is to describe the effective potential seen by valence electrons in terms of a few Fourier components of the lattice periodic potential. Ill C60, instead of linear translational invariance we have spherical symmetry in the empty-lattice limit. And in place of the lattice periodic potential in the solid we have a perturbation of icosahedra} symmetry in C~o. This pseudopotential can be described in terms of only 3 parameters as for the group IV semiconductors. We believe such an approach is complementary to the sophisticated numerical calculations of electronic structure and polarizabilities since it provides an understanding of the whole range of optical properties including absorption spectra and thus facilitates greater interaction between theory and experiment. Secondly, it suggests chemical means of increasing the nonlinearity in some cases. As in the empirical pseudopotential method (Cohen and Chelikowsky, 1988) for semiconductors, NR first make an "empty lattice" description for the C6o molecule and then treat the icosahedral pseudopotential perturbatively. The "empty lattice" in this case has spherical symmetry, and they take
NonlinearOptimal Responseof Fullerenes
85
this zeroth order model to be 60 r-electrons confined in a spherical shell of mean radius R = 3.55/~ and thickness t = 3A, consistent with other estimates of the electron density. The eigenfunctions for this system are (Nair, 1993) ¢=tm(r) = f.t(r)Ytm(O, ~), (7) with
j,(k.,n_) 'k ,"~
f~t(r) = N jt(k~tr) - yl(k.lR_)y~( .t ) ) ,
(8)
where jt and Yl are the spherical bessel and n e u m a n n functions and Ytm(0,~p) denote the spherical harmonics. N is the normalization constant, k.t's are determined by f(R+) = 0, and t ~ = R 4- t/2 are the outer and the inner radii of the shell. The energy levels are given by
E.e = h2k~l
(9)
2m0 The energy levels are well approximated by (Nalr, 1993)
E~ = (hV2m0)
(n27r 2 l(l 4- 1)'~ --h-- + R~ ]
(10)
n and l being the radial and angular quantum numbers. As shown in Fig.l, the n = 1 radial functions are nodeless whereas the n = 2 radial functions have a node near r = R. Since ideally the r-electrons have vanishingly small overlap with the a-electron skeleton, NR choose them to occupy the n = 2, l = 0 to 5 energy levels. The n = 1 levels which lie below the n = 2, l = 0 level have maximum charge density at r = R and are assumed to be filled by the a-electrons. As the (r-electrons are expected to have a much smaller radial spread than the r-electrons, the relative position of the a and r levels is not expected to be accurately depicted by the model potential having a uniform radial width. On the other hand, the total band width occupied by the a-electrons or the 7r-electrons does not depend on this width and is well depicted by the model potential. Later calculations (Yabana and Bertsch) using the spherical jellium model have shown that the core and the a-electrons indeed occupy n = 1, l < 12 levels with l = l l and 12 partially occupied. The 22-fold degenerate (including the spin degeneracy factor), l = 5 level is only partially occupied by l0 electrons so that the r-electron structure is sensitive to all perturbations which split this level. The perturbation due to the presence of the 60-vertex r-electron skeleton has icosahedral symmetry (group lh) which splits the l -- 5 level into 3 levels of tl~, t3~ and h. symmetry (Harter and Weeks, 1989). Since the radial wavefunctions are found to be nearly independent of l, the unperturbed states may be approximated by f~=2(r)Ylm(0, ~). The icosahedral perturbing potential can also be expanded into a series of spherical harmonics. Only those l values will occur in the expansion which contain the identity representation of the icosahedral group Ih. Since the levels close to the Fermi level have either I = 5 or l = 6, it suffices to retain only the I = 6, l0 and 12 components of the potential. The l = 0 term may be ignored because it is a constant. The next non-zero contribution would come from l = 16. Thus to see the effect of the icosahedral perturbation V in the n = 2 subspace one can write,
(l,~lVIl'm') =
A6( lmlV~ll',,~') + A,o(1,nlV, oll'm') + d,:(ImlV,2ll'm')
ill)
where V6, Vm and Vl2 are those linear combinations of I = 6, l0 and 12 spherical harmonics, respectively, which remain invariant under the operations of lh (Nair, 1993). The r-electron energy levels are then obtained by diagonalising the Hamiltonian numerically. A truncated basis set with l ~ 9 is used. The use of a rigid confining shell in the zeroth-order approximation makes the separation between energy levels of different I values somewhat larger than that expected from a more realistic sphericallyaveraged jellium potential (Y~bana and Bertsch). The effect of the icosahedral pseudopotential is
86
K.C. Rustagi et al.
mainly to split the multiplets of a given l. So NR fix the pseudopotential parameters such that the splitting of the l = 5 empty lattice level gives a HOMO-LUMO gap of 1.8 eV and the lowest electric-dipole allowed transition occurs at ~ 3.7 eV. This leaves one parameter free which is chosen to minimize the discrepancy with the next optical absorption peak. Inspire of the simplicity of the starting spherically symmetric potential, the calculated energy levels show reasonable agreement with the valence photo-ionization spectrum (Nair, 1993). The values of the parameters used are A6 = -0.035 Ry, A10 = 0.4 Ry and AI~ = 0.15 Ry. The calculated r-electron density (Fig.2) shows a concentration over the a-bond skeleton. The electron density per solid angle at the centre of a hexagon is less than 2 compared to about 5.5 at the centre of a pentagonal edge and about 6.5 at the centre of a bond shared by two hexagons. These estimates are in very good agreement with a recent LDA calculation (Yabana and Bertsch). It is interesting to note that in the spherical harmonic expansion for the electron density, although the leading correction to the 1 = 0, spherically symmetric term is the l = 6 term, its coefficient is substantially smaller than that of the next term with l = 10. A good approximation to the r-electron density can be obatained by retaining only the 1 = 0 and l = I0 terms which in the present model is 6O p.(r) = 47 - 3"881f2(r)l~v'°(O'~)' (12) where f2(r) is the normalized radial eigenfunction with n = 2 for tile empty lattice problem described above. The weak/-dependence of f2(r) is neglected, as mentioned earlier. There are a number of independent calculations of the electronic structure of Cs0 using the model described above. However, most of them make some unphysical assumptions. Saito et al. (1992) and Savina et al. (1993) take the r-electrons to occupy the n = 1 states of the spherical shell whereby their r-electron density lacks the required antisymmetry about tile nuclear position. The radial relectron density obtained by them is thus unphysical. Saito et al. (1992) also take the mean radius of the shell to be 4.05~ instead of the known value of 3.55~. Further, they apply this model to higher fullerenes of icosahedral symmetry. This is unreasonable as higher fullerenes are expected to have non-spherical structure with the carbon atoms lying at different distances from the centre. Gallup (1991) uses zero-range potentials at the nuclear positions to simulate the potential due to the ioncores and the a-electrons, and estimates its effect on the r-electro,Is using first order perturbation theory. The icosahedral potential is not weak enough to justify such a treatment and as a consequence their r-electron density deviates only marginally from sphericity, in contradiction to the results of more sophisticated calculations. Another simple model for describing the electronic structure of fullerenes is the tight-binding model. Unlike the nearly-free-electron model descibed above, the tight-binding method is more versatile since it can be easily applied to any geometry. A nmnber of tight-binding parametrization schemes for pure carbon structures have been formulated by various authors (Tomanek and Schluter, 1991; Menon and Subbaswamy, 1991; Xu et al., 1992). All these schemes obtained by detailed comparison with ab initio calculations, provide very satisfying results for the structure of many carbon clusters. Even relatively simple molecular dynamics calculations for fullerenes using a tight-binding model have been shown to provide reliable results (Wang et al., 1992). This versatility of the tight-binding method makes it very appropriate for calculating the polarizablity of fullerenes, especially for the higher and substituted fullerenes. In the tight-binding method for Cso the occupied valence electron states and the lowest few excited states are described by the 240 x 240 Hamiltonian matrix in the basis set provided by 240 orthogonal atomic like orbitals - - four (s,p~,p~,pz) at each atonfic site. Since only the nearest-neighbour interag.tions are retained the only parameters involved in the Hamiltonian are the atomic site energies E v and E, which are characteristic of the chemical species occupying the site and the hopping integrals V,,, V,v, Vpv, and Vp~ which are usually scaled to the corresponding interatomic distance (Harrison, 1980). Various tight-binding models used for fullerenes differ in the way in which the parameters are obtained. We describe calculations based on the scheme developed by Tomgnek and Schluter (TS)
Nonlinear Optimal Response of Fullerenes
87
(Tomanek and Schluter, 1991) by detailed comparison with LDA calculations for various crystalline and molecular structures of carbon. Using this model for determining the structure of carbon clusters Cn for n < 60, TS found that the cage geometry of fullerenes becomes more stable than one- and twodimensional structures for n > 20, in very good agreement with the experimental observations. The TS model was also used by Bertsch et al. (1991) to calculate the electronic structure and linear optical response of C6o within the random phase approximation (RPA) and they predicted an unusually large screening correction to the linear response. We defer a detailed description of screening in C6o to the next section. Using the TS model Wang et al. (1991) calculated the nonlinear polarizability of C6o to be X(3} 2 × 10-34esu, in good agreement with the result of NR. Later Ramaniah et al. (1994) used this model to calculate the electronic structure and nonlinear optical properties of boron and nitrogen substituted fullerenes (Cs9N and CsgB) (Rustagi et al., 1993). Since CsgB has 239 valence electrons and C59N has 241, both have partially-filled levels. The lowest gap in both the molecules is the impurity level splitting which is very small but the oscillator strength of this transition is negligible. In C59B the most important new electric-dipole transition is between the two impurity levels split-off from the 2 highest-filled C60 levels of hg and h~ symmetry, which we may refer to as the split-off hg and split-off h~ levels. In C59N the new strong transition is between the impurity levels split-off from the 2 lowest unfilled levels of C6o of tl~ and tlg symmetry. The corresponding transition energies are 0.9 eV and 0.6 eV in C59B and Cs9N respectively. The polarizabilities calculated for C6o using the pseudopotential model (Nair et al., 1991; Rustagi et al., 1993) and for substituted C60 using the tight-binding model (Ramaniah et al. 1994) reveal several interesting points. First we compare the numbers for C6o with those for a one-dimensional r-electron system : fl-carotene with 22 r-electrons delocalized over a zig-zag chain of about 24 bond lengths. For fl-earotene, the calculated values for the polarizabilities, in the free-electron model (Rustagi and Ducuing, 1974) are : c~z~ = 1.8 × l0 -~2 cm 3 and %z~ = 4.8 × l0 -33 esu, where z is the molecular axis. The contributions of c ~ , c~uy, "~. . . . and "/u~yu are expected to be much smaller because the r-electron extension perpendicular to the molecular axis is much smaller. Thus we may approximate (c~) ~ c~:/3 = 60 ~t3 so that (c~)/N~, the linear polarizability per r-electron is nearly the same for the two molecules. In contrast ("/)/N, is much smaller for C60. This is partly because 7 increases steeply with increasing delocalization length and one-dimensional r-electron systems provide a larger conjugation length per r-electron. From this view point tubules would be the preferred r-electron systems. However, the electronic structure of such molecules is still not well understood. Secondly, the hyperpolarizability of C60 is also reduced by the high symmetry of the molecule and consequent selection rules as well as by the fact that the HOMO-LUMO transition is parity-forbidden. This indicates that adding electrons to the tl~ (LUMO) state as in Cs9N or C~0 state or removing electrons from the h~ (HOMO) state as in C59B or C+0 will increase % The tight-binding calculations for CsgB and CsgN indeed confirm this. Tiros as for symmetric cyanines (Mehendale and Rustagi, 1979; Stevenson et al., 1988) perturbation of a r-electron system by chemical substitution appears to be an effective means of increasing the hyperpolarizability of a system if it is much below that of other equally polarizable systems. Further, the lack of inversion symmetry allows the doped molecules to have an electric dipole moment which is negative in CsgB and positive in C59N. The dipole moment is not sensitive to the difference between the long and short C-C bond lengths but does reduce noticeably when boron is moved out in CsgB. However, the values for the dipole moment obtained by Ramaniah et al. (Ramaniah et al. 1994) are about five times larger than those reported in Ref. (Andreoni et al., 1992). Secondly the tight-binding results for c~ and -~ are sensitive to the difference in the long and short bond lengths. In particular, 7 of C60 increases by 46% when the more realistic, experimentally determined, bond lengths are used. The boron displaced structure of C59B is found to have a relatively high nonresonant 7 ~ S x 10-33 esu. This is found to be 30 times larger than that estimated for tim same molecule if the atomic positions are assumed to be the same as in C60.
88
K.C. Rustagi et al.
Finally we note that in doped fullerenes, the hyperpolarizability 13 is large and relatively insensitive to the atomic positions. These values are comparable to the largest reported for organic molecules. Using the density and the local field correction corresponding to the solid C6o, this value of/3 corresponds to a nonlinear susceptibility X(~) of 10-s esu for both CssB and CssN. This will reduce by a factor of ~ 10 by screening but will increase somewhat ~lue to dispersion. Interestingly, while in the one-dimensional case the HOMO-LUMO gap is strongly affected by the bond alternation, for Cs0 the gap is not sensitive to bond alternation.
Screening In the above discussion, screening due to electron-electron interaction has not been considered and all the polarizability values presented are the unscreened or bare values i.e., it has been assumed that the field seen by electrons in a fullerene molecule is the same as the one applied to it. In reality this is not so, because when the electron cloud is polarized, it also modifies the charge density and hence the self-consistent potential seen by each electron. Bertsch et al. (1991) used the RPA neglecting the radial spread in electron density to estimate that for C60 the effect of including this screening correction is to reduce a by a factor of f = (1 + ot/R 3) ~ 6. Nair (Nair, 1993) has also reported a similar calculation using the EPM wavefunctions and found that the screening is very sensitive to the radial spread in the electron density, the screening decreasing with increasing shell thickness. For the linear response he found that f = (1 + aa/R z) to a good accuracy where a is a constant which depends on the shell thickness. For t = 3, he found a = 0.73. It can be much smaller for a more realistic radial density. For 3' the screening correction was found to be slightly smaller than f4. From the measured refractive index (2.25) (Saeta et al., 1992) of C60 thin films, we estimate c~. . . . ca ~ 100 flk 3 using the Claussius-Mossotti relation. With able ~ 200A 3, we estimate the screening factor f to be ~ 2 which implies a reduction of ~t by a factor of 16 from the unscreened values. It may be argued that unless an accurate understanding of the screening correction can be made, no useful estimate of hyperpolarizability can be made. This is only partially true. The screening corrections in Cso and CssB are expected to be similar. Thus, if the unscreened calculations give/3 and 3' for C59B a conservative estimate can be made for the ratio of the hyperpolarizabilities of the two molecules. The effect of screening on the frequency dependent (dynamic) response is more dramatic. Bertsch et al. (1991) have analysed this problem in the RPA and found that the absorption spectra are strongly modified by the screening of the applied electric field by the electrons themselves. All the absorption peaks occur shifted towards higher energies from the energy gaps in the single particle spectra and the oscillator strengths of the low energy transitions are suppressed in favour of the higher energy ones. A strong resonance very similar to the Mie type plasmon resonance in a metal particle characterises the response at higher energies (Bertsch et al., 1991; Steger et al., 1992). These observations point to the similarity of fullerenes to semiconductor quantum dots for which unusual effects in the optical response due to the resonances in the electric field penetration have been predicted (Chemla et al., 1986; Nair and Rustagi, 1987; Ramaniah et al., 1989). Similar classical electrodyanamic treatment of the screening in fullerenes has been attempted by Ramaniah et al. (1992) by modelling fullerenes as spherical or ellipsoidal shells and using the bare polarizabilities as given by tight-binding calculations. These calculations show that screening is responsible for the fact that C70 and higher fullerenes all have only weak absorption bands in the visible and near infrared. This feature of the experimental spectra of higher fullerenes is otherwise surprising as all of them have a low symmetry and their HOMO-LUMO gaps are comparable to that of C60. Extension of these analyses to the nonlinear response would be very interesting. Naively, using the standard analysis of the local field correction to nonlinear susceptibilitites of condensed media (Armstrong et al., 1962), one would expect the screening correction to the n th order polarizability to be given by a product of n linear screening factors corresponding to the (n + 1) frequencies involved.
Nonlinear Optimal Response of FuUerenes
89
This would imply many unusual features in the near-resonant nonlinear optical response of fullerenes. For example, the one-photon and three-photon resonances corresponding to the same pair of singleparticle states would be expected to occur at slightly different frequencies. A detailed analysis of these features is still awaited. Future O u t l o o k
After some initial controversy, reliable estimates of the hyperpolarizability 3' of C6o are now available. However, theoretical estimates are generally smaller than the experimental values and also several experimental results are not yet understood. Still C~0 is just one member of a whole family of new molecules and much more remains to be done. Symmetry properties of/3 and 7 turn out to be rather special. C6o and other molecules of lh symmetry are the only systems for which c~, fl and 7 are isotropic like atoms. Unlike non-inversion symmetric crystallographic point groups, the symmetry group of C70 (Dsh) has fl = 0. We have suggested that harmonic generation in solid C60 and C70 could provide very interesting insights into their intermolecular interactions. Studies on these molecules have also resulted in a better appreciation of screening effects due to the electron-electron interaction. However, much needs to be done on this aspect. From practical considerations, the rather large ~ values anticipated for CsgB, CsgN and CTs appears to hold much promise specially for C7~ as it does not have a dipole moment and thus may easily crystallize in a structure lacking inversion. Finally, it seems fair to conclude that the study of nonlinear optical properties of these molecules with a variety of geometrical shapes and sizes is at the initial stages. With more molecules becoming available, much more exciting developments are anticipated. REFERENCES
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Collective Plasma Excitations in C6o
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binding molecular dynamics. Phys. Rev., B46, 9761. Wang, X.K.,T.G.Zhang, W.P.Lin, S.Z.Liu, G.K.Wong, M.Kappes, R.P.H.Chang and J.B.Ketterson. (1992). AppI. Phys. Lett., 60,810. Wang, Y., and L.-T.Cheng. (1992). Nonlineax optical properties of fullerenes and charge transfer complexes of fullerenes. J. Phys. Chem. 96, 1530. Xu, C.H., C.Z.Wang, C.T.Chan and K.M.Ho. (1992). A transferable tight-binding potential of carbon. J. Phys. Condens. Matter, 4, 6047. Yabana, K., and G. Bertsch, Michigan University preprint. Yang, S.C.,Q.Gong, Z.Xia, Y.n.Zhou, Y.q.Wu, D.Qiang, Y.L.Sun and Z.N.Gu. (1992). Large third order nonlinear optical properties of CT0 fullerene in the infrared regime. Appl. Phys., B 55, 51. Zhang, Z., D.Wang, P.Ye, Y.Li, P.Wu and D.Zhu. (1992). Studies of third-order optical nonlinearities in Cso-toulene and Cs0-toulene solutions. Opt. Lett., 17, 973.
rl=l
0.2 m
c
0
q.-
-.2
Fig. 1. The radial dependence of the eigenfunctions of the spherical shell model.
Nonlinear Optimal Response of Fullerenes
93
t~0.
30.
-w-
I,I
•
20.
t--D t.tJ t'--I --
t.ld t"3
10.
I---
I--< --.J
0.
-10.
-20. - 3 0' .
-2'0.
- 1 '0 .
8.
T10.
10
.
" 30.
LONGITUDE (DEGREE) Fig. 2. The angular dependence of the rr-electron density i]1 Cso. The values indicated denote the n u m b e r of electrons per unit solid angle• (After Nair, 1993)